Semigroups in Algebra, Geometry and Analysis 9783110885583, 3110143194, 9783110143195

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de Gruyter Expositions in Mathematics 20

Editors

Ο. H. Kegel, Albert-Ludwigs-Universität, Freiburg V. P. Maslov, Academy of Sciences, Moscow W. D. Neumann, The University of Melbourne, Parkville R.O.Wells, Jr., Rice University, Houston

de Gruyter Expositions in Mathematics

1

The Analytical and Topological Theory of Semigroups, Κ. H. Hofmann, J. D. Lawson, J. S. Pym fEdsJ

2

Combinatorial Homotopy and 4-Dimensional Complexes, H. J. Baues

3

The Stefan Problem, A. M. Meirmanov

4

Finite Soluble Groups, K. Doerk, T. O. Hawkes

5

The Riemann Zeta-Function, A. A. Karatsuba, S. M. Voronin

6

Contact Geometry and Linear Differential Equations, V. R. Nazaikinskii, V. E. Shatalov, B. Yu. Sternin

7

Infinite Dimensional Lie Superalgebras, Yu. A. Bahturin, A. A. Mikhalev, V. M. Petrogradsky, Μ. V. Zaicev

8

Nilpotent Groups and their Automorphisms, Ε. I. Khukhro

9

Invariant Distances and Metrics in Complex Analysis, M. Jarnicki, P. Pflug

10 The Link Invariants of the Chern-Simons Field Theory, E. Guadagnini 11 Global Affine Differential Geometry of Hypersurfaces, A. -M. Li, U. Simon, G. Zhao 12

Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and Theta Functions, K. Hulek, C. Kahn, S. H. Weintraub

13

Elliptic Problems in Domains with Piecewise Smooth Boundaries, Nazarov, B. A. Plamenevsky

14

Subgroup Lattices of Groups, R. Schmidt

15 Orthogonal Decompositions and Integral Lattices, A. I. Kostrikin, Tiep

S.A.

PH.

16 The Adjunction Theory of Complex Projective Varieties, M. C. Beltrametti, A. J. Sommese 17 The Restricted 3-Body Problem: Plane Periodic Orbits, A. D. Bruno 18

Unitary Representation Theory of Exponential Lie Groups, H. Leptin, J. Ludwig

19

Blow-up in Quasilinear Parabolic Equations, A.A. Samarskii, V.A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov

Semigroups in Algebra, Geometry and Analysis Editors Karl H. Hofmann Jimmie D. Lawson Ernest B. Vinberg

W DE _G Walter de Gruyter · Berlin · New York 1995

Editors Karl Η. Hofmann Fachbereich Mathematik Technische Hochschule Darmstadt Schlossgartenstr. 7 D-64289 Darmstadt, Germany

1991 Mathematics Keywords:

®

Jimmie D. Lawson Department of Mathematics Louisiana State University Baton Rouge, LA 70803-0001 USA

Subject Classification:

Ernest Β. Vinberg Chair of Algebra University of Moscow 119899 Moscow Russia

14Lxx, 20Gxx, 22Exx

Lie groups, symmetric spaces, Lie semigroups, semigroups and big groups, representation theory

P r i n t e d o n a c i d - f r e e p a p e r w h i c h falls w i t h i n t h e g u i d e l i n e s o f t h e A N S I t o e n s u r e p e r m a n e n c e a n d d u r a b i l i t y .

Library of Congress Cataloging-in-Publication Data Semigroups in algebra, geometry, and analysis / editors, Karl H. Hofmann, Jimmie D. Lawson, Ernest B. Vinberg. p. cm. — (De Gruyter expositions in mathematics, ISSN 0938-6572 ; 20) Includes bibliographical references. ISBN 3-11-014319-4 (alk. paper) 1. Semigroups. 2. Lie groups. I. Hofmann, Karl Heinrich. II. Lawson, Jimmie D. III. Vinberg, Ε. B. (Ernest Borisovich) IV. Series. QA182.S45 1995 512'.55-dc20 95-14934 CIP

Die Deutsche Bibliothek - Cataloging-in-Publication Data Semigroups in algebra, geometry and analysis / ed. Karl H. Hofmann ... - Berlin ; New York : de Gruyter, 1995 (De Gruyter expositions in mathematics ; 20) ISBN 3-11-014319-4 NE: Hofmann, Karl Η. [Hrsg.]; GT

© Copyright 1995 by Walter de Gruyter & Co., D-10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing from the publisher. Printed in Germany. Printing: Gerike GmbH, Berlin. Binding: Lüderitz & Bauer GmbH, Berlin. Cover design: Thomas Bonnie, Hamburg.

Preface

In contemporary mathematics semigroups appear naturally in many fields of investigation. Indeed semigroup methodology has become a fundamental tool in areas where no one even suspected a connection fifteen years ago. This methodology has been significantly enhanced and a whole new range of applications opened up by the emergence of a comprehensive new theory resulting from the comparatively recent conjunction of semigroup theory with Lie theory [6, 7, 11, 12, 14], even though the historical roots for such a theory reach deeper [10, 13]. The general mathematical trend of bringing to bear on a specific theory information from quite diverse segments of mathematics is exemplified by developments in modern geometry. The fundamental ties between the theory of Lie groups [1, 2, 3, 4, 5] and geometry have been recognized for over a century since the days of SOPHUS LIE and FELIX KLEIN. The natural concept of symmetry is foundational for geometry, as has become accepted since KLEIN'S Erlangen Program. Geometry and group theory axe, therefore, inextricably linked. It has become increasingly evident that symmetry is no less relevant to physics than to geometry. Yet in physics and other sciences dealing with real world phenomena, one encounters another concept of long standing which until recently was not considered relevant in geometry, namely, that of irreversible processes and phenomena. A suitable mathematical concept to accompany the idea of irreversibility is that of a semigroup. The first genuinely mathematical theory in this direction was the theory of one-parameter semigroups of operators on Banach spaces. Its pioneer was EINAR HILLE (1894-1980), whose enormously influential book of 1948 [8] planted a firm tradition which is still very much alive in the theory of partial differential equations, in probability theory and in other branches of analysis. In as much as these areas all impinge on physics, semigroups did enter this field. The emphasis in "semigroup theory" in the sense of HILLE, however, is on spectral theory and not on the structure theory of the semigroups which occur (with the exception of ergodic theory, in which the concept of a semigroup compactification is now a most appropriate tool). Yet semigroups have a rich theory of their own. It is perhaps astonishing that in physics proper, semigroups modelling irreversibility have not yet become fully developed. The semigroups which occur in this volume, at any rate, share the characteristic that in one way or another they are linked with the theory of Lie groups and Lie algebras and their respresentation theory, and that their applications lie in various directions involving irreversibility—such as causality in the theory of relativity, the geometric theory of control systems, or probability theory.

vi

Preface

The first mathematician to draw attention to a connection between semigroup theory and Lie theory was CHARLES LOEWNER (see [13]), but his work, for reasons not entirely clear, lacked acceptance by the research community and remained largely without resonance . Postulates for a theory of Lie semigroups proclaimed in the late sixties and mid seventies, too, remained without noticeable effect [9]. The theory of Lie semigroups began to take off in the early eighties at different places. In 1980 VLNBERG wrote his paper [17] on invariant cones in Lie algebras, which engendered an entire subculture in the Lie theory of semigroups. OL'SHANSKII began in 1981 [16] a sequence of seminal articles on invariant cones and introduced a class of Lie semigroups which we now call Ol'shanskii-semigroups. Through these papers he inaugurated the study of causal symmetric spaces and semigroup approaches to representation theory. In 1983 there appeared a first overview over the foundations of a general theory of Lie semigroups by HOFMANN and LAWSON [11] in a volume reporting on the Conference on Semigroups in Oberwolfach May 24-30, 1981. By the end of the decade, a monograph on Lie semigroups had appeared [6] (1989). From January 29 through February 4, 1989, Mathematisches Forschungsinstitut Oberwolfach had sponsored a conference on the "Analytical and Topological Theory of Semigroups," whose principal results were collected in a book [12]. The individual chapters were written by various authors so as to provide a self-contained introduction and overview of the latest developments. Further monographs on Lie semigroups appeared more recently [15] and [7] (1993), so that a substantial body of the theory was available in book form by 1993. Yet progress has been rapid and voluminous, and despite some 1500 pages of monograph literature, the recent state of the theory of semigroups in their relation to Lie theory remained unavailable in collected form. The collection before us continues the Oberwolfach concept and presents 14 chapters on "Semigroups in Algebra, Geometry and Analysis". Their authors take stock of the status of semigroup and order theory in the various fields they cover; they lead up to latest results and indicate the directions in which research is likely to develop. Applications and links to neighboring areas are not neglected. From October 10 through 16, 1993 the Mathematisches Forschungsinstitut Oberwolfach hosted a conference under the title "Invariant Ordering in Geometry and Algebra." Invitations to survey particular developments and to write contributions on these had again been extended by the organizers of the conference well before the meeting, and the final concept of the book was agreed upon at the conference. The book has been grouped into the following divisions: 1. Lie semigroups, ordered symmetric spaces, and causality, 2. Invariant cones, Ol'shanskii semigroups, exponential semigroups, 3. Convexity theorems and representation theory, 4. Semisimple Lie groups and semigroups, 5. Applications: Control, 6. Applications: Probability. Part 1 contains a chapter by FARAUT and OLAFSSON on symmetric spaces with causal structure and their harmonic analysis, followed by several chapters con-

Preface

vii

cerning causality on manifolds with cone fields, of which the pseudo-Riemannian geometry of relativity is a special case. HILGERT discusses an effective method from Lie semigroup theory for considering causality on homogeneous manifolds with cone fields. G U T S surveys the Russian work on the foundations of axiomatic relativity and how semigroups play their role in this field, and LEVICHEV gives an extensive report on I. SEGAL'S chronometry and its consequences in physics and cosmology. Part 2 opens with a new and direct approach to the classification of invariant pointed generating cones in Lie algebras by GICHEV; this draws a line from VINBERG'S seminal work on invariant cones in Lie groups to the highly devloped state of the theory today. LAWSON presents a comprehensive theory of Ol'shanskii semigroups in its current state, describing many aspects ranging from their various decompositions all the way through the natural occurences of these semigroups as endomorphism semigroups of geometric objects. HOFMANN and RUPPERT describe the classification of those closed subsemigroups of Lie groups which have a surjective exponential function (and which are reduced in a suitable sense); as is the case in almost all chapters of this book, a great variety of methods had to be developed before this classification problem, which used to be called the "divisibility problem," could be settled. In Part 3 HILGERT and NEEB provide an extensive discussion of the most upto-date results on the convexity theorems orginating from the work of KOSTANT and survey their link with symplectic geometry; they focus on those aspects of the theory relating in one way or another to Lie semigroup theory. N E E B discusses the current state of the theory of holomorphic extensions of unitary representations of Lie groups on the basis of the theory of Ol'shanskii-semigroups; he shows how this approach can be used for the study of the highest weight representations In Part 4 SAN MARTIN reports on his theory of control sets for control systems on flag manifolds of a semisimple Lie group; these control systems are given in terms of subsemigroups of the Lie group. This theory has currently reached a highly developed state. VLNBERG shows how one associates with a connected semisimple complex Lie group a semigroup which may be considered as a completion of the given Lie group. It is an irreducible algebraic variety whose dimension is that of the group, and it may be viewed as describing the structure of the Lie group "at infinity." This chapter is in the spirit of semigroups on algebraic varieties ä la PUTCHA and RENNER (who reported on this theory in [12] four years ago). Part 5 deals with the ever growing interplay between semigroup theory and geometric control theory. The contribution by MITTENHUBER presents the role of Pontryagin's maximum principle in Lie semigroup theory; in particular he demonstrates how this principle is an effective tool in the investigation of the globality of a Lie wedge. This chapter may also serve as an introduction to the entire complex of "Lie semigroup theory and geometric control theory." ZELIKIN discusses the classical optimal control situation from the novel aspect of a Lie group acting on the phase space leaving the cone field of the control system invariant; the focus is on the totally extremal submanifolds of the phase space.

viii

Preface

Part 6 is the shortest chapter. But it points to a still not fully explored area of application of semigroup theory to probability theory. While it is widely known that infinitely divisible laws determine one-parameter convolution semigroups of measures, it is less known that operator decomposable distributions give rise to Lie semigroups which are relevant in the central limit problem. JUREK's chapter introduces the reader to this application of Lie semigroup theory. The subdivision of this book into various groups, of course, must by necessity appear somewhat artificial. Its topics are not sequential but are interwoven in a complicated web of ideas. For instance, invariant cones in Lie algebras do not only occur in GLCHEV's chapter but are pervasive; they play a substantial role in the chapter by FARAUT and OLAFSSON and in everything said about OL'shanskiisemigroups. The chapter on control sets in flag manifolds of semisimple Lie groups by SAN MARTIN could just as well have been placed into Division 5 on geometric control theory. This lack of systematic ordering should not be seen as a defect but as an indication of the complicated network formed by Lie semigroup theory in its present state of development. The topics discussed in this monograph show that semigroup theory in geometry and analysis has attained a rich state of maturity, and that its development rests on fascinating connections with a variety of branches of mainstream mathematics. Literature [1]

Bourbaki, N., Groupes et algebres de Lie, Chap. 1, Paris, Hermann 1960.

[2]

—, Groupes et algebres de Lie, Chap. 2 et 3, Paris, Hermann 1972.

[3]

—, Groupes et algebres de Lie, Chap. 7 et 8, Paris, Hermann 1975.

[4]

—, Groupes et algebres de Lie, Chap. 9, Paris, Masson 1982.

[5]

Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, Boston etc, 1978.

[6]

Hilgert, J., Hofmann, K.H., and J.D. Lawson, Lie groups, convex cones and semigroups, Oxford Science Publications, Clarendon Press, Oxford 1989, xxxviii+645 pp.

[7]

Hilgert, J., and K.-H. Neeb, Lie Semigroups and their Applications, Lecture Notes of Math. 1552, Springer 1993, xii+315 pp.

[8]

Hille, Ε., Functional Analysis and Semigroups, Amer. Math. Soc. Coll. Publ. 31 (1948), xii+ 528 pp.

[9]

Hofmann, K.H., Topological semigroups, History, Theory, Applications, Jahresber. Deutsch. Math.-Verein. 78 (1976), 9-59.

[10]

—, Zur Geschichte des Halbgruppenbegriffs, Historia Math. 19 (1992), 40-59.

Preface

ix

11]

Hofmann, Κ.Η., and J D. Lawson, Foundations of Lie semigroups, in: Lecture Notes in Math. 998, Springer 1983, 128-201.

12]

Hofmann, K.H., J.D. Lawson and J.S. Pym, Eds., The Analytical and Topological Theory of Semigroups, de Gruyter, Berlin, 1990, xii+398 pp.

13]

Lawson, J.D., Historical links to a Lie theory of semigroups, Sem. Sophus Lie 2 (1992) 263-278.

14]

Neeb, K.-H, On the foundations of Lie semigroups, J. Reine Angew. Math. 431 (1992), 165-189.

15]

—, Invariant Subsemigroups of Lie Groups, Mem. Amer. Math. Soc. 104 (1993), viii+193 pp.

16]

Ol'shanskii, G.I., Invariant cones in in Lie algebras, Lie semigroups, and the holomorphic discrete series, Functional Anal. Appl. 15 (1981) 275-285.

17]

Vinberg, E.B. Invariant cones and orderings in Lie groups, Functional Anal. Appl. 14 (1980), 1-13.

Acknowledgements We thank the Mathematisches Forschungsinstitut Oberwolfach for providing firstly the logistics for the organisation of the conference on which the content of this book draws, and secondly for the well-known hospitality during the meeting. To the editors of the series de Gruyter Expositions in Mathematics and to the mathematics editor of de Gruyter Verlag, Dr. MANFRED KARBE, we express our gratitude for publishing this collection in the usual superb form. The editors thank all contributors for having provided IgX-files of their contributions using the macros we supplied. Dr. DIRK MITTENHUBER has invested a great amount of time, effort and expertise to coordinate these and to make everybody's macros compatible with those preferred by our publisher. An editor's life would be so much easier if KNUTH'S ingenious effective and pliable language TßX had not been diversified into a generation of numerous supposedly improved and more user-friendly, but certainly more rigid dialects which are all bound to be used without exception if 14 l^jX-users cooperate. The editors are extraordinarily grateful to KARL-HERMANN N E E B for the continuous, active and effective interest he took in the genesis of this book. We thank him most cordially, but also all other referees who contributed to the final form of the inputs.

Table of Contents

1. Lie semigroups, ordered symmetric spaces and causality Causal semisimple symmetric spaces, the geometry and harmonic analysis, by Jacques Faraut and Gestur Olafsson The halfspace method for causal structures on homogeneous manifolds, by Joachim Hilgert Semigroups in foundations of geometry and axiomatic theory of space-time, by Alexander K. Guts On mathematical foundations and physical applications of chronometry, by Alexander V. Levichev

3 33 57 77

2. Invariant cones, Ol'shanskii-semigroups, exponential semigroups On the structure of Lie algebras admitting an invariant cone, by Victor M. Gichev Semigroups of Ol'shanskii type, by Jimmie D. Lawson Lie groups and exponential Lie subsemigroups, by Karl H. Hofmann and Wolfgang A. F. Ruppert

107 121 159

3. Convexity theorems, representation theory Symplectic convexity theorems, Lie semigroups, and unitary representations, by Joachim Hilgert and Karl-Hermann Neeb 201 Holomorphic representations of Ol'shanskii semigroups, by Karl-Hermann Neeb 241

4. Semisimple Lie groups and semigroups Control sets and semigroups in semisimple Lie groups, by Luiz B. San Martin The asymptotic semigroup of a semisimple Lie group, by Ernest B. Vinberg

275 293

xii

Table of Contents

5. Applications: Control Applications of the maximum principle to problems in Lie semigroups, by Dirk Mittenhuber

Totally extremal manifolds for optimal control problems, by Mikhail I. Zelikin

6. Applications: Probability Lie semigroups and probability: a survey, by Zbigniew J. Jurek List of contributors

1. Lie semigroups, ordered symmetric spaces and causality

Causal semisimple symmetric spaces, the geometry and harmonic analysis /

Jacques Faraut and Gestur

Olafsson

The causal symmetric spaces were introduced by Ol'shanskii ([01s81], [01s82]). The geometry of these spaces is described in terms of invariant cones in Lie algebras. Two main topics in harmonic analysis have been recently developped on these spaces: Hardy spaces and Laplace transform. In the first part of this survey we present the geometry of invariant cones in symmetric Lie algebras, and of causal symmetric spaces. In the second part one considers the Hardy spaces associated with compactly causal symmetric spaces, or symmetric spaces of Hermitian type, and in the third part the spherical Laplace analysis associated with non-compactly causal symmetric spaces, or ordered symmetric spaces.

I. The geometry 1. Causal symmetric spaces Let V be a finite dimensioned real Euclidean vector space with inner product (·|·). Let R + := {A G R | A > 0}. A subset C C V is a a convex cone if for all u,v G C and Α, μ G R + we have Xu + μν G C. The cone C is called non-trivial if C φ —C. Notice that C Φ {0}, and C φ V if C is non-trivial. From now on, if not explicitely stated otherwise, C will always denote a non-trivial closed convex cone. We use the notion Ω for an open non-trivial convex cone. We define

ι)

.-cn-c,

(C) \= C — C = {u - ν I u,v e C}, 3) C* := {u G V I Vt; G C : (υ | u) > 0}. Then V c and (C) are vector spaces and C* is a closed convex cone with C** = C. The cone C* is called the dual cone of C. These constructions are related by 2)

V c * = (C where for a subset U in V we set U1- = {u G V | Vu G U : (u \ v) = 0}.

4

J. Faraut and G. Olafsson

Definition. Let C be a closed convex cone in V. Then C is a generating cone if (C) = V , a proper cone if Y c = {0}, and a regular cone if it is generating and proper. A closed convex cone C is called self-dual if C* = C and pointed if there exists a υ € V such that for all u € C \ {0} we have (u | v) > 0. L e m m a 1.1. Let C be a closed convex cone in V. Then the follomng are equivalent: 1) int(C) is non-empty. 2) C contains a basis ofV. 3) C is generating. L e m m a 1.2. Let C be a non-empty closed convex cone in V. Then the following properties are equivalent: 1) C is pointed. 2) C is proper. 3) C* is generating. Let Cone(V) be the set of closed regular convex cones in V. The set Cone(V) is invariant under C i—• C* as the cone C is regular if and only if C* is regular. Let G be a closed subgroup of GL(V). Then an open or closed cone C C V is invariant if G · C C C. We denote the set of invariant regular cones in V by Conec(V). For a (closed or open) convex cone C we define the automorphism group Aut(C) C GL(V) of C by Aut(C) := {a e GL(V) | a{C) = C}. Then Aut(C) is closed in GL(V). In particular Aut(C) is a linear Lie group. We denote the transpose of a matrix or linear operator a by ί α. Then Aut(C*) =

4

Aut(C).

Let M. be a n-dimensional C°°-manifold. Denote by Tm{M) the tangent space of Μ at the point τη € Μ. and let T(M) be the tangent bundle of M.· The derivative of a differentiate map / : M. —^• Λί at m will be denoted by Tmf: Tm{M) —> T/(m)(Λ/*)· If a Lie group G acts differentiably on A4 we denote by lg the diffeomorphism £g(m) = g -τη. A smooth (analytic) infinitesimal causal structure on Μ is an assignment m — i > C(m) to each point m of M. of a regular cone C(m) in the tangent space at m such that there are an open covering {£7i}te/ of M , smooth (analytic) maps

manifold.

Then

TJa(C))

Cone//(Τ 0 (ΛΊ)) and the set of invariant,

regular

causal

on M. •

Proof. We only have to see that the cone field is smooth. For that we choose a zero neighborhood U C T0(M) and a neighborhood V of ο such that Exp : U —> V is a diffeomorphism. Then { a U } a ^ c is a n open covering of M. and we define ψ α by φα (a E x p Χ , Υ)

= Τ0ία

χ (Y).

Exp

•

Prom now on we will assume that Λί is a connected semisimple symmetric space. This means that Μ = G/H where G is a connected semisimple Lie group, there exists a non-trivial involution r of G such that Η is an open subgroup of the group of fixed points GT. Let θ be a Cartan involution of G commuting with r ([M79]). If φ is an involution of G then we denote by the same letter the derived involution on g. Let us introduce at this point some notation that we will use later on. Let b be a finite dimensional abelian Lie algebra over the field Κ and let V be a finite dimensional b-module. Let π be the corresponding representation of b. If Λ € Κ and Τ e E n d ( V ) we define V ( A , T ) : = { v e V\Tv

=

\v}.

For Χ Ε b we set V ( A , X ) : = ν ( Λ , π ( Χ ) ) . For α € b* — H o m K ( b , K ) we define the root space V ( a , b) by V ( a , b ) : = {υ e V I V X € b : X • υ = a(X)v

}

= p|

V(a(X),X).

X€b

If the role of b is obvious we abbrivate this as Va. Furthermore V b : = V ( 0 , b). Denote the set of roots by A(V, b) : = {a € b* | V a φ { 0 } } . For 0 φ Γ C A ( V , b) we set ν ( Γ ) : = φ α £ Γ V a . If V is a real vector space we set V c : = V C. We define the conjugation o f V c relative to V by u + iv :— u—iv, u, ν € V . Sometimes we also denote this involution by σ.

6

J. Faraut and G. Olafsson Let

:= fl(+l, r), q := β(-1, τ), ί := g(+l, θ) and ρ = fl(-l, θ). Then 0 = ί) Θ q = 6θρ = i)k Θ q/t θ f)p Θ qp

where the subscript k means the intersection with 6 and the subscript p denotes the intersection with p. Furthermore f) is the Lie algebra of Η and t is the Lie algebra of Κ = Ge. Moreover [f),q] C q and Ad(#)q C q. We view q as real Euclidean vector space with the inner product (X\ Y)e -.=

-Β(Χ,Θ(Υ)),

where Β denotes the Killing form of g. Let ο = 1/H. Then q 9 X ~ X 0 e T0(M),

Xo(/)

:= jt f(exp(tX)

· o)|t=0, / e

C°°(M),

defines a linear isomorphism of q onto T 0 {M) such that Exp : q j\4, X >—> exp(X) • ο corresponds to the exponential map T0(M) M.· Notice that the identification of q as T0(M) intertwines the natural action of Η on q, Adq : h

Ad(/i)|q,

and the action H 3 h » T 0 ( 4 ) € GL(T 0 (X))By this we have the identification T(M) — G q, where for a finite dimensional if-module V the vector bundle G x h V —> GjH is defined as (G χ V ) / ~ and (ο, ν) ~ (b, w) 3h e Η such that ah = b and h~l · ν = w. Combining this with Lemma 1.3 gives T h e o r e m 1.4. The map C G(o) induces an bijection between the set of G-invariant causal structures on M. and the set of Η-invariant regular cones in q. E x a m p l e 1. Let G\ be a semisimple Lie group. Let G = Gi χ G\ and define τ by τ(α, b) = (b, a). Then Η = {(α, α) | α e Η} ~ Gx. The map Μ 9 (α, b)H ^ ab~l € Gi is a diffeomorphism intertwining the left action of G on Μ with the action (a, b)-x — axb~l — iar^l{x) of Gι on itself. Here f) = {(Χ, Χ) \ X G gi} ~ 0i and q = {(Χ, —X) I X € 0i} — gi. This second isomorphism induces an isomorphism Conen(q) ^ Cone G l (gi) E x a m p l e 2. Let gi be a semisimple Lie algebra. Let 0 = (0i)c and let r be the conjugation σ. Let Gc be a complex analytic group with Lie algebra 0 and assume that r integrates to an involution on Gc- Let Gi be the analytic subgroup corresponding to gi. Then Gi = (G£) 0 . Thus Gc/G\ is a symmetric space. In this case q = and again the if-invariant cones in q corresponds to the Gi-invariant cones in 0i.

Causal semisimple symmetric spaces, the geometry and harmonic analysis

7

E x a m p l e 3. Let be the hyperboloid {x € R n + 1 | x\ 4*X2 X3 "*"·*· X^j 1 J — 1}· Q+ = S 0 0 ( 2 , n — 1 ) / S 0 0 ( l , n — 1) is a symmetric space where the involution r is given by conjugation by I\ < n = ^

J

w

^ere

n x

identity

matrix. In this case q is isomorphic to R" in which S0 0 (1, η — 1) acts as usual. If C is the inverse image of the forward light cone then Conego„(i,n-i)(q) = {C, — C} for η > 3. E x a m p l e 4. Let Qü be the hyperboloid {a: € R n + 1 \x\-x\ = — 1}· Q- can be realized as the symmetric space S 0 0 ( l , n ) / S 0 0 ( l , n — 1). Again q —so0(i,n—1) and if C is the inverse image of the forward light cone then Cone S O o (i,n-i)(q) = {C, -C} for η > 3.

2. Duality of causal spaces Let M. = G/H be a semisimple symmetric space. For simplicity we will allways assume that M. is irreducible and that there is no non-trivial ideal in 0 contained in f). This implies that [q, q] = f). We say that M. is causal if there exists a Ginvariant cone field on ΛΊ- The first step in the direction of a classification is, cf. [0190] T h e o r e m 2.1. 1) The symmetric space M. is causal if and, only if the space HnK q° q is non-trivial. 2) Μ is causal if and only if Cone//(q) Φ 0. If Η is semisimple and q irreducible as an if-module, then the theorem would be obvious from the following theorem, the different parts of which are due to Kostant, Paneitz and Vinberg, see [Pa81, Se76, Vi82] T h e o r e m [Kostant-Paneitz-Vinberg]. Let G be a connected semisimple Lie group acting on the real vector space V by a representation π. Let Κ C G be a subgroup of G such that π(Κ) is a maximal compact subgroup of π((7) and let Ρ C G be a minimal parabolic subgroup of G. Let (· | ·) be an inner product on V such that 4 π(χ) = π(θ(χ))~1 for all χ e G. 1) There exists a proper G-invariant cone in V if and only if the space of Κ-fixed vectors WK is non-zero. 2) If TT is irreducible, then ConeG(V) Φ 0 if and only if any of the following equivalent conditions are satisfied: a) V * φ 0. b) There exists a ray through 0 which is invariant with respect to P. In this case every invariant pointed cone in V is regular. Furthermore there exists a—unique up to multiplication by (—1)—minimal invariant cone Cm\n ζ

J. Faraut and G. Olafsson

8 ConeGr(V) given by C m i n

= conv(G · u) U {0} = convG·

(R+vK),

where u is a highest weight vector, υχ is a non-zero K-fixed vector unique up to scalar and (u\vk) > 0. The unique (up to scalar) maximal cone is then given by 1 Γ «-'max — — Π* ^min-

The structure of Η and q is clarified by the following theorem, cf. [0190, 0193, OH92] Theorem 2.2. If Jvl is causal then the following holds: 1) q is irreducible as an Ha-module if and only if H0 is semisimple. This is the case if and only ifc\H°nK is one dimensional. 2) If q is reducible as an Η-module, then there exists an element X° G i)p such that the following holds a) Η = ( t f n i O e x p ^ i j l p e x p R X 0 and 3 (fj) = RX°. b) fj = g(0,X°), q = g(+l, X°) φ g(—1, and the subspaces g ( ± l , X ° ) are invariant and irreducible under H. c) 0 ( g ( + l , X ° ) ) =g(-l,X°) and q° = g(+l, X°)KnH

φ g(-l,

X°)KnH

3) Suppose q is irreducible as an Η-module but reducible as an H0-module. Choose X° e f)p as in (2). Let Hx = {h e Η | Ad(/i)g(+l, X°) = g ( + l , X 0 ) } . Then Η ι is a θ-stable normal subgroup of Η and there exists an element k 6 Η such that Ad(fc)g(+l,X°) = g ( - l , X ° ) and Η = HiÜkHx. Furthermore q° is one-dimensional and contained in q^. This theorem together with the Kostant-Paneitz-Vinberg theorem proves Theorem 2.1. For the class in (3) one first applies the KPV-Theorem to g(+l, to construct a minimal invariant cone C* i n in g ( + l , X ° ) . The cones C* := {X + Θ{Υ) \X,Y e C+ i n } and Cp {X - Θ(Υ) \ Χ,Υ € C+in} are both in Cone^(q) and actually Cone//o(q) = {±Cjt, ± C P } . The cone Cp is not ii-invariant if Η φ H\ but Ck is allways if-invariant. Another easy consequence of this is the following lemma. Lemma 2.3. Let Ζ e q° be non-zero. Let Cm;n Cmin is a minimal H0-invariant cone in Cone//o (q).

conv (Ad(ii 0 )R + Z), then

If q° is one-dimensional then either q° C 6 or q° C p. The first case implies that for every C 6 Cone#(q) we have C° Π t φ 0 and C° Π ρ = 0, whereas in the second case it is the other way round C° Π ρ φ 0 and C° Π t = 0. If q° is two dimensional then with Ck and Cp as above we have C£ Π t φ 0 and C% Π ρ = 0 and similarly for Cp with the role of t and p interchanged. Definition.

The irreducible symmetric space M. is called

Causal semisimple symmetric spaces, the geometry and harmonic analysis

9

CC) Compactly causal, if is non-zero. NCC) Νon-compactly causal, if is non-zero. CT) Of Cayley type, if q° is two-dimensional. There is a duality of symmetric spaces interchanging (CC) and (NCC), cf. [KN, 0190, 0190b, 01s82], If g = f) φ q then g c : = I) Θ iq. Let σ be the conjugation on gc relatively to g. Denote by τ the complex linear exstension of r to gc- Then η = τσ is a conjugation on gc and the real form of gc corresponding to η is gc. Let Gc be the simply connected analytic group with Lie algebra g°. Then r integrates to an involution on Gc and the fixpoint group H° is connected. The space is the universal c-dual of Λ4- Let Μ = G/H be the universal covering of M- Here G is a simply connected analytic group with Lie algebra g and Η is the analytic subgroup corresponding to f). It is obvious that Lemma 2.4.

M. is compactly causal if and only if M° is non-compactly

causal.

Example 5. We use here the notation from Examples 1 to 4. Then up to covering Gl ~ Gc/G\ where the identification is given by g° 3 (Χ, X) + i{Y, -Y)^X

+

iY.

The spaces Q+ and Q- are dual to each other.

3. Coverings of causal spaces As the Cayley type spaces show, the existence of non-compactly causal structures on Μ may depend on the fundamental group of .M· In general one can say Lemma 3.1. Assume that rankt = rankg then there exists no causal structure on the symmetric space Ad(G)/ Ad(G) T .

non-compactly

Proof. By [He84] the condition rank! = rankg implies that θ Ε Ad(G). The condition θτ = τθ reads now that θ € Ad(G) T . The lemma follows now because of 0|p = —1. • L e m m a 3.2. 1) Let Z° € q° be non-zero. Then g(0,Z°) — t. In particular Z° is central in t and Z° can be chosen such that ad(Z°) has eigenvalues 0, i and —i. 2) Let Y° e q° be non-zero. Then g(0, Y"0) = f)/t Θ qp and Y° can be choosen such that ady° has eigenvalues 0, 1 and —1. Proof. 1) As Z° is fixed by Ad(iJ Π Κ) it follows that [f)fc,Z°] = {0}. Let now Ye qk. Then [Y, Z°] e f)k and Β ([Y, Z°], [Y, Z0]) = Β (Y, [Z°, [y, Z0]]) = 0.

10

J. Faraut and G. Olafsson

Thus Z° is central in t. The rest of (1) is a well known fact on bounded symmetric domains. (2) follows in a similar way using the c-duality. • The following clarifies the dependence of the causal structures on M. of the connected components of Η. As Η = (Η Π Κ) exp f)p, the minimal cone C m ; n in qjt is generated by A d ( H 0 ) Z ° where by Lemma 3.2 Z° is central in E. But then Z° is fixed by every element in Κ and it follows that C m j n is //-invariant. Thus Lemma 3.3.

A4 is compactly causal if and only if M. is compactly causal.

The situation for the non-compactly causal structure is more complicated. First we need some more notions. Let Y° G q p , Y° ^ 0. By Lemma 3.2 we may choose Y° such that ad(y°) has eigenvalues 0, +1, —1. Let α be a maximal abelian subalgebra of p containing Y°. Then α C q p by Lemma 3.2. Let Δ = A(g, α). As adY° has the eigenvalues 0, ± 1 it follows that Δ is the disjunct union of the set of roots Δ 0 := {α I a(Y°) = 0}, Δ± := {a | α(Υ°) | a(Y°) = ±1}. Furthermore Δ 0 is the set of roots of α in flo := 9(0, Choose a positive system Δβ in Δο· Then Δ + : = Δ + U Ag is a system of positive roots in Δ. Let Μ = Ζκ {a) and Μ* = Ν κ (a)· Let W = be the Weyl group associated to the root system Δ. Then W = M*/M. Similarly we define W0 = W(A 0 ). If m G M* and α e Δ then we define m · a(X) = a(Ad(m _ 1 )X). The group Η is said to be essentially connected if Η = (Μ Π H)H0, cf. [Ba86]. By [0193]: Theorem 3.4. If /A is non-compactly causal, then Μ is non-compactly causal if and only if Η is essentially connected. Proof. Assume that fiA is non-compactly causal. As Η is connected then Ad(/f) = Ad(J7 0 ). Thus G/H0 is non-compactly causal. Let C m i n be the minimal i/ 0 -invariant cone generated by A d ( H 0 ) Y ° . If M. is non-compactly causal let C 6 cone//(q) be such that C m i n C C. Then Y° € C and C is //„-invariant. Let k € Hf)K. Then Ad(fc)Y° G C and WJH ψ · '

Σ ° ke(HnK)/(H0nK)

is Κ Π //-invariant and non-zero. As d i m q = 1 we may assume that Y° is Κ Π //-invariant. As Ad(k)a is maximal abelian in q p we can find a h £ H0 Γ) Κ such that Ad(hk)a — a. Thus we can assume that k G Μ* Π Η. As k • AQ is a set of positive roots in Δο and all such are conjugate by H 0 Π Μ* we can assume that k • A j = A j . As Ad(fc)Y° = Y° it now follows that k • Δ+ = Δ+. But then k G Μ Π Η and it follows that Η is essentially connected. The other direction follows easily as the minimal // 0 -invariant cone is Hinvariant if Η is essentially connected. • The Weyl group W0 is given by (Μ* η H0)/(M Π H0). If Η is essentially connected, then (Μ* Π H0)/{M Π H0) = (Μ* η Η)/{Μ Π Η). Thus

Causal semisimple symmetric spaces, the geometry and harmonic analysis Lemma 3.5.

11

If Μ is non-compactly causal, then WQ = ΝΗ (A) /Zjj(a).

4. The classification In this section we give the classification of the causal symmetric spaces up to coverings. The first case to be considered is the group case from Example 1, cf. [HHL, 01s81, Pa81, Vi82]. A complete list of semisimple causal symmetric spaces appeared for the first time in [01s82] and then in [0190, 0190b]. The compactly causal symmetric spaces, also named symmetric spaces of Hermitian type, appeared also in papers on representation theory [Ma81, 0088, 0091] and were classified by H. Doi in [Do81]. There are many different ways to classify the causal symmetric spaces. Using the duality from Lemma 2.4 it is enough to classify either the compactly causal spaces or the non-compactly causal ones. By Theorem 2.1. and Lemma 3.2 we get Theorem 4.1. Let A4 be an irreducible symmetric space. Then M. is compactly causal if and only if G/K is a bounded symmetric domain and the induced involution τ : G/K —• G/K is anti-holomorphic. This implies that the compactly causal symmetric spaces corresponds up to coverings to the real forms of the complex domain G/K. In turn the conjugations on G/K were classified by H. Jaffee in [Ja75, Ja78]. Another approach is to use Lemma 3.2 part (b) which shows that the non-compactly causal symmetric spaces corresponds to the Z2-graded Lie algebras, which were classified by S. Kobayashi and T. Nagano in 1964, cf. [OH92, 01s82]. We decribe this idea in short, cf. [KS92, OH92]. Let g = 6 θ ρ be a Cartan decomposition of 0. Let ο be a maximal abelian subalgebra of p. Let Δ = A(g, a) be the possibly non-reduced set of roots. Let Δ + be a system of positive roots and let Σ be the set of simple roots. Let

([FHO], Theorem 9.1.)

4. The spherical Laplace transform A

Volterra

A4

χ Μ

kernel on the ordered symmetric space A4 = G / H is a function F on which is continuous on { ( x , y ) \ χ > y } and zero outside this set. One composes two such kernels Fi and F2 via the formula F

l

o F

2

{ x , y ) =

f F J Μ

1

(x,z)F{z,y)ds,

where dz is an invariant measure on A4. This definition makes sense because Μ is globally hyperbolic, i.e., the intervals are compact. With respect to this multiplication the set V(M) of Volterra kernels becomes an algebra, called the Volterra algebra of A4. A kernel F is said to be invariant if F(gx,gy) =

The space

V(A4)#

F(x,y),

V g e G .

of all invariant Volterra kernels is a commutative subalgebra of

V ( M ) .

The spherical CF defined by

Laplace

of an invariant Volterra kernel is the function

transform

C F ( A ) =

[ F(x,x J Μ

0

)e

{ p

-

x

'

A

^

x ) )

dx,

on the set T>(F) of the A G such that the integral converges absolutely. By using 'polar coordinates' this can be written CF(

A) =

F(ax0,x0) 3, and let Οχ, C>2 consist of elliptic cones. Then the actions ct\,a.2 are affinely conjugate and the corresponding affine structures A\, A2 are equivalent. Proof See [7],

•

It is evident that group Gn in the condition of Theorem 3.2 is abelian. For η — 4, Theorem 3.2 yields the uniqueness of the traditional presentation of special relativity, where the light cones in R 4 are taken as parallel. Remark 3.3. The statement of Theorem 3.2 as it was announced in [19] is valid for any pair of normal actions c*i, 3. The relativistic elliptic conal order n-1

\i=1

Px = S y = (ί/ι. · · · I yn) G R n : yn - Xn >

is simultaneously Int-, d- , and ext-homogeneous.

x =

(xi,...,xn),

•

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63

Example 6.3. Let Lx denote the ray with vertex χ and direction vector ξ such that (ξ71)2 = Σ ^ Τ / ί ? 1 ) 2 ) a n d assume η > 3. Then the relativistic conal order Ρχ = < y= (yi, ...,yn)

eR

n

:yn-xn>

η— 1 yZ(yi -Xi)2

\ i=1

>\JLX,

X=

(xi,...,xn)

is Int- and ext-homogeneous, but is not ^-homogeneous.

•

Theorem 6.4. Let Ο be a non-relativistic elliptic conal a (G 3)-invariant order on R 3 , where a is a normal simply transitive affine action of Gz on R 3 . Then Ο is neither Int-, nor d-, nor ext-homogeneous. Proof. See [8, 10]. Example 6.5. formations

•

Let ^ G s V I I q ) denote the non-commutative group of affine trans-

(xi, £2,2:3) >-> (x\ cos(t) + X2 sin(i) + a, —x\ sin(i) + 12 cos(£) + β, x$ + i), t,a,ß e R, and assume η > 3. Now the relativistic elliptic conal order from Example 6.2 is V/io)-invariant, and is simultaneously Int-, d- , and exthomogeneous. • Let Ay be the complete left invariant affine structure on G3 that was introduced by GAVRILOV in [15]. This structure defines a natural simply transitive affine action α Ί on R 3 . In [15] GAVRILOV describes the a 7 (G 3 )-invariant Lorentz metrics with respect to a given smooth structure Ay. Let (·, ·) be such metric in R 3 . A cone Kx with vertex χ in R 3 is called causal, if any ray from Kx is generated by a direction vector ξ belonging to the fixed half of the tangent cone {£ : (ζ, ζ)χ > 0}. An order Ό in R 3 is called affinely causal, if it is defined by a set of elliptic causal cones {Px:xe R3}. Theorem 6.6. Let Ο be the affine causal order with respect to the Gavrilov affine structure Ay. Then for Lie groups G3 of type I,VIq,VIIq of class I (see [15]) the order Ο is simultaneously Int-, d- and ext-homogeneous. The corresponding Lorentz metrics are flat and thus the order Ö is relativistic. In the other cases the order Ö is neither Int-, nor d-, nor ext-homogeneous. Proof. See [16, 17].

•

The relativistic conal homogeneous orders in R n , η > 3, were studied in [21, 22] (see also the review of [1]). Definition 6.7. An order Ο = {Px : χ 6 R n } is said to be connected if χ € Px \ {x}. Otherwise the order Ό is called disconnected.

64

A.K. Guts The following postulate is called the weak Einstein axiom:

(ΑΕ„,) For any x, y e Rn, if y e Px, then Px Π P~ is bounded. Let l(x,y) denote the ray with the origin χ passing through y. The cone ext Px = Uj/€Px\{x} ν) with vertex χ is said to be the exterior cone associated with Px. The exterior cone is a closed convex subset of R n . We say that a cone has an acute vertex or that it is a pointed cone if it does not contain (nondegenerate) straight lines. Now we can formulate the following strong Einstein axiom,·. (AE S ) The exterior cone ext Px is pointed. We use the abbreviation Qx — Px\ {x}. Let if be a convex cone with vertex a. Further let ty denote the parallel translation satisfying ty(a) — y. Now a relativistic disconnected order Ο is said to be Κ-ruled, if ty(K) C Px for every y e Qx, χ e R n . Theorem 6.8. Assume η > 2. Let Κ be as in Definition 6.7 with Int(Ä") ^ 0 and suppose that Ο is a relativistic disconnected d-homogeneous Κ-ruled order on Rn satisfying the axiom (ΑΕ ω ). Then the following assertions are true: 1) Aut(ö) C A f l ^ R ^ _ 2) The order Ο = {Px : χ Ε R n } is ext Pa-ruled, and Aut(O) C Aff (R n ). 3) Axiom (AE a ) holds. 4) The conal order ext Ο = {ext Px : χ G R n } admits an affine group G of ext Ö-automorphisms such that for all χ the stabilizer in χ acts transitively on Int(ext Px). 5) The set Qx is convex and is isolated from the cone ext Px by hyperplanes that cut off a set of finite volume from ext Px. 6) Aut(ö) is a subgroup ofG. Proof. See [24, 25, 26].

•

Thus it is possible to classify relativistic 9-homogeneous disconnected /i-ruled orders that satisfy (ΑΕ ω ). This classification reduces to a classification of Inthomogeneous convex cones in R n , η > 3, and the calculation of the group G. Then dQx is the orbit of the group A u t ( ö ) x C Gx. The group G was calculated by VlNBERG in [22], Chapter III, §2, Proposition 1, and in [23]. Remark 6.9. It is not necessary to assume that Aut(C?) in Theorem 6.8 consists of continuous bijections (see [25, 26]). Example 6.10. We describe the relativistic disconnected d-homogeneous orders in R n for η = 3,4. Let i x : R n —• R n denote the parallel translation defined by tx( 0) = x.

Semigroups in foundations of geometry and axiomatic theory of space-time

65

η = 3

1)

P=

{ x € R 3 : x 3 > ^ J l + x l + x Q y j {(0,0,0)},

Px = tx(P),

and

e x t P = {x € R 3 : x3 > 2)

Ρ = {χ G R 3 : 0:1X22:3 > 1 & χ» > 0 (t = 1 , 2 , 3 ) } U {(0,0,0)}, Px = tx(P),

and

ext Ρ = {χ G R 3 : Xi > 0 (» = 1 , 2 , 3 ) } .

η = 4

3

1)

P={xeR

4

:x

4

> ( l + ^x?)1/2}U{(0,0,0,0)}, i=l

Px = tx(P),

and

3

ext Ρ = {χ e

2)

R4

: x4 > £ x ? ) 1 / 2 } · i=l

Ρ = { χ G R 4 : XiX2X3X4 > 1 & χ» > 0 (» = 1 , 2 , 3 , 4 ) } U { ( 0 , 0 , 0 , 0 ) } , Px = tx(P), ext Ρ = {χ 6

and R4

: x t > 0 (t = 1 , 2 , 3 , 4 ) } .

3) dQ is the set of points which one half of the hyperboloid {(0,X2,X 3 ,X4) : X4 — x | — x | = 1 & £4 > 0} sweeps out when it is moving such that its vertex slides along the hyperbola {(χχ,0,0,X4) : X1X4 = 1 L· χ χ , χ 4 > 0}, and ext Ρ — L χ C, where L is a ray, and C is an elliptic 3-dimensional cone. •

A disconnected order Ο is said to be singular, if the set

2, satisfying the conditions (CI), (C2). Then the order Ο is connected with the possible exception of the singular disconnected order. Proof See [8].

•

We have no example of a singular order. T h e o r e m 6.12. Let Ο be a connected relativistic ext-homogeneous fying the following conditions: 1) Int(cont(P x ,z)) φ 0.

order satis-

2) There is a neighborhood Ux of χ for which Ux Π Px Π Px = {x}. Then Ο is an elliptic conal order. Proof. See [8, 11].

•

T h e o r e m 6.13. Let Ο be a relativistic disconnected order in Mn, η > 2, satisfying the conditions (AE S ), (C2). Then Ο is not Int-homogeneous. Proof See [8, 11].

•

Physical interpretation. The condition of connectedness of order says that the action of an event χ on an event y is transmitted through a continuum of intervening events. In the case of a disconnected order a transmission with a jump in energy-momentum takes place.

7. Dense orders Definition 7.1. An order Ο is called dense if for any x, y e R n , y Ε Px the set Pxf)Pφ {x,y}. We now discuss the question: Which structures have dense orders? A basic affine Lie group Gn is a simply connected solvable Lie group whose algebra Lie has a basis { X i , . . . , Xn} satiying the commutator relations [Xn,Xi]=Xi

(t = l , . . . , n - l ) .

The paper by K.H. Hofmann and J.D. Lawson [2] was influential in pointing out the role of basic affine Lie groups for the theory of Lie semigroups. The orders and the automorphisms of basic affine Lie groups were investigated in [14]. It is

Semigroups in foundations of geometry and axiomatic theory of space-time

67

important to note that the 4-dimensional basic affine Lie group is a transitive subgroup of group of isometries of the stationary de Sitter Universe, which was considered by F . H o y l e and J . N a r l i k a r as a cosmological alternative to the theory of the Big Bang. Theorem 7.2. Let Ο — {Px : χ G G n } be a dense closed left; invariant order in an abelian or basic affine Lie groups Gn, η > 2. Suppose that a minimal Lie semigroup S for which Pe C S satisfies the condition S Π 5 - 1 = {e}. Then Pe is a Lie semigroup. Proof. See [29, 30].

•

Corollary 7.3. Let a be a simply transitive affine action of an abelian Gn on R n by means of parallel translations, i.e., a(Gn) is the group of parallel translations. Suppose that φ is the diffeomorphism (2) and that the order Ο satisfies the conditions of Theorem 7.2. Then the order φ(Θ) — {φ(Ρχ) : Px 6 0} in R n is a relativistic closed conal order. Corollary 7.4. Let a be a simply transitive affine action of the basic Lie group Gn on R" = {χ = ( x i , . . . ,xn) € R n : rri > 0} by means of transformations of the form: (xi,...,xn)(\xi,\x2

+ ai,...,\xn+an-i),

λ > 0,

αι,...,α„-ι 6®.

I.e., the action a defines a non-complete left invariant affine structure on Gn. Suppose that φ is the diffeomorphism (2) and that the order Ö satisfies the conditions of Theorem 7.2. Then the order φ\θ) = {φ(Ρχ) : Px 2.

8. Axiomatization of the three-dimensional pseudo-Euclidean geometry We use the results above for an axiomatic description of pseudo-Euclidean geometry. This objective can be attained by going through the following steps: 1) Equip an abstract group G with a left invariant affine structure; 2) introduce a Lorentz metric (·, ·)0 on the group G without making use of the notion of a smooth tensor field. The first step was taken by Ionin [18]. The structure of a simply connected affine manifold on a set G is a quadruple (G, Γ, Φ, Ψ), where Γ is the set of all affine transformations of the real line R, and the sets Φ C G R , Φ C R G are subject to the following conditions:

68

A.K. Guts

1) For any φ Ε Φ and φ Ε Φ we have φ ο φ Ε Γ. 2) Φ is maximal, i.e., if f:R

—> G but / ^ Φ, then there exists φ Ε Φ such that

3) Φ is maximal, i.e., if /: G —• Μ but / φ Φ, then there exists φ Ε Φ such that 4) For any x, y Ε G, there exists φ Ε Φ such that χ, y Ε (R). 5) For any x,y Ε G,x φ y, there exists ψ Ε Φ such that φ(χ) φ φ{ν). In these terms an affine transformation h: G —» G is defined as a map for which ψ ο ho φ ε Γ for all φ Ε Φ and φ Ε Φ. Any set (R), φ Ε Φ is called a line. A ray with origin χ is a set 0(M+), where φ Ε Φ, R+ = {t Ε Μ : t > 0}, and φ{0) = χ. The dimension dim G of an affine manifold G is defined in the obvious way. An affine structure (G, Γ, Φ,Φ) is called left invariant if G is a group and each left translation L:x i—• ax is an affine transformation. A left invariant affine structure (G, Γ, Φ,Φ) is called normal, if for any x, y Ε G and t Ε T, where Τ is a maximal abelian normal subgroup of G, there exists unique ζ Ε G such that φ(ζ) — φ(y) — φ(^(χ)) — φ{χ) for every φ Ε Φ. A point α Ε Μ C G is called an inner point of Μ if for any φ Ε Φ, such that a e 0 ( R ) there exist α, β Ε R, α φ β, for which α Ε φ((α,β)) and φ((α,β)) C Μ. The set of all inner points for Μ is denoted Int(M). Hence we have the topology in (G, Γ, Φ, Φ) which is not a priori connected with any kind of order in G. Now we take the second step in the following way. Let G be an abstract group equipped with a left invariant affine structure (G, Γ, Φ, Φ). Suppose that G has a left invariant order Ö — {Px : χ Ε G} satisfying the following conditions: A O i . Ρ = Pe is a cone with acute vertex e. A 0 2 . The affine hull of Ρ is equel to G. A 0 3 . The set Ρ is closed, i.e., G\P

= Int(G \ P ) .

AO4. The cone Ρ is elliptic, i.e., for any two x, y Ε Ρ, χ, y φ e, χ φ y, there exists an affine transformation / Ε A f f ( G ) such that /(e) = e, f(x) = y, m

= p-

AO5. The order Ο is Int-homogeneous. We have the following Theorem 8.1. Let G be an abstract group equipped with the three-dimensional normal left invariant affine IONIN structure {G,T, Φ,Φ) and left invariant order Ό satisfying the conditions AO\ -AOThen G admits the structure of the three-dimensional connected simply connected solvable Lie group with complete left invariant affine structure, generated by the IONIN structure (G, Γ, Φ,Φ), and left invariant flat Lorentz metric (·,•) such that in some global affine chart xi,x2,x3 3 X = (X1,X2|®3).

(Ξ, O X =

SIFC^C*,

i,k—1

9ik = COUSt.

Semigroups in foundations of geometry and axiomatic theory of space-time

69

Hence the order Ό is G-invariant and causal with respect to metric (,), and the group G is simply transitive subgroup of the group of all isometries of pseudoEuclidean space (G, (·,·)). Proof. See [8, 19].

•

Theorem 8.1 shows that the pseudo-Euclidean geometry in R 3 can be defined without the assumption that the Lie group G of invariance of the order Ό is abelian. This is the usual condition in axiomatic theories of relativity (see [1]).

9. Axiomatization of the special theory of relativity The system of axioms which we present below is based on the definition of spacetime given by A.D. Alexandrov (see [1]). According to this definition, space-time is the set of all events in the universe A4, abstracted from all their characteristics except those determined by relations of action of some events on others. In other words, the space-time structure of the universe is nothing else but its cause and effect structure taken in the corresponding abstraction. 9.1. Initial notions. M. will denote a set, x, y,... elements of Λ4, and Px, Py,... subsets of Λ4. We set Ό — {Px : χ G ΛΊ}. Finally, Aut(C) is a subset of Mm. The space-time of the special theory of relativity is a structure (M,0,Aut(0)). 9.2. Axioms of order. Αχ. The family of sets Ο = {P x : χ € ΛΊ} gives a nontrivial order on ΛΛ, i.e., 1) Ρχ φ {x}, 2) the conditions ( 0 1 ) - ( 0 3 ) hold. Αι. The order Ό is dense. 9.3. Axioms of topological and group structures. Let Β be a family of sets of Μ of the form (Px Π P~) \ Axy, where Axy is the union of all linear ordered intervals Pa Π Ρ\, C Px Π Py,b € Pa such that either a — χ or b = y. Consider on M. the topology T< with subbasis B. A3. (Λ4,Τ R n is constant. Any such family ( / i ) · · · ι fn) is called a chart.

3) A transformation of coordinates corresponding to two charts is smooth (i.e., a C°° map). Let Tx — { / € Τ : x G dom(/)}. The vector space of all linear functionals X: Tx —• R satisfying X(fg) = (Xf)g(x) + f(x)(Xg) is the n-dimensional tangent vector space TxAi. If / € Tx then we obtain an element df(x) G Τ*ΛΊ given by (.df(x), X) = fix). If A C Tx then a function / G Tx \ A is said to be a boundary element of A if there exist fk G A,k = 1,2,... for which limfe-^ dfk(x) = df{x) (with respect to usual topology of T*M). A function / G Tx is called locally isotonic in a point χ (resp.: locally antitonic in a point x) if there exists a neighborhood of χ in which α X b implies f(a) < f(b) (resp.: f(a) > f(b)). A function / G Tx is locally isotonic in a domain if / is locally isotonic in each point χ of this domain. The set of all locally isotonic functions of T x is denoted by T^. Let / G Tx . A function f € Tx is said to be strongly isotonic in a point χ if there exists a neighborhood U C T*M of df(x) such that for all g Ε J-x the relation dg(p) G U implies g G . APz. (ΛΊ,Τ^) has the smooth structure T. AP4. For every χ Ε Μ the set of all strongly isotonic functions in χ is not empty. AP5. If / G T x is a boundary element of the set of strongly isotonic functions and of the set of strongly antitonic functions in x, then df(x) = 0. APq. For any point χ Ε Μ the set Qx is a pseudogroup of diffeomorphisms g:U —• V with U, V C M) satisfying the following conditions: 1) g(x) = x.

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2) If fv is strongly isotonic in χ, then the function fu = fv°g is also strongly isotonic in χ for any g 6 Qx. 3) If fu and fv are boundary elements in a point χ of the set of functions which are strongly isotonic in z, but if fu and fv are not strong isotonic in χ themselves, then there is a g € Qx for which dfu = dfydg{p) Theorem 10.2. Suppose that the system of axioms AP\-AP§ hold. Then can be equipped with a pseudo-Riemannian structure (·,·)χ of signature (H— ... —), unique up to a conformal factor, such that (Μ,Τ^,Τ, (·,·)χ) is a time-oriented manifold. Proof. See [36].

•

11. Causal categorical theory of the space-time In Sections 9 and 10 we gave axiomatizations of special and general relativity. We see that the systems of axioms of these theories differ significantly. The system of axioms for special relativity contains fewer primary notions and relations, is simpler, and leads directly to the ultimate goal. In the case of general relativity the difficulty lies in the introduction of a smooth structure. Does a unified way of axiomatization of these different physical theories exist? An affirmative answer is provided by the language of topos theory [40] which gives a unified way of axiomatization of the special and general relativity, the axioms being the same in both cases. Selecting one or another physical theory amounts to selecting a concrete topos. In this section we give a topos-theoretic causal theory of space-time that is the analog of the set-theoretic axiomatization of pseudo-Euclidean geometry from Section 8. Let £ be an elementary topos with an object of natural numbers, and let R t be the object of continuous real numbers [39]. An affine morphism a : Κχ —> R τ is a finite composition of morphisms of the form 1r t >

® o ( A x 1 r t ) o j,

φ ο ( 1 Κ τ χ μ ) ο j,

where Θ, are the operations of addition and multiplication in R τ respectively, λ, μ are arbitrary elements in Rτ, and j : R t — 1 x R t is an isomorphism. Let Γ be the set of all affine morphisms from R t to Rt· An affine object in £ is an object a together with two sets of morphisms: Φ C Homf(Rr, α),

Ψ C Hom f (o, R t )

such that the following conditions hold: 1) For any φ G Φ, φ € Φ we have ψ ο φ e Γ. 2) If / G Hom£(R T , a) \ Φ then there exists φ € Ψ such that φ ο f —* a, where χ : 1 —> a is an arbitrary element, such that: 1) χ e px. 2) If y G px, then ζ G py implies ζ G px. The order (Ρ, {p x }) is denoted as Ό. A morphism / : a —> a is called affine, if φο f οφ G Γ for any φ Ε Φ and φ G Φ. We denote the set of all affine morphisms by Aff(o). Let A C AfF(a) consist of pairwise commuting morphisms. An order Ο is invariant with respect to A if for any px,py there exists gxy G A such that gxy opx = Py

A morphism / : α —> α preserves an order (9, if for each px there exists py such that / ο px = py. The collection of all morphisms preserving an order Ö that is invariant with respect to A is denoted by A u t ( ö ) . Let ! : R τ —> 1, and let R + be the subobject of the object R T consisting of those t for which 0 < t (see definition of order in Ry in [39]). A ray is a morphism λ: R+

RT

a,

where φ G Φο C Φ, and for any φ G Φο there is no χ : 1 —• a such that φ = ίο!. An order Ö is called conic if 1) for every y G px there exists a ray Λ C px such that x,y G Λ, and 2) χ is the origin of λ, i.e. if μ is a ray and y G μ C Λ, μ φ Λ, then χ φ μ. An order Ο has acute vertices or is pointed if for each px there does not exist φχ G Φο such that φχ C px. An order Ο is complete, if for any element ζ : 1 —> α and px there exist different elements ux,vx : 1 —• a and φ Ε Φο such that z,ux,vx Gφ and ux,vx G px. An element u G px is called extreme if there exists φ G Φο for which u G φ, but y £ φ for all y Epx,y^u.

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A conic order Ο is said to be strict if, for each nonextreme element u G px, and t i e p u v / u , and each ray Λ with origin u such that t> € λ, there exists an extreme element w € λ, and w Ε px. Definition 1 1 . 2 . An affine object a with an order Ö, which is complete, strict, conic, has an acute vertex, and is invariant with respect to Λ is said to be Lorentz if for each χ : 1 —» a and each extreme elements u,v G px, where u , v / i there exists a f Ε A u t ( ö ) such that f ou = v, f οχ = χ. Theorem 11.3. A Lorentz object in the topos Set is an affine space admitting a pseudo-Euclidean structure defined by a quadratic form Xq — xf, where η is finite or equal to oo, and Aut(C) is the Poincare group (see [3]). A Lorentz object in the topos Τ ο ρ ( Λ ί ) is a fiber bundle over M. with fibers equipped with an affine structure and a continuous pseudo-Euclidean structure of finite or infinite dimension. It is quite possible to take not only the topoi Set, Β η ( Λ Ί ) , or Τορ(ΛΊ), but also any others that have an affine object. The existing categorical determination of the set theory and determination of Τορ(ΛΊ) between elementary topoi gives the possibility to speak about the solution of problem of categorical description of the theory of relativity. Theorem 11.4. If £ is a well-pointed topos satisfying the axiom of partial transitivity with a Lorentz object a, then £ is a model of set theory Ζ and a is a model of the special relativity. If € is α topos defined over Set that has enough points and satisfies the axiom (SG) (see [41],) mth a Lorentz object a, then £ is a topos Τορ(Λ / 1) and a is a model of the general relativity.

References [1]

Guts, A.K., Axiomatic Theory of Relativity. Russian Math. Surveys 37 (1982), 41-80.

[2]

Hofmann, K.H., Lawson, J.D., The local theory of semigroups in nilpotent Lie groups. Semigroup Forum 23 (1981), 343-357.

[3]

Abdrahimova, N.R., Guts, A.K., Shalamova, N.L., The synthetic theory of affine Lorentz manifolds and ordered Lie groups. Soviet Math. Dokl. 38 (1989), 583587.

[4]

Abdrahimova, N.R., The classification of affine orders on 3-dimensional connected simple connected solvable Lie groups. IX All-Union geometrical Conference. Kishenev, 1988. - P.3 (Russian).

[5]

Abdrahimova, N.R., Guts, A.K., The description of affine orders on threedimensional solvable Lie groups. All-Russian Institute of Scientific and Technical Information, Moscow, Paper No. 1467-B94, 1994, 35pp. (Russian).

[6]

Yamaguchi, S., On complete affinely flat structures of some solvable Lie groups. Mem. Fac. Sei. Kyuchu Univ. Ser. A 33 (1979), 209-218.

Semigroups in foundations of geometry and axiomatic theory of space-time

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Guts, A.K., The uniqueness of Abelian affine chronogeometry. Siberian Math. J. 34 (1993), 84-86 (Russian). Shalamova, N.L., The orders and order automorphisms in homogeneous affine manifolds. Candidate dissertation, Novosibirsk, Institute of Mathematics (1992) (Russian). —, The chronogeometry of solvable Lie groups. Ail-Union Institute of Scientific & Technical Info (VINITI), Moscow, (1989), paper No. 4196 - B89 (Russian). —, The homogeneous affine left invariant order in solvable Lie groups. All-Union Institute of Scientific & Technical Info (VINITI), Moscow (1991), paper No. 1925 - B91 (Russian). —, The externally homogeneous order in affine space. All-Union Institute of Scientific & Technical Info (VINITI), Moscow (1991), paper No. 1926 - B91 (Russian). Shaidenko, A.V., On the problem of mappings of families of cones. Siberian Math. J. 20 (1978), 118-125. Alexandrov, A.D., Mappings of ordered spaces. Proc. Steklov Inst. Math. 128 (1972), 1-21. Guts, A.K., Mappings of an ordered Lobachevsky space. Siberian Math. J. 27 (1986), 347-361. Gavrilov, S.P., The left invariant metrics on solvable simple connected 3-dimensional Lie groups. Theory of Relativity and Gravitation. Kazan University, 22 (1985), 31-64 (Russian). Guts, A.K., Ermakova, E.N., The homogeneous affine causal orders on solvable three-dimensional Lie group. III Ail-Union Pontriagin school, Kemerovo, University, 1990. - P. 22 (Russian). —, The homogeneous affine causal orders on solvable three-dimensional Lie group. All-Union Institute of Scientific & Technical Info (VINITI), Moscow (1993), paper No. 1841 - B93 (Russian). Ionin, V.K., One method of definition of affine structure. Geometr. Sbornik, Tomsk 23 (1982), 3-16 (Russian). Guts, A.K., Shalamova, N.L., The affinelly ordered Lie groups and axiomatization of pseudo-euclidean geometry. Dokl. Russian Acad. Sei. 332 (1993), 283-285 (Russian). Guts, A.K., Automorphisms of local order structures on hyperbolic manifolds. Siberian Math. J. 28 (1987), 740-742. Alexandrov, A.D., Cones with transitive groups. Soviet Math. Dokl. 10 (1969), 1460-1463. Vinberg, E.B., Theory of homogeneous cones. Trans. Moscow Math. Soc. 12 (1963), 43-103. —, Theory of homogeneous cones. Trans. Moscow Math. Soc. 13 (1965), 56-83. Guts, A.K., Groups of order automorphisms of affine space and their discontinuous extensions. Soviet Math.Dokl. 32 (1985), 537-541.

A.K. Guts —, The order and space-time structures on homogeneous manifolds. Doctor Math. Dissertation, Novosibirsk, Institute of Mathematics (1987) (Russian). —, The disconnected order in affine space and its automorphisms. All-Union Institute of Scientific & Technical Info (VINITI), Moscow (1992), paper No. 3427 - B92 (Russian). Shaidenko, A.V., Mappings almost preserving cones. Proceedings of the Institute of Mathematics, Siberian Branch, Russian Acad, of Sei. 9 (1987), 151-153 (Russian). —, Mappings with condition on distortion of cones. Ail-Union Conference on Geometry at the large, Novosibirsk, Institute of Mathematics, 1988. - P. 132 (Russian). Guts, A.K., Dense order in the Lobachevsky space. Siberian Math. J. 29 (1988), 684-686. Levichev, A.V., Some conditions under which pre-order is defined by cone. Siberian Math. J. 22 (1981), 116-126 (Russian). —, The methods of investigation of causal structure of homogeneous Lorentz manifolds. Siberian Math. J. 31 (1990), 39-54 (Russian). Hilgert, J., Hofmann, K.H., Lorentzian cones in real Lie algebras. Monatsh. Math. 100 (1985), 183-210. Hilgert, J., Hofmann, K.H., On the causal structure of homogeneous manifolds. Math. Scand. 67 (1990), 119-144. Hilgert, J., Invariant Lorentzian orders on simply connected Lie groups. Ark. Mat. 26 (1988), 107-115. Neeb, K.-H., Conal orders on homogeneous spaces. Invent. Math. 104 (1991), 467-496. Pimenov, R.I., Axiomatics of general relativistic and Finsler space-time by means of causality. Siberian Math. J. 29 (1988), 133-143 (Russian). —, Anisotropic Finsler extension of general relativity as order structure. Komi Scientific Center, Syktyvkar, 1987 (Russian). —, The foundation of a theory of the temporal universe. Komi Scientific Center, Syktyvkar, 1991 (Russian). Stout, L.N., Topological properties of the real numbers object in a topos. Cahiers Topologie Geom. Differentielle Categoriques 17 (1976), 295-326. Goldblatt, R., Topoi. North-Holland Publishing Company, 1979. Johnstone, P.T., Topos theory. Academic Press, 1977.

On mathematical foundations and physical applications of chronometry Alexander V. Levichev

1. Motivation and introduction By way of introduction I begin with justifying the statement "The chronometric theory by I. SEGAL is the crowning accomplishment of special relativity" which I made in the title of my earlier survey article [Le93]. The term "world" which we shall use below is close to the term "space-time" [SaWu, p. 27]; however, it does not assume that we fix a particular Lorentzian metric tensor field from the conformal class [Se76, GuSt]. Since the present article is dedicated mostly to the conformal compactification Mo of the Minkowski world Mo and its universal coverings, I shall not go into the general definitions specifying space-times and causality. I begin with Newtonian world N, namely, a 4-dimensional affine space equipped with a "Newtonian causal structure" [Se76, p. 23]. The latter is defined as the family { J + : χ € Ν } of closed half-spaces with parallel boundary hyperplanes. An "event" χ belongs to its "future set" J + . The symmetry group S is the 11dimensional Galilean group (including scaling) [GuSt]. The group S yields the Euclidean geometry of absolute 3-space. Next, Minkowski world Mo is defined as 4-dimensional affine space, but its causal structure { J + : χ £ Mo} consists of elliptic convex cones such that J + is obtained from J + by parallel translation ζ ι—• ζ + y — x. The symmetry group Ρ is the 11-dimensional Poincare group (included scaling). It was H . MINKOWSKI who insisted on the "absolute" status of space-time (instead of that of space). He also raised the question of transforming the structures involved into less degenerate ones (for his "anti-deformation" thesis, see e.g. the introduction of [0rSe]). The well-known result by A . D . ALEXANDROV (see e.g. [Gu] for exact references on the subject of this and other matters discussed in this section), later rediscovered in a weaker version by E.C. ZEEMAN, originated the method of deriving the geometry from the causal structure. The usage of Ρ instead of the "standard" 10-dimensional Po is motivated by this very method when applied to Mo- It is worth noting that matters are quite different with the symmetry group of Newtonian world: there are many transformations of Ν which preserve the causal structure even though they do not belong to S.

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Finally we discuss briefly the main aspects of chronometry. Its world Μ consists of the Einstein space-time Ε as the underlying conformal manifold. The metric on Ε is dt2 — ds2, where t is time and ds2 is the Euclidean on S 3 induced by the standard immersion of § 3 into R 4 . A future time direction in Ε is fixed, and a "future cone" appears in every tangent space of M . In Μ one can define "future sets" [Se76] in a fashion similar to the previous cases. This structure gives rise to the symmetry group G which is now the universal covering of SU(2,2). It acts globally on M . These and other notions will be defined in the next section in a greater detail. The Minkowski world is conformally imbedded into Μ via the "Cayley transform". The radius R of the space § 3 does not depend on the chosen metric from this conformal class, i.e., the metric in which it is calculated. In other words, R is a conformal invariant [Se82]. It is convenient to use natural chronometric units in which R, the speed of light c, and the Planck constant h are equal to 1. We deote by Κ the 7-dimensional Einstein isometry group. A subgroup of a Lie group is said to be essentially compact if its image under the adjoint representation is compact. Now Κ is a maximal essentially compact subgroup of G. It consists of translations in time and rotations in space (—• 2.2) 1 . Several features are important in applications. I shall indicate them without defining explicitly the mathematical definitions of the notions involved. I shall reproduce a small piece from [Se91] almost without changes. The chronometric energy Η is the generator of time in E. Relative to any point of observation in M , the Minkowski world Mo is imbedded P-covariantly, and the relativistic or Minkowski energy HQ is the generator of time in Mo relative to the Lorentz frame in Mo, which, at the point of observation, osculates the frame defined by the space-time splitting in E. For any unitary positive-energy representation of G, the corresponding Einstein energy exceeds the Minkowski energy by an amount that vanishes infinitesimally but increases with the spatial support of the state in question in terms of the appropriate quantum mechanical consideration. The inertial mass of a cosmologically long-lived particle is represented in accordance with Mach's principle as its interaction energy with the cosmic backround and is correspondingly only K-invariant, implying approximate local Po-invariance of its rest mass. Additional background on chronometry is given in SEGAL'S book [Se76] and [PS-I,II; P-III, P-IV, 0rSe, Se86]. In these articles the physical particles have been modelled, in accordance with the thrust of decades of theoretical investigation in this area, by induced bundles (—> 4.1) over causally oriented space-times. Let me now conclude with the justification of the expression "crowning accomplishment of special relativity". Firstly, the conformal group is semisimple, in constrast with the Poincare group. Hence it cannot be regarded as resulting through a contraction process from a non-isomorphic Lie group of the same dimension. Secondly, it arises as maximal local causal group of the special relativistic world in which only the 11-dimensional Poincare group can be globally realized. 1

I shall try to make the presentation as self-contained as possible. Cross-references in the text are indicated by arrows —>.

On mathematical foundations and physical applications of chronometry

79

When compared with other theories based on the world of special relativity or particular space-times of general relativity, chronometry has other preferable features, we mention only a few: —the absence of the fixed Lorentzian structure which seems to be connected with a concrete metric observer [Se76] in the world under consideration, —a better unification of elementary particles (—• 6.1 — 6.3), —the existence of leaking (6.1,6.3) which gives kinematic explanation of several decays (—> 6.2,6.3). In discussing chronogeometry it is worthwhile to mention that there are exactly four 4-dimensional real Lie algebras which admit an invariant nondegenerate form of Lorentzian signature [GuLe, Le86']. Such a form is well-known to correspond to a biinvariant metric on the Lie group in question. The above and several other facts (see [Se76, Se86, GuLe, Le86, Le86', Le93] and references therein) support the conclusion that Μ is the "basic world" of nature and that it is, together with the Minkowski space-time, one of the most important ones in the applications. Summing up we note that chronometry is derived from very general considerations of causality, stability, and symmetry. Therefore, it is somewhat abstract, and its empirical implications call for further development. Indeed I consider it as one of my goals in the present survey to convince the specialists in relativity that chronometry is an effective point of departure for cosmology and that they should take part in its implementation and further development. Chronometry, like special relativity and quantum mechanics, may initially appear contradictory to accepted doctrine. But its application to extragalactic astronomy (—• 7, [SeNi] and references therein) has shown that it is capable of precise and detailed predictions regarding the cosmic redshifi (—• 7) and other directly measured quantities, in spite of its lack of adjustable cosmological parameters. Remark. I use the opportunity to mention that the proof of Lemma 2 in [Le86'] should be slightly modified. Notably, the subspace S occuring in this article need not be a subalgebra. I am indebted to A . K U Z E M C H I K O V who has pointed this out to me.

2. Synthetic geometry 2.1. The "Hermitian" model of the Minkowski world Fix the following representation of Pauli matrices:

The set of all 2 by 2 Hermitian matrices is denoted by H; each X from Η has a unique decomposition X =

xma.τη·

(1)

A.V. Levichev

80

Here and henceforth we assume the Einstein summation rule. (We allow also the use of lower indices for coordinates of the vectors under consideration. Let an orthonormal coordinate system with basis eo, ej, β2, e^ be chosen in the Minkowski world Mo- The map from Mo to Η which takes χ = xmern into X via equation (1) is a linear bijection. Then detX = xq2 — x\2 — Χ22 — Χ32· Let the future cone in Mo of an event χ be denoted by J+. Then the causal relation y Ε J+ in Mo holds if and only if the matrix Υ — X is positive semidefinite [Se76]. The restricted 10dimensional Poincare group Po is the semi-direct product Po = Η * Λ0 of the vector group Η and the restricted 6-dimensional Lorentz group Λο· Its universal covering P 0 equals H x S L(2, C), where (F, L) e H x SL(2, C) acts in Η by Η h-» LHlT+F. The simply connected Poincare group including scaling is denoted by P . It is the semi-direct product of Η by the group Λ = R1 χ SL(2, C), and the element (F, (t, L)) e Ρ acts on Η as Η elLHLT + F with t € R 1 and (L, F) € P 0 . The conventionally defined 7-dimensional Lorentz group including scaling is denoted by A. It is a well known fact that Λ 0 = SL(2, (—X°, X) in Mo is a typical example of an anticausal

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transformation. If Minkowski world is represented by H, then the group of all causal and anticausal transformations on it is denoted by G + ( H ) . It is generated by the connected component GQ (H) of the identity together with To and space reversal P 0 : ( x ° , f ) ^ ( Λ - x ) . Note G+(H)/Gjj"(H) ^ Z 2 x Z 2 . P + (respectively, G + ) denotes the group generated by Ρ (respectively, G) and these discrete symmetries. _ We now define P + and G + : By a basic pair of discrete symmetries (P,T) at a point Η of Η we mean an ordered pair of causal, respectively, anticausal transformations on Η of the form Ρ = S~1PoS,

Τ = 5_1Γ05,

where S is a transformation in Ρ that carries Η into 0 and PQ and To are the transformations on H: PQ : Η ^ txH - Η,

T 0 : Η ^ Η - tr H.

Since PQ and To are causal and anticausal, respectively, the same must be true of Ρ and T. Note that Ρ2 = 1 = T2,PT = TP. It follows that a basic pair of discrete symmetries at a given point is unique within conjugation by a causal transformation connected to the identity that leaves the point fixed. Ρ and Τ generate F = Z 2 x Z 2 . The semi-direct product of Ρ with F gives P + . The action of F as a group of automorphisms in Ρ is canonically extended to an action on Ρ and the^semi-direct product of F with Ρ relative to this action forms the universal cover P + . The natural projection of P + onto P + is independent of the choice of a basic pair and of a base point H. The notion of a basic pair of discrete symmetries at a point is now canonically extended to U(2). Introduce the following transformations of U(2): P0 : U

(detU)U~\

T0 : U ^

(detU^U.

A basic pair of discrete symmetries at a point V of U(2) is defined as an ordered pair of Ρ = S'iPoS,

Τ = S~1ToS,

where S is a transformation in G that carries V into 1. Such a basic pair intertwines with one on H, when V is in the range of the Caley map (—• 2.2). It follows from this that the basic pair of discrete symmetries at a point of U(2) is unique within conjugation by an element of G leaving the point fixed. One gets G + as the semidirect product of F with G, where the action of F on G is defined by a canonical extension to G of the action of F on G just defined. Corollary 2.1.4. G + acts on Μ causally or anticausally, extending the action of G given in Theorem 2.1, and with isotropy subgroup P + . Moreover, there is a causal equivalence between an open orbit of this isotropy group and Η that intertwines the respective actions of P+ on them, canonically identifiable with that given in the theorem. •

On mathematical foundations and physical applications of chronometry

85

In the proof in [PS-I] the β (—> 2.2) is extented to P + with the conservation of the intertwining relation of the theorem. The χ of the theorem extends similarly. Note that Corollary 2.1.2 does not extend. The center of G + is generated by 2 ζ and η. Due to its failure to commute with P , the element ζ is no longer in the center. In terms of the connected group, the result may be stated as follows. Corollary 2.1.5. The elements ζ and η of the center of G are both invariant under T, and η is invariant under P, but Ρ~*ζΡ = ζη. • As a consequence we note: R e m a r k 2. Every causal or anti-causal transformation on Mo corresponds to a unique such transformation on Μ that agrees on Mo regarded as imbedded in Μ causally, with the given transformation.

2.4. More on chronometric geometry The map (e l t , V) euV : U( 1) x SU(2) -» U(2) is a double covering. Denote the domain manifold by M^2^. It is equipped with an (infinitesimal) causal structure and the parametrization (ti_i, UQ) χ («1,142,^3,^4) subject to the condition that +UQ = u\ -\ bti4 = 1. Here is the presentation of a general element in U(2): (u_ χ + iu0)(iuiai

+ iu2a2 + iu3a3 + u4).

When Mo is embedded into U(2) via the Cayley map then the coordinates X0,XI,X2, X3 in Mo agree with the um(m = 0,1,2,3) within terms of second order in the xm, (—• 3.3). Note that although from a Minkowskian standpoint, (7r,l) appears infinitely distant in time and ( 0 , - 1 ) appeares infinitely distant in space, from the point of observation (0,1) which corresponds to the origin in Mo, they both cover the same point —1 of U(2); the space-time separation in Μ is only infinitesimally the same as that in MoFour presentations of the infinitesimal causal symmetries are given in Table I of [SeJa]. It is presupposed there that the real projective quadric q-1 + Qo - Qi - 92 - 4 - 94 = 0, where the qm parametrize a point in projective 5-space has a unique causal structure invariant under the group of projectivities that leave it fixed (within reversal) which is locally 5 0 ( 2 , 4 ) and is equivalent to U(2) [Se76]. The vector fields on U(2) corresponding to the operators TSQSDM

-

Qmds

are consequently infinitesimal causal symmetries and, when lifted to are denoted as Lsm, forming the entries of Table I, column 1 (here and further on, the row e of the six elements em equals (1,1,—1,-1,—1,-1)). Column 2 gives the

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expressions for the Lsm as linear combinations of the Xm where the latter are the generators of one-parameter groups of transformations lifted from the action U ·—• Uexp(itam) on U(2). Expressions for the Xm in terms of the Lsk are -Xo = £-10, Xi — Lu - L23, X i = •£'24 — £31, = ^34 — £12· In the third column of Table I the Lsm are antirepresented as concrete matrices in su(2,2), with commutation relations (3.1) below. The fourth (and the last) column of Table I expresses Lsm as vector fields in Mo. The flat limit of the Lsjn is seen from column 4 by replacing xm by xm/R in which R is the "radius of the universe" S 3 in laboratory units, then rescaling Lsm appropriately, and finally forming the limit as R-* 00.

3. Analytic geometry 3.1. Preliminaries It is convenient to make an explicit distinction between a generator of the group G and the corresponding vector field on M . Generators of the abstract group will be denoted by boldface letters; corresponding vector fields on Μ by the same nonbold letter. Note that the mapping Χ η I from Q to the space of vector fields on Μ is an anti-representation of the Lie algebra Q. Thus, in terms of the 50(2,4)-generators Lj m , the commutation relations [L im ,L m fc] -- -e m Lifc

(1)

are opposite in sign compared with the ones between the corresponding vector fields Li m • [Lim, Lmk]

' emLik·

If the L a m are regarded as elements of su(2,2) as in part of Table 1 of [SeJa], they satisfy the set (1) of commutation relations.

3.2. Infinitesimal causal symmetries in polar coordinates Table I is supplemented by a presentation of vector fields Lsm in polar coordinates (which are convenient when regarding M) in Table II of [PS-I]. These coordinates have the following relations to the u m : elt =

+ zuq,

«ι = sin ρ sin 0 cos

u2 = sin ρ sin 0 sin

On mathematical foundations and physical applications of chronometry U3 = sin ρ cos Θ,

U4

87

= cos p;

here 0 < θ, φ < π, 0 < φ < 2ττ.

3.3. Relations between the x m and the um The

standard imbedding of

Mo into Μ takes an ( X Q , X I , X 2 , X S ) to (£, U), where

1 = p( 1 — x2/4), Uj = pxj,U4 = p( 1 + x1 / 4), 2

ρ = ((1-χζ/4)

2

ο _1/2 t + XQ ) , e = u_i + iuo,

U = M4 + tiifci + U2&2 4-

— π < ί < π,

and bm stands for (—• 2.1). This mapping, followed by the covering map of Μ onto M ( 2 ) (whose coordinates are the um) is one-to-one. A point of M ' 2 ) corresponds to a point of M o if and only if U-1+U4 > 0; the Minkowski coordinates are recovered by the equation xm = 2u m (u_i + U4) - 1 . The function ρ is strictly positive on M o and is extended smoothly to M^ 2 ) by ρ — (u_i + u 4 )/2, and then to Μ via polar coordinates (—> 3.2).

3.4. Expressions for the rightand left-invariant symmetries The above introduced vector fields Xm are left-invariant on U ( 2 ) though generating right translations on it. In Table III of [PS-I] the corresponding generators Ym of left translations are presented (as well as the Xm) as linear combinations of the L s k and as concrete vector fields in polar coordinates. Their commutation relations are given therein.

3.5. Actions of relevant vector fields on the u m Table I V in [PS-I] gives Lum for various vector fields L involved.

3.6. Basic flat and inverted generators Conformal inversion in M o

is defined as the map χ 4x/x 2 : M o \ { 0 } —• Mo- This map extends uniquely to the everywhere-defined smooth map U 1—» —U/detU on U(2). A corresponding map on Μ is ambiguous within an element of the center of G . In [PS-I] this element is standardized and causal inversion is defined on Μ as the map (t, V) 1—• (—t, —V). Conformal inversion on M 0 carries the dm into vector fields dm that are sometimes called "special conformal (infinitesimal)

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transformations" when extended by continuity to be everywhere defined on MoThe generators of the Lie algebra of G that correspond to the vector fields dm and dm on Mo are called the basic flat and inverted generators, and denoted as T m and T m , respectively. In these terms the relation L _ i m = T m — T m (m = 0,1,2,3) stands. In Table V of [PS-I] these eight generators are expressed in terms of the L . m and in terms of the dm.

3.7. Metrics, measures, forms The following objects are introduced as standard in the corresponding section of [PS-I]: The flat metric on Mo, the curved metric on M, the flat measure in Mq, the curved measure in M , and so on.

3.8. Enveloping algebra relations Let us use the following notations for designated elements of the universal enveloping algebra S of the Lie algebra Q: Τ

— Li-14,

Τ

—

Γ¥ΐ2 πρ2— AfT 5.1) is considered.

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5. Spaces of scalar representations 5.1. Scalar representations of SU(2, 2) Scalar representations are representations which are induced by one-dimensional representations of an isotropy subgroup. By Section 2.2 the isotropy subgroup of the point ρ = (π, 1) in U(2) is isomorphic to P = H * (R 1 χ SX(2,C)). A one-dimensional representation of G p has the form Rw(X({t,L),F))=exp(ut) for a unique complex number ω, where χ is the isomorphism of Ρ onto G p introduced in Section 2.2. A representation induced from Εω is said to have conformal weight ω. R e m a r k 1. This notion plays an important part in the studies of conformal bundles. A representation β of Ρ is said to be of (conformal) weight w, w being a given complex number, in case R(S\) = AWI for λ real and positive, S\ denoting the transformation χ • Xx in the Minkowski space. T h e o r e m 5 . 1 [PS-I]. In terms of the left parallelization the scalar representation of weight ω takes the form U(g):4>y->i7 with η(Ζ) = \det(C'Z +

D')\~",p(g-lZ)

• In the remainder of this section the scalar representation is treated in terms of the flat parallelization. Left- and flat-parallelized sections of an (abstract) section Φ are denoted by φ and ψο· I recall that G stands for SU(2,2) and denote its left action on U(2) by Z>-^gZ. T h e o r e m 5.2 [PS-I]. Assume that Ζ is obtained from h € Η via the Caley transformation (—* 2.2), and

Then the relation φ0(Ζ)=ρωφ(Ζ)

On mathematical foundations and physical applications of chronometry holds with ρ — (u_i + u^)/2.

The scalar representation = (det (^ψ

υ(9)(φ0)(Ζ)

+

93

of weight ω is defined by (Q-'Z).

•

5.2. Covariance of wave operators T h e o r e m 5 . 3 . [PS-I]

In the scalar representation

of weight ω — 1 the equality

[dU{Lc + 1), dt/(S)] = —2u-\u^dU{Lc

+ 1)

holds. If ωφ 1, the left hand side is not the product of a function with d£/ u l (L c + 1 ) .

•

C o r o l l a r y 5 . 3 . 1 . (5.3.2 of [PS-I]) dU(Lf)

=p2dU(Lc

+ l)

for ω = 1.

(1) •

This means, in particular, that the space S admits an interesting invariant subspace that may be correlated with solutions of the wave equation. On the one hand, it consists precisely of all sections annihilated by the flat wave operator dU(Lf). Note that this operator acts on all of M , and not merely on the submanifold Mo, on which it coincides with the usual d'Alembertian. On the other hand, it can be described equivalently as the space of sections annihilated by dU(Lc + 1). The relation (1) is referred to in [Se87] as an example of a bundle-invariant property. It is also noted there that L c + 1 does not quite correspond to the usual wave operator in the Einstein universe but does give the temporal evolution that is the main^purpose of the wave operator to define. This evolution is a special case of the G-transformational properties. The situation is similar for the Dirac, Maxwell, and so-called higher spin equations, which (in their "massless" forms) correspond to irreducibly invariant, unitarisable subsjgaces of the section spaces of bundles induced from other representations of Ρ that are trivial on the translations, and are holomorphic on the homogeneous Lorentz group cover, realized as SL(2,C), and have a uniquely determined conformal weight [Se87]. For the w = 1 scalar bundle the following hermitian forms are of particular importance: ((a,b»

= J ((dU( L c ) + 1 )a)bd4u,

(a, b) = —i I äb + i I ab.

(2)

(3)

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A.V. Levichev

In the expression above, Ά stands for XQÜ, integration in (2) (respectively, in (3)) is over Μ (respectively, over § 3 ); the sections a, b are supposed to be left-parallelized. In (3) each of the a, b satisfies an additional condition of annihilation by the wave curved operator. T h e o r e m 5.4.

((·, ·)) and (·, ·) are invariant under the representation U of G.D

One of the next important steps is intertwining the (global) forms (2) and (3) with the usual (local) forms on Minkowski space, expressed in the flat parallelization, and determination of the curved and flat energies in terms of Cauchy data (the reader is referred to Theorem 5.6 of [PS-I]). The proof that these hermitian forms actually become definite on distinctive positive- and negative-energy invariant subspaces (—• 5.3) is not reproduced here. It is mentioned in [PS-I] that such a proof seems to require a special (K-finite) basis or the use of Fourier transforms and certain integral special function identities.

5.3. Factors of composition series Among conformal scalar bundles those of weights w = 1 (—> 5.2) and w = 2 (—• 6.2) are of greatest importance. The determination of factors and the order in which they occur in the corresponding composition series are important in applications (-6.1).

It is possible to treat the situation in terms of the corresponding representations of the Lie algebra Q on the K-finite vectors ([PS-I, pp. 135-138], where the "restricted" section space Ε of the scalar conformal bundle of weight w — 1 is introduced). This Ε equals the direct sum of all Ea, the evolved Ag[0, 2) issues from the concrete basis considerations which we skip. It is convenient to set Ea+2 = Ea for all real λ. The factors in case w = 1 are the following: T h e o r e m 5.5. dU is irreducible on each Ea with λ^Ι. Ει has two minimal invariant subspaces W+ and W-. The vector space W+ + W_ is the kernel of dU(Lc + 1) (or equivalently dU(Lf)) in E. The chronometric Hamiltonian Η = idU(X.o) is positive (negative) on W+ (respectively, W-). The restriction of U to (the closures of) W+ and are unitary, with the unitary structures (·,·) and — (·, ·), respectively, (defined in Section5.2). Ei is the sum (not direct) of the invariant subspaces V+, V, VL. Moreover, V Π V+ — W+, V Π V_ = W-, and V+ ΓΊ V_ = 0. The factor spaces V+/W+, V-/W-, and V/(W+ + W_) are irreducible and unitarizable, with the unitary structures —((·, ·)), —((-, ·)), and ((·, ·)), respectively. The spaces V+, V_, and V have no Q-invariant complements for W+ W-, and W+ + , respectively. • For w = 2 the space E2 is the only subspace of Ε which is not irreducible. E 2 equals the direct sum of the spaces V + , V, and V_ which are irreducible and

On mathematical foundations and physical applications of chronometry

95

invariant. The chronometric Hamiltonian Η is positive (respectively: negative) on V+ (respectively: on V_). For the proofs see [PS-I], [Mo90], and references therein. There the subspaces involved are determined explicitly in terms of the K-finite basis. The latter is labelled by quantum numbers (—> 6.4) associated with a system of subgroups 0 ( 2 ) C 0 ( 3 ) C 0 ( 4 ) of K, together with the 0 ( 2 ) subgroup generated by Xo- This basis is used also in the treatment of higher spin representations, and is important in physical applications. The determination of factors of scalar bundles can be used later for representations of spannor (—• 6.2) and plyor (—• 6.3) bundles as has been already done in case of two-dimensional chronometry [0rSe].

6. Elementary particles associations 6.1. The general viewpoint I . E . SEGAL introduces in [Se91] the notion of a clan (consisting of all fields on Μ

having designated transformation properties under G) and considers the fermionic (—> 6.2) and bosonic (—> 6.3) ones. He describes the part of his program as an extension of Wigner's classical formulation of relativistic particles as irreducible unitary positive-energy representations of the Poincare group Po to one in which the causal group G of Μ is substituted for the F V SEGAL emphasizes the fact that the bundle or, equivalently, the transformation group aspect, is no less essential than the pure group representation aspect, and physically more fundamental. Prom here on the use of some terminology from elementary particle physics seems inevitable: see my Remark 3 in Section 6.4. The spatiotemporal labelling of vectors in the induced representation spaces is necessary for the concept of local interaction to be meaningful, and effectively necessary for the treatment of the closely related issue of causality. An example is the difference between the natural models for the electron ve and muon νμ neutrinos that emerge [P-IV, PS87, Se91]. We discuss these notions more explicitly. The starting point is an induced representation U of G (—* 4.1) with V as the representation space. Only representations with a composition series are considered [Se91], i.e., with a maximal chain of invariant subspaces 0 C So C Si C • · · C S n = V .

(1)

A subquotient of U is defined as the corresponding representation on the quotient space S / T between invariant subspaces Τ C S C V . The factors are those subquotients that are irreducible, i.e., for which Τ is a maximal invariant subspace of S. SEGAL distinguishes between an exact particle, which is represented by a vector in the clan and corresponds to a free physical state and a reduced particle, a theoretical entity which is extracted from a clan by formation of subquotients. The factors define the elementary particle spectrum; the stable spectrum consists of those factors that are unitary and have a one-sided frequency spectrum (i.e., the one-sided

96

A.V. Levichev

spectrum of the chronometric Hamiltonian Η (—• 3.8)). The (chronometric) energy of a particle in the (normed) state / is defined as ( H f , / ) where (·, ·) is the positive-definite Hermitian form in the corresponding factor. Although there will in general be many inequivalent (non-conjugate) chains (1), the factors are unique as group representations. Notwithstanding the lack of uniqueness for the maximal chain, there are nontrivial constraints on the order in which the factors occur, corresponding to the order of inclusion of the corresponding invariant subspaces. SEGAL remarks [Se91] that this contrasts greatly with the entirely arbitrary order in which the factors occur in the case of a fully decomposable representation, as in conventional theory. Thus, in the chronometric fermion clan (—> 6.2), the exon χ appears as a bottom invariant subspace or factor, and the electron e as a top factor; in the middle are the muon and the electron neutrino factors, in that order. In the boson clan (—> 6.3), the photon appears as a bottom factor, above which are the bare versions of W and Z. Corresponding to any given chronometric clan is a relativistic free particle family consisting of the direct sum of the stable factors, restricted to the conventional Poincare group Po and fixed in mass (—• 6.2). Because of indecomposability, the action of Po on the clan mixes up the factors, and so is quite different from the relativistic action of Po on the direct sum of the stable factors. In other words, the chronometric free temporal evolution gives rise to apparent particle production within the frame of the relativistic limit. It is called [Se91] indecomposable production, to distinguish it from Lagrangian production of the conventional type; both are causal and covariant. Since indecomposable production is absent in the relativistic limit of the chronometric theory, it appears as a weak interaction in conventional terms. But there are also Lagrangian interactions between neutrinos and other particles (—• 6.4), which would be classified as weak in the relativistic theory.

6.2. The fermion clan The corresponding representation is induced (—» 4.1) to G (—• 2.2) from the spannor representation Σ of Ρ (—> 2.1). The representation Σ is defined as the direct sum of Σ + and of Σ~, where

and Σ - may then be defined, within equivalence, as either the parity transform Ρ ([P-IV]) of Σ + or the complex conjugate representation. The matrix Τ above stands for et^2L where (t,L,F) € P , see Section2.1. R e m a r k 1. I drop the index d = 2 from the appropriate notation of [P-IV], d being the degree of the spannor. Each of the two inducing representations is defined in C 4 . In C 8 = C 4 φ C 4 the discrete symmetries C,P,T act as well [P-IV, Theorem 16.3.1]. The eight-

On mathematical foundations and physical applications of chronometry

97

dimensional spin representation Ξ of G is introduced in [P-IV]. It figures in the following useful statement (where the spannor bundle stands for the corresponding induced bundle (—• 4.1)).

The spannor bundle is (bundlewise) the tensor product of the scalar bundle (- 5.1,) of weight w = 2 with the spin representation of G. •

T h e o r e m 6 . 1 . (Corollary 16.4.4 of [P-IV])

The rigorous mathematical derivation of the corresponding elementary particles seems to be still absent in the literature. A fundumental attempt has been made in [P-IV] but later it was noted [Se91] that "the composition series shown in [P-IV] is inexact and should be replaced by that indicated in [Se91]". The latter presents the fermion clan as the direct sum of a stable subspace and a tachionic subspace (i.e., one with the both-sided unbounded frequency spectrum). The stable subspace is the direct sum of a positive-frequency (—• 6.1.) representation F+ (which is "indecomposably built" into the representation induced from Σ + ) with its complex conjugate F~. Physical assignments for the reduced (—> 6.1) particles are given in Table 1 of [Se91]; those for antiparticles are obtained by interchanging the left and right spins [PS-II]. There are exactly four factors (hence, the fermionic clan includes four particles and four antiparticles), they correspond to the exon x, muon νμ and electron ve neutrinos, and electron e [Se0r]. This physical particle assignment is depicted in Column I of the mentioned table. The assignment is determined by the massive/massless character of the particle and the vanishing/nonvanishing of its interaction with the photon (—* 6.4). The massive (respectively: massless) means in the context that the Gelfand-Kirillov dimension equals four (respectively: three), see Column VII of the table. The second column is the bare (chronometric, intrinsic) mass of the particle expressed in chronometric units. It is defined as the minimum of the chronometric energy (—* 6.1) in the corresponding factor and equals 3 / 2 for the neutrinos, 5 / 2 for exon and electron; this is the contents of Column II. It is remarked in [Se91] that the minimum of the Minkowski energy vanishes and that this bare mass is far below the level of physical observability, since the proton (—> 6.4) relativistic mass mp « 10 40 . This latter notion is introduced as follows. The starting point is an exact particle represented by an eigenstate of the chronometric Hamiltonian Η (—• 3.8). Let M 2 denote the usual relativistic mass operator TQ — Tf2 — T| — T|. This operator and Η act on the representation space; then the state in question has to be an approximate eigenstate for e-itHM2eitH

of the same narrow-width eigenvalue, for a nongenerically long time interval. Thus in particular, [M 2 ,H] should have expectation value 0. The states with such constraints appear likely to exist and their relativistic masses be computable from them. One of the next quantum numbers is the height of the particle which seems to correspond to the order of inclusion of the corresponding invariant subspace. The four just mentioned particles have heights from 1 to 4, in the same order.

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R e m a r k 2. The spannor section space of the two-dimensional chronometry has been treated in [0rSe]. It has much in common with the physical four-dimensional situation. The investigation of the composition series exploits effectively the analog of the above stated Theorem 6.1.

6.3. The boson clan The boson clan is induced (—> 4.1) from the particular 15-dimensional indecomposable representation (see [P-IV]) of the Poincare group P . The corresponding field (i.e., section of the induced bundle or vector in the space of induced representation) is referred to as ply or (see [P-IV] and several later publications). Let me reproduce the corresponding information from [P-IV]. To characterise the plyor representation infinitesimally (see Lemma 17.1.1. of [P-IV]) it is convenient to use 8 x 8 matrices wm, m = — 1 , 0 , . . . , 4, therein introduced. The two subspaces P+, P_ of conformal weight 1 (—> 5.1) include photons as reduced vector particles. P+ is defined by the basis (w-ι — w^)w9l and P - is similarly defined by (W-i — wi)wswjwk\ all indices have values from 0 to 3. The bases are chosen in the spin space (—>4.1). The corresponding subspaces of the conformal weights 0 and —1 are similarly defined by their bases in the spin space [P-IV]. In the statement below π stands for the infinitesimal plyor representation and the generators T m (of time and space translations in the Minkowski world Mo) have been distinguished earlier (—> 3.8). Theorem 6.2. The total space of plyors is indecomposable under the action of P. On restriction to the scale-extended Lorentz group (—> 2.1), it decomposes as direct sum (of w = 1 , 0 , - 1 subspaces) shown in Table 17.1.1 of [P-IV]. Its subspace of weight — 1 leaks nontrivially into that of weight 0, and that of weight 0 into that of weight 1, while the latter subspace is Ρ-invariant. (The exact meaning of leaking is that, e.g., the w — 0 subspace is taken by the operators ir(Tm) into the w = 1 one; etc.) •

6.4. The chronometric fermion-boson interaction The exposition of this section is mainly extracted from [Se91]. I add references to make the reading easier. R e m a r k 3. There are several standard notations and terminology from elementary particle physics somewhere above and in the remainder of this section. The reader is to consult one of the numerous books on the subject (see, e.g., [Derd] and references therein).

On mathematical foundations and physical applications of chronometry

99

The interaction Lagrangian Li is the essentially unique G-invariant coupling of the boson clan with the local bilinear fermion clan current (bilinear current here stands for the section of the tensor product of the bundle with its dual [PS-II]). Charged particles are either electrons (—> 6.2) or composites with electrons. (A proton p, e.g., is chronometrically modelled as ρ = χ + e + + fe)· If / denotes the fermion and A the boson state, where A is represented canonically by a matrix on the fermion spin space [P-IV], then

where the inner product is the invariant [P-IV] one in the fermion spin space at χ, and for the measure d$x see Section3.7. The bosons have weights dual to those of the fermion currents: since these are of weights 3/2 + 3/2,3/2 + 5/2,5/2 + 5/2 (Table 1 of [Se91]), the boson weights are 1,0, and —1 (the sum of the three weights must equal 4, [PS-II]). The weigts 0 and —1 are well defined only in the relativistic limit; w = — 1 states leak (—• 6.3) under the action of G into w = 0 states, and w — 0 states leak similarly into the w — 1 subspace, which is G-invariant. Corresponding to the three different types of currents just indicated, there are three different types of interactions, in terms of relativistic limit. (i) Two w = 3/2 fermions and a w = 1 boson: The two w = 3/2 fermions are electrons and neutrinos. The w = 1 bosons include the photon, at the bottom of the subspace, and distinct candidates for the bare W = Wo and the Z, the former in a neutral form (the physical W+, W~ being composites of Wo with electrons and other particles). All three reduced particles have distinct quantum numbers that play a role comparable to the gauge degrees of freedom in the standard model. Charges of the w — 3/2 particles are included automatically in the form of the Lagrangian; e.g., the neutrino-photon integrated interaction vanishes, as a consequence of the transformation laws (or equivalently, the Dirac and Maxwell equations). The neutrino interactions with the Wo and the Ζ are nonvanishing and parallel those of e with the latter, providing a form of weak isospin. (ii) A w = 3/2 fermion, a w = 5/2 fermion, and a w = 0 boson: This is not readily characterized in relativistic terms but seems to underlie lowenergy-electron and top-neutrino ("top" - in terms of the chain (1), see Section 6.2) interactions with baryons and light mesons. The large nucleon to physical electron mass ratio appears to give this interaction a strong appearance in relativistic terms, although in bare chronometric terms it appears formally as approximately symmetric between the e and the x. The w — 0 sector includes a natural candidate for the neutral pion, whose decay into two photons may derive primarily from the leaking of the w = 0 bosons into the w — 1 subspace. The decay into neutral pions of the K° may be of similar character. (iii) Two w = 5/2 fermions and a w = — 1 boson: This interaction appears as purely strong in relativistic terms. The stable reduced elementary boson in this sector shows mixing of two relativistically invariant components and would be expected to leak into w — 0 bosons, among other possi-

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A.V. Levichev

ble decays. This suggests identification with the K°, but the mixing shown by the B° and the D°, together with their decay products, suggests they may be higher forms of the K° via the above proposed mechanism. The top positions of the e and the K° in their respective clans should facilitate this mechanism. The conformal weight sum constraint suppresses decay of the K° into μ+μ~ but allows K° —> xx. There are several other claims in [Se91] as regards chronometric description of elementary particles characteristics and interactions. SEGAL argues, in particular, that all relativistically "internal" symmetries may originate in the interplay between the quantum numbers associated with the maximal subgroups Κ and Ρ of G in the chronometric clans and thus be of an ultimately geometrical character.

7. Chronometry and extragalactic astronomy It is outlined in [Se 91] (and is seen in the examples of several previous sections) that the chronometric treatment of microscopic phenomena (elementary particles, interactions, quantization) is not superradical in comparison with the standard model. The situation in extragalactic astronomy is quite different since in its main predictions it contrasts greatly with the nowadays most accepted FriedmanLemaitre cosmology. I.E. SEGAL collaborates in the subject with J . F . NICOLL. During the past decades more than twenty papers appeared in astronomy and physics journals in which the predictions of the chronometric theory and systematic astronomical observation were compared in detail [Se91, SeNi, and references therein]. One particular difference betweeen the two cosmological theories is the chronometric redshift-distance relation ζ = tan 2 (r/2),

(1)

where τ is the distance in radians on the sphere § 3 that represents space (—>1). The equation (1) has been obtained in [Se76] at the classical quantum mechanical level under particular assumptions on the photon wave function. In [SZ] this law is rederived on a mathematically more rigorous basis for a photon of localized spatial support. The unitary representation of the conformal group on the Hilbert space of normalizable photon wave functions is applied, in the Schrödinger and Heisenberg representations. The analysis shows also the existence of photon states of cosmic spatial support that are not redshifted at all, as time evolves. Recall the crucial idea in deriving the law (1): According to chronometry, the "true" Hamiltonian is the operator Η (—> 3.8) corresponding to the advance of chronometric time t (—> 2.2), while direct laboratory observations of the energy yield only the scale-covariant component HQ (—> 3.8) of H. This component does not commute with Η and so is not conserved; after an elapsed chronometric time s, it is represented, in the Heisenberg picture, by the operator H0(s) =

e~i9HHQei3H.

On mathematical foundations and physical applications of chronometry

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The redshift ζ is defined so that 1 + ζ is the factor by which the special relativistic energy is reduced in the state in question. Chronometry explains, intrinsically, why the redshift is "red", though it may appear to lack the intuitive simplicity of a Doppler explanation for the redshift, which has become so familiar as to appear almost axiomatic to some astrophysicists. The distance r in (1) is not an observable quantity, but the purely geometric relations between apparent luminosity and distance (as well as other observed quantities, such as angular diameter) permit the distance to be eliminated and purely observable relations derived. These relations are then tested on the large samples of galaxies, quasars, and radio sources. Remarkably good agreement between predictions and observation is found. Chronometry leads to a much better fit with observation than do Priedman-Lemaitre models with their two free parameters qo and A. In addition, a number of anomalies within the Friedman-model of cosmology are simply eliminated: the apparent superrelativistic lateral velocities of a number of sources, and extraordinary luminosity and apparent evolution of quasars. The cosmic background radiation is not necessarily indicating the "big bang" but is predicted as the temporally homogeneous equilibrium photon gas established by the diffusion and scattering of electromagnetic radiation around the physical space S 3 in accordance with energy conservation. In [Se91] it is also argued that the mechanism of indecomposable production (-+ 6.1,6.3) Ve

-> Ve + l/μ +

ΰμ

may contribute to the solar neutrino deficiency caused by the attrition of the number of ve particles in flight due to the conversion into νμ pairs that are unable to revert to v e . On the other hand, the inverse process can proceed on a comparable scale only by Lagrangian rather than indecomposable production. As regards gravitation, chronometry says that there is no special force of gravity as such: it is simply the totality of the scale-contravariant [Se86], or super-relativistic, components of the energies associated with forces that also act microscopically (—> 6.4). It is stated in [Se91] (see references therein) that cosmic ray observations have been indicative of a neutral extremely long-lived hadron-like particle coming from Cygnus X-3 and from Hercules X-l. The chronometric exon χ (—ν 6.2) is a theoretical counterpart for these particles. Its relativistic mass (—• 6.2) varies, in principle. If it is of the order of the neutron mass, confusion between χ and τι could be a factor in the many revisions in the estimated neutron lifetime in recent decades and an anomaly in neutron scattering [Sla] but observations on Hercules X-l suggest that it may be light enough to be confused with a neutrino, if produced in high-energy collisions. Further cosmic ray observations are needed, but conclusive identification of the cygnet with the exon will depend on an observation of the latter in accelerator experiments. It should be possible to produce it in energetic electron-nucleon or nucleon-nucleon collisions.

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References [Derd]

Derdzinski, Α., Geometry of Elementary Particles, Proc. of Symp. in Pure Math. 54 (1993),157-171.

[Gu]

Guts, A.K., Axiomatic Theory of Relativity, Russian Math. Surveys 37 (1982), 41-80.

[GuLe]

Guts, A.K., Levichev, A.V., On the foundations of relativity theory, Soviet Math. Dokl. 30 (1984), 253-257.

[GuSt]

Guillemin, V., Sternberg, S., Symplectic techniques in physics, Cambridge University Press, Cambridge 1984.

[Ki]

Kirillov, A.A., Elements of the Theory of Representations, Grundlehren Math. Wiss. 220, Springer Verlag, New York-Berlin-Heidelberg 1976.

[Le86]

Levichev, A.V., Chronogeometry of an electromagnetic wave given by a biinvariant metric on the oscillator group, Siberian. Math. J . 27 (1986), 237-245.

[Le86']

—, Several symmetric spaces of general relativity theory as solutions of EinsteinYoung-Mills equations, in: Group theoretical methods in physics. Proceedings of the III. International Seminar, Yurmala, May 1985, vol. 1, 145-150. Nauka, Moscow 1986.

[Le87]

—, Prescribing the conformal geometry of a Lorentzian manifold by means of its causal structure, Soviet Math. Dokl. 35 (1987), 452-455.

[Le93]

—, Chronometrie theory by I.Segal is the accomplishment of special relativity, Izv. Vyssh. Uchebn. Zaved. Fiz. 36 (1993), 84-89.

[Mo90]

Moylan, P., Scalar representations of the conformal group, Preprint 1990.

[0rSe]

0rsted, B., Segal, I.E., A Pilot Model in Two Dimensions for Conformally Invariant Particle Theory, J . Funct. Anal. 83 (1989), 150-184.

[PaSc]

Parthasarathy, K.R., Schmidt, K., Positive Definite Kernels Continuous Tensor Products, and Central Limit Theorems of Probability Theory, Lecture Notes in Math. 272, Springer Verlag, New York-Berlin-Heidelberg 1972.

[PS-I]

Paneitz, S.M., Segal, I.E., Analysis in space-time bundles,I. General considerations and the scalar bundle, J. Funct. Anal. 47 (1982), 78-142.

[PS-II]

Paneitz, S.M., Segal, I.E., Analysis in space-time bundles,II. The spinor and form bundles, J. Funct. Anal. 49 (1982), 335-414.

[P-III]

Paneitz, S.M., Analysis in space-time bundles,III. Higher spin bundles, J. Funct. Anal. 54 (1983), 18-112.

[P-IV]

Paneitz, S.M., Segal, I.E., Vogan, D.A., Analysis in space-time bundles,IV. Natural bundles deforming into and composed of the same invariant factors as the spin and form bundles, J. Funct. Anal. 75 (1987), 1-57.

[SaWu]

Sachs, R.K., Wu, H., General Relativity for Mathematicians, Springer Verlag, New York-Berlin-Heidelberg, 1977.

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Segal, I.E., Mathematical Cosmology and Extragalactic Astronomy, Academic Press, New York, 1976.

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—, Chronometrie cosmology and fundamental fermions, Proc. Nat. Acad. Sei. USA 79 (1982), 7961-7962.

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—, The physics of extreme distances and the universal cosmos, in: The Quantum Theory of Time and Space, 121-137. Hanser Verlag, München, 1986.

[Se87]

—, Induced bundles and nonlinear wave equations, in: Group Representations, Ergodic Theory, Operator Algebras, and Mathematical Physics, 199-224; Springer-Verlag, New York 1987.

[Se91]

—, Is the Cygnet the quintessential baryon? Proc. Nat. Acad. Sei. USA 88 (1991), 994-998.

[SeJa]

Segal, I.E., Jacobsen, H.P., 0rsted, B., Paneitz, S.M., Speh, B., Covariant chronogeometry and extreme distances, II. Elementary particles, Proc. Nat. Acad. Sei. USA 78 (1981), 5261-5265.

[SeNi]

Segal, I.E., Nicoll, J.F., Comparative statistical study of the chronometric and Friedman cosmologies based on the Palomar bright quasar and other complete quasar surveys, The Astrophysical Journal 300 (1986), 224-241.

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Segal, I.E., 0rsted, B., Paneitz, S.M., Vogan, Jr., D.A., Explanation of parity nonconservation, Proc. Nat. Acad. Sei. USA 84 (1987), 319-323.

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Slaus, I., Akaishi, Y., Tanaka, H., Neutron-Neutron Effective Range Parameters, Phys. Rep. 173 (1989), 257-300.

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Sternberg, S., On Charge Conjugation, Commun. Math. Phys. 109 (1987), 649679.

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Sviderskii, O.S., Oscillator Parallelization of the Induced Scalar Bundle, Siberian Math. J . (1995), to appear.

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Segal, I.E., Zhou, Z., Maxwell's equations in the Einstein universe and the chronometric redshift, Preprint 1993.

2. Invariant cones, Ol'shanskii-semigroups, exponential semigroups

On the structure of Lie algebras admitting an invariant cone Victor M. Gichev

Invariant cones in Lie algebras arise in important ways in various areas of research discussed in this volume. The classification of Lie algebras which admit an invariant cone is now complete. The final step was taken in [6]. Basic special cases were analyzed by VINBERG [9] (1980), PANEITZ [7] (1981), N E E B [4] (1988), SPINDLER [8] (1988), GICHEV [1] (1989). SPINDLER [8] (1988) gave a universal construction for Lie algebras supporting invariant cones in terms of symplectic representations. In 1989, HILGERT and HOFMANN [2] formulated and proved necessary and sufficient conditions for a cone in a Lie algebra to be invariant reducing the classification of invariant cones in Lie algebras to that of cones in euclidean "configuration spaces" via intersections with Cartan subalgebras; cones in configuration spaces were discussed by ZIMMERMANN [10] in 1990. The monographs by SPINDLER [8] (1988), HILGERT, HOFMANN and LAWSON [3] (1089) and by N E E B [5] (1993) summarize what was known on invariant cones and the Lie algebras generated by them. This article contains a remarkably short and elementary introduction to the structure theory of Lie algebras admitting an invariant cone. It contains an approach to these algebras which continues and further develops the methods used by the author in [1] (1989) for the solvable case. Initially, the article was designed to be the preliminary portion of a paper dealing with the problem of the globality of a given invariant cone. So a large part is devoted to material which is needed for its solution. Most of the results of the article are known. The uniqueness part of Theorem A and the explicit construction of the Poisson representation in Theorem Β appear to be new. I am grateful to K.-H. Neeb and K.H. Hofmann who suggested to include the article in this book.

1. Introduction In this section, we give some definitions and introduce notations which will be used throughout the article. We formulate the two essential theorems.

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1.1. All cones considered in this paper are assumed to be convex. A cone C in a finite dimensional real vector space V is called pointed if it is closed and C Π (—C) = 0; it is called generating if Int(C) φ 0, where Int means "interior". A closed cone C is pointed (generating) if and only if the dual cone C* in the dual space V* given by C* = {X 6 V* : A(x) > 0 for all χ ι Qi which define the multiplication. This was first accomplished by KARLHEINZ SPINDLER [8] in 1 9 8 8 in terms of a universal canonical construction and resumed by K A R L - H E R M A N N N E E B [6] in ( 1 9 9 4 ) . These authors showed that Q\ could be equipped by a skew-symmetric bilinear form which is nondegenerate and unique up to a constant multiplier in every real irreducible submodule, and gave an explicit description of all irreducible submodules. 1.5. The leading example of a nonreductive Lie algebra admitting an invariant cone is the Lie algebra of polynomials of degree < 2 with the Poisson brackets as a multiplication. Recall that a nondegenerate skew-symmetric bilinear form ω in a real vector space W defines the Poisson brackets of functions u and υ by {it, υ}(ιυ) = u>(Jdwu, Jdwv)

(1.1)

where J : W* —• W is defined by , χ) = —ζ(χ) for all χ G W. The space C°°(W) of real-valued infinitely differentiate functions is a Lie algebra with respect to { , } and its subspace ^ ( W ) of polynomials of degree < 2 clearly is a subalgebra. Vector fields of the type J du, u e ^ ( W ) are exactly those affine vector fields which annihilate ω. Hence the Lie algebra sp(W) is isomorphic to the subalgebra Q^W) C Vz(W) of quadratic forms, and the space V\(W) of polynomials of degree < 1 is an ideal isomorphic to the Heisenberg algebra Ήη (dim Hn = 2n + l . d i m W = 2n). The center of Vk(W), k = 1,2, consists of constant functions. The group Ad(7>2(W)) may be identified with the semidirect product of the group Sp(W) and the vector group W via the natural linear action of the first group in the second. There is a natural affine action α of this group in the space W such that a(w)u — u + w for w € W and a(A)u = Au for A € Sp(W), and the corresponding representation in P2(W) by change of variables is the representation Ad. This consideration shows that ^(VV) and Q2(W) admit invariant cones consisting of nonnegative functions; let C m ,„ denote this cone in ^ ( W ) . The cone Cm3x of functions in ^ ( W ) bounded from below is convex, generating, Ad("P2(W))-invariant, but not closed in general, and its edge C m a x Π (—C max ) consists of constant functions (so it is one-dimensional). The gradation of Theorem A for ^(VV) is as follows: 7>2(W) = Q 2 (W) Θ Qi(W) Θ QO(VV)

where, for k = 0,1,2, Qy. = Ö2-fc(VV), and Qfc(W) is the space of polynomials of degree k.

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Theorem B. For any nonreductive real Lie algebra Q with an invariant cone C and a gradation of Theorem A, the space Gi may be equipped with a nondegenerate skew-symmetric bilinear Ad(C?o)-invariant form in such a way that there exists a homomorphism Ρ ·. G Ρ2(GI) of graded Lie algebras such that p{C) C C m a x and p(Gi) = Qi(Si). An explicit construction for ρ given in Section 4.1 shows that the intersection of kernels of these representations is included in the maximal semisimple ideal of G (see Lemma 4.2). Suppose that Go = sl(2, R) and ad is irreducible in G\- A short calculation shows that the only case for the assertion of Theorem Β to be satisfied is dim Gi = 2, i.e., ad must be the natural representation of sl(2,R) in R 2 . The consideration of sl(2, R)-subalgebras of G corresponding to noncompact roots shows that a highest weight of the representation ad in Gι must takes values —1, 0, 1 on noncompact roots. This very restrictive condition allows us to classify Lie algebras admitting an invariant cone (see [6], Theorem IV. 1).

2. Preliminaries We start with a simple geometric lemma (recall that we use notations and assumptions of Section 1.3). Lemma 2.1. Let χ € G, y € C, and suppose that k is α nonnegative integer such that aAk(x)y Φ 0, ad fc+1 (x)j/ = 0; set ν = ad k {x)y, and, for k > 0, u = ad k ~ 1 (x)y. Then (1) k is even and υ 6 C; (2) if k > 0 then u e TVC. Proof Since C is closed, v=

lim f ^ ' y e C , t—»-+00

and

lim (-t)-kea^ix)y t—• — oo

= (~l)kv

e C.

So k cannot be odd. To prove (2), note that for s = j , υ + su + 0(s2) e C. Note that there is no need to suppose in Lemma 2.1 that C is generating; using this assumption, we immediately obtain Corollary 2.2. For any χ € G, if adfc(x) φ 0 and ad^+^x) = 0 then (1) k is even and adfc(a:)C C C; (2) if k > 0 then ad k ~ 1 (x)G C T£C, where £ = ad k {x)GLemma 2.1 has many algebraic implications.

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111

Proposition 2.3. For a Lie algebra Q admitting an invariant cone C, the following conclusions hold: (1) the center Ζ of Q contains every abelian ideal of Q; (2) the center of the algebra Q/Z is trivial; (3) if Q is nilpotent then it is abelian; (4) M n C c Z ; (5) [λί,λί\ C Z; in other words, λί is at most a two-step nilpotent ideal in Q; (6) ζ ( κ ) η π = ζ(λ!)ηλί= z. Proof. (1) If an ideal I is abelian and χ G I then ad 2 (ζ) = 0, thus ad(x) = 0 and xeZ. (2) If x+Z belongs to the center then ad(x)C? C Ζ and ad 2 (x) = 0, so ad(x) = 0. (3) is an immediate consequence of (2). (4) The linear span of λίπ C is an Ad(£7)-invariant linear space and so an ideal. By (3) this ideal is abelian thus it is contained in Ζ by (1). (5) Let I be the greatest number of elements of λί with nonzero product. Consider the ideal generated by all products of the length > fc; clearly, Z* is an ideal in Q. If 2k > I then Jjt is abelian; if / > k then the center of λί does not contain Ik- If / > 2 then there exists k such that 2k > 1,1 > k, and k > 1, so Ik is proper and we have a contradiction with (1). (6) Since Z(K) η TZ and Z(N) Π Μ are abelian ideals in Q, (1) implies the nontrivial inclusions Z(Tl) Π Tl C Ζ and Z(N) Γ\ Ν C Ζ. By (5), k = 2 is the only case in Corollary 2.2 for χ £ Λ/"\ Ζ. So we have Corollary 2.4. For any χ G Ν", χ £ Ζ (1) ad 2 (χ) Φ 0; (2) ad2(x)CcC; (3) a.d(x)G C T£C where S = ad2(x)öCorollary 2.5. If Q is not reductive (1) λί is not abelian; (2) C(1Z^{0}.

then

Proof. (1) Otherwise, λί = Ζ{λί) Π λί = Ζ by (6). Since the center of Q/Z is trivial and TZ ^ Ζ by the assumption, we have a contradiction with the inclusion [G, K\ c λί. (2) By Proposition 2.3, (5), ad2{x)Q Q Z, so the assertion is true by Corollary 2.4, (1) and (2). 2.6. Another geometric tool is the averaging procedure. Let K. C Q be a compactly embedded subalgebra, σ be the invariant measure on Κ — closAd(/C) with the total mass 1. Set AKx = / kxda{k). JK

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The averaging operator AK is the Ad(JC)-invariant projection onto the space of fixed points Z{K). Clearly, AKF C F for every closed convex if-invariant set FcG. Lemma.

Let K. be a compactly embedded subalgebra ofQ. Then Z(fC)nC = AKC.

Proof. Clear. Lemma 2.7. A subalgebra K. C Q is compactly embedded into Q if and only if Z{K) η Int(C) ^ 0. Proof. If χ G Z{K) Π Int(C) then the set (x — C) Π (C — x) is Ad(£)-invariant bounded neighborhood of the zero in Q, so closAd(JC) is compact. To prove the converse, note that AKx € Int(C) for any χ € Int(C). Proposition 2.8. For a regular h G Int(C), let C be a corresponding Cartan subalgebra, i.e., C = U£LX kerad^/i)· Then (1) C = Z(h); (2) C is a compactly embedded abelian Cartan subalgebra of Q; (3) C n M = Z ; (4) N=Z®[h,N}. Remark. Note that each compactly embedded abelian Cartan subalgebra intersects Int(C) by Lemma 2.7. Proof. (1) By Lemma 2.7, h is compact. Therefore, ad(h) is semisimple and C = kerad (h) = Z(h). (2) C is nilpotent being a Cartan subalgebra, so (2) is a consequence of (1) and Lemma 2.7. (3) For χ e Z(h) Π Λ/", a linear operator ad(x) must be semisimple (since χ is compact) and nilpotent, so ad(x) = 0. (4) follows from (1) and (3) because h is compact. 2.9. By Proposition 2.8, every ideal in the short flag Ζ C Μ C TZ C Q has a complementary Ad(C)-invariant subspace. Thus we have Ad(C)-invariant decompositions tf=W®Z,

(2.1)

n = A@M,

(2.2)

Q = S@ n . (2.3) Our aim now is to investigate the uniqueness of them. Note that Ad(C) - invar iance of the space X is equivalent to the inclusion [C, AjCA 1 . Lemma.

W is the unique Ad(C)-invariant subspace of Af complementary to Z.

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Proof. By Proposition 2.8, W = [Λί, h] for any h £ Int(C) Π C. Lemma 2.10. The space A in (2.2) may be chosen as any subspace ofCr\1Z complementary to Ζ. Then A is an abelian compactly embedded subalgebra of Q and C Π TZ = Α Θ Ζ.

(2.4)

Proof. The action of the group Ad(C) on the factorspace TZ/Af is trivial because [G, H\ C J\f. So any invariant subspace of ΊΖ complementary to Μ consists of fixed points and A C Z(C) = C. Thus A is compactly embedded. The last assertion follows from Proposition 2.8, (3), the first is clear. 2.11. The decomposition (2.3) is not unique if S considered as a subspace of Q. We shall prove the uniqueness when S is supposed to be a subalgebra. Then S must be a semisimple component in the Levi decomposition. It is not true in general that for any Cartan subalgebra C there exists an Ad(C)-invariant semisimple Levi complement. The simplest example is a semidirect product of sl2 with s^, considered as a vector space, via ad. Proposition. There exists only one Ad(C)-invariant semisimple of Q satisfying (2.3).

subalgebra S

We need a lemma to prove the proposition. 2.12. By Proposition 2.3, the algebra Q = Q/Z is centerfree, and the ideal Ν — M/Z in G is abelian (so ~ denotes the canonical homomorphism Q —• Q/Z). Let B=[Z(A), Z(A)). Lemma.

The algebra Β is centerfree and has an abelian radical.

Proof. Note that C C Z(A), so [C, Z(A)] C Z(A). Thus [C, B] C Β and Β is Ad(C) -invariant. Since [A, Q) C TZ and A is compactly embedded, we have 1Z+Z(A) = Q. Hence any Levi semisimple component for the subalgebra Z(A) is a Levi complement for 7Z in G, and the same is true for the subalgebra B. In particular, 7Z + Β = GLet ρ : G —• Β be a projection commuting with Ad(C). Since ΊΖ + Β = G, kerp may be chosen as any Ad(C)-invariant subspace of 7Z complementary to 1Z Π Β. Then pTZ C 7Z. Evidently, the set of Ad(C)-fixed points is p-invariant, hence pC C C. Therefore, C ΓΊ Β = pC. Let CB = C Π Β. Denote by J the radical of B. Thus B/Jis semisimple and J= ΒΠΊΖ. In fact, JC Μ since [G, ö ] n Ä C ^ . By Proposition 2.8, (3), CnJC Ζ ; it follows from (2.4) that pA C Z) and we have the decomposition C = A® C,Β·

(2.5)

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Note that Ad(C) acts on without fixed points by Proposition 2.8, (4). This means that Z(C) Π J r = 0. Therefore, (2.5) implies that Z(CB) η J=

0.

(2.6)

Thus Β is centerfree. The ideal J is abelian by Proposition 2.3, (5). There is a natural affine action of Β in J. To construct it, consider the projection κ. : Β —* J whose kernel is a Levi semisimple complement S. Then, for u 6 J, ad(u)k — 0 and nad(u) = ad(u); for s G S, ad(s)/c = « a d ( s ) . So e -ad(«) Ä e ad(u) = K

+

ad(u)

and the desired action is the action on the orbit of κ by conjugations; the orbit could be identified with J since Β is centerfree. Note that the isotropy subgroup of the point κ is Ad(»S). Proof of Proposition 2.11. Let us keep the notation of Lemma 2.12. Fix a semisimple Levi complement S in Β and consider the affine action α of A d ( ß ) in J constructed above. Being compact, the group c l o s a ( A d ( C s ) ) has a fixed point in J\ moreover, this point is unique by (2.6). So CB is included to the unique semisimple Levi complement conjugate to S by J. We may assume that CB C S. Then the inverse image of S under ~ is Ad(C B )-invariant. It follows from (2.5) and the definition of Β that it is Ad(C)-invariant. Since the kernel of ~ is Z , its semisimple part S is uniquely determined and Ad(C)-invariant. Thus we prove the existence of a complementary subalgebra in (2.3). To prove the uniqueness, note that the inclusion [C, (exp X)h: C χ Η —> S is a homeomorphism, and a diffeomorphism if that is appropriately defined for this context, see [16]. In particular, (X, h) 1—> (exp X)h: C° χ Η —> 5° is an analytic diffeomorphism, where S° is the interior of S in G. (iii) £(S) = C θ g+, a symmetric wedge. We denote S by Γ#((?, C). For the case that Η is connected (and hence equal to G + ), it is clear from (i) that S is determined by C, and we denote it by r(G, C) in this case. In both cases we often omit G, if G is clear from context. If, in addition, the cone C is pointed and satisfies C — C = g_, then the Ol'shanskii semigroup r(G, C) will sometimes be called the proper Ol'shanskii semigroup associated to C and G, to distinguish this special case. • Remark 2.2. (i) The invariance of the cone C under Ad(//) implies that it is g+-invariant, since G+ C H.

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(ii) Condition (iii) of Definition 2.1 can be derived directly from the first two (see e.g. [16]). Hence existence theorems depend only on establishing the first two conditions. (iii) The cone C is uniquely determined from any corresponding Ol'shanskii semigroup S. Indeed, it follows directly from part (iii) of Definition 2.1 that C = £(S , )ri0_. Hence for fixed G, C and T(C) uniquely determine each other, provided the latter exists. • In the case that Η is connected, properties (ii) and (iii) of Definition 2.1 allow one quickly to deduce Proposition 2.3. The Ol'shanskii semigroup T(G, C) associated to C and G is strictly infinitesimally generated. • We next present the major existence theorems for Ol'shanskii semigroups for the data G and C. The ground-breaking existence theorem was the original one of Ol'shanskii [27] in the semisimple case, and the later improvements continue to rely heavily on his insights and methods. For Theorems 2.4-2.7 we assume the following setting: (g, r ) is a real symmetric Lie algebra, C C g_ is a g+-invariant cone with real spectrum, G is a connected Lie group with Lie algebra g, and G+ is the analytic subgroup of G generated by the exponential image of in G. The first theorem may be found in [16] or [12]. Theorem 2.4. If the involution τ on g integrates to G, then G+ is the identity component of the fixed-point set (hence closed) and the following two conditions combined are necessary and sufficient for property (ii) of Definition 2.1 to obtain: (a) if Ζ 6 ^(g) Π (C — C) satisfies exp Ζ = e, then Ζ = 0; (b) if 0 / X e C Π 3(g), then the closure o/exp(RX) is not compact. In the case (a) and (b) hold, then (i)-(iii) of Definition 2.1 hold, andS = exp(C)G+ is an Ol'shanskii semigroup. • Remark 2.5. The theorem remains valid if G+ is replaced by any closed subgroup Η such that G+ C Η C GT for which C is Ad(i?)-invariant. • Theorem 2.6. If G is simply connected, then τ integrates to G, and the conditions of the previous theorem are satisfied. The fixed point set GT of the involution is connected and hence equals G+, and the Ol'shanskii semigroup T(C) = exp(C)G r exists. • The existence of the Ol'shanskii semigroup for simply connected G with involution of complex conjugation was first established by Dörr [4], although now it can be derived as a corollary from Theorem 2.4 [16]. The fact that the fixed-point group is connected in the simply connected case is non-trivial (see Theorem 3.4 of Chapter IV of [21]). The following mild sharpening [16] of Ol'shanskii's original

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result establishes (under appropriate hypotheses) the existence of the Ol'shanskii semigroups, even when the involution may not integrate to G. T h e o r e m 2.7. Suppose that (a) the center $ C g + (e.g. g is semisimple) and G+ is closed, or (b) j n ( C —C) = {0}. Then the Ol'shanskii semigroup F(G,C) = e x p ( C ) G + exists for the data C and G. •

In the case of complex Lie groups these results specialize to T h e o r e m 2.8. Let Gc be a connected Lie group with Lie algebra 0c = 0 + iSi equipped with the involution σ of conjugation. If either (a) Gc is simply connected, or (b) 0 is semisimple, then

C «-> r(»C) ~ £ ( r ( i C ) ) defines a one-to-one correspondence between the pointed generating invariant cones in 0, the Ol'shanskii semigroups of Gc with non-empty interior and with group of units G+, and the generating Lie wedges W C j c with W i l - W = 0. Furthermore, any infinitesimally generated semigroup S φ Gc with non-empty interior and with C(S) Π — C ( S ) = 0 is an Ol'shanskii semigroup of this type. • T h e o r e m 2.9. Let Gc be a complex group unth Lie algebra 0c = 0 + iß> ^ G+ be closed and generated in Gc by exp(0), and let C be a generating invariant cone in g with C Π — C a compactly embedded subalgebra. Then the Ol'shanskii semigroup F(Gc, C ) exists if g is semisimple or Gc is simply connected. •

Theorem 2.9 follows from Proposition 1.4 and Theorems 2.6 and 2.7. The semigroup exp(iC)G+ is called a complex Ol'shanskii semigroup. In this context, we will generally denote the complex Ol'shanskii semigroup by T(C) instead of r(iC). Observe that the classical tube domains R n + iC C C n are examples of complex Ol'shanskii semigroups for the abelian groups R n . We give next a basic non-abelian example. Example 2.10.

Let 0 = sl(2, R) and let C :

= { ^ = ( c

!

A

) W < 0 ,

b

> 0

•

The cone C is one-half of the Lorentzian double-cone of matrices where the CartanKilling form is non-positive (the Cartan-Killing form in this case is given by k(X,Y) = 4Tr(XY)). It is an invariant cone since it is the closure of one of the two components of the set where the Cartan-Killing form is strictly negative, and each of these components is invariant under the adjoint action. The corresponding complex Ol'shanskii semigroup T(C) lives in S1(2,C) and is given by r(C) - Sl(2,R)(expzC) = (expiC) S1(2,R). • We close this section with the presentation of a class of Ol'shanskii semigroups which exhibits both abelian and non-abelian aspects.

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Example 2.11. Let G be a connected Lie group with Lie algebra g, let the finite dimensional real vector space be a G-module, and let C be an invariant cone in V. Then C x G is an Ol'shanskii semigroup in the semidirect product V x G ; the latter is an involutive group with (V*\G) T — G, where r(v,g) = (—v,g). As a special case, one can consider the group of positive reals (under multiplication) acting on the reals by multiplication and leaving invariant the positive cone of real numbers. One obtains a half-plane Ol'shanskii semigroup in the non-abelian Lie group of dimension 2. •

3. Polar decompositions Given an invertible matrix A over the complex numbers, the positive definite hermitian matrix AA* has a unique positive definite square root P, the matrix θ := P~lA is then unitary, and the decomposition A — ΡΘ is called the polar decomposition of A, since this construction restricted to the non-zero complex numbers yields the standard polar decomposition of complex numbers. The polar decomposition of matrices is a special case of a more general polar decomposition which is of fundamental importance in the theory of semisimple and reductive Lie algebras and Lie groups. Given a Cartan involution on a semisimple Lie algebra, the involution integrates to a corresponding involution at the group level, and this group involution gives rise to a polar decomposition at the group level, which generalizes the polar decomposition of matrices. If the semisimple group has finite center, then the fixed point set of the involution is a maximal compact subgroup, and this compact subgroup plays the role of the group of unitary matrices in the polar decomposition. For more general involutions at the Lie algebra level, one faces the problem of trying to find a decomposition at the group level corresponding to the involution. Although one does not expect nor obtain a global decomposition for the entire group, yet under rather general conditions one is guaranteed such a decomposition for at least a "large" subsemigroup of the group. The fact that the semigroup setting is the appropriate one led to this decomposition being overlooked until recent years, except in certain special cases. In the semisimple case, this semigroup decomposition was first worked out in a seminal paper of G.I. Ol'shanskii [27], and hence semigroups admitting this type of decomposition or factorization are referred to as Ol'shanskii semigroups. Let us now be more specific: Let (g, r) be a symmetric Lie algebra, let C C g_ be a cone with real spectrum, let G be a closed Lie group with Lie algebra g, and suppose that the Ol'shanskii semigroup S = r(C, G) exists for the data C, H, and G. Lemma 3.1. For s G S, set s* := h_1 expX if s = ( e x p X ) h . Then (i) s* = s - 1 if and only if s G Η and s* — s if s G exp C; (ii) (expX)* = e x p ( - T ( X ) ) for X G 0+ U C.

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129

1 for s Ε S. Hence (iii) If τ integrates to G and Η C GT, then s* = (r(s)) s ι—• s*: S —> S agrees with the restriction of the adjoint involution to S. (iv) If Η is connected (i.e., Η = G+ and S — r(C),), then s i—• s*: S —> S is a continuous antiautomorphism of order 2 on S, whether or not τ integrates to G. •

Definition 3.2. (i) Each s Ε S admits a unique (Ol'shanskii) polar decomposition s = (exp X)h, Χ Ε C, h 6 H. The left-right dual also holds. We restrict our attention to the left polar decomposition and drop the adjective "left". (ii) The mapping s s*: S —• S defined in the preceding lemma is called the adjoint involution of S in the case that it is indeed an antiautomorphism. Part (iii) demonstrates that this does not introduce ambiguity in terminology. • The next proposition (see [16]) gives a computational approach to the Ol'shanskii polar decomposition that is analogous to that for finding the standard polar decomposition of a matrix. Proposition 3.3. Suppose that τ integrates to G and Η C GT. Then the polar decomposition is uniquely determined for s Ε T ( C , H) as s = e x p ( X ) h Ε e x p ( C ) H , where X is the unique member of C such that exp(2X) = ss* and h — e x p ( - X ) s .

•

A similar computation allows one to determine membership in T ( C ) in certain particular cases. Proposition 3.4. If the involution τ integrates to G and GT is connected, then g Ε T ( C ) if and only if gg* Ε exp(C). This holds in particular for the case that G is simply connected. Proof. If gg* Ε exp(C), then one verifies directly that h := (exp(—l/2)X)g, where exp(X) = gg*, satisfies r(/i) = h, and thus h Ε GT. Hence g — exp(l/2)X-h is in the Ol'shanskii semigroup. • Recall that for a semigroup S the free group on S consists of a group G(S) and a homomorphism t: S —> G such that for any homomorphism ß:S —> Η into a group H, there exists a unique homomorphism a: G(S) —• Η such that a ο l = β, i.e., any homomorphism from S into a group "extends" uniquely to a homomorphism from G{S) to the group. The homomorphism t:S —> G is an embedding if and only if S can be embedded in some group. For an Ol'shanskii semigroup viewed strictly as a semigroup apart from its containing group, its free group embedding gives a canonical Lie group in which it lives (see Section VII. 3 of [11] or Section 3.5 of [12]). Theorem 3.5. Let S = T ( C ) be an Ol'shanskii semigroup in some connected Lie group G. Then the free group G(S) on S is a Lie group which is a covering

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group of G and the embedding of S into G lifts to an embedding into the covering group G(S). Furthermore the involution τ on q integrates to G(S). • The preceding theorem (from [16]) allows us to assume in many circumstances that the adjoint involution on T(C) is the restriction of g • r(g~1). Proposition 3.6. Let S = exp(C)H be an Ol'shanskii semigroup in G, where C has non-empty interior C° in g_. Then in G, the interior S° of S is a dense semigroup ideal in S, and S° = exp(C 0 )if. In the special case that S = T(C) is a complex Ol'shanskii semigroup in Gc, then S° is a complex manifold, the multiplication on S° is holomorphic, and the adjoint involution is antiholomorphic on S°. Proof The last assertion follows from Theorem 3.5, since when conjugation lifts, g* = ( g ) - 1 , and the others are straightforward. • We illustrate the application of the machinery of Ol'shanskii polar decompositions to show that it both generalizes and yields new derivations of well-known classical results. Example 3.7. (Matrix polar decomposition) Let g = M n (F) denote the Lie algebra of η χ η matrices over F, where IF is R or C. The matrix antiautomorphism A A* (the transpose for R and conjugate transpose for C) is an involutive algebra antiautomorphism and gives rise to the symmetric algebra structure t(A) = —A* on g. For IF = R (resp. IF = C) the space g_ is the space of symmetric (resp. hermitian) matrices and g+ is the space of skew-symmetric (resp. skew-hermitian) matrices. Let G = GLn(F) be the group of invertible η χ η matrices. Then τ integrates to G and is given by r(g) = (g*)'1. For F = R (resp. F = C) the group GT is then the set of orthogonal (resp. unitary) matrices, and the fixed point set of the adjoint involution is the set of invertible symmetric (resp. hermitian) matrices. We apply Theorem 2.4 with C = g_ the set of symmetric (resp. hermitian) matrices. Since symmetric and hermitian matrices have real spectra, we have that ad X has real spectrum for each X G C by Lemma 1.5. The appropriate invariance condition on C is easily verified. Since 3 Π C = 3 Π g_ is the set of diagonal real scalar matrices, conditions (a) and (b) are also satisfied. Hence the conditions of Theorem 2.4 all hold. Since g = g + + C and —C = C = C°, the semigroup S := (exp C)GT is actually an open subgroup containing GT, the orthogonal or unitary matrices. The set S is thus an open and closed subgroup meeting every component of G (since GT does) and hence must be all of G. Thus every element of G can be written uniquely in the form (expX)U, where X is symmetric (resp. hermitian) and U is orthogonal (resp. unitary), the polar decomposition. We remark that analogous developments apply to the case that F = H, the quaternions. In particular, every invertible matrix over HI admits a unique polar

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decomposition of the form (exp X)U, where X is quaternionic hermitian and U is quaternionic unitary. • Example 3.8. (Cartan involution and polar decomposition) Let g be a reductive Lie algebra, and let g = t + p be a Cartan decomposition, where ί is a maximal compact subalgebra. This is well known to be a symmetric decomposition. Since 0 is reductive, it is the direct sum of the center 3 and the commutator algebra [0>fl]> which is a semisimple ideal. By maximality of 6, we conclude j C t. For X € p, a d X restricted to 3 is trivial. The images of t and p in 0/3 induce a Cartan decomposition of this semisimple algebra; it is then standard that a d X induces on this quotient algebra a transformation which is symmetric with respect to the positive definite symmetric form defined from the Cartan-Killing form and the Cartan involution. It follows that ad X has real spectrum. Let G be any connected Lie group with Lie algebra 0. Since the analytic subgroup Κ generated by exp t is the identity component of the inverse image under the adjoint representation of the compact subgroup generated by e a d l , we conclude that Κ is closed. We now apply Theorem 2.7 with C = p to conclude S := (expp)K is a subsemigroup of G. Since p = —p, S is a group, and since 0 = ί + ρ, S is open. Hence S = G. Thus again by Theorem 2.7 each g € G can be written uniquely in the form (exp X)k for X € p and k Ε K. We remark that if Κ is indeed compact (which it is for semisimple groups with finite center), then it follows immediately from the decomposition that it is maximally compact, and hence that the maximally compact groups are connected. • Ol'shanskii polar decompositions may sometimes be used to establish that certain subsemigroups of an Ol'shanskii semigroup are again Ol'shanskii semigroups. We consider an important such case. Let Q be a subset of the general linear group Gl(n,C). Let g%j{\ < i,j < n) denote the matrix elements of g G Gl(n, C), and let Xij(g) and yij(g) be the real and imaginary parts of gij. The subset Q is called a pseudoalgebraic subset of Gl(n, C) if there exists a set of polynomials { f b } i n 2n 2 arguments such that g € Q if and only if ρ 6 Gl(n, C) and Pf,(... Xtj(g), Vij(g), ·· ·) = 0 for all P^. Using properties of pseudoalgebraic subsets, one can show [16] Theorem 3.9. Let (G,r) be an involutive Lie group and a subgroup o/Gl(n,C), and let Η C GT be a closed subgroup containing the identity component of GT. Let C C 0_ be an Ad Η-invariant cone with real spectrum. If S := (expC)-if is an Ol'shanskii semigroup and F is a subgroup of G which is τ-invariant, a pseudoalgebraic subset, and meets S, then SDF = exp(C Π C(F)) (F Π Η) is also an Ol 'shanskii semigroup and the factorization on the right is the Ol 'shanskii polar factorization (where C(F) denotes the Lie algebra of F). •

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4. Contraction semigroups for indefinite forms Subsemigroups of Lie groups arise in a variety of contexts and have varied descriptions or definitions. In certain circumstances one would like to determine that a semigroup S under consideration is an Ol'shanskii semigroup. It is generally rather straightforward to compute the tangent wedge £(S), and to determine whether it is a symmetric wedge, and hence the tangent wedge for an Ol'shanskii semigroup that lies inside of the original semigroup S. What often turns out to be a much more difficult computation is showing the reverse containment, that every member of the original semigroup S belongs to the Ol'shanskii subsemigroup. In the case of involutive groups, such a determination can sometimes be made by constructing a polar-like decomposition for members of the semigroup S, which then turns out to be the Ol'shanskii decomposition of the previous section. Such a program can be carried out, for example, for general types of contraction semigroups. The type of contraction semigroup which we consider was apparently first studied by V. P. Potapov [32], who called its members ./-contractions. He established that each such operator has a unique polar decomposition into a product Τ = RQ of a J-symmetric matrix R with positive real eigenvalues and a J-unitary matrix Θ. This semigroup was studied in more detail by Brunet and Kramer in [3], who were able to establish that it was indeed an Ol'shanskii semigroup (although that terminology was not used there), and rederived the factorization of Potapov as the Ol'shanskii polar decomposition for this semigroup. Further details for the following analysis may be found in [16] or Section 8.5 of [12]. Example 4.1. (Contraction semigroups) Let V be a complex finite dimensional vector space equipped with a pseudohermitian form ·/(-, ·)• Each linear operator Τ G End(V) determines an adjoint operator T" satisfying J(Tx, y) = J(x, T*y) for all x, y G V. The mapping T h T ' is an involutive algebra antiautomorphism on End(V). The corresponding Lie algebra involution τ(Α) —A^ on g := End(V) has for g+ the set Uj of J-skew-symmetric operators and for the set of Jsymmetric operators. One verifies that g_ = ig+ = iuj, and that τ is complex conjugation with respect to g+. The involution τ integrates to A i—> ( A " ) - 1 on Aut(V). The closed subgroup Uj := {T G Aut(V): J(Tx,Tx) = J(x,x), Vre G V} of J-unitary operators is the set of fixed points for this involution. We consider the semigroup S-

:= {T g Aut(V) : J(Tx,Tx)

< J(x,x)

for all χ e

V},

the semigroup of length decreasing or contraction operators. We list standard properties about this semigroup. (1) The subgroup Uj of J-unitary operators is the (unique) maximal subgroup of S^. (2) The tangent wedge of the semigroup S- is given by C(S-)

: = {X = {X

G End(V) : exp(tX) Ε S- for all t > 0} G End(V) : J ( ( X + Χ*)υ,ν)

< 0 for all ν G V } .

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(3) Since Uj acting by inner automorphisms on Aut(V r ) leaves invariant S-, the adjoint action of Uj on End(V) leaves invariant the tangent wedge C(S-), and hence the intersection iC :=£(£*)

n 0 _ ={X:X

= X\ J(Xv,v)

< 0 for all ν

eV},

i.e., iC consists of all J-symmetric negative semidefinite operators. Note that C(S-) = iC 4- 94. is a symmetric wedge, and that C = {X euj.X

= J(iXv,v)

< 0 for all ν e V }

is an invariant cone in u j = g + . (4) Using methods of linear algebra or the fact that C is invariant, one can verify that iC has real spectrum and that conditions (a) and (b) of Theorem 2.4 are satisfied (see e.g. [16]). Hence T ( C ) := exp(iC)J7j is a complex Ol'shanskii semigroup. It follows directly that T ( C ) C S-. (5) Using the principal results about factorization of J-contractions from Potapov [32] or from Section 3 of [3], one derives the other inclusion. This analysis depends heavily on a spectral theory for contractions, results which do not extend to arbitrary linear operators. Indeed the Ol'shanskii polar factorization does not extend to the whole group Aut(V), as it does for positive definite forms (Example 3.8). We summarize: Theorem 4.2. The contraction semigroup S- for the pseudohermitian form J ( · , · ) on the complex space V is a complex Ol'shanskii semigroup of the form (expiC)Uj, where iC consists of all J-symmetric negative semidefinite matrices in End(F) and Uj C A u t ( V ) is the subgroup of J-unitary operators. • Analogous results hold for pseudo-euclidean forms on real vector spaces. Example 4.3. Let V be a finite-dimensional real vector space and J = ( , ) be a pseudo-euclidean form on V. Then End(V) is a symmetric Lie algebra, with involutive algebra antiautomorphism given by Τ ι—• the adjoint with respect to the given form. The contraction semigroup S^ : = {T e End(V) : (Τχ,Τχ)

< (χ, χ) for all χ e V } ,

is an Ol'shanskii semigroup, and each Τ 6 S- has a unique factorization in the form Τ = (exp A)U, where A is in the cone C of J-symmetric negative semidefinite operators and U is a member of the group of J-orthogonal operators preserving the bilinear form. Proof. The bilinear form extends to a nondegenerate pseudohermitian form Be on Vc = V + iV (see Remark V.4.61 of [11]) given by Bq(V + iw,v' + iw') = {v,v')

-I- (w,w') + i(w,v')

-

i(v,w').

Let Τ £ S-. One verifies directly that the complex extension of Τ to Vc is also a contraction on Vfc. The real case now follows as a consequence of the complex

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case, via Theorem 3.9, since the contractions of V may be identified with those of Vc leaving V invariant. • Remark 4.4. There are corresponding semigroups of expansions S- in each of the preceding cases which are also Ol'shanskii semigroups, and have analogous factorizations in terms of positive semidefinite operators. Indeed one has that S> = ( S ^ ) " 1 . • We remark that analogous results for the dual case of the semigroup of expansive matrices have appeared in the context of geometric control theory in [1] for the special case of a positive definite form. The computations are much simpler in this case since one has available globally the standard polar decomposition of matrices (Example 3.8). We seek to extend the factorization of Example 4.1 to as large a semigroup as possible (see Section 8.5 of [12]). Example 4.5. Let V be a complex finite dimensional vector space equipped with a nondegenerate pseudohermitian form J(·, ·). We consider the compression semigroup S+ = 5(Ω+) := {Τ e Aut(V): Τ ( Ω + ) C Ω + } where Ω+ := {v e V: J(v,v) We set C m a x := {X € uj: J(v,v) = 0 => J(iXv,v) the group of J-unitary transformations. Then

> 0}.

> 0}, and again let Uj denote

5+ = 5(Ω+) = exp(iC max )J7j is an Ol'shanskii semigroup with tangent wedge £(S+) = iC m a x + 0+, where g+ = uj is the Lie algebra of J-skew-symmetric operators, and g_ Π C(S+) = iC m a x , the J-symmetric operators in C(S+). We note that S+ is not a proper Ol'shanskii semigroup since iCmax contains all real scalar multiples of the identity I. The proof of the containment S+ C exp(iC m a x )i/j is difficult and requires considerable machinery (see Chapter 8 of [12]). We will see later that the semigroup S+ is a maximal subsemigroup of Aut(V) in the sense the only subsemigroup properly containing it is Aut(V) itself. It then follows that C m a x is a maximal invariant (non-pointed) cone in uj. Note that the expansion semigroup S- is a subsemigroup of S+, and so the Ol'shanskii polar factorization of the latter extends the former.•

5. Invariant cones Invariant pointed cones have been rather intensely studied and classified (see, for example, [29], [30], [31], Chapter III of [11], [36], and for the most extensive and upto-date treatment [24]). Indeed, the fact that they give rise to complex Ol'shanskii semigroups is a major motivation for their careful study and classification. One follows here the general strategy in the theory of Lie algebras of reducing the prob-

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lem to a question about Cartan subalgebras. The presence of a pointed generating invariant cone in 9 guarantees that g contains a compactly embedded abelian Cartan algebra t, i.e., one for which the closure of exp(adt) in Aut(g) is compact, and these are the appropriate Cartan subalgebras to consider in this context. One of the major results is the following (see [37], [31], Theorem 7.5 of [36], or Theorem 7.27 of [12]): T h e o r e m 5.1. (The uniqueness and reconstruction theorem for invariant cones) Let C be an invariant, pointed generating cone in a finite dimensional real Lie algebra g, and let t be a compactly embedded Cartan subalgebra. Then C° = (e adfl ) · algint(tnC). in particular, C is uniquely determined by its intersection with t, since C is the closure of C°. • The uniqueness and reconstruction theorem has an interesting and useful reformulation in the context of Ol'shanskii semigroups. This reformulation provides a second decomposition (besides the polar) of a complex Ol'shanskii semigroup, a decomposition which may be viewed as an analogue of what is often called a Cartan decomposition of a semisimple Lie group (see e.g. Chapter IX, Theorem 1.1 of [8]). T h e o r e m 5.2. Let C be a pointed invariant generating cone in a finite dimensional real Lie algebra g, and let t be a compactly embedded Cartan subalgebra of g. Let gc = g + ig be the complexification of g and let Gc be a connected Lie group xuith Lie algebra gc· Let Go be the analytic subgroup generated in Gc by expg. If Gq is closed and T(C) := exp(iC)Go is a complex Ol'shanskii semigroup in Gc, then r ( C ) ° = Go exp i(algint(C D t))Go· In particular, the right-hand side is an open dense subsemigroup o/T(C). Proof. Let S := T(C) and let s €. S°. Then s = exp(iX)h for some X e C° and h Ε G0 by Definition 2.1(ii). By Theorem 5.1 there exists Ad(i/) e Ad(G 0 ) = (e adfl ) such that Ad(s)(F) = X for some Y e algint(tfl C). It follows that A d ( g ) ( i Y ) = iX, and thus exp(iX) = g(expiY)g~1. Thus s = g(expiY)(g~1h), which completes one inclusion. For the reverse inclusion, note that algint(t Π C) C C° by Theorem 5.1, and thus Go exp i(algint(C η t))G 0 C G 0 (expiC°)G 0 C G0S° C S°, where the last two inclusions follow from Theorem 3.6.

•

We remark that it is not the case in general that T(C) = Go expi(C Π t)GoIndeed Example 2.10 provides a counterexample. In the remainder of this section let g denote a finite dimensional real Lie algebra which contains a compactly embedded Cartan algebra t. Associated to the Cartan subalgebra tc in the complexification gc is a root decomposition as follows. For a linear functional λ 6 t£ we set gc :=

€ g c : (VY € t c ) [ Y , X ] = A(F)X}

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and Δ

A(sc, tc) := {Λ 6

\ {0} : g£ φ {0}}.

Then 0C = tc θ 0 flc, λ€Δ A(t) C iR for all λ G Δ and σ(β£) = 0£ λ , where σ denotes complex conjugation on 0c with respect to g. Let t D t denote a maximal compactly embedded subalgebra; t always exists and is uniquely determined by t. Then a root is said to be compact if 0c ^ We write Ak for the set of compact roots and Δ ρ for the set of noncompact roots. A subset τ C Δ is called a parabolic system of roots if there exists an element Χ G it such that τ = {α G Δ: a(X) > 0}. A positive system A+ is a parabolic system with Δ + Π —Δ+ = 0 . The Wey I group associated to t is the group We := NG(t)/ZG(t)

=

NK(t)/ZK(t)

which coincides with the Weyl group of the compact Lie algbra t. A positive system A+ is said to be t-adapted if Δ+ is invariant under the Weyl group. The Lie algebra g is said to have cone potential if for every non-compact root a and for every non-zero element Xa G g£ we have that [ Χ α , Χ α ] Φ 0. For a positive system Δ + of roots we define the cone Cmax := c m a x (A+) := {X e t: (Va G A+)ia(X) > 0} and Cmin := c mi „(A+) := cone{i[X a ,X a ]:a G A+,Xa

G gg},

where for a subset A of a vector space cone(A) denotes the smallest closed convex cone containing A. For a cone C in a vector space V we write C* {v G V*: u(C) C R + } for the dual cone. The next theorem is the major one on the general existence of invariant cones (see Chapter III of [11], [36], [24], or Section 7.2 of [12]). T h e o r e m 5.3. Let Q be a finite dimensional real Lie algebra which contains a pointed generating invariant cone Co· Then there exists a compactly embedded Cartan algebra t C g and a t-adapted positive system of roots Δ + satisfying (i) Cmin C Co η t C C maxi (ii) the invariant pointed generating cones containing cmin are in one-to-one correspondence via C +-> C Π t with those cones in t which (a) are pointed and generate t, (b) contain c m i n , (c) are contained in c m a x , and (d) are invariant under the action of the Weyl group We; (iii) if g is semisimple, then there exist largest and smallest pointed generating invariant cones C m a x and C m i n containing cmin and these satisfy C m a x Π t = Cmax and Cmin Π t = Cmin. •

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R e m a r k 5.4. Let 0 be a finite dimensional Lie algebra containing a pointed generating cone, and let Gc be a connected Lie group with Lie algebra 0c = 0 + ig for which the subgroup Go generated by expg is closed. Theorems 5.2 and 5.3 establish a program for generating all complex Ol'shanskii semigroups of the form exp(iC)Go in Gc- Fix a compactly embedded Cartan algebra t of q. Then for any C-adapted positive system of roots Δ + , and any cone c C t satisfying (a)-(d) of Theorem 5.3(ii), there results an Ol'shanskii semigroup of the form Go exp(ic)Go, provided the complex Ol'shanskii semigroup exists for the data Gc and C, where C is pointed, generating, invariant, and satisfies C Π t = c. This will always be the case if Gc is simply connected or g is semisimple. Thus in either of these latter cases, such cones c parameterize the complex Ol'shanskii semigroups. •

6. Hermitian simple algebras The preceding results on invariant cones can be sharpened somewhat in the semisimple setting. Recall that a simple Lie algebra is called hermitian symmetric or simply hermitian if the corresponding symmetric space is hermitian; this is equivalent to the condition that a maximal compact subalgebra has a nontrivial center (which then turns out to be one-dimensional). Α semisimple Lie algebra is hermitian if each of its simple ideals is hermitian. The reader is referred to [35], [37], or [11] for the next result. T h e o r e m 6.1. (The Kostant-Vinberg theorem) Α semisimple Lie algebra g contains a pointed, generating invariant cone if and only if it is hermitian. • Let g be a hermitian simple Lie algebra, let t be a maximal compact subalgebra, and let Bg(·, ·) be a positive-definite form on 0 given by Bq(X,Y) — —κ(Χ, Θ(Υ)), where κ is the Cartan-Killing form and θ is the Cartan involution which fixes For any wedge W C g, we define the dual wedge W * by W * : = {X e g: ΒΘ(Χ, Y) > 0, VY G W } . If t is a Cartan subalgebra of 0 contained in Ϊ, then we also consider the dual in t of wedges contained in t, where the dual is calculated with respect to the restriction of Bq to t. It should be clear from context which dual is under consideration. Let Ζ £ Ε span the center of Ϊ, and let t be a compact Cartan subalgebra of 0 contained in 6. Then Ζ e t, by maximal commutativity of t, and there exists a t-adapted positive system of roots Δ + such that ia(Z) > 0 for all α Ε Δ + . The following conditions hold for the hermitian simple Lie algebra 0 (see [30]): T h e o r e m 6.2. Let g be a hermitian simple Lie algebra. Then (i) Cmin contains Ζ in its algebraic interior, and c m a x = (Cmin)*, where the dual is taken in t.

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(ii) Each pointed generating invariant cone in g contains either Ζ or —Z in its interior. (iii) For each pointed generating invariant cone C, we have (Cflt)* = C*flt, where the first dual cone is computed in t and the second in g, both with respect to Β,Μ(iv) The cone C m i n is the smallest invariant cone containing Ζ and C m a x = (C mi „)*. • The next theorem appears in [16]. Theorem 6.3. Let g be a simple Lie algebra over R that does not admit a complex structure. Let W be a Lie wedge in 0c, W φ —W, such that the edge go : = W Π — W is isomorphic to g. Then the following conclusions hold: (i) The subalgebra go is a real form for gc, and for Χ, Y 6 go, the mapping σ(Χ + iY) = X — iY on gc, conjugation with respect to go, is α Lie algebra involution and is the only Lie algebra involution of gc with go C g + φ gc(ii) IfiC : = W n igo, then C is a pointed invariant cone in go, and iC has real spectrum. (iii) The Lie algebra g is hermitian. (iv) If β Q Qo is an isomorphism, then β extends to an automorphism of gc defined by β(Χ + iY) = ß(X) + iß(Y) for X,Ygq. If ß{Ci) = C, then β carries the symmetric wedge g + zCi to W = go -MC. (v) Let GQ be a connected Lie group with Lie algebra gc· Then (expiC)Go is a closed strictly infinitesimally generated Ol 'shanskii semigroup with subtangent wedge W . Thus every Lie wedge in gc with edge isomorphic to g is a symmetric Lie wedge, and the subtangent wedge for an Ol 'shanskii semigroup in Gc, which it (strictly) infinitesimally generates. • Example 6.4.

Consider the pseudohermitian form on C 2 given by J((w1,w2),(zi,z2))

=WIT[-W2z5.

Then su(l,l), the J-skew-symmetric matrices of trace 0, form a subalgebra of sl(2,C) that is isomorphic to the subalgebra sl(2,R), as is well-known and easily checked. Applying Theorem 6.3, we can find a Lie algebra automorphism β on sl(n, C), carrying sl(2, R) to su(l, 1). It must then carry the sl(2, R)-invariant cone C = C m a x = Cmin of Example 2.10 to an su(l, l)-invariant cone. Since Sl(2, C) is simply connected, β lifts to S1(2,C) and must carry the complex Ol'shanskii semigroup exp(iC) Sl(2, R) of Example 2.10 to one with group of units SU(1,1). Since the invariant cone C is unique up to ±1, it follows that the image semigroup must either be the Ol'shanskii semigroup of ./-contractions or ./-expansions that have determinant 1, by Theorem 6.3 and the results of Section 4. •

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7. Maximal and minimal Ol'shanskii semigroups We consider the two extreme cases of proper complex Ol'shanskii semigroups. Knowledge that the maximal Ol'shanskii semigroup is a maximal semigroup is often helpful in determining whether certain semigroups are Ol'shanskii semigroups. In particular this information can often establish the difficult inclusion we encountered in Section 4 of showing that a certain semigroup is a subsemigroup of an Ol'shanskii semigroup. Definition 7 . 1 . Let g be a semisimple Lie algebra containing a pointed generating invariant cone, and let Gc be a connected Lie group with Lie algebra gcA maximal resp. minimal complex Ol'shanskii semigroup T(C) in Gc is one for which C is a maximal resp. minimal pointed generating invariant cone in g, or, equivalently (from Theorem 5.3), one for which C Π t = c m a x resp. C Π t = c m i n for some compactly embedded Cartan algebra t and some t-adapted positive system Δ + . By Theorem 2.8, maximal and minimal Ol'shanskii semigroups always exist.• P r o p o s i t i o n 7.2. Let g be a hermitian simple Lie algebra, and let Gc be a connected Lie group with Lie algebra gc· Then there are precisely two maximal (resp. minimal) complex Ol'shanskii semigroups in Gc containing expg, and these two are inverses of each other. Proof. It follows from (iv) of Theorem 6.2 that there exist only two minimal invariant cones, and it follows immediately that one is the negative of the other. By duality, there are exactly two maximal invariant cones, which must also be negatives of each other. The proposition follows, since the existence is guaranteed by Theorem 2.8. • Recall that a subsemigroup S of a Lie group G is said to be maximal or a maximal semigroup if S is not a subgroup and there is no subsemigroup strictly between S and G. Hilgert and Neeb have shown that in the setting of hermitian simple algebras, maximal Ol'shanskii semigroups are maximal semigroups (see Chapter 8 of [12]). Actually, a little more is true (see [17]): Theorem 7.3. Let g be a hermitian simple Lie algebra, let Gc be a connected Lie group with Lie algebra gc, and let S φ Gc be a subsemigroup of Gc containing G, the subgroup generated by expg. If G is not open in S, then S contains a minimal Ol'shanskii semigroup and is contained in a maximal Ol'shanskii semigroup. In particular, a maximal Ol'shanskii semigroup is a maximal semigroup. • R e m a r k 7 . 4 . The machinery developed thus far can often be very effectively used for studying a particular semigroup S and determining whether it is equal to some complex Ol'shanskii semigroup r ( C ) . In particular suppose that S is closed, contains G, the subgroup generated by exp g, and contains a sequence converging to e in S \ exp g. Then S contains a minimal Ol'shanskii semigroup, and thus has dense interior. Since S is contained in a maximal Ol'shanskii semigroup, it follows

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from Theorem 5.2 that each member of S° can be written as gi(expiX)gi, where c m a x C t, a fixed compact Cartan subalgebra of g. Since S is the closure of all such elements, it follows that S is determined by its intersection with exp(zt). In particular, S — T ( C ) if and only if S Π exp(zt) = exp(i(C Π t)). A similar equivalence holds for containment in either direction. These computations at the Cartan algebra level are generally much more manageable. •

9i)92 € G and X €

E x a m p l e 7 . 5 . We recall the example T ( C ) = exp(iC) Sl(2, R) in S1(2,C) appearing previously in Example 2.10. Since C is the only invariant cone in sl(2,R) (up to ± 1 ) , this is the only Ol'shanskii semigroup containing S1(2,R) and must be a maximal semigroup by Theorem 7.3. Similar remarks apply to the isomorphic variant of this example given in Example 6.4 with group of units SU(1,1). It follows that the semigroups of Ji,i-contractions and Jij-expansions are maximal semigroups of Sl(2, C). • We remark that the Ji t i-contraction semigroup of the preceding example was introduced under another guise in [6] as a tool to aid in the study of certain physical problems in multiperipheral dynamics. Certain properties of this semigroup and its representations are established there. Among other things, a direct proof is given that the contraction semigroup is a maximal semigroup.

8. su(p, q) We return to the consideration of semigroups associated to a form J ( · , •), but this time restrict our attention to linear operators of determinant 1. Since every nondegenerate pseudohermitian form is equivalent to one of the standard forms given by

we may assume that J = JPtQ for a pair (ρ, q) with ρ 4- q = n. We consider the subgroup U(p, q) of Gl(p + q, C) of complex matrices which preserve the pseudohermitian form J on C n and the hermitian simple subgroup SU(p, q) consisting of those matrices of determinant 1. We want to apply the results of the earlier sections to the corresponding hermitian simple Lie algebra su(p, q). We first note that

A maximal compact subalgebra is given by

Semigroups of Ol'shanskii type

141

and a compact Cartan subalgebra t of su(p, q) is given by t =

Β

0 \

n

n

€ t:B,C

diagonal

Note that tc consists precisely of the diagonal matrices of trace 0 in gl(n, C). Define functional ε* G by

(0 « " {

fei, Ci_p,

for i = 1, for i = ρ + 1 , . . . ,n.

In the following we write d i a g ( i i , . . . , tn) for the diagonal matrix with entries £ i , . . . , i n . Then by results of Paneitz [30], the minimal and maximal invariant cones in the hermitian simple algebra su(p, q) are determined by the following cones in the Cartan algebra t: Cmin = {c = diag(iAi,... ,ίΧρ,ισι,..

.,iaq):tic

= 0, (Vi, j)Aj < 0 < Cj}

and Cmax = {c = d i a g ( i A i , . . . , ί λ ρ , ί σ ι , . . . , i a 9 ) : t r c = 0, (Vi, j)aj - A, > 0}. One observes that for c = diag(zAi,..., i\p, ϊ σ χ , . . . , iaq) G c m a x , we have c G c m i n if and only if ic is positive semidefinite with respect to J. It follows that C mmin i n = {X € su(p, q): J(iXv,v)

> 0 for all ν G C n }

since this is an invariant cone whose intersection with t is that this agrees with the computation in [30] that c

min

m i n

=

cmin.

One verifies directly

{X € SU(p,q):^i(Xv,v) > 0, W G C"},

where 2l(u, v) := — Im J(u, v) is a sympletic form. We observe that sl(p +

we conclude from o jm = j'm ο (π Ο restf ( 5 v v ) ), that j» is an isomorphism. Hence the group of units H(Sw) is connected and simply connected ([23]). Therefore the restriction H(Sw) —• H(Sw) = GHQ is the universal covering mapping. We write G" for the analytic subgroup of H(Sw) corresponding to the Lie algebra Q. This is a covering group of G. Since the semigroup Sw Q Gc is invariant under J h e involution g g*y this involution lifts to a continuous involution s ·—* s* on Sw which is holomorphic on the interior int := g - 1 ( i n t S W ) . Let D C H(Sw) be a discrete central subgroup which is invariant under the involution on Pick d € D. Then we consider the automorphism Id', s > dsd~l on of Sw- Since Id fixes ^i(S) C H(Sw), it induces the automorphism S\v which is trivial, because it is trivial on the subgroup G, it is holomorphic, and Gc is connected. Hence Id is trivial because it is the unique lift of the identity mapping of S to S. We conclude that D is a discrete central subgroup of Sw· In [12, 3.20] it is shown that the quotient semigroup Sw/D is well defined and that the quotient mapping Sw S w / D is a covering morphism of monoids. Definition 13.2. Let g be a real Lie algebra, W C g a generating invariant wedge, and Sw as above. An extended complex Ol'shanskii semigroup r(g, W, D) is a semigroup isomorphic to Sw/D, where D C Sw is a discrete central subgroup invariant under the involution. Note that these are precisely those discrete central subgroups of the simply connected group associated with the Lie algebra 0 + i)c which are invariant under the corresponding complex conjugation. We also write Γ(07 W) for r(g, W, {1}). •

Semigroups of Ol'shanskii type

155

T h e o r e m 13.3. Let S = Γ(β, W, D) be an extended complex Ol'shanskii semigroup. Then the following assertions hold: (i) The dense semigroup ideal int(S') : = i n t ( S w ) / D is a complex manifold and the multiplication on int(5) is holomorphic. (ii) The involution s i—• s* induced on S is antiholomorphic on int(5). (iii) H(S) = H/D, where Η is the simply connected Lie group associated with the Lie algebra g + H(W)c. (iv) The subgroup XJ(S) : = { s € S: s*s = 1 } is a subgroup with Lie algebra g. ( v ) S = H(S)Exp(iW) = /rwU(S)0Exp(iWO, where Hw = ( e x p H ( W ) C ) is the subgroup corresponding to the ideal H(W)c in £(H(S)) and Exp: g + iW —• S is the exponential function of S. If W is pointed, then U(S) = H(S) is connected and the decomposition S = H(S) E x p ( i W ) is topologically a direct product decomposition. (vi) S= (Exp C(S)). •

References [1]

R.W. Brockett, Lie algebras and Lie groups in control theory, in: Geometric Methods in System Theory, D.Q. Mayne and R.W. Brockett, Eds., Reidel, Dordrecht, 1973, 43-82.

[2]

M. Brunet, The metaplectic semigroup and related topics, Rep. Math. Phys. 22 (1985), 149-170.

[3]

M. Brunet and P. Kramer, Semigroups of length increasing transformations, Rep. Math. Phys. 15 (1979), 287-304.

[4]

N. Dörr, On Ol'shanskii's semigroup, Math. Ann. 288 (1990), 21-33.

[5]

N. Dörr, Cartan-Algebren und Ol'shanskii-Keile in symmetrischen Algebren, Dissertation, Technische Hochschule Darmstadt, 1991.

[6]

S. Ferrara, G. Mattioli, G. Rossi, M. Toller, Semi-group approach to multiperipheral dynamics, Nuclear Phys. Β 53 (1973), 366-394.

[7]

I.Μ. Gel'fand and S.G. Gindikin, Complex manifolds whose skeletons are semisimple real Lie groups and analytic discrete series of representations, Funktsional. Anal, i Prilozhen. 11 (1977), 19-27.

[8]

S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, Orlando 1978.

[9]

J. Hilgert, A note on Howe's oscillator semigroup, Ann. Inst. Fourier 39 (1989), 663-688.

[10]

J. Hilgert, Applications of Lie semigroups in analysis, in: The Analytical and Topological Theory of Semigroups, Editors: K.H. Hofmann, J.D. Lawson, J.S. Pym, de Gruyter, Berlin 1990, 27-50.

[11]

J. Hilgert, K.H. Hofmann, and J.D. Lawson, Lie Groups, Convex Cones, and Semigroups, Oxford University Press, Oxford 1989.

Lie-

J.D. Lawson J. Hilgert and K.-H. Neeb, Lie Semigroups and their Applications, Lecture Notes in Math. 1552, Springer, Heidelberg 1993. R. Howe, The oscillator semigroup, in: The Mathematical Heritage of Hermann Weyl, Proceedings of Symposia in Pure Mathematics 48, American Mathematical Society, Providence 1988, 61-132. K. Iwasawa, On some types of topological groups, Ann. Math. 50 (1949), 507557. J.D. Lawson, Computing tangent wedges in semigroups, in: Kochfest, J. Hildebrand, ed., Baton Rouge 1986, 63-67. J.D. Lawson, Polar and Ol'shanskii decompositions, J. Reine Angew. Math. 448 (1994), 191-219. J.D. Lawson, Maximal Ol'shanskii semigroups, J . Lie Theory 4:1 (1994), 17-35. J.D. Lawson, Free local semigroup constructions, Monatsh. Math. (1995), to appear. C. Loewner, On some transformation semigroups invariant under Euclidean or non-Euclidean isometries, J . Math, and Mech. 8 (1959), 393-409. C. Loewner, Collected Papers, L. Bers, ed., Contemporary Math., Birkhäuser, Berlin 1988. O. Loos, Symmetric Spaces I, W. A. Benjamin, New York 1969. M. Lüscher and G. Mack, Global conformed invariance in quantum field theory, Comm. Math. Physics 41 (1975), 203-234. K.-H. Neeb, On the fundamental group of a Lie semigroup, Glasgow Math. J. 34 (1992), 379-394. K.-H. Neeb, Invariant subsemigroups of Lie groups, Mem. Amer. Math. Soc. 499, American Mathematical Society, Providence 1993. K.-H. Neeb, Holomorphic representation theory and coadjoint orbits of convex type, Habilitationsschrift, Technische Hochschule Darmstadt, 1993. K.-H. Neeb, Holomorphic representation theory. I, Math. Ann. 301 (1995), 155181.

G.I. Ol'shanskii, Invariant cones in Lie algebras, Lie semigroups, and the holomorphic discrete series, Funct. Anal. Appl. 15 (1982), 275-285. G.I. Ol'shanskii, Convex cones in symmetric Lie algebras, Lie semigroups and invariant causal (order) structures on pseudo-Riemannian symmetric spaces, Soviet Math Dokl., Amer. Math. Soc. Transl. Ser. 26 (1982), 97-101. G.I. Ol'shanskii, Invariant orderings on simple Lie groups, the solution to Vinberg's problem, Funct. Anal. Appl. 16 (1982), 311-313. S.M. Paneitz, Invariant convex cones and causality in semisimple Lie algebras and groups, J . Funct. Anal. 43 (1981), 313-359. S.M. Paneitz, Determination of invariant convex cones in simple Lie algebras, Ark. Mat. 21 (1984), 217-228.

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V.P. Potapov, The multiplicative structure of J-contractive matrix functions, Trudy Moskov. Mat. Obshch. 4 (1955), 125-236. English translation: Amer. Math. Soc. Transl. Ser. 15 (1960), 131-243. W.F. Reynolds, Hyperbolic geometry on a hyperboloid, Amer. Math. Monthly 100 (1993), 442-455. L.J.M. Rothkrantz, Transformatiehalfgroepen van niet-compacte hermitische symmetrische Ruimtem, Dissertation, Univ. of Amsterdam, 1980. I.E. Segal, Mathematical Cosmology and Extragalactic Astronomy, Academic Press, New York 1976. K. Spindler, Invariante Kegel in Liealgebren, Mitt. Math. Sem. Glessen 188, 1988. E.B. Vinberg, Invariant cones and orderings in Lie groups, Funct. Anal. Appl. 14 (1980), 1-13.

Lie groups and exponential Lie subsemigroups Karl H. Hof mann and Wolfgang A.F.

Ruppert

We shall summarize in this chapter the fundamentals of the theory of exponential or, equivalently, divisible Lie semigroups. The proofs will generally be omitted; they appear in the monograph [15] which also contains a detailed exposition of the technical facets of the theory. *

1. Introduction An element s in a semigroup S is said to be divisible if for every natural number η € Ν there exists an element s(n) € S such that s(n)n = s. If S is written additively then the last equation becomes η • s(n) = s, which is the formal equivalent of s being divisible by η in the elementary sense. A semigroup is said to be divisible if all of its elements are divisible. Divisibility plays a key role in the structure theory of abelian groups, in the solution of Hilbert's fifth problem, and in the structure theory of compact topological semigroups. The unifying idea behind is to introduce 'linear' coordinates via one-parameter subgroups or semigroups, in other words: to define an exponential function. Clearly, if s lies on a one-parameter subsemigroup then it is divisible; a great many theorems establish the converse in various contexts. In the realm of Lie groups the most general statement of this type seems to be M C C R U D D E N ' S result that every divisible element in a connected Lie group lies on a one-parameter subgroup ([17], 1981). In particular we know that a connected Lie group is divisible if and only if it is exponential, that is, if its exponential map is surjective. Let us also recall that all Lie groups with compact Lie algebra are exponential, and that a simply connected solvable Lie group is exponential if and only if its exponential map is injective. The Lie group G is called weakly exponential if the exponential image is dense in G. It is well known that not all connected Lie groups are weakly exponential, *

We thank the Deutsche Forschungsgemeinschaft, the Alexander von Humboldt Founds tion, the Vereinigung von Freunden der Technischen Hochschule Darmstadt, the Technische Hochschule Darmstadt and the Universität für Bodenkultur Wien for their support of the research on this project which extended over several years.

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let alone exponential. For instance, the group Sl(2, R ) is not weakly exponential: the exponential image expsl(2, R ) misses the nonvoid open set of all matrices g Ε Sl(2, R ) with strictly negative entries. By a result of HOFMANN and MUKHERJEA [11] (1978) a connected Lie group is weakly exponential if and only if, in current terminology, every Cartan subgroup ([5]; [10], Section 4) is connected. (This follows from [4], Corollary 18, p. 34, and [5], Proposition 1.2, p. 140.) In the present survey we consider divisibility of closed subsemigroups S of a connected Lie group G. Similarly to the group case S is divisible if and only if it is exponential, i.e., S = exp W, where W is the Lie wedge {a; € g | e x p R + · χ e S } of S. This fact was shown by HOFMANN and LAWSON [8] in 1983. We have shown in [15] that the closure of a divisible subsemigroup of a Lie group is weakly exponential. However a divisible subsemigroup need not be contained in the exponential image of the Lie algebra, so the closure of a divisible subsemigroup need not be divisible (cf. 1.3 below). Let us list a few examples. 1.1. Examples in connected abelian Lie groups. (Fig. 1) It is trivial that every abelian Lie semigroup is exponential. As a prototype we consider the real half-line R + . Slightly less trivial examples are depicted in Figure 1.

Fig. 1. Lie subsemigroups of the direct product G = (R/Z) X R. The Lie wedge of the exponential Lie subsemigroup at the left is the wedge {(x,y) € 0 = R 2 I 0 < 0.2x < y < 0.6x}, the Lie wedge for the semigroup at the right is {(x, y) 6 g = R2|y>|x|}.

1.2. Examples in the two-dimensional affine group. (Fig. 2) The Lie algebra of the affine group G = (R, +)>Ί (]0, oo[, ·), the so-called ax + 6-group, is the nonabelian Lie algebra in two dimensions. This Lie algebra is metabelian (i.e., g " = { 0 } ) , and all operators adx|[g,g] are diagonalizable. Every half-space in g is bounded by a one-dimensional subalgebra (hence is a Lie wedge) and its exponential image is a closed subsemigroup. All these subsemigroups are exponential and so is the intersection of every two of them. Thus every wedge in g is the Lie wedge of a closed divisible subsemigroup of G. If such a semigroup S is not a half-line then it is absolutely closed: If φ is a continuous homomorphism mapping S injectively into a Lie group G then 4 (R_|_, ·). The one parameter subgroups of G are exactly the lines passing through the identity (0,1). It is easy to see that the product ab of any two elements o, 6 € G always lies in the convex cone generated by the line segments connecting the identity with α and b. Thus every half space is an exponential subsemigroup of G and so are all intersections of half spaces.

(0,0)

1.3. The half-space in Mot. (Fig. 3) Let g be the motion algebra, g = {(ζ, α) | β R} with Lie brackets [(C,a), (£',«')] = (ΐαζ' - ΐα'ζ,Ο), and let G be the associated simply connected Lie group, G = {(z,t) | z Ε C, t € R} with multiplication (z,t)(z',t') = (ζ+β**ζ',ί+Ϋ). Then the half-space semigroup G+ = {(z, t) £ G 11 > 0} is not the exponential image of its Lie wedge W = {(ζ, α) | a > 0}, since exp g does not meet the points (c, 2kir) with k € N, c Φ 0. Nevertheless we have G+ — exp W (Fig. 3). Note that G+ contains the dense divisible subsemigroup Gj" = {(z, t) e G 11 € Q, t > 0, }. The Lie wedge W is a half-space bounded by the unique two-dimensional ideal in g, hence is invariant (under the adjoint action). G+ is not absolutely closed. ζ e C,a

Fig. 3. T h e half s p a c e S = M o t . Here G = Mot is the point set C χ R with multiplication (z,t)(z',t') = (ze-u' +eitz',t + t/), and S = C χ R+. Every one parameter subgroup of G either is contained in the normal subgroup C X {0} or else maps onto one of the curves {(zsini, t) | t £ R}, 2 € C. Thus the exponential image in G misses the sets {(c,kn) | c 6 C*}, k 6 N. The half space semigroup S is weakly exponential, but not exponential.

1.4. The semigroup Sl(2) + . (Fig. 4) Let g = sl(2, R), realized as the Lie algebra of all real matrices with trace 0, and define sl(2,R) +

c

—a

a

G R, 6, c > 0

}

(this is the set of all matrices m in sl(2, R) such that exp([0,1] · m) has nonnegative entries). Then for any Lie group G with Lie algebra g the exponential image exp W is a closed subsemigroup of G, hence an exponential Lie semigroup. Moreover, all these semigroups exp W are isomorphic. We write Sl(2) + for the exponential image of W in the group G = Sl(2, R). It can be shown that, similarly to the semigroups

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in Example 1.2, Sl(2) + is absolutely closed. Thus embedding G = S1(2,R) as a dense nonclosed subgroup into some reductive Lie group we get an example of a closed divisible subsemigroup which generates a nonclosed analytic subgroup.

Fig. 4. T h e standard Lorentzian cone and semialgebras in 0 = sl(2, R). We write g as the span of three elements H, P, Q with [Η, Ρ] = Ρ , [Η, Q] = - Q and [P, Q] = H. The standard Lorentzian cone is the null set {Λ" € fl | k(X, X) = 0} of the Killing form, it is invariant under the adjoint action. Its 'upper' part generates the invariant wedge Wj. The Lie wedge W+ = R · Η + R + · Ρ + Ε · Q of the semigroup Sl(2, R ) + of matrices with nonnegative entries is a semialgebra.

Fig. 5. Exponential and non-exponential subsemigroups of t h e simply connected group G = SL(2, R). In the parametrization of [3] the group G is defined on the space R^ in such a way that the one parameter subgroups axe planar curves. The exponential image expsl(2,R) (above left) is invariant under rotations about the z-axis, it definitely does not cover all of G. The connected subsemigroup S corresponding to the set S1(2,R)+ of all matrices in S1(2,R) with nonnegative entries (above middle) lies in expsl(2, R), it is a non-invariant exponential Lie subsemigroup of G, (Note that its Lie wedge is a Lie semialgebra.) The exponential image of the upper half of the invariant wedge {X € sl(2, R) | k(X, X) < 0} generates an invariant subsemigroup which is not exponential (above right).

Lie groups and exponential Lie subsemigroups

163

1.5. It can be shown that if S is any closed proper subsemigroup of a Lie group G with Lie algebra g — sl(2, E) such that the Lie wedge W of S has inner points in β then the following assertions are equivalent: (i) S is weakly exponential. (ii) W is a semialgebra and contained in some conjugate of Sl(2) + . (iii) S is exponential. Note that by [3], p. 110 each semialgebra in sl(2,R) satisfying (ii) above is the intersection of at most four half-space semialgebras. 1.6. An example in SO(3) χ R. Let G = SO(3) χ R and write g as the direct sum g = so(3) φ R (where R has trivial brackets). Let ||·|| be the operator norm of gl(3, R). For every k 6 so(3) the operator norm of k coincides with the spectral radius of k, that is, for nonzero k we have ||A;|| = |A|, where Λ is one of the two nonzero eigenvalues of k. For any t > 0 we define B(t) = {k € Κ | ||fc|| < i}, these sets are 0neighborhoods and invariant under the adjoint action of SO(3). The natural action of SO(3) on the unit sphere § 2 = {p e R 3 | ||p|| = 1} is effective and for every peS2 the orbit (exp.B(i))p is the 'scullcap' C(p, t) = {q e § 2 | ||p-g|| < 2| sin Conversely, we have B{t) = {k € so(3) | (exp k)p € C(p,t) for all ρ G § 2 }. Moreover, for any pair t i , t 2 6 R + and any ρ € S 2 we get (expß(ii))(expß(i 2 ))p = (ex P j B(ii))C( P ,i 2 ) =

(J

C(q,t1) = C(p,t1+t2)

= (exp5(ti + i 2 ))p,

qeC(p,t2)

and we conclude that expß(ii + f 2 ) = (exp(ß(ii))(exp(jB(i 2 )). In other words, the map t exp B(t) defines a one-parameter subsemigroup of the semigroup of all closed subsets of SO(3). (In the language of compact topological semigroups, it maps onto a 'nil-thread'.) Now the wedge W = {x + t\ xEt, te R + , with ||x|| < i} is a pointed invariant cone, and its exponential image S = exp W is closed. Since t ·—> B(t) is a one-parameter semigroup of subsets, S is a semigroup. Thus S is an exponential subsemigroup and invariant under all inner automorphisms of G. • Note that with the exception of the half-space in Mot (which is only weakly exponential) all of these examples are exponential, hence divisible. We ask whether (and if so, in which sense) these examples provide a 'zoo,' containing a typical representative of every species. 1.7. T h e classification problem. groups of connected Lie groups.

Classify all weakly exponential subsemi-

Since the closure of a divisible subsemigroup is weakly exponential the classification problem essentially also includes the classification of all divisible subsemigroups. We shall see later that after factoring the largest normal subgroup all

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weakly exponential closed subsemigroups are exponential, hence divisible. (Note that factoring the maximal normal subgroup of the half-space in Mot leaves us with a copy of R + , which is obviously exponential.) One might conjecture that the classification of divisible closed subsemigroups is easier than the task we set up in 1.6. In fact, however, the weakly exponential case is needed also in this (formally) more special case. Thus without loss of technical convenience we shall treat the more general case right from the beginning. Let us now look for the local version of divisibility. 1.8. Definition. A subsemigroup S of a Lie group G is called locally divisible if there exist arbitrarily small neighborhoods U of the identity such that for every element 5 G U Π S and every η G Ν there is an element s(n) lying in U Π S such that s(n)n = s. The crux of this definition is that the 'n-th root' s(n) can be taken in U η 5" too. 1.9. Example. Let G — R 2 / Z 2 and write p:R2 G for the quotient homomorphism. Set S = p ( R + · (1, \/2))· Then S is algebraically isomorphic to R + and thus each element in S has unique roots of all orders in S. We let UQ denote the identity neighborhood p(\ — )· Now let 0 < ε < \ and set Ue = ρ(]—ε, ε[ 2 ). Since S is dense in G we find an r G R + and natural numbers m and η such that |r(l, >/2) - (m + §)| < §. We set χ = p(r(l,\/2)) e S. It then follows that χ φ Uq and that 2x G Sf~)Ue. Thus there are arbitrarily small elements in S whose (unique!) square root in S is outside of the fixed zero neighborhood Uq. • The one-parameter semigroup S in this example is uniquely divisible but not locally divisible. It is not clear that this cannot occur when S is a closed divisible subsemigroup of a Lie group. The difficulty illustrated in Example 1.8 is caused by the presence of compact elements in the closure of S. Local divisibility is linked with special types of Lie wedges. Recall that a wedge W in a Lie algebra is called a Lie semialgebra, or semialgebra for short, if it is a local semigroup with respect to the Campbell-Hausdorff multiplication: we can find a zero neighborhood Β in g such that Β * Β is defined and such that (W Π Β) * (W Π Β) C W. Note that if the wedge W is invariant (i.e. W = eadxW for all χ G g) then it is a semialgebra. Semialgebras are precisely the Lie wedges of exponential local semigroups. 1.10. P r o p o s i t i o n . ([7], [3], IV.1.14 and IV.1.32) A closed subsemigroup of a Lie group is locally divisible if and only if its Lie wedge is a semialgebra. • The structure of semialgebras is fairly well understood, a full classification of them is now available through the work of ANSELM EGGERT [2]. Therefore, the classification problem becomes amenable via the classification of semialgebras, provided the answer to the following question is 'yes':

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1.11. T h e divisibility problem. Is every closed weakly exponential subsemigroup of a connected Lie group locally divisible (or, equivalently, is the Lie wedge of a closed weakly exponential subsemigroup a Lie semialgebra)? It is not difficult to check that in the above examples 1.1-1.6 all Lie wedges are semialgebras, yielding evidence that the answer to question 1.11 is "yes." This conjecture is further supported by the following result: T h e o r e m . (HOFMANN and LAWSON [8]; 1983) If the group of units of an exponential subsemigroup S is trivial (= {1}) then the Lie wedge of S is a Lie semialgebra. • 1.12.

At first glance this theorem seems to lead us quite near a definite affirmative answer to the divisibility problem, thus providing also a favorable vantage point for attacking the classification problem. However, the really challenging aspects of the problem do not show up if either S has a trivial group of units or G does not contain compact elements (cf. Example 1.8). Thus Theorem 1.12 is still tantalizingly far from establishing the conjecture. So far, our investigations indicate that problems 1.7 and 1.11 are interwoven very closely: the solution of the divisibility problem intrinsically requires at least a partial knowledge about the solution of the classification problem, and vice versa. Moreover, it appears that in order to prove that the Lie wedge of a divisible semigroup is a semialgebra one cannot avoid solving the general divisibility problem.

2. Proof strategies I: quotients and intersections 2.1. In order to reduce our task to the investigation of more special situations — and more handy pieces of the structures we want to study— there are two natural options: (i) passing to quotient objects SN/N and G/N, where Ν is a suitable closed normal subgroup of G; (ii) replacing G by a suitable analytic subgroup G\ and S by the intersection S η Gi. Clearly, each of these two operations can be expected to be useful only if the resulting new subsemigroup is closed and exponential [weakly exponential] whenever the old one is closed and exponential [weakly exponential]. Moreover, there should be a fairly obvious procedure translating structural information about the new semigroups into information about the old ones. In case (i) we have no troubles showing that SV = SN/N is exponential [weakly exponential] if S is. But in general the quotient SN/N will fail to be closed (a well known phenomenon for cone subsemigroups of a vector group), and the transfer of information on S/N and G/N back to S and G is far from straightforward. Nevertheless everything runs smoothly if Ν is contained in S. The following proposition

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shows that for determining whether the Lie wedge of an exponential [or weakly exponential] subsemigroup is a semialgebra (the solution of the divisibility problem 1.11) we may safely replace S by S/N, where Ν is the largest normal subgroup of G contained in S. On the level of Lie algebras this means that we may assume that W is ideal free: {0} is the only ideal of g contained in W. 2.2. Proposition. Let S be a closed subsemigroup of a Lie group G and let Ν be a closed normal subgroup of G with Ν C S. We write η for the Lie algebra of N. (i) The quotient morphism K:G —* G/N maps S onto a closed subsemigroup SN of G/N. Further, the induced morphism £(«) : g —• g/n maps the tangent wedge W of S onto the tangent wedge WN of SN(ii) The following statements (a), (b) are equivalent: (a) W is a semialgebra. (b) WN is a semialgebra. (iii) Of the following statements (c) implies (d): (c) S is weakly exponential [respectively, exponential]. (d) SN is weakly exponential [respectively, exponential]. • The implication (c)=»(d) in 2.2(iii) suggests to adopt the assumption "W is ideal free" likewise in connection with the classification problem 1.7, thereby leaving aside less important or obvious structural features. Properly speaking, in these notes we shall give a solution of the classification problem only under this reduction. We shall not address the problem of characterizing divisibility of a closed semigroup S in terms of its largest normal subgroup Ν and the related quotient SN·

The half-space semigroups in Mot show that in 2.2 the implication "SN divisible divisible" need not hold, even if Ν is divisible. It does hold, however, in the following, very special case: 2.3. Proposition. Let S be a closed subsemigroup of a Lie group G and let Ν be a closed normal subgroup of G with Ν C S. Then the quotient morphism K:G —* G/N maps S onto a closed subsemigroup SN of G/N. Suppose that the following hypotheses are satisfied: (a) SN is exponential. (b) Ν is exponential. (c) g is a vector space sum c + ri such that for any χ G c and y e η we have [x,y] e R - y . Then S is exponential. •

The restriction to the case where W is ideal free will turn out to be vital in dealing with the basic 'porcupine technique' which we will introduce in Section 5.

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Let us now turn to option 2.1(ii), the use of 'probing sections.' In this case the new subsemigroup Si will be closed if S is closed; possibly we have to replace the topology inherited from G by the intrinsic Lie topology of G\. Also, as S\ is a subsemigroup of S, the correspondence between assertions on S\ and assertions on S is immediate. The main obstacle, however, is the fact that the intersection of a closed subsemigroup S with an arbitrary analytic subgroup G\ need not be exponential [weakly exponential] if S is. Or, to put it into algebraic terms, an element s Ε S Π G\ may have roots in S but none in G\. Thus we should like to work only with probing analytic subgroups G\ which satisfy the following 'hit and stay' condition: (*) If a one-parameter subgroup of G hits G\ in an element g φ 1 then it must be contained in G\. Lemma 2.5 below will lend good arguments for the feasibility of the probing section approach, under restrictions similar to (*). For the sake of notational convenience we postpone its formulation, recalling first some basic definitions and introducing additional notation. 2.4. Definition. (i) A point g EG is said to be regular in G if the dimension of the nilspace of the operator Ad(c/) — 1 equals rankg, the dimension of the Cartan subalgebras in the Lie algebra g of G. The set of regular points in G is denoted Reg(G). (ii) An element χ e g is called regular in g if the nilspace of ado: is a Cartan subalgebra of g. The set of regular elements in g is denoted Reg(g). (iii) An element χ Ε g is called exp-regular if none of the non-zero eigenvalues of ad a: is contained in Ζ · 2πί; or equivalently, if any exponential function on g is nonsingular at x. We write reg exp for the set of all exp-regular points in g. It is known (cf. [4]) that exp re is regular in G if and only if χ lies in Regg Π reg exp. Recall that an analytic subgroup is said to be of maximal rank in G if its Lie algebra contains a Cartan subalgebra of g. 2.5. L e m m a . ([3], p. 461) Suppose that x,y Ε £ ( G ) = g are elements with exp a; = exp y. Then the following assertions hold: (i) If the exponential map is nonsingular at either χ or y then χ and y commute. Thus exp(x — y) = 1 and therefore χ differs from y only by an element which is either zero or generates a circle group. (ii) If the exponential map is nonsingular at χ and χ is a regular element in g then y lies in the Cartan subalgebra generated by x. (iii) If G\ is an analytic maximal rank subgroup of G with exp χ Ε Reg(G) Π G\ then expR · y C G\. • Maximal rank subgroups are automatically closed; obviously G itself and each exponential image of a Cartan subalgebra is a maximal rank subgroup. In a maximal rank subgroup the G-regular points form an open and dense subset, and every

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Cartan subalgebra of its Lie algebra is a Cartan subalgebra of g. A subgroup Μ is a maximal rank subgroup if and only if it contains regular points of G and its rank equals the rank of G. By 2.5(ii) the above hit and stay condition (*) is satisfied for maximal rank subgroups if we consider only points g Ε Gι which are regular in G. Wanting to apply the information of 2.5, we therefore have to modify condition (*), contenting ourselves with the following, slightly weaker one, which is satisfied by all maximal rank subgroups: (**) If a one-parameter subgroup of G hits G\ in a regular element ρ ^ 1 then it must be contained in G\. The class of maximal rank subgroups is indeed the 'right' class of probing subgroups to work with. The only restriction is that the probing subgroup should pass well through the interior of S, say m Π int e W φ 0 . (Note that this involves in particular that W has nonempty interior in g.) 2.6. T h e o r e m . (The maximal rank theorem) Suppose that the Lie algebra m of a maximal rank subgroup Μ of G meets the interior in g of the Lie wedge W of a weakly exponential subsemigroup S. Then the intersection S Π Μ is weakly exponential, too. • (Actually it suffices that 1 € Μ Π int S, but this condition is checked easily only if m meets the interior of W.) Note that the theorem does not say that S(~)M is exponential whenever S is exponential. Thus even if we are interested only in exponential subsemigroups we still have to consider also weakly exponential ones. For the proof of the maximal rank theorem one first observes that in a maximal rank subgroup Μ the G-regular elements are open and dense, so Reg(G) Π Μ Γ) S is open and dense in S Π Μ. It remains to settle the natural question whether the set of G-regular points in S Π Μ (or at least some dense subset thereof) is contained in the exponential image of W Π m. On the other hand Theorem 2.6 can be applied only if W is generating in g, i.e., has interior points in g. Thus we also need a result assuring us that we do not do harm to generality when assuming W — W = g. We shall deal with both of these questions in the next section.

3. Regularity and the intrinsic embedding theorem The first result of this section, while giving a sweeping answer to the questions at the end of the previous one, is of independent interest by itself, and is basic for the whole theory. The main idea of its proof is to use inner automorphisms to

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reformulate everything in terms of a fixed Cartan subalgebra— a Cartan subalgebra and its representations contain much of the structural information on the Lie algebra g, yet are much more manageable. The proof also involves the topological structure of the space of all closed subsemigroups of a Lie group (Chapter VII of [15], where further references are given), one has to show that the limit of one-parameter subsemigroups of S is contained in expVT, provided it contains a G-regular point. We omit further details. 3.1. Theorem. Every G-regular point in a weakly exponential subsemigroup S of a connected Lie group G is contained in the exponential image of the Lie wedge W ofS. •

3.2. Exercise. Deduce from 3.1 that in a nilpotent Lie group G every weakly exponential closed subsemigroup is exponential. Since in a connected Lie group the regular elements form an open and dense subset the following fact (known under various guises in geometric control theory) shows that S contains plenty of them if W Lie generates g. 3.2. Theorem. ([3], p. 380) Let W be the Lie wedge of a closed subsemigroup S of a connected Lie group G and suppose that (a) S is the smallest closed subsemigroup of G containing exp W; (b) the smallest Lie subalgebra of g containing W is g itself Then the interior points of S in G form a dense ideal of S. • If in addition S is weakly exponential then also W has interior points in g (so that the interior of W with respect to g is dense). More generally we have the following theorem, which is interesting for its own sake, too: 3.3. Theorem. (The interior point theorem) Let W be a wedge in the Lie algebra Q of a connected Lie group G. Then W has interior points in g if one of the following conditions holds: (a) The interior of exp W in G is nonvoid. (b) W is the Lie wedge of a closed subsemigroup S of G and the interior of exp W in G is nonvoid. • The above interior point theorem is the analogue for weakly exponential subsemigroups of the well known fact that a Lie semialgebra has interior points in the Lie algebra it generates (or, equivalently, W — W is a Lie algebra whenever W is a Lie semialgebra). At this stage of the theory one would like to pass to the analytic subgroup G(S) generated by S in G (or by the Lie subalgebra generated by W in g) and to consider S as a closed subsemigroup of G(S) with dense interior.

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The problem is that G(S) need not be closed in G, even if S is closed in G— remember that we encountered this situation in Example 1.3. By now, the results so far presented do not allow us to conclude that S is weakly exponential with respect to the intrinsic Lie topology of G(S). It is well conceivable that exp W might be dense in S with respect to the topology of G but not with respect to the intrinsic topology of G(S). Our next theorem settles this question in a very satisfactory fashion. For its proof one has to show first that in a dense analytic subgroup G\ of a Lie group G the G-regular elements are dense with respect to the intrinsic Lie topology of Gi. The assertion that the topologies induced on S by G(S) and G are the same is not straightforward; it is actually not necessary for the sequel (very fortunately so, since its proof uses a fact available only after the classification theorem is established). 3.4. Theorem. (The intrinsic embedding theorem) Suppose that G is a Lie group and that S is a closed subsemigroup with Lie wedge W. We write A for the analytic group generated in G by exp W and A^ for the same group, but endowed with its intrinsic Lie group topology. Further, let Τ denote the smallest closed subsemigroup of A^ containing exp W. Then the following assertions hold: (i) If Τ is weakly exponential in then S is weakly exponential in G. (ii) If S is weakly exponential then S = T, and Τ is weakly exponential in ALje. Also, W-W = a. (iii) Moreover, if S is weakly exponential then the topologies of S and Τ coincide. • In particular, if S is exponential then Τ is exponential. Thus replacing G by j4.Lie we can always pass to an isomorphic copy of S whose Lie wedge has inner points in g.

4. Reduced weakly exponential subsemigroups Recall that a Lie semialgebra is said to be reduced (in the Lie algebra g) if it is ideal free and generating. Now we know by Theorem 2.2 that for deciding whether the Lie wedge W of a weakly exponential subsemigroup is a semialgebra we may factor out the maximal normal subgroup contained in S, thus assuming that W is ideal free. By the results of the previous section we also may replace G by the analytic subgroup G(S) generated by S, so that W becomes a generating wedge. This justifies the following definition. 4.1. Definition. A closed weakly exponential subsemigroup S of a Lie group G is called reduced if (i) its Lie wedge W is ideal free, i.e. does not contain a nonzero ideal of the Lie algebra g of G] (ii) W has inner points in g.

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Given a weakly exponential subsemigroup S in a Lie group G it is easy to pass to a reduced one. This is a straightforward consequence of the material in the previous sections. 4.2. Proposition. Let S be a weakly exponential submonoid of a Lie group G and let A be the analytic subgroup generated in G by exp W. Denote by the same group A but endowed with its intrinsic Lie group topology. Suppose that Ν is the union of all closed normal subgroups of Aue which are contained in S. Then the following assertions hold: (i) Ν is a closed normal subgroup of A^ and the quotient map κ: A^ —> A^/N maps S onto a reduced weakly exponential subsemigroup SN of A^/N. (ii) The tangent wedge of SN is the image of the tangent wedge of S under the induced quotient map α —> o/n between the respective Lie algebras. (iii) W is a semialgebra if and only ifWx (iv) If S is divisible then so is K,(S).

Wf η = £(£/v) is a semialgebra.

Proof, (i) By Theorem 3.4 S is weakly exponential in Ai,[e and α = £[^Lie] is W — W. Thus W has inner points in α and S has inner points in ^Lie· (See 3.2.) Clearly

0, o 2 +

be

4

The exponential image exp fC is not a semigroup, it generates the semigroup +

SK+ = exp(/C+) U expN · zexp ^ j ^

a2 + bc> 0

,

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which is closed in G. The exponential image of g (hence, a fortiori the exponential image of fC) misses the nonvoid open set e x p N · zexp ^ j

^

a2 + be > o j ^

Thus S)c+ is not weakly exponential and so is every subsemigroup containing it. It is known that every Lie semialgebra in sl(2, R ) contains whenever it contains a nonzero compact element. •

Let us now give a survey over those cognates of ose which we need for the definition of test algebras. These are introduced in a unified way, with the aid of the following lemma. 6.2. L e m m a . Assume that b is a real Lie algebra, spanned by five vectors u, v, x, y, z, subject to the following relations: [u, ν] = 0,

[u, χ] = χ,

[v,x]=y,

[u, y] = y,

[v,y] =

[x, y] = z,

-x,

[x, z] = 0

[y,z} = 0. We write η for the ideal R-rr + R - y + R - z and m for the ideal R · ν + η. Then we have the following assertions: (i) [u, z] = 2 · z, and [υ, ζ] = 0. (ii) If the vectors u, v, x, y are linearly dependent then χ — y = ζ = 0 and hence b = R - u + R - u is abelian. (iii) If x, y, ζ are linearly independent then the algebra η is isomorphic with the three-dimensional Heisenberg algebra. (iv) Ifv,x,y,z are linearly independent then m is isomorphic with the oscillator algebra ose. def

(v)

Each Cartan subalgebra of b is conjugate to I) = R - u + R - υ . Also, reg(b) = b \ m if ζ φ 0 and reg(b) = b \ η otherwise. (vi) Ifb is not abelian then its proper ideals are the subspaces { 0 } , R - z , all vector subspaces containing n. Every ideal of m is also an ideal ofb. (vii) For any X G R the proper ideals o/R · (λ · u + ν) φ η are { 0 } , R · ζ, η. (viii) / / u / 0 then the sets comp b and comp b are given by the following formulas: comp b = { Λ · e adb i/1 b G b, Λ e R } = R · { υ + λ · χ 4- μ • y - (Λ 2 4- μ2) • ζ | λ, μ G R } . γ Comp

Γ comp b U R - z , \ c o m p b U n = R - i / + n,

In particular, comp K m t / u / 0 .

if ζ ^ 0, if ζ = 0. •

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6.3. Definition. (i) The algebra spanned by five linearly independent elements u,v,x,y,z as in 6.2 will be denoted by extosc. For A e R we write osc(A) for the subalgebra R - ( A - u + v) + n o f extosc. (ii) The algebra spanned by four linearly independent elements u,v,x,y, and ζ — 0 as in 6.2 will be denoted by extmot. For λ 6 R we write mot(A) for the subalgebra R · (A · u + ν) + η of extosc. (iii) We define Β to be the class of all Lie algebras which are isomorphic to one of the following: {0}, so(3), sl(2,R), extosc, osc(A), extmot, mot(A),

where A € R.

6.4. Remarks. (i) We have the isomorphisms osc(0) = ose and mot(0) = mot. Each member of the class Β is isomorphic to an ideal of either extosc or extmot. (ii) For any proper nonzero ideal j of extosc the quotient extosc / j is either abelian or isomorphic to extmot. If j is a proper nonzero ideal of extmot then extmot / j is abelian. (iii) Let A € R. For any proper nonzero ideal j of osc(A) the quotient osc(A) / j is either abelian or isomorphic to mot(A). If j is a proper nonzero ideal of mot(A) then mot (A) / j is abelian. (iv) By (ii), (iii) above every homomorphic image of an algebra in Β either also belongs to Β or is abelian. (v) Every connected Lie group with Lie algebra mot(A) or osc(A) for some Α φ 0 contains a unique maximal compact subgroup (namely {1}). 6.5. Definition. A test algebra is a Lie algebra which is an ideal direct sum α Θ b where α is abelian and b belongs to the class Β of Definition 6.3. A test group is a connected Lie group G whose Lie algebra is a test algebra. We shall denote the analytic groups (expo) and (expb) by A and B, respectively. • We observe that a test algebra is reductive if it is either abelian or contains a copy of sl(2,R) or a copy of so(3). In a reductive test algebra the summand α is uniquely determined as the center of g, and the summand b is uniquely determined as the Levi complement. All nonreductive test algebras are solvable and in such algebras the summand b can be choosen so as to contain any given regular element. 6.6. Definition. A weakly exponential subsemigroup S of a connected Lie group is called trivial if it contains the commutator subgroup G' of the analytic subgroup G = (S,S~1) it generates. • We note that a weakly exponential semigroup with inner points in G is trivial if and only if its Lie wedge contains the commutator algebra g' of g = £(G), i.e, is a trivial semialgebra in the sense of [3], p. 130, II.4.11. 6.7. Definition. A Lie wedge W is called a test wedge if (a) W — W is a test algebra 9, and (b) either 0 is solvable or there is a nonzero compact element of the Levi complement b which lies in intb(b Π W).

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A test semigroup is a weakly exponential semigroup whose Lie wedge is a test wedge. • The crucial result about test semigroups is the following: 6.8. Theorem. (The testing theorem)

All test semigroups are trivial.

The proof of this theorem requires considerable effort and is omitted. It is essentially a mixture of rather explicit (but nontrivial) calculations and porcupine arguments. In the next section we shall apply the triviality of test subsemigroups to show that the Lie algebra of a Lie group supporting a reduced weakly exponential subsemigroup is an Eggert algebra.

7. O c c a m ' s razor' It is now time to produce Occam's razor' as promised in the previous sections. In combination with the results about test algebras this tool will enable us to unravel the structure of Lie groups supporting reduced weakly exponential closed subsemigroups. The key idea of this device is to apply a variant of the familiar pointing procedure ([3], II.4.3, p. 127) of the theory of Lie semialgebras in a new way and for a new purpose. 7.1. Lemma. Let W be a wedge in a Lie algebra g and let i denote the largest ideal of g contained in W. Then for any χ 6 intW there is a neighborhood Ux of 0 in q such that for every symmetric neighborhood U of 0 contained in Ux the intersection Wu d = p | e a d u W C W u et/ is a wedge with the following (i) x 0}, 0

0

ω

We now collect the most immediate consequences of the nonexistence of 'mixed' roots for a certain Cartan subalgebra. The proof is straightforward and therefore omitted. 8.9. Proposition. Suppose that J) is a Cartan subalgebra of a Lie algebra g, such that all roots with respect to fjc are either real-valued or purely imaginary-valued. Then the following conclusions hold: 0)

0 = ί)θ0Λ.θ0Ω+·

(ϋ) [0A r ,0 Ω+ ] = {0}. (iii) [) and suppose that the following hypotheses are satisfied: (a) f) is a Cartan subalgebra meeting the interior ofW = £(S). (b) No root Λ with respect to f)c satisfies A(f)) C R. Then gj) is compact. Proof. Define g* d = g/j, G* d= G/J, and f)* = ψ . Further W* = W/) and S* = S/J. Then by 2.2(iii) the semigroup 5 * is weakly exponential and ( a * ) f)# is a Cartan subalgebra of g * meeting the interior of = ( b # ) No root Λ with respect to f)* satisfies A ( f ) # ) C R. ( c * ) S1* is reduced. Now Theorem 10.8 applies to and shows that g * is compact. This is the assertion. • We are now very close to the solution of the divisibility problem. This is seen from the following proposition, whose hypothesis (comp) will be shown to hold in general, by courtesy of Corollary 10.9.

10.10. P r o p o s i t i o n . Let G be a Lie group containing a reduced weakly exponential subsemigroup S with Lie wedge W. Suppose that i) is a Cartan algebra meeting the interior of W and that g = s ® D ® ! is a decomposition according to Theorem 9.5. Assume the following hypothesis: (comp) The algebra Ε has a unique largest compactly embedded subalgebra. Then the following conclusions hold: (i) W is a semialgebra. (ii) W decomposes in the form

w η s = (w η s i ) θ · • · e (w η s n ) where the Sj = sl(2,R) are the simple components of s. (iii) The commutator = [t>,0] = jja* is an ideal of g and the base ideal of g is the direct sum δ' φ 3 of ö' with the center 3 of g. (iv) Set W i n v = W Ö T ö 7 , W sec = r\{LP(W0) I ρ e C\W0) and t>' % TP(W0)}. Then the wedge Winv is the smallest invariant wedge containing Wq, the wedge Wgec is an intersection of half-space semialgebras containing A = V, and WO - Winv Π W s e c .

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Proof, (i) We let c denote the largest compactly embedded subalgebra of 6. Now [s + 3, t] = { 0 } and comp δ = { 0 } . Then compg = (comps) φ c. But c o m p G ( g ) C comp g. Then Theorem 5.8 applies and shows that W is a semialgebra. Assertions (ii), (iii) and (vi) follow from well established properties of reduced Lie semialgebras (EGGERT's results 4.20 and 4.29 in [2], p. 76 and 79).

11. The answer Now here is the first main result on the structure of reduced weakly exponential subsemigroups. 11.1. Theorem. Let G be a connected Lie group with a reduced weakly exponential closed subsemigroup S. Then the following assertions hold: (i) g is an ideal direct sum 0=51φ···φ5λ:φϊ>φί

(ii) (iii) (iv) (v)

such thatSj = sl(2, R ) , j — 1 , . . . , k, 5 is diagonally metabelian without center, and I is a compact ideal. I contains the Α-radical and the center of Q. The radical of g is Ό φ j ( t ) and there is a unique Levi complement Si φ · · · φ sk®t'. W = £(S) is a Lie semialgebra. I f , conversely, g is a Lie algebra satisfying (i) then the simply connected Lie group with Lie algebra g contains a reduced weakly exponential closed subsemigroup.

Proof, (i) Prom Theorem 8.13 we know that g is an Eggert algebra and by Theorem 9.5 that g is the ideal direct sum g = s φ D φ t, where ( a ) s = Si φ · · · φ Sfc is a direct sum of ideals isomorphic with sl(2, R ) ; ( β ) 0 = 1 ) ι θ 0λ· is a centerfree diagonally metabelian Lie algebra; and f = f ) 2 ® 0 Ω + , where f)2 is a Cartan subalgebra of ? with imaginary-valued roots, containing the center of g. Let f)o denote a Cartan subalgebra of s not containing any compact elements. The (7)

Eggert decomposition is constructed in such a way that f) l)o Φ ill Φ 1)2 is a Cartan algebra of g meeting int W. The subalgebra m = [) + fc = f ) o © f h ® * i s a maximal rank subalgebra meeting int W . Let Μ = (expm). Then by the maximal rank theorem 2.6 VF Dm is the Lie wedge of a weakly exponential semigroup S Π Μ. Let j denote the largest ideal of m contained in W Π m and J the group generated by it. We apply Corollary 10.9 and find that m/j is compact. Also, it is not difficult to see that j Π H(W) is an ideal in g. Since S is reduced its Lie wedge W must be ideal free, so j Π t = { 0 } . Hence t = ^φϊ is compact. This finishes the proof of (i). (ii) By Theorem 9.6(iv) the ideal t contains the Δ-radical ( β Ω + ) .

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(iii) The radical χ adapts to the direct ideal decomposition and therefore is Ö Θ (r Π t). Since 6 is a compact Lie algebra by (i), its radical is its center. Thus τ = ί θ3(ϊ). Since g = s ® i ® t i s a n ideal direct sum, Levi complements for t = Ö φ 3(6) adapt to the decomposition. The Levi complement of 3(C) in t is t'. Thus there is a unique Levi complement s ® f . Assertion (iv) follows at once from (i) and Proposition 10.10(i), the proof of assertion (v) is omitted. • Combining Theorem 11.1 and Proposition 4.2 we immediately get the solution of the divisibility problem. 1 1 . 2 . T h e o r e m . Let G be a Lie group and S a weakly exponential with Lie wedge W. Then W is a Lie semialgebra.

subsemigroup •

For reduced weakly exponential closed subsemigroups S we can say more about the structure of W. We shall summarize this information in Theorem 10.7 below, but we need some preparation before. 1 1 . 3 . L e m m a . Let g = s(BQ®t be an Eggert algebra decomposed 9.5. Suppose that t is a subalgebra satisfying (a) e is ideal free. (b) e = (e Π Si) φ · · • φ (e Γ) Sfc) φ (e φ (δ θ t)). (c) e n t = { 0 } . (d) t is a compact algebra. Then e is diagonally metabelian (cf. Remarks 9.4/ Suppose, in addition, that the following condition is also (e) For all j = 1 , . . . , k we have dim(s:7· Π e) < 1. Then e Π s is central in e.

as in

Theorem

of

hyperplane

satisfied:

Finally, suppose that we have the follomng condition: (f) e Π (0 φ f) = / Π J where I is an ideal and J is an intersection subalgebras. Then e Π (D φ t) C 5 Φ 3(6)·

•

Now we recall that the semialgebras in sl(2,R) are classified in [3], pp. 109-110 as follows: Let Q = sl(2,R). Then every plane [X, 0] for k(X) = 0 is a tangent plane to the standard double cone and bounds two half-space semialgebras in g. Every semialgebra in g is the intersection of such half-space semialgebras. Note that the intersection of any family of semialgebras is a semialgebra. Also recall from 1.5 that the semialgebras in g which are Lie wedges of weakly exponential semigroups are intersections of conjugates of s l ( 2 , R ) + .

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We retain the notation of Theorem 11.1, we recall ö' = [ö, 0] = g^*, and for a wedge W in g we set, firstly, WQ = W Π (D φ C) and, secondly, W inv = WO + Ö',

and

11.4. T h e o r e m . Let G tial subsemigroup S with the following conclusions (i) There is a canonical

Wgec = F]{LP(W0)

| ρ e C^Wq)

and ö' %

TP(W0)}.

be a connected Lie group with a reduced weakly exponenLie wedge W. Decompose g as in Theorem 11.1. Then hold: decomposition

w = (Si Π W) Θ · · · Θ (afc (Ί W) Θ (W Π (D Θ C)) for which the following information is available: (a) Wo = WinvfiWsec, further Wmy is the smallest invariant wedge containing Wo, and Wsec is an intersection of half-space semialgebras containing A = t'. (β) Under the isomorphism Sj = sl(2, R), the semialgebras Sj Π W correspond to semialgebras contained in conjugates o/sl(2, R ) + C sl(2,R). (7) dimH(WDSj)

then we have p t (Ad (G).X) = conv(W t .X) + C m i n for all X e int t (W Π t).

•

Paneitz' proof of Theorem 2.6 depended on the linear Kostant convexity theorem and some ad hoc estimates for the elements of the adjoint orbit. Now it is an immediate corollary of Theorem 2.2 upon the identification between g and g* via the Killing form. Paneitz' approach to the classification of invariant cones was taken up by Olafsson ([42]) in the slightly more complicated situation of invariant cones in symmetric Lie algebras. More precisely, given a semisimple Lie algebra g and an involution σ on g, one considers the decomposition g = f) + q of g into the ±1eigenspaces of σ (cf. [22]). The aim is to classify the closed convex cones W in q which are invariant under the inner automorphisms of g coming from elements of [). In this case the existence of such cones also puts severe restrictions on the structure of the Lie algebra and the involution. The role of t is now played by a suitable maximal abelian subspace ο of q. Moreover, one also has a minimal cone which intersects ο in a cone c m i n . A key ingredient in Olafsson's classification of the invariant cones in q by their intersection with α is the following analogue of Paneitz' convexity theorem ([42]). T h e o r e m 2.7. (Olafsson's convexity theorem) Let pa: q —> α be the orthogonal projection w.r.t. the Killing form and g = t + p α σ-invariant Cartan decomposition. Further let WH be the Weyl group associated to a via the normalizer ofainKDH, where Κ and Η are the adjoint groups belonging to I and f). If now W is a closed Η-invariant convex cone in q containing Cmjn Π α, then pa(Ad(H).X)

= conv(WH.X)

+ (Cmin η a)

for all X e i n t e ( W Π α ) .

•

Olafsson's proof of this theorem consists of a clever adaptation of the Paneitz approach to the presence of an involution. Similarly to the group case it is possible to derive Theorem 2.7 from Theorem 1.2. The idea is to consider the involution τ := —σ*: g* g*. We do that in a more general situation. Let σ be an involutive automorphism of the Lie algebra g. Then g = I) + q, where f) = {X e g: σ(Χ) = X )

and

q = {X G g: σ(Χ) =

-X}.

Accordingly we have a direct decomposition of g* as g* = f)* θ q* = q x Θ f)"1. Suppose that t C g is a compactly embedded Cartan algebra which is invariant under σ. Then t = t+ Θ t_, where t+ := t Π fj, t_ := t Π q and t* = φ t l . We consider an element / e t * . Then r ( / ) = / and r is an antisymplectic involution

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of the coadjoint orbit Of. If / G t* is admissible, then the projection pt_ :Of —> t*_ is proper (cf. [19, Lemma 5.22]). For the following we fix a simply connected Lie group G with Lie algebra g and lift the automorphism σ of g to an involutive automorphism σα of the group G. Then Η := (exp f)) is the connected component of the group of fixed points of σα in G. T h e o r e m 2.8. g, and g = f) + q embedded Cartan is pointed, and f

(cf. [19]) Let g be a Lie algebra, σ an involutive isomorphism of the corresponding decomposition. Further let t C g be a compactly algebra, A+ a t-adapted positive system such that the cone G t!_ Π C£, in . Then

pt._(Ad*(H).f)

= pr_ (Of) = convp r _ ( W t . / ) + cone(iAj" | t _),

where Δ + = {a € Δ+: ( 3 7 € Wt)(3Xa

e g£) {/,Ί.[Χα,Χα})

< 0}-

•

Assume in addition that t C g is a compactly embedded Cartan algebra such that dim t_ is maximal. Then pt*_ ( A d * ( H ) . f ) = conv(VV./) + cone(iA+ | t _), where Δ+ = {a G A J : ( 3 7 e W t )(3X Q G g£) < / l 7 . [ * « , ^ « ] ) < 0} and VV is the subgroup of Gl(t_) generated by the reflections in the hyperplanes which are non-zero restrictions of compact roots to t_. So far we have simplified the formula for the set of extreme points of the image of an H-orbit. If / G int then Af = Δ+ and therefore the limit cone of the image is given by cone(A+ |t_), the cone generated by the restrictions of the non-compact roots to t_. Thus we have Pt* (Ad*(H).f)

= conv(VV./) + cone(iA+ | t _).

If, in addition, g = t is a compact Lie algebra, then Δ+ = 0 and therefore p t . ( A d * ( H ) . f ) = conv(VV./) is the linear convexity theorem of Kostant in the version for symmetric spaces of compact type. In the theory of symmetric spaces it is often convenient to pass from the symmetric Lie algebra (g, l), σ) to the c-dual Lie algebra (g c , f), σ°), where g c = f) + iq C gc and σ°(Χ + ιΥ) = X — iY. To explain how Olafsson's convexity theorem follows from our results, we assume from now on that g is a semisimple Lie algebra endowed with an involutive automorphism σ. We choose a Cartan involution θ commuting with r ([24, p. 153]) so that we obtain a Cartan decomposition g = C -I- p and a direct decomposition g = i)e + f)P + qi + q P ·

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We choose a maximal abelian subspace ο C q p . Then g c = f)t + i)p + iqt + iqp and t° = f)t + ic\p is a maximal compactly embedded subalgebra of gc. Therefore ia is contained in a Cartan subalgebra t of lc which is invariant under σ° and which satisfies t = te Θ ia, where te = t Π t. We consider g and g c as subalgebras of the complexification gc- Let Β denote the Cartan Killing form of gc and κ = Im Β its imaginary part. Then /c is a real symmetric invariant bilinear form on gc- The restriction to the real form g c is therefore 0, so that we can identify (gc)* with the subspace if) + q of gc- More explicitly, we have an isomorphism φ: if) + q —• (gc)* satisfying ψ(Υ)(Ζ) = κ(Υ, Ζ) which is equivariant with respect to Ad(G c ). Let X € q. Then ψ(Αά(Η).Χ)

= Αά*(Η).ψ(Χ)

C (iq)*

so that the calculation of pa(Ad(H).X) reduces to the determination of the image of the set Ad* (Η).ψ(Χ) under the restriction mapping to (ία)* = φ {a). To meet the assumptions we have made above, one needs that g c is a quasihermitian Lie algebra, i.e., a sum of compact and Hermitian simple ideals. This is clearly satisfied if either σ = θ is a Cartan involution (in this case g c is compact) and in the setting of Olafsson's theorem. In this case every maximal compactly embedded subalgebra u of g c has full rank so that the Cartan subalgebras of u are compactly embedded Cartan algebras in g c . Let g = t + ρ a Cartan decomposition compatible with the decomposition g = f) + q , a C q p a maximal abelian subspace, t C t + ip a Cartan subalgebra containing ia, and Δ + a C-adapted positive system of roots of gc with respect to tc- We set Σ := {α | α : α Ε Δ} \ {0} and define Σ ρ and Σ* accordingly. Then WH is the Weyl group of the restricted root system Σ*;, and Σ+ is the set of non-zero restrictions of positive non-compact roots. We define Cmax

:= - ( Σ + r = {Xea·.

(Va e Σ+) a(X)
G a such that B(Aa>,Y) = a'(Y) for all Υ € a. Then ^ - 1 (α>ηβ(ίΣ+)) = - cone{j4a': a ' € Σ+} = CminThe following theorem follows directly from Theorem 2.8 and the above considerations. Theorem 2.9. (cf. [19]) Let g be α semisimple Lie algebra, σ an involutive isomorphism of g, g = fj + q the corresponding decomposition, and g = I + ρ a compatible Cartan decomposition. Further let a C qp be a maximal abelian subspace, t C 6 + ip a Cartan subalgebra containing ia, and Δ + a t-adapted positive system of roots of gc with respect to tc such that X 6 int c m a x holds for the associated system Σ + of positive restricted roots. Then pa(Ad(H).X)

= conv (WH.X)

+ cmin,

J. Hilgert and K.-H. Neeb

212

and Wh is the subgroup of Gl(a) generated by the reflections in the hyperplanes which are non-zero restrictions of compact roots to a. • One should note two things at this point: If (g, f)) is an irreducible symmetric Lie algebra of regular type, i.e., gc is a Hermitian simple Lie algebra or the complexification of a simple Hermitian Lie algebra, then Theorem 2.9 is Olafsson's convexity theorem. If g is a semisimple Lie algebra and θ = σ a Cartan involution, then Ap = 0 and therefore Theorem 2.9 simply states that pa(Ad(K).X)

= conv(WH-X)

VX G α

which is Kostant's linear convexity theorem referred to after Theorem 2.8. For completeness and later use we record a result which illustrates the relation between invariant cones in g and cones in t: Proposition 2.10. (cf. [18]) Let g be a Lie algebra with cone potential which is not compact semisimple, t C g a compactly embedded Cartan algebra, and Δ+ a t-adapted positive system such that Cmis pointed. Let X G C m a x . Then (i) There exists a pointed generating G-invariant cone W C g with X G W. I f , in addition, X G intC m a x , the cone W may be chosen in such a way that X G int W. (ii) Let τ: g —• g be an involutive automorphism and suppose that int C m a x contains (—r)-fixed points and that τ(Χ) — —X. Then the cone W from (i) can be chosen to be (—τ)-invariant. •

3. Harmonic analysis on ordered spaces and non-linear convexity theorems It is a well known fact that in the study of spherical functions for a non-compact Riemannian symmetric spaces G/K one encounters integrals with phase functions of the form k i—• , where G = KAN is an Iwasawa decomposition, L\ KAN —> α the corresponding Iwasawa projection, k G K,a G A and Λ G α£· It is then useful to have control over L(aK) for all a € A. The non-linear convexity theorem of Kostant gives just that ([21]): Theorem 3.1. (Kostant's non-linear convexity theorem) Let WK be the Weyl group associated to α via the normalizer of a in Κ. Then L(aK) = c o n v ( W K . L ( a ) ) for all a G A.

•

Similar things come up in the context of ordered symmetric spaces. These are non-Riemannian symmetric spaces G/H with a causal structure given by a Ginvariant cone field in the tangent bundle. Such a cone field is determined by its

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213

value in the tangent space of a single base point i.e., by the choice of a cone in q = g/fj which is invariant under the action of H. Thus the causal structures are closely related to the invariant cones in q as they were discussed by Olafsson ([43], cf. also Section 2). The special structure of symmetric Lie algebras admitting such cones shows that one has an Iwasawa like decomposition HAN C G which no longer is valid for all of G but only for an open subset. More precisely, let (g, fj) be a symmetric pair of regular type and g = t + po a Cartan decomposition which is invariant under the involution defining I). Then any maximal abelian subspace α of po Π q is maximal abelian in po as w e ll as in q. Fix a compactly embedded Cartan algebra t of g c containing ia and a f-adapted positve system of roots Δ + . As before this yields a positive system Σ + of restriced roots. Let g = T + A + η and G = KAN be the corresponding Iwasawa decompositions. Using the decomposition with respect to the restricted roots it is not hard to show that g = f) + α 4- η. Moreover one sees that HAN is open in G and the multiplication map Η Χ ΑΧ Ν —» HAN is a diffeomorphism. Therefore we have an Iwasawa like projection LH : HAN —• a, han > log a. It is possible to define spherical functions on the positive domain of the ordered space which is of the form H A + H / H , where A+ denotes the exponential image of the positive Weyl chamber a + of a. It is then essential to have control over LH(O,H) for α € A+. To this end one has the following theorem. Theorem 3.2. (cf. [39]) Let WH be the Weyl group associated to a via the normalizer of a in Κ Π Η. Then LH{aH) = conv{W H -L H (a)) + (C min η α) for all a G A+ \ {1} (whenever G/H is irreducible).

•

This theorem can be formulated in such a way that it actually is a generalization of Theorem 3.1. Note here that the cosets aH are, in contrast to the cosets aK, non-compact. It should also be noted here that there is a similar non-compact convexity theorem due to van den Ban which we do not describe explicitly since so far it has not come up in the context of Lie semigroups. As was mentioned in the introduction, there have been attempts to prove nonlinear convexity theorems using the general symplectic convexity theorems. So far these attempts do not reproduce Theorems 3.1 and 3.2 in all cases. On the other hand the approach via Poisson geometry yields convexity theorems especially for non-semisimple groups which are not covered by those theorems.

4. Poisson semigroups A bivector field Λ on a connected Lie group G defines on G the structure of a Poisson Lie group if and only if A(l) = 0 and the Lie derivative is left invariant whenever A" is a left invariant vector field on G (cf. [26]).

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If now (G, Λ) is a Poisson Lie group, then one finds that L#A only depends on Λ"(1). Therefore we have a linear function ε A : T\ (G) = 0 —• 0 ® 0 which we also denote rfA(l). Taking adjoints we obtain a skew-symmetric mapping 4 : 0* x 0* ^ 0*.

(X »

(άΑ(1)(Χ),(ξ,η))).

This map defines a Lie algebra structure on g* (cf. [25]). We equip the vector space 0 X 0 * with the bracket [(Χ, ξ), {X\ ξ')] = ([Χ, X'} + ad* ξ.Χ' - ad* ξ'.Χ, [ξ, £'] + ad* Χ.ξ' - ad* Χ'.ξ), where ad* Χ.ξ = —ξ ο ad Χ . This bracket defines a Lie algebra structure on 0 χ 0* and this Lie algebra is called the dual extension of 0 and 0*. It is denoted 0 txi g*. is The bilinear form J((X, £)> {Yv)) '·= Ή{Χ) + invariant for 0 cxi 0*. Let 0 and b be Lie algebras. Then we say that (0, b) is a Manin pair if there exists a Lie algebra 0 isomorphic to the vector space 0 x b such that 0 and b are Lie subalgebras of Ö, and a non-degenerate symmetric bilinear form J on 5 such that (Ml) 0 and b are isotropic. (M2) J is invariant under the group (e a d a ). In this case we also write D = 0 txi b and call Ö a twilled extension of g and b. Let (0, b) be a Manin pair and D := 0 μ b the corresponding twilled extension. We write pg: D —ι• 0 and Pb: D —* b for the projections of Ö onto 0 and b according to the direct decomposition D = 0®b. We also recall the linear isomorphism ψ: g —> b* defined by J(X,Y)

= (Φ(Χ),Υ)

VXeg,Ye

b.

Let ß be a connected Lie group with L ( B ) = b such that the adjoint representation of b on D integrates to a representation Ad: Β —> Aut(ö). We write Adß(6) for Ad(ft) | b . Note that tpopg oAd(6)| fl = Adß(6)oV

V6 e B.

T h e o r e m 4.1. (cf. [18]) Let (0, b) be a Manin pair of Lie algebras and Β a connected Lie group with L(jB) = b such that the adjoint representation of b on ί = 0 txi b integrates to a representation of Β. Then the bivector field Κ on Β defined by (dP;\b)A(b))^V)

= (£,AdB(&)Pb A d i f t " 1 ) ^ " 1 ^ ) ) ,

where pb' B —• Β denotes the right multiplication Poisson Lie group on Β.

by b, defines the structure of a •

Similarly any connected group G with Lie algebra 0 is a Poisson Lie group whenever the adjoint representation of 0 on ί integrates to a representation of G because the definition of a Manin pair is symmetric in 0 and b.

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215

We recall the bundle map A" : T*(B) -* T{B) defined by (A»(0,t/> = A(e,i/) P r o p o s i t i o n 4.2.

V6 e Β,ξ,η

G Τζ{Β).

(cf. [18]) For Y G ί and X G g we define the vector field XY(b) :=dpb(l)pb Ad(6)y

and write ξχ for the left invariant 1 -form on Β with £γ(1) = φ{Χ)· = Χχ and the mapping g -» V(B),

X~X

Then

x

is a homomorphism of Lie algebras. If G is a connected Lie group with L(G) = g, then the orbits of the local Gaction on Β which is defined by the above homomorphism of Lie algebras are the symplectic leaves of Β. • The local G-action on Β is called the dressing action. Let D be a connected Lie group containing Β as the subgroup corresponding to b and also a subgroup G corresponding to g. From now on we assume that the mapping G χ Β

D,

(g,b)~gb

is a diffeomorphism onto the open subset GB of D. We denote the projections which constitute the inverse of this map by pb'· GB —> Β and pc- GB —• G. Note that this assumption implies that the subgroups G and Β of D are locally closed, and therefore closed. Moreover the proof of Proposition 4.2 is easy in this case: Let pa and ρ β denote the corresponding projections with PG(gb) = g and pB{gb) = b. Then g.b := pe(bg~1) defines an action of G on Β and for X G g the corresponding vector field is given by υ (6exp(tX)) t=o • Β A dt

t=oPB

(bexp(tX)b-1)

b = dPb{l)Pb Ad(b)X.

Consider the subsemigroup Bpoi := {b eB-.bG C GB} of B. We call it the Poisson semigroup in B. Now let r : D —• D be a Lie group involution which leaves G invariant such that the corresponding involution of the Lie algebra Ö which we also denote by r satisfies J(tX,TY)

=

-J(X,Y)

for all X , Y G D.

L e m m a 4.3.

(cf. [18]) If Β is invariant under τ, then τ* A = —A.

•

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216

An interesting class of examples comes from the dual extensions, where g* is an abelian Lie algebra. Let G be a Lie group with the involutive automorphism τ, and Ö := g*x g the corresponding dual extension. We set D := g*x G and extend r to an involution of D by τ(ξ,9)

:={-τ*ξ,τ(9))·

Then a straightforward calculation shows that r is an involutive automorphism of D, that Β := g* is invariant under r, and that

This means that all assumptions we made are satisfied for this class of examples. Suppose now that τ is an involution on D such that Β is invariant. The Poisson semigroup 5p 0 i is invariant under r , and for g Ε G and b 6 -Bp0i we have that r(g.b) = r(g).r(b). In particular τ permutes the G-orbits in Bpol and if a G-orbit contains a fixed point for τ, then it is invariant under r . Let Η :— GQ denote the connected component of the identity in the group of r-fixed points in G. Let further b 6 Bp0i with r(6) = b. Then is a submanifold of the symplectic leaf Lb through b which is invariant under Η and its connected components are the orbits of Η in Lb (cf. [18]) . Let now again be g a real Lie algebra containing a compactly embedded Cartan algebra t. Further let Gc denote the simply connected complex Lie group with Lie algebra L(G) = gc- Then, for G := (expg), we have that G = G"c = {g e Gc : σ(9) = g}, where σ denotes complex conjugation on Gc since the group on the right hand side is connected (cf. [24, p. 171]). We set α := it,

η := ^ ^ g£

and

b : = a + n = nxa.

α€Δ+

We set A := exp a, Ν := expn, and Β := exp b. P r o p o s i t i o n 4.4. (cf. [18]) The following assertions hold: (i) ο and η are subalgebras of g such that g c = 0 ® a e n = g®b is a direct vector space sum. (ii) The group Β is closed, simply connected, and Z(B) = exp (iZ(g)). (iii) The mapping G χ Α χ Ν —» Gc,

{g, a, n)

is a diffeomorphism onto an open subset of Gc·

gan

•

Symplectic convexity theorems, Lie semigroups, and unitary representations

217

In view of the preceding result, we have an analytic mapping (4.1)

L : GAN —> α,

gan Η-» log α.

In general the set GBG is strictly larger than the set GB, i.e., the set GB is not right invariant under G. We need an additional hypothesis on g in order to have that (g, b) is a Manin pair: Recall that a finite dimensional real Lie algebra g is called orthogonal Lie algebra if there exists a non-degenerate symmetric invariant bilinear form κ on g. We note here that the class of orthogonal Lie algebras contains the reductive Lie algebras, and in particular the compact Lie algebras. It also contains an interesting class of solvable Lie algebras called oscillator algebras which fit into our context. Proposition 4.5. (cf. [18]) Let g be an orthogonal Lie algebra with compactly + embedded Cartan algebra t, Δ the set of roots and Δ C A a positive system. Then g carries the structure of a Lie bialgebra such that the dual Lie algebra g* is isomorphic to b := b(A + ) := it Φ 0 By duality b also carries

the structure

g£.

of a Lie bialgebra

such that b* = g.

Proof Fix an invariant symmetric bilinear form Κ on g and write KQforits complex bilinear extension to gc, i.e., for Ζ = X + iY, Ζ' — X' + iY' with Χ, Χ', Υ, Υ' e g we have KC(Z,Z')

= Κ(Χ,Υ)

- K(Y,Y')

+ ΐ(Κ(Υ,Χ')

+

K(X,Y')).

We set J ( Z , Z')

:= Im KC{Z,

Ζ') = Κ{Υ, Χ')

+ Κ(Χ,

Υ').

This is a non-degenerate symmetric invariant bilinear form on gc· It is clear that g is isotropic. We claim that also b is isotropic. Let Χ Ε a. Then a d X is skew symmetric with respect to « c and therefore «c(flc>0c) = {0} for α + /3 ^ 0. It follows that «c(b,b) = Kc(a,a) = {0} because ig is also isotropic. Thus b is isotropic with respect to J . We now know that (g, b) is a Manin pair. • T h e o r e m 4.6. (cf. [18]) Let g be an orthogonal pactly embedded Cartan algebra and A+ a positive be a simply connected Lie group such that L{B) structure of a Poisson Lie group such that the Lie

Lie algebra containing a comsystem of roots. Further let Β = b(A + ). Then Β carries the algebra of the dual group is g.D

The Poisson semigroup in this case consists of all elements b € Β with bG C GAN = GB. We set Apol :— Α Π Bpol. Moreover Bp0i is the set of all elements in Β such that the symplectic leaf through this element is a whole global G-orbit.

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J. Hilgert and K.-H. Neeb

Thus, if Β φ ßpoi, the group Β provides an example of a Poisson Lie group where the dressing action, i.e., the local action of G on Β cannot be extended to a global action. The subsemigroup Bpol of Β is the maximal subset of Β with the property that the dressing action restricted to this set extends to a global action of G. Let ψ-.g —> b* be the isomorphism coming from J. Then φ(ί) — [α, b]"1 = α* and therefore ψ*(ο) = t*, where t* is viewed as the subspace [t, g]-1 of g*. Proposition 4.7. (cf. [18]) Letpß'GB —> Β be the projection gb i—• b. Then t.b = pß(bt~l) = tbt~l defines a Hamiltonian group action Τ χ Β —> Β with moment mapping ψ* oL : Β = AN

t*,

an ^ ip*(loga).

•

Let X e C m a x . Then one can show that the symplectic leaf La through α exp iX in Β is closed and the restriction of L to La is a proper mapping. Theorem 4.8. (cf. [18]) Let g be an orthogonal Lie algebra with cone potential, t C g a compactly embedded Cartan algebra, and A+ a I-adapted positive system such that Cmin is pointed. Let X € C m a x , a := expiX and suppose that the group e a d 1 is closed. Then a £ ^4ρ0ί and the set L(aG) is a convex set given by L(aG) = conv(W t .logo) + Ca, where Ca = cone{[X a ,X a ]:a G Δ+, (3 7 e Wt)(-y.a)(Z)

> 0, Xa €

flg}.

•

To clarify the scope of the preceding non-linear convexity theorem, we have the following characterization of the orthogonal Lie algebras with cone potential. For more details on this class of Lie algebras such as decompositions into indecomposable constitutents and the description of those, we refer to Section 4 in [18].

Theorem 4.9. Let g be a Lie algebra with cone potential. (a) Then g is orthogonal if and only if it is a direct sum of a reductive Lie algebra with a compactly embedded Cartan algebra and an orthogonal solvable Lie algebra with cone potential which has isotropic center. (b) A solvable Lie algebra with cone potential is orthogonal with isotropic center if and only if there exists a linear isomorphism I = 3 φ 3* and if Ta G 3, α G Δ are the elements representing the linear functionals ia on j*, then [ 0 W, gW] = RTa> where gH := g η (gg θ 0 ^ α ) . • Finally we consider the case of an involution r : g —• g of g which leaves κ invariant. We denote the complex antilinear extension of r to gc also by r . We assume that t is invariant under r so that α is also invariant under r . Let g = f) + q

Symplectic convexity theorems, Lie semigroups, and unitary representations

219

be the decomposition into r-eigenspaces with eigenvalues 1 and —1. Then the subalgebra g c = f) + iq is the subspace of fixed points for τ on gc· We write t+:=tnf),

t_:=tDq,

α_ := α Π ü) = it+,

a + : = a η iq = it_.

Note that r acts on Δ via τ.α = a or,

and

r(gg) = g£α.

Since every root is real on α we have in particular that (τ.α) | α = (α ο τ) |a. One can show that the positive system Δ + is invariant under r if and only if there exists a regular element XQ G a + such that Δ+ = {oG A:a(Xo)

> 0}.

This shows the limits for the applicability of the approach via Theorem 1.2. A necessary condition on the involution τ on g is that the subspace a+ contains regular points. This condition is equivalent to 3 fl (t_) = t or to the condition that 30 (t_) is abelian. Let rt_: t* —* t* be the restriction map. Then ψ* οp a + = η_ οψ*. Now we assume that Β is r-invariant. Then we have an Xo £ a+ such that + Δ = {α G Δ:α(Χο) > 0}. But then 3i q (a + ) = a + and in particular o + is maximal abelian in iq. We have already seen that the action of Τ — exp t on Β is Hamiltonian. If α € Apoi is r-invariant, i.e., contained in exp(a+), then the symplectic leaf G.a C βρ 0 ; is invariant under r , and τ induces an antisymplectic involution on this symplectic manifold. Now we are interested in the restriction of this action to the subgroup T_ = exp(t_) which anticommutes with the involution r on B. T h e o r e m 4.10. (cf. [18]) Let logo := Ζ € (iC m a x ) Π α+. Then the image of the map pa+ ο L: (GT)o.a —> a+ is given by c o n v ( W . Z ) + (Ca Π β+), where Ca = c o n e { [ X a , X a ] : a e Δ+, (3 7 e We) 7 .a(Z) > 0,Xa

e

and W is the subgroup o / G l ( t I ) generated by reflections in the hyperplanes which belong to compact roots not vanishing on t_. •

5. A general non-linear convexity theorem Let g be an admissible Lie algebra and gc complexification. We write G c for the associated simply connected complex Lie group and G for the analytic subgroup corresponding to the subalgebra g of gc· We have seen in the preceding section that for every compactly embedded Cartan algebra t C g and each positive system Δ + of roots, we obtain subgroups

J. Hilgert and Κ.-Η. Neeb

220

Ν = exp »ι, A = exp(it) of Gc such that the multiplication mapping G χ Αχ Ν —> Gc is a diffeomorphism onto the open subset GAN of Gc, so that we have a generalized Iwasawa projection L: GAN —> α,

gan

log a.

We have also seen how to describe the set L(aG) for a Ε exp(iC max ) by using Poisson semigroups and dressing actions. Here we had to make the assumption that β is an orthogonal Lie algebra. As follows from Theorem 4.9, this is a rather severe restriction. It implies that the radical of g is a direct summand which is a solvable orthogonal Lie algebra with cone potential. So we simply recover the result of [38] together with a corresponding result for solvable groups. In this section we decribe how one can use symplectic geometry methods in a completely different way to obtain the inclusion L(aG) C conv(Wt- log a) + iCmin without any further assumption on the type of the Lie algebra. Moreover, we will see that the set L(aG) is not always convex. Let r denote the radical of g and s C g a t-invariant Levi complement. Then we set Δ Γ : = {α € Δ: gg C t c }

and

Aa : = { a G Δ: g£ C s c } ·

Since g is admissible, it follows from [27] that Δ is the disjoint union of Ar and As. Note also that Ar C Δ ρ and Δ* C A , , We start with a simple oberservation which rests on an explicit computation: Lemma 5.1. Then

Set ηΓ : = ©λ€Δ + 0c

ant

^ 1st α Ε A, Χ Ε n r , and η = exp(X + X).

L(an) = log α - ]-[X, ( l - Ad(a" 2 )) X}. Ζ

•

Applying the lemma to elements in exp(zCmax), we get Proposition 5.2.

If a Ε exp(iC max ) and R is the radical of G, then L(aR) — log a +

(iv, 0). Then D is a derivation of u because q(iv, w) + q(v, iw) =

Vu, w € C2.

0

Hence g := ux R is a Lie algebra with respect to [(ν,α,Ο,ίυ',α',ί')]

= {itv' - it'ν, q(v, υ'), 0)

(cf. [27, 11.21]). The subalgebra t = {0} χ R 4 χ R is a compactly embedded Cartan algebra and mc decomposes into the two root spaces m + := {υ 1 — iv ® i : υ € m}

and

m~ := {υ 1 + iv ® i : υ e m},

where m + = for λ(1) = i. Setting Δ + := {Λ}, we find that the cone C m i n is generated by the elements ® 1 -I- i f ® i), (ν ® 1 + iv ® i)] ® 1 — iv ® i), (ν 1 + iv ® i)] =2i[v ® 1, iv ® z] = — 2[v, iv] = 2q(iv,

v).

On the other hand q(iv,v)(X)

Let Xo

= Re(iv,X.v)

=

—(ιΧ.ν,ν).

^Q

° ^ = i l e u ( 2 ) . Then q(iv,v)(Xo) = (v,v) > 0. Hence g has cone potential and C m i n is pointed with XQ e int We claim that the set C := Χ}· X € m + } is not convex. Since R + C = C, we consider the set C 0 : = {CGC:

(C,XO) = 1}.

Then Co — {q(iv,v): ||υ||2 = 1}. The group SU(2) acts transitively on the unit sphere in C 2 and an easy calculation shows that the mapping q: C 2 —> u(2)* is equivariant with respect to the SU(2) action on C 2 and the coadjoint action on u(2)*. Hence CQ C u(2)* is a two-dimensional sphere and therefore not convex. The general idea behind this example is that q is a moment mapping for the two-dimensional unitary representation of U(2) on C 2 (cf. Section 6). Such moment

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mappings do often have convex images but not always. The example above is the one with non-convex image with the lowest dimension (cf. [1]). • Let Τ = exp t C G. We say that a functional ω Ε t* is G-integral if there exists a character χ: Τ S 1 with χ (exp X) = eiwfor all X Ε t (cf. [31]). Let ω Ε G^ i n be a G-integral element for which λ = ίω Ε it* is dominant integral for , i.e., λ([Χ α , Xa}) > 0 for Xa e flg and α € A f j c f . [16, p. 180]). We define Σ := {a Ε Δ: (VX € > 0}. Then Σ C Δ is a parabolic system with Σ Π Δ = Δ + whenever ω Ε int and we define the subalgebra ρ = 0c· We write Ρ = (expp) for the analytic subgroup of Gc corresponding to p. Then Ρ is closed, PC\G = G w by Theorem 1.3 in [17] and we obtain a complex structure on ΟΩ = GJGLJJ by embedding ΟΩ as the open orbit G.XQ of the base point XQ in the complex homogeneous space Gc/P (cf. [17, Th. 1.3]). Note that ο + η C ρ might be strictly smaller than p because it may happen that e c C p. We briefly recall the basic definitions concerning homogeneous vector bundles. Let Μ = G/H be a homogeneous space of G and V a vector space on which Η acts by the representation τ:Η —> G1(V). Then we obtain an action of Η on G x V via h.(g,v) := (gh~l, r(h).v) and the space of i/-orbits is denoted G x η V and called a homogeneous vector bundle. We write [5, υ] for the element of G x Η V which corresponds to the orbit of (g,v) in G x V and note that G acts from the left on G XJJ V by g\g',v] := [gg',v]. If G is a complex group, Η is a complex subgroup and the representation τ is holomorphic, then the corresponding vector bundle is holomorphic. Since ω is G-integral, we find a holomorphic character χ: Ρ -> C*

with

x(exp X) =

for X e p. Thus we obtain two homogeneous holomorphic line bundles: the line bundle Ε := G Xc u C x and the line bundle E' := GQ Xp C x , where C x denotes the complex numbers considered as a P-module with respect to the holomorphic character χ. The bundle Ε embeds as the open subset E' \G.X0 of E'. Let q\ G x C —• Ε denote the quotient mapping which identifies the elements (g, z) and (gh-1 ,x(h)z) for h € Gu. We define a function h on Ε by h([g, z]) := |z| 2 for g Ε G, Ζ G C. We write EQ C Ε for the complement of the zero section. The following proposition is the key observation (cf. [35]). P r o p o s i t i o n 5.4. σ(ϊΧ) on Ε by

For Χ Ε ig let τηχ := sup(ö li ,, iX) and define the vector field

&(iX)(p) := jt t=0

ex

P(~tiX)-P

for ρ Ε Ε. Then ^(dlogh(p),ia(iX){p)) for all ρ Ε Eq.

< mx •

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Let S C Gc denote the compression semigroup of the open G-orbit G.xο in Gc/P, i.e., S = {ge Gc--gG.x0 C G.x0} (cf. [17], [22]). Then S acts holomorphically on the bundle E' and since S leaves G.x ο invariant, the action on E' leaves the subbundle Ε invariant. We recall that S C GP = GAN (cf. [17, Prop. II. 7]), so that L is defined on S. Writing s = gan one finds that \logh([s,l))

=

(KL(s)),

where Λ = ίω. Fix g € G and X 6 zC m a x . We set F(t) := logh(exptX.[g, 1]). Then e x p M + X C S and therefore exp tX.[g, 1] = [exp tXg, 1] G EQ for all t > 0. Hence we can use Proposition 5.4 to see that F'(t) = (d(logh), Ia(iX))([exptXg,l])


g*,

V

/

is called the moment mapping of this representation. In this setting Φ can be interpreted as a moment mapping for the "Hamiltonian" action of G on the infinite dimensional Kahler manifold Ρ (Η) (cf. [23], [26], [48]). The main problem which arises in the infinite dimensional case is that the Hamiltonian functions rι Μ -

/wr η (idir(X).v,v) vPO(M) = — M

are only defined on the domain of the generally unbounded essentially self-adjoint operator idir(X). • Since the operators idn(X) are essentially self-adjoint (this follows from the density of the space of analytic vectors), the closed convex hull of the spectrum of the operator ΐάπ(Χ) is given by the closure of the range of the Hamiltonian function φ(Χ), whence supSpec (id-K{X)) = sup^(X)(P(H°°)) = sup(X,Φ(Ρ(Ή°°))).

(6.1)

This property is an essential link between the theory of unitary representations and symplectic geometry. It permits us to study the spectrum of the operators idn(X), Χ Ε g by the invariant subset Φ(Ρ(Ή°°)) C g*. Actually this observation

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is a key ingredient in the theory of holomorphic extensions of unitary representations as described in [39]. According to (6.1), the upper (and therefore also the lower) bound of the spectrum of an operator ίάπ(Χ) only depends on the closed convex hull of the image of the moment map Φ. Therefore it is natural to look at a smaller subset of which yields the same convex hull under the moment map and another question is whether the closure of the image is already a convex set. Let us write / π for the closure of the image of Φ which is called the moment set of π. We say that the representation π is convex if this set is convex. The abelian case is relatively easy to handle (cf. [1] and [33]). Proposition 6.2. Every continuous unitary representation of a connected abelian Lie group G is convex. More precisely, if X G fl ivith m = sup(Jf, then for every [υ] G Ρ(Ή°°) with (Χ, Φ([υ])) = m the vector ν is an eigenvector for idn(X) with eigenvalue τη. • The first one who studied the above moment map systematically was Wildberger (cf. [51], [52]). He has shown that an irreducible representation π of a simply connected nilpotent Lie group is convex and that the moment set in this case is the closure of the convex hull of the coadjoint orbit associated to π by Kirillov's orbit method. This result has been generalized by Arnal and Ludwig in [1] to the case of general connected solvable groups where the moment set Ι π is the closure of the convex hull of the set Ρ(π) C g* which corresponds by Pukansky's theory to the unitary equivalence class of the representation π (cf. [1, Th. 13], [4, VI]). In this section we are mainly concerned with the "opposite" side of the theory, namely with unitary representations associated with highest weight modules (cf. [39]). Definition 6.3. Let Δ+ C Δ be a positive system. (a) Let V be a gc-module. For a G we set Va := {ν e V : (VX G t c )X.v = a(X)v}. This space is called the weight space of weight a and α is said to be a weight of V if Va Φ {0}. We write Vv for the set of weights of V. (b) Let V be a gc-module and ν G V. We say that ν is a primitive element of V of weight Λ with respect to Δ + if 0 φ ν G Vx and QQ.v = 0 holds for all a G Δ + . A gc-module V is called a highest weight module with highest weight Λ (with respect to Δ + ) if it is generated by a primitive element of weight λ. The up to equivalence unique irreducible highest weight module of highest weight Λ is denoted L( A). (c) An irreducible highest weight module L(Λ) is said to be unitarizable if there exists a unitary representation (πχ,Τί) of the simply connected Lie group G with L(G) = g such that L(Λ) is isomorphic to the gc-module H K of Ä"-finite smooth vectors in Ή (cf. [39]). In this case (πχ,Ή) is called a unitary highest weight representation of G. •

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For more details on highest weight representations we refer to Section II in [30] or to [39]. The following theorem ([31]) associates to each highest weight representations a strictly admissible coadjoint orbit. Theorem 6.4. Let (πχ,Η) be a highest weight representation of the connected, Lie group G with discrete kernel and highest weight λ with respect to the positive system, Δ + . Then Λ is dominant integral with respect to and strictly admissible.

• Applying the convexity theorem for coadjoint orbits (Theorem 2.2) to the orbit Oix associated by Theorem 6.4 to a highest weight representation and some standard information on the set VL{\) of t-weights in HK = L(λ), we get Proposition 6.5. (The moment set for Τ) Under the assumptions of Theorem 6.4, we have for the restriction ttJ of πχ to T: Ιπτ = zconvVx = conv(Wt.cj) - cone(iA+) = p t . ( Ι π χ ) = p t . (C>iA).

•

So far we know for each coadjoint orbit in the moment set of πχ that the projection to t* is contained in the closed convex set Pt'(Oix). To profit from this information one needs a result such as Proposition 2.4. Combining Proposition 2.4(ii) and (iii) with Proposition 6.5, one obtains that / π λ C conv ö{\ holds for every highest weight representation πχ and that the convex hull of the highest weight orbit is closed. This shows already that πχ is convex if and only if the moment set fills out the convex hull of the highest weight orbit. To obtain a characterization of the convex highest weight representation one has to look at the smallest ΑΓ-invariant subspace Ή+ of H K which contains a vector of highest weight. Then the representation πχ of Κ on this space is an irreducible representation of Κ with highest weight Λ and in particular finite dimensional. For / e t * let Of := Ad *(K).f,Ff := conv Of, and Ff := conv Of. Applying the second part of Proposition 6.2 to an element X € i n t C m a x Π i(t), one finds that (i) $ -1 (-FiA) = Ρ ( Ή ^ ) and (ii) Inx r\F$ = Φ(Ψ{7ίξ)). As a consequence of (ii), we see that the convexity of πχ implies the convexity of If G is a simple Hermitian Lie group and (πχ,Η) is a representation in the holomorphic discrete series, then this observation generalizes the principal result in [47] to general highest weight representations. For irreducible representations of compact groups one has the following criterion due to Arnal, Ludwig and Wildberger ([1], [52]): Theorem 6.6. Let t be a compact Lie algebra, t C g a compactly embedded Cartan algebra, Δ^ a positive system of roots and Λ € it* dominant integral.

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Further let πχ denote the irreducible representation of Κ with highest weight A. Then the following are equivalent: (1) is convex. (2) (λ,α) Φ 1 for all simple roots α € . • To prove that the convexity of π^ actually implies the convexity of πχ is harder. In addition to a careful analysis of the convex Ä"-types contained in πχ, the geometric ingredient needed here is a consequence of Proposition 2.5, more precisely, the following two conditions are equivalent: (1) /,rA is convex. (2) Ι π χ Π t* is convex. As a final result we have ([33]): Theorem 6.7. (Convexity theorem for highest weight representations) The following conditions are equivalent: (i) The representation πχ of G is convex. (ii) The representation πχ of Κ is convex. (iii) (λ, ά) φ 1 for all simple compact roots a. (iv) Ιπχ = conv Oix. • Example 6.8. (a) For more details concerning the following example we refer to Section II in [34]. If t)n is the (2η + 1)-dimensional Heisenberg algebra and 0 = R), then the metaplectic representation π is a highest weight representation of the simply connected group G = Hn>\ Sp(n,R)~and the highest weight is given by λ = ε - pr, where ε e t * n ( t n s p ( n , R ) ) \ Δ+ = {a e Δ + : 0 £ e (f) n )c}, and pr := \ Σ ω 6 δ + α· Since pr is invariant under the Weyl group, we have (pr, ά) = 0 for all a G , so that Theorem 6.7 shows that π is convex and Ι π Π t* = iX - cone(iA+). (b) We discuss the example g = su(2,1) which is the smallest non-compact Lie algebra which permits non-convex highest weight representations. For the classification of the corresponding highest weight modules we refer to [8]. Since su(2, l ) c = s[(3, C), we can identify the diagonal matrices in sl(3, C) with tc, so that we identify it* with {(xi, 12,^3) £ R 3 : x i + 1 2 + 1 3 = 0}. We write ει,ε2 and ε3 for the basis vectors. Then = {ει - ε 2 }, Δ+ = {ει - 63,£2 - £3}, a n d ß = ει - £3 is the highest root. The half sum of the positive roots is ρ = ε\ — ε3 = β and we set c := 5(ει +62 - 2ε3) which spans j(t)*. Let Λη := ^ ( n - 2 , - 2 n - 2 , n + 4) = -2( + wy, υ

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where 7 = ^(1, —2,1) spans kerβ. Those functionals which correspond to unitaryhighest weight representations are given by λ η + ζζ, where η e No and z < 2 for η = 0 and z < 1 for η > 0. The representations of the holomorphic discrete series are those with z < 0. Note also that η = 0, z = 2 yields the trivial one-dimensional representation. Since ζ € the convexity condition for λ η + ζζ is (λ η + ζζ, ει - ε 2 ) = (λ η , εχ - ε 2 ) = n(-y, ελ - ε 2 ) = η ^ 1. Therefore the highest weights of non-convex representation are i ( - l + z,-4 + z,5-2z),

z < 1.

Ο

If s is the reflection corresponding to ει — ε 2 , then λ η - s.Xn := η(εχ - ε 2 ). This shows that ττχ is an irreducible representation of Κ = R χ SU(2) of dimension η + 1 which is convex if and only if η + 1 Φ 2. For η = 1 the image of the moment mapping is Ad*(Ä").(iA n ) which is a 2-sphere in t*, whence not convex. •

Representations and the non-linear convexity theorem In this section we show how the old proof for the non-linear convexity theorem becomes more transparent with the material presented in Section 5. We first consider finite dimensional representations. Let J a non-degenerate pseudo-Hermitian form on the finite dimensional complex vector space V. For X € gl(V) we write X" for the adjoint of X with respect to J, i.e., J(X.v,w)

= J(v, XKw)

Vv,weV.

Let P(V) denote the projective space of V, [υ] the one-dimensional subspace spanned by the non-zero vector v, and

This set decomposes into Ω + := {[υ] € Ρ ( ^ ) : J(v,v)

> 0}

and

Ω_ := {[υ] 6 P(V) : J (ν,ν) < 0}.

The set Ω carries a natural pseudo-Kähler structure such that the action g. [υ] := [g.v] of the pseudo-unitary group U( J) on Ω is Hamiltonian and

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229

is the corresponding moment mapping (cf. [16, Ch. 8]). Note that this setting generalizes the setting of Hilbert spaces, where Ω+ = 1P(V) and the Kahler structure on projective space is the Fubini-Study metric. Now suppose that G is a connected Lie group and that 7r:G —> G1(V) is a complex representation on V. We say that π is pseudo-unitary if 7r(G) C U(«/), i.e., if the action of G on V preserves the form J. In this case we have a moment map λ η * Φ : Ω —• g ,

η

/ν

.J(d*(X).v,v)\ )• J M

We recall that every finite dimensional complex representation of a quasihermitian connected semisimple Lie group G can be turned into a pseudo-unitary representation ([16, Th. 8.30]). Suppose that (ττχ,Υ) is an irreducible pseudo-unitary representation of the reductive quasihermitian Lie group G with highest weight λ. Let [u\] be a highest weight vector with respect to a t-adapted positive system Δ + . Then [υχ] G Ω ([16, p. 229]) and we may assume that v\ and J are normalized with J ( υ χ , υ χ ) — 1. Let Μ :— G.[t>>] denote the orbit of the highest weight ray in Ω+ and set ω := Φ([υλ]) = iX. Then ω G G^ in because λ([Χ α ,Χ α ]) > 0 holds for Xa G gg, a G Δ+ which in turn follows from the fact that λ is dominant integral with respect to Δ + . The G-orbit Μ is a complex submanifold of Ω and since Φ induces a diffeomorphism of Μ onto Ο ω , we can think of Μ as the coadjoint orbit O w . Then the line bundle EQ = G ΧΑΩ C* can be obtained as the set π - 1 ( Μ ) , where π: V"\{0} —» P(V), υ •—> [υ] is the canonical projection. The mapping q: GxC* —> E0 is then given by q(g, ζ) = zn(g).v\ and the function h (cf. Proposition 5.4) is simply h

([9,z\) = M2 =

J(zn(g).vx,ZTr(g).v\),

where J is the pseudo-Hermitian form. Let π: Gc —> G1(F) denote the canonical extension of the representation to Gc- Then the whole complex group acts on V, but it does not leave the set EQ invariant. This set is mapped into itself by the semigroup S = G exp(zW/max), where W max C g is a maximal invariant cone containing the center (cf. [22]). We have &(X)(v) = —άπ(Χ).υ and — ϊσ(Χ)(υ) = ίάπ(Χ)υ for X G g and ν G V, so that we find for the derivative of the function rflog h the simple formula l

Δ

2.

J[V,V)

This makes it very transparent that the supremum of the Hamiltonian function Ν η-• V{X) on the coadjoint orbit ΌΩ — $(G.[ua]) plays an important role for the behaviour of the function h on the line bundle EQ.

-(dlogh(v),ia(X)(v))

J. Hilgert and K.-H. Neeb

230

Another class of examples has been used in the original proof of the convexity theorem for the reductive case in [38] and it has been generalized to the general case of admissible groups in [28]. Here one has the following situation. We have a functional λ = ίω € which is the highest weight of a unitary highest weight representation (πχ,Τί) of the group G. This representation extends holomorphically to the interior of the semigroup S which we have already encountered above (cf. [39]). Let v\ denote a highest weight vector. As in the finite dimensional case we can realize the line-bundle Eq = G Xgu C* as the set C*7r\(G).vx via the mapping q: G x C* -» H,

(g, ζ) ^

zirx{g).vx.

The norm function is h([g,z}) = \\q(9,z)\\2

=

\\z*x{g).vx\\2

and we also have \(d\ogh{v),id*x{X).v)

=

=

*([«])(*)>.

Here the constant τηχ := s u p ^ ^ , , X) turns out to be τηχ =

sup HeP(w«)

(ίάπχ(Χ).ν,υ)^ τ ) = supSpecid7TA(A) \viv)

(cf. Proposition 6.2, Theorem 6.7). The selfadjoint operators ίάπχ(Χ), X e C m a x are bounded from above so that we have another interesting interpretation of the numbers τηχ.

7. Geometric character formulas Let g be a finite dimensional real Lie algebra. We have already seen that every coadjoint orbit possesses a natural G-invariant symplectic structure given by Ω„(ι/ ο ad Χ, ι/ ο ad Y) =

v([X,Y\)

for all ν Ε Of and Χ,Υ e q. Note that i^oadg = T„(Of) is the tangent space in v. Suppose that 2 η = dim O f . Then the symplectic structure Ω defines the 2n-form β := ( 2 π)"η!^ η w ^ich defines a measure μ©, on Of which is called the Liouville measure. Let us assume that g contains a compactly embedded Cartan algebra t and write Τ = exp t for the associated analytic subgroup of a connected group G with L(G) = g. The action of Τ on Of is Hamiltonian with moment map Φ =pt.:Of

—> t*,

ν >-> v\t.

For the sake of applications to representation theory and in particular to character formulas, one is interested in the image Φ*μο / of the Liouville measure on

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231

t* which is defined by Φ * μ ο , ( Ε ) = μ

σ

, ( φ -

1

( Ε ) )

for every Borel subset E C t* and in particular in its Fourier-Laplace transform. Note that information on the set Φ(0/) which is more or less the support of the measure Φ*μο{ can be obtained from convexity theorems, but these do in general not give enough information to calculate the measure. The first result in this direction was the formula of Duistermaat and Heckman for the image of the Liouville measure under the moment mapping for Hamiltonian torus actions on compact symplectic manifolds ([7]). Recently Prato and Wu obtained a similar formula which applies to non-compact symplectic manifolds under certain properness conditions on the moment mapping ([45]). In this section we will explain how the results of Prato and Wu can be used to obtain a general formula for admissible coadjoint orbits which specializes to (1) the Duistermaat-Heckman formula in the case of compact Lie algebras (this formula had already been obtained by Harish-Chandra ([11])), (2) the Prato-Wu formula for strictly admissible coadjoint orbits for Hermitian simple Lie algebras, and (3) a well known formual for the Fourier transform of Gauss kernels for the Lie algebra g = f)n>i£p(n,R), where f)n is the (2η + l)-dimensional Heisenberg algebra. The formula for Φ*μοί can be used to obtain a character formula of RossmannKirillov-type for those unitary highest weight representations of the group G which arise as holomorphically induced representations. Let V be a finite dimensional real vector space and μ a non-zero Borel measure on V*. We define the L a p l a c e t r a n s f o r m of μ to be the function

We set e x ( a ) : = ea on V * . Note that £(μ)(ζ) > 0 for all χ e V since μ φ 0. We say that μ is a d m i s s i b l e if the set Ό μ := {χ € V : £(μ)(χ) < oo} is non-empty. If μ is admissible, then there exists an χ Ε V such that e x • μ is a finite measure and since e x is bounded away from 0 on every compact set, it follows in particular that μ is a Radon measure and therefore σ-finite. We recall that if int ϋ μ φ 0, then the function £(μ) has a holomorphic extension to the tube domain int D ß + i V given by

and that £ { μ ) is a convex function on Όμ. We note that if μ is a finite measure on a bounded set, then Ό μ = V so that we even obtain a holomorphic function on the complex vector space VcIf μι and μ2 are two admissible measures on V * , then we define the convolution μι *μ2 := add* (μι μ2), where add: V * χ V * —> V*, ( α , β) •-» α + β is the addition

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function. Then one checks easily with Fubini's theorem for σ-finite measures that £ ( μ ι * μ2) = £ ( μ ι ) £ ( μ 2 )

(7.1)

holds pointwise on V with the convention coo = ooc = oo for c > 0. It follows in particular that Ζ) μι * μ2 = Όμι Π Dß2 and that μι * μ 2 is admissible if the sets Dßl and D ß 2 have points in common. If μ = 6a is a point measure on V*, then C(6a) = e a : x ~ e a ( l ) and therefore C(6a * μ) = e Q £(/i) gives the Laplace transform of a shifted measure. Let Μ be a symplectic manifold of dimension 2η, Τ a torus and Τ χ Μ —> Μ a Hamiltonian torus action with moment mapping Φ: Μ —• t*. For X e t we write φ(Χ) := (Φ(·), X) for the associated Hamiltonian function and um = Φ*(μΛί) for the Borel measure on t* which is the push-forward of the Liouville measure under the moment map. The Laplace transform of this measure which is given by C{uM){X)

for X G D„m

= [ evWW JΜ + it C tc, or, equivalently, by C{uM){iX)

= f e«*·*™ JΜ

άμΜ{πί)

άμΜ(m)

for X € t - iDUM. We start with a general result on Hamiltonian torus actions on symplectic manifolds. It is a slight adaption of Theorem 2.2 in [45] (cf. [41, Th. II.6]). T h e o r e m 7.1. Let Μ be a connected symplectic manifold of dimension 2η and Τ a torus acting on Μ in a Hamiltonian fashion with moment map Φ:M —> t*. Suppose that (i) Φ is proper, (ii) Φ(Μ) contains no affine lines, and that (iii) the set MT of Τ-fixed points is finite. For ρ € MT let Vp := Vtp(M)-l/tp(M) ^ denote the set of weights for the L T-action on the symplectic vector space TP(M)- /TP(M), and for a G Vp write ma for the complex dimension of the corresponding weight space. Then βί(Φ

(p),Z>

for each element Ζ Ε tc satisfying (a) Ζ e t + i int (lim Φ (Μ))*, and (b) Ζ is M-regular in the sense that a(Z) Φ 0 holds for all a Ε UpgMT ^p·

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233

We know already that all assumptions made in Theorem 7.1 are satisfied if Μ is an admissible coadjoint orbit. So we can combine it with the information we have on admissible orbits to obtain Theorem 7.2. (Duistermaat-Heckman formula for admissible coadjoint orbits) Let / 6 t* be an admissible element such that span Of = 0* and Δ+ a i-adapted positive system such that f € Cmin- Then C{uot){iZ)

= Σ 7ew,

(7.2)

- — r Πα€Δ+