Grassmann and Stiefel Varieties over Composition Algebras (RSME Springer Series, 9) 303136404X, 9783031364044

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RSME Springer Series 9

Marek Golasiński Francisco Gómez Ruiz

Grassmann and Stiefel Varieties over Composition Algebras

RSME Springer Series Volume 9

Editor-in-Chief Maria A. Hernández Cifre, Departamento de Matemáticas, Universidad de Murcia, Murcia, Spain Series Editors Nicolas Andruskiewitsch, FaMAF - CIEM (CONICET), Universidad Nacional de Córdoba, Córdoba, Argentina Francisco Marcellán, Departamento de Matemáticas, Universidad Carlos III de Madrid, Leganés, Madrid, Spain Pablo Mira, Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Cartagena, Spain Timothy G. Myers, Centre de Recerca Matemàtica, Barcelona, Spain Joaquín Pérez, Departamento de Geometría y Topología, Universidad de Granada, Granada, Spain Marta Sanz-Solé, Department of Mathematics and Computer Science, Barcelona Graduate School of Mathematics (BGSMath), Universitat de Barcelona, Barcelona, Spain Karl Schwede, Department of Mathematics, University of Utah, Salt Lake City, UT, USA

As of 2015, RSME - Real Sociedad Matemática Española - and Springer cooperate in order to publish works by authors and volume editors under the auspices of a co-branded series of publications including advanced textbooks, Lecture Notes, collections of surveys resulting from international workshops and Summer Schools, SpringerBriefs, monographs as well as contributed volumes and conference proceedings. The works in the series are written in English only, aiming to offer high level research results in the fields of pure and applied mathematics to a global readership of students, researchers, professionals, and policymakers.

Marek Golasi´nski • Francisco Gómez Ruiz

Grassmann and Stiefel Varieties over Composition Algebras

Marek Golasi´nski Faculty of Mathematics and Computer Science University of Warmia and Mazury Olsztyn, Poland

Francisco Gómez Ruiz Departamento de Álgebra, Geometría y Topología Universidad de Málaga Málaga, Spain

ISSN 2509-8888 ISSN 2509-8896 (electronic) RSME Springer Series ISBN 978-3-031-36404-4 ISBN 978-3-031-36405-1 (eBook) https://doi.org/10.1007/978-3-031-36405-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

Differential geometry is a part of geometry that studies spaces, called differentiable manifolds, where concepts like the derivative make sense. Differentiable manifolds locally resemble ordinary Euclidean space, but their overall properties can be very different. It is a wide field that borrows techniques from analysis, topology and algebra. Algebraic geometry is a complement to differential geometry. Substantially, it is a field of mathematics, which combines abstract algebra (always commutative algebra) with geometry. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. For matrix manifolds, being algebraic varieties over reals, there is a natural representation of their elements in the form of matrix arrays. The subject of the authors’ long cooperation consists mainly of an algebraic approach to matrix manifolds. Their obtained results have been published in joint papers [11–19] as the following examples show: Theorems 4.16, 4.20, 5.46, and 5.50 of the present monograph are generalizations of [11, Proposition 1.2] and [12, Theorem 2.3]. Both papers [11] and [12] have been published 20 years ago (on 2002). The prerequisites are basic courses in linear algebra, algebraic geometry, differential geometry and Riemannian geometry. The first author wishes to express his gratitude to the University of Warmia and Mazury in Olsztyn (Poland) for supporting his three-month visit to Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, España. He is particularly grateful for the opportunity to work there with the second author and would like to acknowledge with gratitude the help given by all his colleagues and friends from Departamento de Álgebra, Geometría y Topología, Universidad de Málaga. The second author is indebted to the Nicolaus Copernicus University in Toru´n (Poland) and the University of Warmia and Mazury in Olsztyn (Poland) for supporting several stays at their institutions. Olsztyn, Poland Málaga, Spain

Marek Golasi´nski Francisco Gómez Ruiz v

About the Book

In the meaning of classical differential geometry, matrix manifolds are manifolds for which there is a natural representation of elements in the form of matrix arrays. Similarly, in algebraic geometry, one can consider matrix varieties and many algebraic varieties are manifolds. This monograph deals with such objects, s e.g., Grassmannians .Gn,r (A), Stiefel .Vn,r (A) and flag .Fn;r (A) varieties for 1 ,...,rs .A = K, C(K), H(K) and .O(K). All of them are matrix varieties. We point out that [4] and [31] are devoted to real algebraic geometry. Furthermore, [22] investigates manifolds of idempotent matrices over reals. The aim of this monograph is to provide tools to exploit matrix varieties over Pythagorean formally real fields in order to develop efficient matrix arguments for their investigations. The presentation of the book is reasonably self-contained since we have assumed familiarity only with basic knowledge of algebra, differential geometry, Riemannian geometry and some rudiments of algebraic geometry. This monograph is organized into six chapters which we now briefly summarize chapter by chapter. The main goal of Chap. 1 is to introduce some notation and terminology. Its main result is, of course, the generalized Frobenius-Hurwitz’s Theorem presented as Theorem 1.15, and conditions (1.1), which actually look very weak, imply quite surprisingly Frobenius Theorem. We present a complete detailed proof of that theorem. This justifies why it is quite natural our blanket assumption all along the monograph of K being a Pythagorean formally real field and the four cases: .K, C(K), H(K) and .O(K). Chapter 2 takes up the systematic study of a generalization of the exceptional compact Lie groups .G2 and .F4 to groups .G2 (K) and .F4 (K) provided K is Pythagorean formally real field. The main result stated in Theorem 2.48 says that any Hermitian .3 × 3-matrix .A ∈ Herm3 (O(K)) can be transformed to a diagonal form by some element of .F4 (K) for K being a formally real closed field. Some results on .F4 (K) have been already published in [20]. Next, we apply the T. Miyasaka and I. Yokota’s work [32] to show a natural polynomial group monomorphism of .U3 (H(K)), the group of unitary .3×3-matrices over .H(K), into .F4 (K) to present another proof of that result. We point out that most known proofs (see, e.g., [1, Proposition 16.4]) of that result for .K = R vii

viii

About the Book

are via analytical methods, make use of the compactness of .F4 and reduce to a contradiction. But, the two proves of Theorem 2.48 presented here are algebraic, direct and constructive as well. Furthermore, we show in Corollary 2.51 that the K-algebra of invariant homogeneous polynomials on .Herm3 (O(K)), under the natural action of .F4 (K), is a polynomial algebra on the characteristic coefficients .C1 , C2 , C3 . Chapter 3 introduces and analyses some properties of the classical manifolds of Stiefel, Grassmann and flag manifolds over the field of reals, .R, the field of complex numbers, .C, the skew field of quaternions, .H and, except if otherwise said, the octonion division algebra, .O. Then, the Grassmanian .G3,1 (O) = G3,2 (O) called the Cayley plane or octonionic projective plane and denoted by .OP 2 is investigated. Theorem 3.35 shows that the exceptional Lie group .F4 acts transitively on the Cayley plane .G3,1 (O), Propositions 3.41 and 3.45 show that .F4 is the group of isometries of .G3,1 (O) and Corollary 3.50 identifies the Cayley plane with the homogeneous quotient .F4 /Spin(9). Chapter 4 generalizes Stiefel, Grassmann and flag manifolds, defined in Chap. 3, to what we call here i-Stiefel, i-Grassmann and i-flag manifolds. This “i” comes from idempotent. Those manifolds do not seem to have being enough studied in the literature. In particular, they do not have even a name. As in Chap. 2, .A denotes the field of reals, .R, the field of complex numbers, .C, the skew field of quaternions, .H and, occasionally, the octonion division algebra .O. Our main results are Theorems 4.16 and 4.20 which identify i-Grassmannians and i-flag manifolds with the total tangent spaces of the corresponding Grassmannian and flag manifold. Chapter 5 uses previous chapters to define and extend previous notions and results presented there to matrix varieties over a more general division algebras .A: .K, the complex K-algebra .C(K), the quaternion K-algebra .H(K) or the octonion K-algebra .O(K) where K is a formally real Pythagorean field. In particular, we extend the classical definitions of Riemannian, Hermitian, symplectic and Kähler manifolds. We prove Theorems 5.46 and 5.50 which generalize, respectively, Theorems 4.16 and 4.20 to this more general situation. Chapter 6 deals with the more closely study of the Riemannian structure of classical matrix manifolds introduced in Chaps. 3 and 4. Here K is a Pythagorean formally real field, .A = K, C(K), H(K) in Sects. 6.1, 6.2, 6.3, 6.4, 6.5 and .A = O(K) in Sects. 6.6, 6.7, and 6.8. In particular, in Sects. 6.1–6.4, we present the Stiefel Riemannian submersions, obtain explicit formulas for the Riemann curvature tensor field, sectional curvatures, second fundamental tensor field. It is shown that .C(K)- Grassmanians are Kähler manifolds and .A- Grassmannians are Einstein even in this generalized setting.

About the Book

ix

The second fundamental tensor field is used to obtain the geodesics in Grassmanians .Gn,r (A) and i-Grasmannians .Idemn,r (A): the geodesic with the origin A and unitary initial velocity B is given as solution of the differential equation σ  (s) = 2(In − 2σ (s))σ  (s)2

.

with initial conditions .σ (0) = A and .σ  (0) = B. Of course, to obtain the solution σ (s) =

.

1 (In − e2s(In −2A)B (In − 2A)) 2

the field K has to be complete since we need convergency of the exponential .eX . Section 6.5 makes use of the Stiefel submersion to get the volumes of .AGrassmannians for .A = R, C, H. Sections 6.6 and 6.7 study the Riemannian geometry of the Cayley plane .G3,1 (O) and its generalization to .G3,1 (O(K)) for any Pythagorean formally real field .K. In particular, we extend and compute explicit formulas for the Riemannian curvature tensor field, the sectional curvatures and the second fundamental tensor field. We extend definition and show that .G3,1 (O(K)) is an Einstein manifold. The second fundamental tensor field is used to obtain the geodesics in .G3,1 (O(K)): the geodesic with origin A and unitary initial velocity B is given as a solution of the differential equation σ  (s) = 2(σ  (s)2 − tr(σ  (s)2 ))σ (s).

.

The solution, for which we need K being complete, is given in Proposition 6.23 by σ (s) = cos(2s)A +

.

sin(2s) B + sin2 sB 2 . 2

Finally, we discuss in Sect. 6.8 the volume of the Cayley plane .G3,1 (O), and 3! 8 Theorem 6.30 shows that .vol(G3,1 (O)) = 11! π . Certainly, this result has been already stated in the literature, e.g., [23]. But no direct proof is known to the authors. Then, the general formula .vol(G3,1 (A)) = ( d2 −1)! d π ( 3d 2 −1)!

for .d = 2, 4, 8 and .A = C, H, O, respectively, is derived.

Contents

1

Algebraic Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 K-Algebras with Involutions: Composition K-Algebras . . . . . . . . . . . 1.2 Generalized Frobenius-Hurwitz’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Matrices over K-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Background on Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Natural, Order and Zariski Topologies on M(A) . . . . . . . . . . . . . . . . . . . .

1 1 12 15 36 43

2

Exceptional Groups .G2 (K) and .F4 (K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Cross Products and the Exceptional Group G2 (K) . . . . . . . . . . . . . . . . . 2.2 Automorphisms of Hermn (O(K)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Exceptional Group F4 (K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 49 75 81

3

Stiefel, Grassmann Manifolds and Generalizations . . . . . . . . . . . . . . . . . . . . . . 3.1 Stiefel Varieties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Flag Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 112 129 163

4

More Classical Matrix Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 4.1 i-Grassmannians and i-Stiefel Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 4.2 i-Flag Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

5

Algebraic Generalizations of Matrix Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 5.1 Varieties of Idempotent Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 5.2 Atlas on Varieties of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

6

Curvature, Geodesics and Distance on Matrix Varieties . . . . . . . . . . . . . . . . 6.1 The Stiefel Submersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Ricci Tensor and Einstein Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Geodesics of G(A) and Idem(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Volume of Gn,r (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Riemannian Geometry of the Cayley Plane. . . . . . . . . . . . . . . . . . . . . . . . . .

251 252 256 267 272 284 286

xi

xii

Contents

6.7 6.8

Ricci Tensor and Einstein Structure of the Cayley Plane . . . . . . . . . . . 295 Volume of the Cayley Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

A Definitions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Multiplication Table in O(K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 ∼A,m,n -Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Hermitian and Skew-Hermitian Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 Inner Product on M(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.7 Rank of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.8 Groups of Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.9 Classical Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.10 Group Monomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.11 Stiefel Varieties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.12 Non-compact Stiefel Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.13 Grassmann Varieties (Classical) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.14 Grassmann Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.15 Stiefel Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.16 Stiefel Varieties as Homogenous Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.17 Grassmann Varieties as Homogenous Spaces . . . . . . . . . . . . . . . . . . . . . . . A.18 Flag Varieties (Classical) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.19 Flag Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.20 Stiefel Maps Over Flag Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.21 i-Grassmann Varieties (Classical) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.22 i-Grassmann Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.23 i-Stiefel Varieties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.24 i-Stiefel Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.25 i-Stiefel Varieties as Homogenous Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . A.26 i-Grassmann Varieties as Homogenous Spaces . . . . . . . . . . . . . . . . . . . . . A.27 i-Flag Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.28 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.29 Tangents and Normals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

305 306 306 307 308 308 308 309 309 310 310 310 311 311 312 313 313 314 315 316 317 318 318 319 319 320 320 321 322 322

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

Chapter 1

Algebraic Preliminaries

The main goal of this chapter is to introduce some notation and terminology. We assume that the reader is more or less familiar with the basic concepts of algebraic geometry and linear algebra. Typical references are: [25, 27, 35, 36] and [38].

1.1 K-Algebras with Involutions: Composition K-Algebras We make mainly use [35], [37], and [38] to present some prerequisites to K-algebras with involutions. While proofs of many results are in the indicated sources, we present them for the completeness of this chapter. Before launching into proofs simple examples are served to illustrate presented results. Let K be a commutative field. A left K-algebra is a left K-linear space .A endowed with a K-bilinear product · : A × A −→ A.

.

This product is not assumed to be commutative nor associative. But, we assume that it has a left unit .1 = 0. Notice that the map K −→ A

.

given by .λ → λ · 1 for .λ ∈ K is a monomorphism of unitary rings: • .λ1 + λ2 → (λ1 + λ2 ) · 1 = λ1 · 1 + λ2 · 1 for .λ1 , λ2 ∈ K; • .λ1 λ2 → (λ1 λ2 )·1 = λ1 (λ2 ·1) = λ1 (1·(λ2 ·1)) = (λ1 ·1)(λ2 ·1) for .λ1 , λ2 ∈ K; • if .λ · 1 = 0 for a non-zero .λ ∈ K then .λ−1 (λ · 1) = 1 leads to a contradiction .1 = 0.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Golasi´nski, F. Gómez Ruiz, Grassmann and Stiefel Varieties over Composition Algebras, RSME Springer Series 9, https://doi.org/10.1007/978-3-031-36405-1_1

1

2

1 Algebraic Preliminaries

Therefore, from now on we identify the field K with its image in the left Kalgebra .A. In a similar way, we define a right K-algebra .A and again we identify K with the image of a monomorphism of unitary rings .K → A given by .λ → 1 · λ for .λ ∈ K. A division algebra is a K-algebra .A in which for any .x ∈ A and any non-zero  .y ∈ A there exists precisely one .z ∈ A with .x = yz and precisely one .z ∈ A such  that .x = z y. An alternative K-algebra is a K-algebra .A in which the following system of identities is satisfied: (xy)y = x(y 2 ) (right alternative);

.

(x 2 )y = x(xy) (left alternative)

.

for any .x, y ∈ A. Next, the map [ −, −, −] : A × A × A −→ A

.

given by [x, y, z] = (xy)z − x(yz)

.

for any .x, y, z ∈ A is called the associator of .A. If .A is an alternative K-algebra then its associator is an alternate map: given .x, y, z ∈ A, we have [x, x, z] = x 2 z − x(xz) = 0, [x, y, x] = (xy)x − x(yx) = 0,

.

[x, y, y] = (xy)y − xy 2 = 0. Furthermore, any alternative K-algebra .A is flexible, i.e., (xy)x = x(yx)

.

for all .x, y ∈ A. In fact, .x 2 z+y 2 z+(xy)z+(yx)z = (x+y)2 z = (x+y)((x+y)z) = x 2 z +y 2 z +x(yz)+y(xz) implies .(xy)z +(yx)z = x(yz)+y(xz) and taking .z = x we get .(xy)x = x(yx). A pair .(A, −) is called a left K-algebra with an involution provided .A is a left Kalgebra with a left unit element .1 = 0 and a map .− : A → A, called a conjugation, such that: 1. .x¯¯ = x; 2. .x + y = x¯ + y; ¯ 3. .xy = y¯ x¯ for all .x, y ∈ A;

1.1 K-Algebras with Involutions: Composition K-Algebras

3

4. .λx = λx¯ for any .λ ∈ K and .x, y ∈ A. Observe that .1¯ = 1. In fact, since 1 is the left unit, we have 1¯ = 1 · 1¯ = 1 · 1¯ = 1¯¯ = 1.

.

In particular, using the identification of K with the image of the monomorphism K → A given by .λ → λ · 1 for .λ ∈ K, we have .λ¯ = λ · 1 = λ · 1¯ = λ · 1 = λ. Next, we get that 1 is also a unit on the right:

.

x · 1 = 1 · x¯ = x¯¯ = x

.

for all .x ∈ A. Consequently, 1 is a unit on both sides. Further, .xλ = x¯¯ λ¯ = λx¯ = λx¯¯ = λx for .λ ∈ K and .x ∈ A. We also have that .x(λμ) = (xλ)μ for .λ, μ ∈ K and .x ∈ A. In fact, .x(λμ) = x(μλ) = (μλ)x = μ(λx) = μ(xλ) = (xλ)μ. Next, if .A is a left alternative K-algebra, .x, y ∈ A then .x(y 2 ) = y¯ 2 x¯ = y( ¯ y¯ x) ¯ = = (xy)y. This implies .x(y 2 ) = (xy)y and so .A is an alternative K-algebra. y(xy) ¯ Consequently, .A is a flexible K-algebra as well. Recall that a field K is called formally real provided .−1 is not a sum of squares in K, or equivalently, no equation of the type .x12 + · · · + xn2 = 0 has a nontrivial solution in K for .n ≥ 1. In particular, the characteristic .char(K) = 0. Formally real fields are those fields admitting an order. We recall here the definition: a field K is called an ordered field if it is given a subset P of nonzero elements of K (we say that P is the subset of positive elements) such that .P , −P , {0} is a partition of .K, P contains the square of any non-zero element of .K, and P is closed by addition and multiplication of any of its elements. It is clear that any ordered field is formally real and conversely any formally real field admits a (not necessarily unique) ordering: just consider the set of all sums of non-zero squares and use the lemma of Zorn to obtained a set P of positive elements. A formally real field K is called closed if there is an ordering on K that does not extend to an ordering on any proper algebraic extension of K. Furthermore, a field K is called Pythagorean provided that the sum of two of its squares is a square or, equivalently a sum of any finite number of its squares is a square. Certainly, the field .R of reals and any algebraically closed field K are Pythagorean. If a and b are elements of an ordered field .K, we write .a < b and .b > a if .b − a ∈ P and for .λ ∈ K, we write .|λ| = λ if .λ ≥ 0, and .|λ| = −λ, if .λ ≤ 0.

4

1 Algebraic Preliminaries

Now, let K be a field and .(A, −) a left alternative K-algebra with an involution, and such that: (1) x + x¯ ∈ K for any x ∈ A;

.

(2) if n ≥ 1 and x1 , . . . , xn ∈ A then

n 

xk x¯k = y 2 for some y ∈ A;

k=1

(3) if n ≥ 1 and x1 , . . . , xn ∈ A with

n 

xk x¯k = 0 then x1 = · · · = xn = 0.

k=1

(1.1)   In particular, for .λ1 , . . . , λn ∈ K with . nk=1 λ2k = 0 = nk=1 λk λ¯ k , we get .λ1 = · · · = λn = 0. Consequently, the field K is formally real. Since .(A, −) a left alternative K-algebra with an involution, we know that .A is an alternative K-algebra as well. Under the conditions above, we show: Proposition 1.1 (1) (2) (3) (4) (5)

K = {x ∈ A; x = x}; ¯ there is a K-isomorphism .A ≈ K ⊕ A−1 , where .A−1 = {x ∈ A; −x = x}; ¯ if .x ∈ A−1 then .x 2 ∈ K; if .x 2 ∈ K for some .x ∈ A then .x = x¯ or .−x = x; ¯ for any non-zero .x ∈ A−1 there are .μ ∈ K and .y ∈ A−1 with .y 2 = −1 such that .x = μy and this presentation is unique up to the signs of .μ and y.

.

Proof (1): if .x ∈ A with .x¯ = x then, by (1.1)(1) of .(A, −), we get .2x ∈ K. Since .ch(K) = 0, we deduce that .x ∈ K. (2): the K-isomorphism .A → K ⊕ A−1 is given by .x → ( x+2 x¯ , x−2 x¯ ) for .x ∈ A. (3): if .x ∈ A−1 then .x 2 = −x x. ¯ Hence, .x 2 = −x x¯ = −x x¯ = x 2 and, in view 2 of (1.1)(1), we get .x ∈ K. (4): let .x 2 ∈ K and .x = x1 + x2 with .x1 ∈ K and .x2 ∈ A−1 . If .x1 = 0 then 2 = x2 + x2 + x x + x x = .x = x2 ∈ A−1 . If .x1 = 0 then consider .x 1 2 2 1 1 2 2 2 2 x1 + x2 + 2x1 x2 . Since .x2 ∈ K, we derive that .2x1 x2 = x 2 − x12 − x22 ∈ K. Then, .x1 = 0 implies .x2 ∈ K and consequently .x2 = −x2 = x2 . This implies .x2 = 0 and so .x = x1 ∈ K. (5): if a non-zero .x ∈ A−1 then .x x¯ = −x 2 and, in view of (1.1)(2), we have .x x ¯ = z2 for some non-zero .z ∈ A. Hence, by (3), we get .z2 = −x 2 ∈ K and, in view of (4), we have that .z ∈ K or .−z = z¯ . For .−z = z¯ , we have .x x ¯ + z¯z = 0 and (1.1)(3) leads to .x = z = 0, contrary to .x = 0. This implies

1.1 K-Algebras with Involutions: Composition K-Algebras

5

that .z ∈ K. Taking .y = z−1 x and .μ = z, we have .x = μy with .y 2 = −1 and .y ¯ = z−1 x¯ = −z−1 x = −y. Let now .x = μy = μ y  with .μ, μ ∈ K and .y 2 = (y  )2 = −1. Then, .−μ = and .μyy  = −μ and so .y  y, yy  ∈ K. Hence, .yy  = yy  = y  y and so 2  2     .μ = (μ ) . Consequently, .(μ − μ )(μ + μ ) = 0 implies .μ = μ or .μ = −μ . This   leads to .y = y or .y = −y and the proof is complete.  μ y  y

Notice that a direct consequence of Proposition 1.1(1) and (1.1)(3) is that the map .N : A → K, given by .N(x) = x x¯ for .x ∈ A, is a nondegenerate quadratic form whose associated K-bilinear symmetric form

−, − : A × A → K

.

is given by . x, y = 12 (x y¯ + y x). ¯ Furthermore, in the proof of this proposition the condition that .A is alternative is not needed. In general, a composition K-algebra .A for a field K (with the characteristic .char(K) = 2) is defined as a not necessarily associative algebra over K with identity element 1 such that there exists a nondegenerate quadratic form .N : A → K which permits composition, i.e., such that N(xy) = N(x)N(y)

.

for .x, y ∈ A. See [38, Chapter 1] by Tonny A. Springer and Ferdinand D. Veldkamp. Given .x ∈ A, we define its real part Re(x) =

.

x + x¯ 2

Then, Proposition 1.1 leads to: Corollary 1.2 (1) (2) (3) (4)

x x¯ = xx ¯ for any .x ∈ A; Re(xx  ) = Re(x  x) for any .x, x  ∈ A; any non-zero element of .A is invertible; if . nk=1 xk x¯k = y 2 for some .y ∈ A then .y ∈ K. In particular, K is Pythagorean field.

. .

Proof (1): if .x = λ + μy with .λ, μ ∈ K and .y ∈ A−1 with .y 2 = −1 then .x x¯ = ¯ (λ + μy)(λ − μy) = λ2 + μ2 = xx. (2): let .x = λ + μy, .x  = λ + μ y  with .λ, λ , μ, μ ∈ K and .y  , y ∈ A−1 with .y 2 = (y  )2 = −1. Then, .xx  = λλ + λμ y  + λ μy + μμ yy  and .x  x = λ λ+λ μy +λμ y  +μ μy  y. This implies .xx  = λλ −λμ y  −λ μy +μμ y  y

6

1 Algebraic Preliminaries

and .x  x = λ λ − λ μy − λμ y  + μ μyy  . Consequently, Re(xx  ) = λλ + μμ

.

yy  + y  y = Re(x  x). 2

(3): if .x ∈ A then .x = λ + μy for some .λ, μ ∈ K and .y ∈ A−1 with .y 2 = −1. Then, .x x¯ = λ2 + μ2 with .λ2 + μ2 = 0 provided .x = 0. Then, .x −1 = 2 + μ2 )−1 x. (λ ¯ n (4): if .  x x ¯ = y 2 then .y 2 ∈ K. Hence, .y¯ = y or .y¯ = −y. For .y¯ = −y, we k k k=1 n get . k=1 xk x¯k + y y¯ = 0 and so .x1 = · · · = xn = y = 0. This implies that .y ∈ K.   Let .x1 , . . . , xn ∈ K. Then, . nk=1 xk2 = nk=1 xk x¯k = y 2 for some .y ∈ K. If n  n 2 . k=1 xk = 0 then . k=1 xk x¯ k = 0 and so .x1 = · · · = xn = 0. Consequently, K is formally real Pythagorean field and the proof follows.  Notice that Corollary 1.2(3) says that .A is division (alternative) algebra. Next, we state: Corollary 1.3 (1) .x(x −1 y) = y = (yx −1 )x for all .x, y ∈ A with .x = 0. In particular, .A has no zero divisors; (2) .x(xy) ¯ = (x x)y ¯ = (y x)x ¯ = y(xx) ¯ for all .x, y ∈ A; (3) .(A, −) is a division and composition K-algebra; (4) .(xy)−1 = y −1 x −1 for all non-zero .x, y ∈ A. Proof (1): let .x = λ + μz = 0 with .λ, μ ∈ K, .z2 = −1 and .z¯ = −z. Then, x −1 = (λ2 + μ2 )−1 (λ − μz)

.

and .x(x −1 y) = (λ + μz)((λ2 + μ2 )−1 (λ − μz)y) = (λ2 + μ2 )−1 (λ2 y − (λμ)(zy) + (λμ)(zy) − μ2 (z(zy)) = .(λ2 + μ2 )−1 ((λ2 + μ2 )y) = y. In the same way, we show that .(yx −1 )x = y. Next, let .xy = 0 for .x, y ∈ A. If x is non-zero then .x −1 (xy) = 0 and so −1 = 0 and so .x = 0 as well. .y = 0. If y is non-zero then .(xy)y −1 −1 (2): notice that .x = (x x) ¯ x¯ and then use (1). (3): if .x, y ∈ A and y is non-zero then for .z = y −1 x and .z = xy −1 , in view of (1), we have .yz = y(y −1 x) = (yy −1 )x = x and .z y = (xy −1 )y = x(y −1 y) = x. Let .x = yz1 = z2 y for some .z1 , z2 ∈ A. Then, .y(z − z1 ) = 0 = y(z − z2 ) and so .y −1 (y(z − z1 )) = 0 = y −1 (y(z − z2 )) implies .z = z1 and .z = z2 . Now, we show that .A is a composition algebra with quadratic form N given above, i.e. .N (x) = x x¯ for all .x ∈ A.

1.1 K-Algebras with Involutions: Composition K-Algebras

7

Thus, we have to show that .(xy)(xy) = (x x)(y ¯ y) ¯ for all .x, y ∈ A. First, notice that we may take .x ∈ A−1 . In fact, if .x = λ + x  with .λ ∈ K and  .x¯ = −x then ¯ , x¯ = λ − x  , xy = λy + x  y, xy = y¯ x¯ = λy¯ − yx

.

(xy)(xy) = (λy + x  y)(λy¯ − yx ¯  ) = λ2 y y¯ − λ(y y)x ¯  + λ(y y)x ¯  − (x  y)(yx ¯ )

.

= λ2 y y¯ − (x  y)(yx ¯ ) and (x x)(y ¯ y) ¯ = (λ2 − (x  )2 )(y y) ¯ = λ2 y y¯ − (x  )2 (y y). ¯

.

Thus, we have to prove that .(x  y)(yx ¯  ) = (x  )2 (y y) ¯ for all .x  ∈ A−1 and .y ∈ A. If .y = μ + y  with .μ ∈ K, .y¯  = −y  then .y¯ = μ − y  and (x  y)(yx ¯  ) = (μx  + x  y  )(μx  − y  x  )

.

= μ2 (x  )2 − (μx  )(y  x  ) + μ(x  y  )x  − (x  y  )(y  x  ) = μ2 (x  )2 − (x  y  )(y  x  ) and, (x  )2 (y y) ¯ = (x  )2 (μ2 − (y  )2 )

.

= μ2 (x  )2 − (x  )2 (y  )2 . Therefore, it is enough to show that for all .x  , y  ∈ A−1 we have (x  )2 (y  )2 = (x  y  )(y  x  ).

.

Since .x  = λx  and .y  = μy  for some .λ, μ ∈ K and .(x  )2 = (y  )2 = −1, we may assume that .(x  )2 = (y  )2 = −1. Further, we may choose .z with .(z )2 = −1 and .x  z = −z x  . In fact, if           .x y + y x = 0 then we take .z = y . If .x y + y x = 0 then first notice that             −1 y  .   .x y + y x = x y + x y ∈ K and then take .z = x + 2(x y + y x )   2  Since, .z = −z , we may assume as above that .(z ) = −1. Next, notice that .y  = αx  + βz for some .α, β ∈ K. Then (x  )2 (y  )2 = 1 = α 2 + β 2

.

and (x  y  )(y  x  ) = (−α + βx  z )(−α − βx  z ) = α 2 − β 2 (x  z )2 .

.

8

1 Algebraic Preliminaries

But, (x  z )2 x  = (x  z )((x  z )x  ) = (x  z )(−(z x  )x  ) = (x  z )z = −x 

.

and so .(x  z )2 = −1. This implies that a (x  y  )(y  x  ) = α 2 + β 2 = 1 = (x  )2 (y  )2 .

.

(4): if .x, y ∈ A are non-zero then .x −1 = (x x) ¯ −1 x¯ and y −1 = (y y) ¯ −1 y. ¯ Hence, in view of (3), we have y −1 x −1 = (x x) ¯ −1 (y y) ¯ −1 y¯ x¯ = ((xy)(xy))−1 (xy) = (xy)−1

.

and the proof is complete.  Remark 1.4 A different proof of the above corollary could be given by using the beautiful result by E. Artin, [35, Theorem 3.1]: Theorem 1.5 (E. Artin) The subalgebra generated by any two elements of an alternative K-algebra .A is associative. A New Proof of Corollary 1.3 Given elements .x = λ + μy and .x  = λ + μ y  of   2 2 = 1, let .B be the subalgebra of .A generated .A, with .λ, λ , μ, μ ∈ K and .y = y  by the elements .1, y and .y . Then, .B is a unital and, by Theorem 1.5, associative Ksubalgebra of .A. Since .x¯ = λ − μy and .x¯ = λ − μ y  , we get .x −1 = (λ2 + μ2 )−1 x¯ and .x −1 = (λ2 + μ2 )−1 x¯ . Consequently, we have that all these elements lie in .B and items (1), (2), (3) and (4) follow easily.  Remark 1.6 In [38, Chapter 1] it is shown that any composition K-algebra is an algebra with an involution, where the conjugation is defined by .x¯ = 2 x, 1 − x for .x ∈ A. Furthermore, we show: Lemma 1.7 (1) K and .A−1 are orthogonal; (2) if .x, y ∈ A−1 then . x, y = 0 if and only if .xy = −yx or equivalently, .xy ∈ A−1 ; (3) . x, x = 1 if and only if .x x¯ = 1. In particular, for .x ∈ A−1 we have . x, x = 1 if and only if .x x¯ = 1 or equivalently, .x 2 = −1; (4) if .V ⊆ A−1 is a K-linear subspace of .A−1 and .dimK V = d − 1 with .d ≥ 2 then there exists an orthonormal basis .1, e2 , . . . , ed of .K ⊕ V . Proof (1)–(3): those are obvious.

1.1 K-Algebras with Involutions: Composition K-Algebras

9

(4): since K is a formally real Pythagorean field, by the Gram-Schmidt orthonormalization procedure there exist .1, e2 , . . . , ed such that .eα2 = −1, α = 2, . . . , d, .eα eβ = −eβ eα for all .α = β and .e¯α = −eα for all .α and the proof is complete.  Next, we state: Lemma 1.8 (1) . zx, zy = (z¯z) x, y = xz, yz for all .x, y, z ∈ A; (2) (Flip law) . vx, zy + vy, zx = 2 v, z x, y = xv, yz + yv, xz for all .x, y, v, z ∈ A; (3) . xz, y = x, y z¯ , zx, y = x, z¯ y for all .x, y, z ∈ A. Proof (1): .x, y, z ∈ A. Then, using Corollary 1.3(4), we get (z¯z)((x + y)(x + y)) = (z(x + y))(z(x + y)).

.

This implies .(z¯z)(x x¯ + y y¯ + 2 x, y) = (zx)(zx) + (zy)(zy) + 2 zx, zy and so (z¯z) x, y = zx, zy.

.

Similarly, we show (z¯z) x, y = xz, yz.

.

(2): it follows from (1) by replacing z by .z + v. (3): it is clear for .z ∈ K, so we may take .z ∈ A−1 . Then, taking .v = 1 in (2), we have . zx, y + zy, x = 2 z, 1 x, y = 0. This implies

zx, y = x, (−z)y = x, z¯ y.

.

Further, . xz, y + yz, x = 2 z, 1 x, y = 0 implies

xz, y = x, y z¯ 

.

and the proof follows.  Notice that Lemma 1.8(3) for .y = 1 yields . xz, 1 = x, z¯ , zx, 1 = x, z¯ . This leads to Re(xz) = Re(zx) = x, z¯ .

.

10

1 Algebraic Preliminaries

Then, we derive . (xy)z, 1 = xy, z¯  = x, z¯ y ¯ = 1, (¯zy) ¯ x. ¯ Consequently, we get Re(xy) = Re(yx) and Re((xy)z) = Re(x(yz))

.

(1.7)

for any .x, y, z ∈ A. Notice that (1.7) implies: Re(z(xy)) = Re((xy)z) = Re((zx)y) = Re((yz)x) = Re(x(yz)) = Re(y(zx))

.

for any .x, y, z ∈ A. Now, we derive Moufang laws for .(A, −), a K-algebra with an involution satisfying all properties stated above. Corollary 1.10 If .x, y, z ∈ A then: (1) .((xy)x)z = x(y(xz)) equivalently .x(yz) = (xyx)(x −1 z); (2) .(xy)(zx) = (x(yz))x; (3) .((yx)z)x = y((xz)x) equivalently .(yz)x = (yx −1 )(xzx). Proof It is clear that (3) follows from (1) by the conjugation and flexibility properties. (1): we begin, by proving y(xz) ¯ = 2 x, yz − x(yz). ¯

.

(∗)

In fact, for all .v ∈ A, we have

y(xz), ¯ v + x(yz), ¯ v = xz, yv + yz, xv = xz, yv + xv, yz

.

= 2 x, y z, v. The first equality above follows from Lemma 1.8(3), and the last one follows from Lemma 1.8(2) (Flip law). If now, we replace .y¯ by y in .(∗) and multiply on the left by .x, we get x(y(xz)) = 2 y, xxz ¯ − y( ¯ x(xz)) ¯ = (2 y, xx ¯ − (x x) ¯ y)z. ¯

.

Taking .z = 1 in .(∗), we get ¯ x(yx) = 2 y, xx ¯ − (x x) ¯ b,

.

and so, by the flexibility property, we have ((xy)x)z = x(y(xz)).

.

1.1 K-Algebras with Involutions: Composition K-Algebras

11

(2): by Lemma 1.8(2–3), we have

(xy)(zx), v = xy, v(x¯ z¯ ) = 2 x, v y, x¯ z¯  − x(x¯ z¯ ), vy =

.

2 x, v xy, z¯  − (x x) ¯ ¯ z, vy = 2 x, v y, x¯ z¯  − (x x) ¯ ¯ zy, ¯ v

.

= 2 yz, x x, ¯ v − (x x) yz, ¯ v. Thus, replacing z by 1, we have

(xy)x, v = y, x x, ¯ v − (x x) ¯ y, ¯ v.

.

Next, replacing y by yz in the above, we have

(x(yz)x, v = yz, x x, ¯ v − (x x) ¯ yz, ¯ v.

.

Then, we derive that

(xy)(zx), v = (x(yz))(1 · x), v.

.

This implies (xy)(zx) = (x(yz))x

.

and the proof is complete.  Now, we present examples of composition K-algebras provided K is a formally real field. Example 1.11 For a field K, we define .C(K) = K ⊕Ki with .i 2 = −1, the complex K-algebra which consists of elements .α0 + α1 i, the conjugation α0 + α1 i = α0 − α1 i

.

and the multiplication (α0 + α1 i)(α0 + α1 i) = (α0 α0 − α1 α1 ) + (α0 α1 + α1 α0 )i

.

for .α0 , α0 , α1 , α1 ∈ K. Notice that .C(K) could have zero divisors in general. For instance, take .K = C, the field of complex numbers. For .A = C(K), we define .H(K) = C(K) ⊕ C(K)j with .j 2 = −1, the quaternion K-algebra which is no longer commutative, consists of elements .α0 + α1 j , the conjugation α0 + α1 j = α0 − α1 j

.

12

1 Algebraic Preliminaries

and the multiplication (α0 + α1 j )(α0 + α1 j ) = (α0 α0 − α1 α1 ) + (α0 α1 + α1 α0 )j

.

for .α0 , α0 , α1 , α1 ∈ C(K). For .A = H(K), we define .O(K) = H(K) ⊕ H(K)l with .l 2 = −1, the octonion (or Cayley) K-algebra which is no longer associative but alternative, consists of elements .α0 + α1 l, the conjugation α0 + α1 l = α0 − α1 l

.

and the multiplication (α0 + α1 l)(α0 + α1 l) = (α0 α0 − α1 α1 ) + (α1 α0 + α1 α0 )l

.

for .α0 , α0 , α1 , α1 ∈ H(K).

1.2 Generalized Frobenius-Hurwitz’s Theorem Frobenius Theorem [10], proved by Ferdinand Georg Frobenius in 1877, characterizes finite-dimensional associative division algebras over the real numbers. The classical Hurwitz’s Theorem generalizes Frobenius Theorem and is a theorem of Adolf Hurwitz (1859–1919) originally proved in 1898 and then published [28] posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions, or the octonions. Such algebras, sometimes called Hurwitz algebras, are examples of composition algebras. The theory of composition algebras has subsequently been generalized to arbitrary quadratic forms and arbitrary fields (see, e.g., [2]). Furthermore, write .N (K) the set of all natural numbers n such that a division algebra (not assumed to be associative) of dimension n over K exists. Then, the paper [5] proves that .N (K) = {1} if K is algebraically closed, .N (K) = {1, 2, 4, 8} if K is real closed, and .N (K) is unbounded if K is neither algebraically closed nor real closed. We aim to state a necessary condition for a K-algebra being isomorphic to one of K-algebras: .K, C(K), H(K) or .O(K).

1.2 Frobenius-Hurwitz’s Theorem

13

Let .(A, −) be a K-algebra with an involution and all properties as in Sect. 1.1. Given a K-linear subspace .V ⊆ A, write .V ⊥ for its orthogonal complement with respect to . −, −. Then, we state: Proposition 1.12 Let .A ⊆ A be a finite dimensional subalgebra of .A with  .dimK A = d + 1 ≥ 2 and assume that .u ∈ / A . Let .A be the minimum subalgebra  of .A containing both .A and .u. Choose an orthonormal basis .1, e1 , . . . , ed of .A , i.e., .eα2 = −1, e¯α = −eα for .α = 1, . . . , d and .eα eβ = −eβ eα for .1 ≤ α, β ≤ d. Clearly, we may take then    .ed+1 ∈ A so that .A is the minimum subalgebra of .A containing .A and .ed+1 with 2  ⊥ .e d+1 = −1 and .ed+1 ∈ (A ) , i.e., .e¯d+1 = −ed+1 and .ed+1 eα = −eα ed+1 , α = 1, . . . , d. Then .1, e1 , . . . , ed+1 , e1 ed+1 , . . . , ed ed+1 is an orthonormal basis of .A ; in particular .dimK A = 2 dimK A . Proof Since, .eα ed+1 = ed+1 eα = −eα ed+1 , we have .eα ed+1 ∈ A−1 . Then, by using Lemma 1.8(3) for the first equal above, we have

eα ed+1 , eβ  = ed+1 , e¯α eβ  = − ed+1 , eα eβ  = 0 for 1 ≤ α, β ≤ d.

.

Next, using Moufang laws when needed, we have: (eα ed+1 )ed+1 = −eα , ed+1 (eα ed+1 ) = −er+1 (ed+1 eα ) = eα for α = 1, . . . , d;

.

(eα ed+1 )2 = −(eα ed+1 )(ed+1 eα ) = −1;

.

(eα ed+1 )(eβ ed+1 ) = −(ed+1 eα )(eβ ed+1 ) = −(ed+1 (eα eβ ))ed+1 =

.

((eα eβ )ed+1 )er+1 = −eα eβ ;

.

(eβ ed+1 )(eα ed+1 ) = −eβ eα = eα eβ

.



and the proof is complete.

With the same hypotheses and notations as above, using Moufang laws, we can show: (q1 + q2 ed+1 )(q1 + q2 ed+1 ) = q1 q1 − q¯2 q2

.

+(q2 q1 + q2 q  1 )er+1 for q1 , q2 , q1 , q2 ∈ A . Further, we have q1 + q2 ed+1 = q¯1 − q2 ed+1 .

.

14

1 Algebraic Preliminaries

In particular, (q1 + q2 ed+1 )(q1 + q2 ed+1 ) = q1 q¯1 + q¯2 q2 .

.

Remark 1.13 (1) Observe that .A is commutative if and only if .d = 0. (2) If .A is non-commutative then .A is non-associative but alternative. Proof (1): since .(qed+1 )ed+1 = −q, .ed+1 (qed+1 ) = −q, ¯ and so, if .A were commuta tive, then .q = q¯ for all .q ∈ A, i.e., .A = K. (2): if we choose .q, q, ∈ A with .qq  = q  q then q(q  ed+1 ) = (q  q)ed+1 = (qq  )ed+1 .

.

 Proposition 1.14 If .A  A with .dimK A ≥ 2 then .A is associative. Proof To simplify notations, write .u = ed+1 . Then, for all .x, y, v, z ∈ A , we have: (x + yu)(a + yu) = x x¯ + y y, ¯

.

(v + z)(v + zu) = v v¯ + z¯z,

.

(x + yu)(v + zu) = xv − z¯ y + (zx + y v)u. ¯

.

∗ = ((x + vyu)(v + zu))((x + yu)(v + zu))

.

= (xv − z¯ y)(xv − z¯ y) + (zx + y v)(zx ¯ + y v) ¯ = (xv)(xv) + (¯zy)(¯zy) − (xv)(yz) ¯ − (¯zy)(v¯ x) ¯ + (zx)(zx)

.

+ (y v)(y ¯ v) ¯ + (zx)(v y) ¯ + (y v)( ¯ x¯ z¯ ). On the other hand, Corollary 1.3(4), we have ∗ = (x x¯ + y y)(v ¯ v¯ + z¯z)

.

and so, we must have .(xv)(yz) ¯ + (¯zy)(v¯ x) ¯ = (vx)(v y) ¯ + (y v)( ¯ x¯ z¯ ), i.e.,

xv, z¯ y = zx, y v. ¯

.

1.3 Matrices over K-Algebras

15

But, by Lemma 1.8(3), we have . z(xv), y = (zx)v, y for all .x, y, v, ∈ A , and so z(xv) = (zx)v

.

for all .x, v, z ∈ A . Therefore, .A is associative and the proof follows.



Now, the following result generalizing Frobenius–Hurwitz’s Theorem is clear as a consequence of all previous ones: Theorem 1.15 If a left alternative K-algebra .(A, −) with an involution satisfies (1.1) then K is a formally real Pythagorean field and .d = 0, 1, 3 or 7. Further: if .d = 0 then .A ≈ K; if .d = 1 then .A ≈ C(K); if .d = 3 then .A ≈ H(K); if .d = 7 then .A ≈ O(K). Proof Let .(A, −) be a K-algebra with a involution with all hypothesis above. If dimK A = 1 then certainly we have an isomorphism .A ≈ K. If .dimK A > 1 then, by Proposition 1.12, there is a commutative subalgebra .A1 ⊆ A with .dimK A1 = 2 with an isomorphism .A1 ≈ C(K). If .dimK A = 2 then .A1 = A. If .dimK A > 2 then, by Proposition 1.12, there is a noncommutative subalgebra .A2 ⊆ A with .dimK A2 = 4 with .A1 ⊆ A2 and an isomorphism .A2 ≈ H(K). If .dimK A = 4 then .A2 = A. If .dimK A > 4 then, by Proposition 1.12, there is a nonassociative subalgebra .A3 ⊆ A with .dimK A2 = 8 with .A2 ⊆ A3 and an isomorphism .A3 ≈ O(K). If .dimK A = 8 then .A3 = A. If .dimK A > 8 then, by Proposition 1.14, there is an associative K-subalgebra .A4  A with .dimK A4 = 8 This contradiction shows .d = 0, 1, 3 or 7 exhaust all possible values and the proof is complete.  .

In the rest of the book, given a commutative field K with the involution .− = idK , a K-algebra .(A, −) with involution is either the field K, .C(K), .H(K) or .O(K).

1.3 Matrices over K-Algebras Let .M(A) denote the set of matrices .A = (aαβ ) over .A, where .α and .β are natural numbers, and the sets .{β; aαβ = 0} (respectively .{α; aαβ = 0}) for a fixed .α (respectively .β) are finite. Given a matrix .A = (aαβ ) over .A, also write .Aαβ for its element .aαβ . Observe that the matrix sum and multiplication in .M(A) are well defined: for .A = (aαβ )

16

1 Algebraic Preliminaries

and .B = (bαβ ) in .M(A), the entries .(A + B)αβ = aαβ + bαβ and .(AB)pq is given by .(AB)pq = α apα bαq . Then, .M(A) is a unitary K-algebra with respect those two operations and the obvious K-structure. We write .Mn,− (A) for the K-linear subspace of .M(A) of all matrices .A = (aαβ ) with .aαβ = 0 with .α > n and .M−,n (A) for the K-linear subspace of .M(A) of all matrices .A = (aαβ ) with .aαβ = 0 with .β > n. Then, we set .Mm,n (A) = Mm,− (A) ∩ M−,n (A), and instead of .Mn,n (A) we write .Mn (A). We set also .M(A) = ∪n≥1 Mn (A). We identify .Mn,− (A) with the space of matrices with n rows and infinite columns, .M−,n (A) with the space of matrices with infinite rows and n columns, .Mm,n (A) with the space of matrices with m rows and n columns. The conjugation and transposition lead to K-linear involutions .M(A) → M(A) given by .A → At and .A → A¯ for .A ∈ M(A), respectively. The transposition restricts to K-isomorphisms Mn,− (A) → M−,n (A), M−,n (A) → Mn,− (A), Mm,n (A) → Mn,m (A)

.

and the conjugation restricts to K-isomorphisms Mn,− (A) → Mn,− (A), M−,n (A) → M−,n (A), Mm,n (A) → Mm,n (A).

.

We have .(AB)t = B t At and .AB = A¯ B¯ for matrices over .A = K, C(K). But none of those relations hold, in general, for .A = H(K), O(K). Instead, the following relation always holds (AB)t = B¯ t A¯ t

.

for matrices over .A = K, C(K), H(K), O(K). Now, recall that: any .x ∈ C(K) can be written uniquely as .x = a + bi, where .a, b ∈ K; any .x ∈ H(K) can be written uniquely as .x = a + bj with .a, b ∈ C(K); any .x ∈ O(K) can be written uniquely as .x = a + bl with .a, b ∈ H(K). Then, any matrix .X ∈ M(A) can be written uniquely as .X = A + Bt with .t = i, j, l and .A = C(K), H(K), O(K). Hence, for any .X = A + Bt ∈ Mm,n (A)   A B ˜  we have the adjoint .2m × 2n-matrix .X = A + Bt = . −B¯ A¯ It is easily checked that: if .X, Y ∈ Mm,n (A) with .A = C(K), H(K), O(K) then: • .I˜n = I2n ,  • .X + Y = X˜ + Y˜ , t X¯ t ) = X˜ ; • .( if .X, Y ∈ Mm,n (A) with .A = C(K), H(K) then  = X˜ Y˜ . • .XY

1.3 Matrices over K-Algebras

17

Unfortunately, the operation .∼ cannot be extended to .M(A). Therefore, given a matrix .X = A + Bt ∈ M(A) with .t = i, j, l and .A = C(K), H(K), O(K), we define: ⎧ ⎪ apq , for α = 2p − 1, β = 2q − 1, ⎪ ⎪ ⎪ ⎨a¯ , for α = 2p, β = 2q, pq .(χA (A + Bt))αβ = ⎪ bpq , for α = 2p − 1, β = 2q, ⎪ ⎪ ⎪ ⎩ ¯ −bpq , for α = 2p, β = 2q − 1 for .A = (aα,β ), B = (bα,β ). Observe that .χA (X) for .X = A + Bt ∈ Mm,n (A) with .A = C(K), H(K), O(K) does not coincide with .A + Bt except for .m = n = 1. One can easily also show that: if .X, Y ∈ M(A) with .A = C(K), H(K), O(K) then: • .χA (In ) = I2n , • .χA (X + Y ) = χA (X) + χA (Y ), t • .χA (X¯ t ) = χA (X) ; if .X, Y ∈ M(A) with .A = C(K), H(K) then: • .χA (XY ) = χA (X)χA (Y ). But, .χO(K) (XY ) = χO(K) (X)χO(K) (Y ) for any .X, Y ∈ M(O(K)) in general. This is already false for .1 × 1-matrices: for .X = (i) and .Y = (j l) we have .χO(K) (XY ) = χO(K) (X)χO(K) (Y ). Now, we relate .X˜ and .χA (X) for .X ∈ M(A) with .A = C(K), H(K), O(K). To do that, given a permutation .σ of a set with n elements, consider the .n × n-matrix .Jσ defined as follows: (Jσ )αβ = δσ (α)β

.

for .1 ≤ α, β ≤ n. One can easily show that: • .Jσ σ  = Jσ Jσ  ; • .Jσ −1 Jσ = Jσ Jσ −1 = In . Now, consider the permutation .σ2n of a set with 2n elements given as follows:  .

for .α = 1, . . . , n.

σ2n (2α − 1) = α σ2n (2α) = α + n.

18

1 Algebraic Preliminaries

Then, in the sequel we need: Lemma 1.16 If .A = C(K), H(K), O(K) and .X ∈ Mm,n (A) then ˜ χA (X)Jσ2n = Jσ2m X.

.

Proof Given .X = A + Bt ∈ Mm,n (A) with .A = (aα,β ), B = (bα,β ) we have: ⎫ apq for α = 2p − 1, β = q; ⎪ ⎪ ⎬ bpq for α = 2p − 1, β = q + n; ˜ αβ = (Jσ2m X) = ⎪ −bpq for α = 2p, β = q; ⎪ ⎪ ⎪ ⎭ ⎩ apq for α = 2p, β = q + n ⎧ ⎪ ⎪ ⎨

(χA (X)Jσ2n )αβ

.

and the proof follows.  Further, we define the map τ : M(A) −→ M(A)

.

 given by .τ (A) =

 1 0 for .A ∈ M(A). 0A

Hermitian and Symmetric Matrices For .A = K, C(K), H(K), O(K), we define Hermitian, resp. symmetric matrices, Herm(A) = {A ∈ M(A)| A¯ t = A}, resp. Sym(A) = {A ∈ M(A)| At = A}

.

and Herm(A) = Herm(A) ∩ M(A) =



Hermn (A),

.

n≥1

where .Hermn (A) = Herm(A) ∩ Mn (A); Sym(A) = Sym(A) ∩ M(A) =



.

Symn (A),

n≥1

where .Symn (A) = Sym(A) ∩ Mn (A). Further, we can also consider skew-Hermitian matrices Sk(A) = {A ∈ M(A)| A¯ t = −A}

.

1.3 Matrices over K-Algebras

19

and Sk(A) = Sk(A) ∩ M(A) =



.

Skn (A),

n≥1

where .Skn (A) = Sk(K) ∩ Mn (A). We also define skew-symmetric matrices Sksym(A) = {A ∈ M(A)| At = −A}

.

and Sksym(A) = Sksym(A) ∩ M(A) =



.

Sksymn (A),

n≥1

where .Sksymn (A) = Sksym(K) ∩ Mn (A). Notice that .Hermn (K) are the symmetric matrices and .Hermn (A) are K-linear spaces with .

dimK Hermn (A) = d

n(n − 1) + n, 2

where .d = dimK A for .A = K, C(K), H(K), O(K), respectively. It is easily checked that χA (X¯ t ) = χA (X)t .

.

Therefore, we have induced injective K-linear maps χC(K) : Herm(C(K)) → Herm(K),

.

χH(K) : Herm(H(K)) → Herm(C(K))

.

and χO(K) : Herm(O(K)) → Herm(H(K)).

.

Trace Recall that .M(A) ⊆ M(A) is the subring of .M(A) given by matrices .A = (aαβ ), where .α and .β are natural numbers, and all entries .aαβ are zero with the possible exception of a finite number of them.

20

1 Algebraic Preliminaries

Notice that the canonical inclusion maps Mn (A) → Mn+1 (A)

.

given by .A →

  A0 for .A ∈ Mn (A) and .n ≥ 1 lead to 0 0  .

n≥1

Mn (A) = M(A).

The trace operation yields a map tr : M(A) → A

.

¯ = tr(A). such that .tr(At ) = tr(A) and .tr(A) We also have tr(AB) = tr(BA)

.

for matrices over .A = K, C(K), but this equality does not hold in general for A = H(K), O(K). Instead, in view of (1.7), we always have for the real part of the trace:

.

Re tr(AB) = Re tr(BA) and Re tr((AB)C) = Re tr(A(BC))

.

for .A, B, C ∈ M(A) with .A = K, C(K), H(K), O(K). ¯ = If .A ∈ Herm(A) with .A = C(K), H(K), O(K) then .tr(A) = tr(A¯ t ) = tr(A) tr(A). Consequently, .tr(A) ∈ K. Furthermore, let .A, B ∈ Herm(A). Then, tr(AB) = tr(AB) = tr(BA).

.

In fact, .tr(AB) = tr(A¯ t B¯ t ) = tr(BA ) = tr(BA) = tr(BA). It is easy to check that: t

˜ and Re tr(χA (A)χA (B)) tr(χA (A)) = 2Re tr(A) = tr(A),

.

= Re tr(χA (AB)) = ˜  = Re tr(A˜ B) Re tr(AB)

.

for .A, B ∈ M(A) with .A = C(K), H(K), O(K).

(1.14)

1.3 Matrices over K-Algebras

21

From this it follows that  = tr(BA)  tr(χA (AB)) = tr(χA (BA)) = tr(AB)

.

for all .A, B ∈ M(A) with .A = C(K), H(K), O(K).

Inner Products on M(A) For .A = K, C(K), H(K), O(K), we define {−, −} : M(A) × M(A) −→ A

.

and

−, − : M(A) × M(A) −→ A

.

given by .{A, B} = tr(AB) and . A, B = tr(A¯ t B) = {A¯ t , B}, respectively, for .A, B ∈ M(A). Clearly, the maps .{−, −} and . −, − coincides with . −, − on .Herm(A) = Herm(A) ∩ M(A). For instance, if .Eαβ denotes the matrix with 1 at place .αβ and 0 otherwise, we have 1 {Eαα }α=1,...,n ; { √ (Eαβ + Eβα )}1≤α 0 such that .Bd (x) ⊆ U. Following mutatis mutandis the proof of [33, Theorem 32.2], we show: Proposition 1.48 The space X with the d-topology is Hausdorff and normal.

1.5 Zariski Topologies on M(A)

45

d Proof Let .x, y ∈ X with .x = y and . = d(x,y) 2 . Then .x ∈ U = B (x), .y ∈ V = d B (y), and the open sets U , and V do not intersect. This implies that the space X is Hausdorff. Suppose now that .A, B ⊆ X are disjoint closed subsets of X. For each .a ∈ A choose .ra > 0 such that the ball .Brda (a) does not intersect B. Similarly, for each d .b ∈ B choose .rb > 0 such the ball .Br (b) does not intersect A. Define b

U=



.

B(a,

 rb ra ) and V = B(b, ). 2 2 b∈B

a∈A

Then U and V are open subsets containing A and B, respectively; we assert that they are disjoint. For, if .z ∈ U ∩ V then z ∈ B dra (a) ∩ B drb (b)

.

2

2

for some .a ∈ A and .b ∈ B. b The triangle inequality applies to show that .d(a, b) < ra +r 2 . If .ra ≤ rb then d .d(a, b) < rb so that the ball .Br (b) contains the point a. If .rb ≤ ra then .d(a, b) < ra b so that the ball .Brda (a) contains the point b. Neither situation is possible and the proof is complete.  Observe that if we choose an ordering on .K, so as to be an ordered field, then we get a K-distance on .M(A) given by .d(A, B) = ||A − B|| and so we have the corresponding d-topology on .M(A), which is Hausdorff and normal by the above proposition. We call this d-topology now the order topology on .M(A). Further, observe that d B (A) ⊆ B|| (A) = {X ∈ M(A); ||X − A|| < ||}.

.

In particular, the natural topology is finer than any order topology. On the other hand, it is clear that natural and order topology coincide on .M(A) if K is a formally real closed field. We summarize some properties relative to the natural and ordering topologies on .M(A) as follows: Proposition 1.49 Suppose that .M(A) is endowed with either the natural or with any ordering topology, then: (1) .Mn (A) is closed, for any .n ≥ 1 with respect to the natural or any ordering topology; (2) sum and product on .M(A) are continuous for the ordering topology; (3) the projections .xpq : M(A) → A, sending .A ∈ M(A) to its pq-entry, is continuous, both for the natural or for any ordering topology and, in particular, polynomial maps are continuous for any ordering topology;

46

1 Algebraic Preliminaries

(4) a map .f : X → Mn (A), where X is a topological space and n a natural number, is continuous if and only if the corresponding maps .fpq = xpq ◦ f : X → A are continuous, both for the natural or for any ordering topology; (5) the inverse map .GL(A) → GL(A), is continuous, for any ordering topology and for .A = K, C(K) or H(K). In particular, .GL(A) is completely regular. Proof (1): Suppose .A ∈ / Mn (A) and choose .m > n such that .A ∈ Mm (A). Therefore,  A=

.

A11 A12 A21 A22



with .A11 ∈ Mn (A), .A12 ∈ Mn,m−n (A), .A21 ∈ Mm−n,n (A) and .A22 ∈ Mm−n,m−n (A). Further, since .A ∈ / Mn (A), we know that at least one of the three matrices .A12 , A21 , A22 is non zero. Take . 2 = A12 , A12  + A21 , A21  + A22 , A22 , and so . = 0. .B (A) ∩ Mn (A) = ∅. In fact, if .X ∈ Mn (A) we have

X − A, X − A = X − A11 , X − A11  +  2 .

.

This shows that .X ∈ / B (A). Thus .Mn (A) is closed for the natural topology. On the other hand for any ordering topology we have .||X − A|| ≥  and so this shows that .Mn (A) is closed for the ordering topology. (2): Take .A, B in .M(A) and . > 0 in .K. Then by Corollary 1.20 (2), for any       .A , B ∈ M(A) such that .||A − A|| < 2 and .||B − B|| < 2 , we have ||A + B  − (A + B)|| ≤ ||A − A|| + ||B  − B|| < .

.

If we use now Corollary 1.20 and choose .δ > 0 such that ||A − A|| < δ, ||B  − B|| < δ,

.

then we have ||A B  −AB|| = ||A B  −A B+A B−AB|| ≤ ||A ||||B  −B||+||B||||A −A|| =

.

||A − A + A||||B  − B|| + ||B||||A − A|| < δ 2 + δ(||A|| + ||B||) < ,

.

 }. if we take, for instance, .δ < min{1, 1+||A||+||B|| (3): To show that .xpq : M(A) → A is continuous at .A ∈ M(A), with respect to the natural topology, suppose . ∈ K, . = 0, such that .A ∈ B (A), i.e.,

 2 − A − A, A − A = λ2

.

1.5 Zariski Topologies on M(A)

47

for some .λ = 0 in .K. Then,   − a ) =  2 − A − A, A − A + A − A, A − A  2 − (apq − apq )(apq pq

.

  −a )= − (apq − apq )(apq pq

λ2 +



.

  − a ) = μ2 (aαβ − aαβ )(aαβ αβ

αβ=pq

for some .μ = 0. This shows the continuity of .xpq with respect to the natural topology. To show that .xpq : M(A) → M(A) is continuous at .A ∈ M(A), with respect to any ordering topology, suppose . ∈ K, . > 0 such that .||A − A|| < , then |bpq − apq |2 ≤ ||B − A||2 <  2

.

and so, .|bpq − apq | < . In particular, by using (2), polynomial maps are continuous with respect to any ordering topology. (4): If .f : X → M(A) is a continuous map, then by (3), the maps .fpq = xpq f : X → A are continuous for .1 ≤ p, q ≤ n. Conversely, given .x ∈ X, consider a ball .Bε (f (x)) ⊆ M(A). Then, by the continuity of .fpq : X → A there are open subsets .Upq ⊆ X with .x ∈ Upq and .fpq (Upq ) ⊆ B nε (fp,q (x)) for balls .B nε (fp,q (x)) ⊆ A and .1 ≤ p, q ≤ n.   Hence, .x ∈ 1≤p,q≤n Upq and .f ( 1≤p,q≤n (Upq ) ⊆ Bε (f (x)). (5): It is also clear because for .A = K or C(K) the inverse could be written using determinants and we use (4). For .A = H(K), we use the map .χ from Sect. 1.3 and then (4). 

The Zariski Topology on M(A) We have the K-Zariski topology on .Mn (K): for an ideal I of the polynomial ring K[xpq ], with infinite number of variables .xpq , .p, q ≥ 1. We define .V (I ) as the set of matrices .X ∈ Mn (K) such that .p(X) = 0 for all polynomials .p ∈ I. Those sets .V (I ) are the closed sets of the K-Zariski topology on .Mn (K). Then, we have the K-Zariski topology on .

Mn (C(K)) = Mn (K)2 , Mn (H(K)) = Mn (K)4 , and Mn (O(K)) = Mn (K)8 .

.

48

1 Algebraic Preliminaries

In case .A is .C(K), we also consider the .C(K)-Zariski topology on .Mn (C(K)), by considering ideals I of .C(K)[xpq ] and defining .V (I ) as the set of matrices .X ∈ Mn (C(K)) such that .p(X) = 0 for all .p ∈ I. We endow .M(A) with the colimit topology with respect to the sequence of inclusions M1 (A) ⊆ M2 (A) ⊆ · · · ,

.

i.e., a subset .C ⊆ M(A) is closed provided .C ∩ Mn (A) is Zariski closed in .Mn (A) for all .n ≥ 1. It is clear by using Proposition 1.49(1),(3), that the Zariski topology is coarser than any order topology on .M(A), and so coarser than the natural topology as well.

Chapter 2

Exceptional Groups G2 (K) and F4 (K) .

.

This chapter takes up the systematic study of a generalization of the exceptional compact Lie groups .G2 and .F4 on groups .G2 (K) and .F4 (K) provided K is Pythagorean formally real field. The main result stated in Theorem 2.48 says that any Hermitian .3 × 3-matrix .A ∈ Herm3 (O(K)) can be transformed to a diagonal form by some element of .F4 (K).

2.1 Cross Products and the Exceptional Group G2 (K) Cross Product in Ln Let L be any commutative field and write .u, v = u1 v1 + · · · + un vn ∈ L, for u = (u1 , . . . , un ) and .v = (v1 , . . . , vn ) in .Ln . The map

.

−, − : Ln × Ln → L

.

is clearly L-bilinear symmetric and non-degenerated. For .w1 , . . . , wn−1 ∈ Ln we define the cross product .w1 × · · · × wn−1 ∈ Ln by .

det(w1 | · · · |wn−1 |w) = w1 × · · · × wn−1 , w, for w ∈ Ln , n ≥ 3,

where .(w1 | · · · |wn−1 |w) is the matrix in .Mn (L) with columns .w1 , . . . , wn−1 , w. In particular, if .e1 , . . . , en denotes the canonical basis of .Ln then w1 × · · · × wn−1 =

n 

.

α=1

w1 × · · · × wn−1 , eα eα =

n 

det(w1 | · · · |wn−1 |eα )eα .

α=1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Golasi´nski, F. Gómez Ruiz, Grassmann and Stiefel Varieties over Composition Algebras, RSME Springer Series 9, https://doi.org/10.1007/978-3-031-36405-1_2

49

2 Exceptional Groups .G2 (K) and .F4 (K)

50

Therefore, if .wβ = (w1β , . . . , wnβ ) ∈ Ln , for .β = 1, . . . , n − 1, and .(w1 × · · · × wn−1 )β denotes the .β-th component of .w1 × · · · × wn−1 , we have 

(w1 × · · · × wn−1 )β =

.

(σ )wσ (1)1 · · · wσ (n−1)n−1 , β = 1, . . . , n,

σ ∈Sn ,σ (n)=β

where .Sn denotes the permutation group of the set .{1, . . . , n}. Example 2.1 If .n = 3, .w1 = (a1 , a2 , a3 ), w2 = (b1 , b2 , b3 ) then w1 × w2 = (a2 b3 − a3 b2 , a3 b1 − a1 b3 , a1 b2 − a2 b1 ).

.

Properties (1) The cross product n−1

×n−1 : Ln × · · · ×Ln → L,

.

for n ≥ 3, is L-multilinear and skew-symmetric; (2) wα , w1 × · · · × wn−1  = 0 for α = 1, . . . , n − 1; (3) det(w1 | · · · , wn−1 |w1 × · · · × wn−1 ) = 0 if and only if w1 , . . . , wn−1 are Llinearly dependent or, equivalently, if and only if w1 × · · · × wn−1 = 0; (4) w1 × · · · × wn−1 = 0 if and only if w1 × · · · × wn−1 , w1 × · · · × wn−1  = 0; (5) w1 × · · · × wn−1 , w1 × · · · × wn−1  = det(wα , wβ 1≤α,β≤n−1 );  (6) the map f : Vn,n−1 (L) = {X ∈ Mn,n−1 (L); Xt X = In−1 } → SOn (L), given by f (w1 | · · · |wn−1 ) = (w1 | · · · |wn−1 |w1 × · · · × wn−1 ) is a L-polynomial bijection of degree n − 1 with the inverse polynomial of degree 1, where (w1 | · · · |wn−1 ) is the matrix in Mn,n−1 (L) with columns w1 , . . . , wn−1 . Remark 2.2  (L) to avoid conflict with our notation (1) We have used the notation Vn,n−1 Vn,r (A) = {X ∈ Mn,r (A); X¯ t X = Ir } for the particular case r = n − 1, A = C(K) and L = C(K). (2) Observe that the cross product ×n−1 is a binary operation only for n = 3 and then we write × instead of ×2 .

2.1 The Exceptional Group G2 (K)

51

We prove now properties (5) and (6), all others being easily checked. Write wn = w1 × · · · × wn−1 = (w1n , . . . , wnn ),

.

⎞ w11 · · · w1,n−1 w1n ⎜ . .. .. ⎟ .W = ⎝ . . . . ⎠ wn1 · · · wn,n−1 wnn ⎛

and let G be the matrix in Mn (L) whose αβ-entry is wα , wβ  = w1α w1β + · · · + wnα wnβ , 1 ≤ α, β ≤ n.

.

Thus, G = W t W.

.

Therefore, .

det G = (det W )2 = det(w1 | · · · |wn−1 |wn )2 = wn , wn 2 .

Property (2) yields .

det G = det(wα , wβ 1≤α,β≤n−1 )wn , wn  = wn , wn 2 .

This proves (5) whenever wn , wn  = 0. But, if wn , wn  = 0, then, (3) and (4) give detwα , wβ 1≤α,β≤n−1 = 0. Observe that, in particular, if wα , wβ  = δαβ for 1 ≤ α, β ≤ n − 1, then w1 , . . . , wn−1 , wn = w1 ×· · ·×wn−1 is an orthogonal basis with det(w1 | · · · |wn ) = 1, and this proves (6).

The Group Aut(Ln , ×n−1 ) Let .Aut(Ln , ×n−1 ) be the group of L-linear automorphisms .ϕ of .Ln such that ϕ(w1 × · · · × wn−1 ) = ϕ(w1 ) × · · · × ϕ(wn−1 )

.

for any .w1 , . . . , wn−1 in .Ln .

2 Exceptional Groups .G2 (K) and .F4 (K)

52

Then, the next result holds. Proposition 2.3 (1) .SOn (L) ⊆ Aut(Ln , ×n−1 ) for .n ≥ 3; (2) if .n = 3 and L is any commutative field, or .n > 3 and L is a Pythagorean real field then .SOn (L) = Aut(Ln , ×n−1 ). Proof Let .e1 , . . . , en be the canonical basis and observe that e1 × · · · × eˆα × · · · × en = (−1)n−α eα , α = 1, . . . , n,

.

where .eˆα means that .eα is deleted. Suppose first that .ϕ ∈ SOn (L). Then, (−1)n−α = (−1)n−α ϕ(eα ), ϕ(eα ) = ϕ(e1 × · · · × eˆα × · · · × en ), ϕ(eα )

.

and n−α  ϕ(e1 ) × · · · × ϕ(e det(ϕ(e1 )| · · · |ϕ(en )) α ) × · · · × ϕ(en ), ϕ(eα ) = (−1)

.

= (−1)n−α . Therefore,  ϕ(e1 ) × · · · × ϕ(e α ) × · · · × ϕ(en ) = ϕ(e1 × · · · × eˆα × · · · × en )

.

for any .α = 1, . . . , n. Thus, ϕ ∈ Aut(Ln , ×n−1 ).

.

Conversely, suppose now .ϕ ∈ Aut(Ln , ×n−1 ) and write .ϕ(eα ) = wα , .α = 1, . . . , n. Then, for .α = β, we have wα , wβ  = (−1)n−α w1 × · · · × wˆ α × · · · × wn , wβ  = 0

.

and wα , wα  = (−1)n−α w1 × · · · × wˆ α × · · · × wn , wα  =

.

(−1)n−α det(w1 | · · · |wˆ α | . . . , wn |wα ) = det(w1 | · · · |wn ) = det ϕ.

.

Write .λ = det ϕ, and so we have that .w1 , . . . , wn is an orthogonal basis of .Ln with .wα , wα  = λ, for .α = 1, . . . , n.

2.1 The Exceptional Group G2 (K)

53

But, property (5) gives λ = det(w1 | · · · |wn ) = w1 × · · · × wn−1 , w1 × · · · × wn−1 

.

= det(wα , wβ 1≤α,β≤n−1 ) = λn−1 . Therefore, λn−2 = 1.

.

This finishes the proof of the proposition for the particular case of .n = 3, because then .λ = 1. Now, suppose that L is a Pythagorean real field. Then, the roots in L of the polynomial .Xn−2 − 1, are either 1 for even .n, or 1 and .−1 for odd .n. But, since .λ = det ϕ is a sum of squares in L and so a square, we conclude that .λ = 1. Therefore, for a Pythagorean real field L we have .Aut(Ln , ×n−1 ) = SOn (L) and the proof is complete.  Remark 2.4 It is not true in general that we have equality .SOn (L) Aut(Ln , ×n−1 ). Check for instance [6, Proposition 3.4].

=

Cross Product in K 3 Suppose now that .L = K ⊆ H(K) and let .x = λ + x1 i + x2 j + x3 k, .y = μ + y1 i + y2 j + y3 k be elements of .H(K) with .λ, μ, x1 , x2 , x3 , y1 , y2 , y3 in .K. Then, we have xy = λμ−x1 y1 −x2 y2 −x3 y3 +(x2 y3 −x3 y2 )i +(x3 y1 −x1 y3 )j +(x1 y2 −x2 y1 )k =

.

Re(xy) + (x2 y3 − x3 y2 )i + (x3 y1 − x1 y3 )j + (x1 y2 − x2 y1 )k.

.

Identify now .K 3 with .H(K)Re=0 and so if x and y have real part .0. Thus, .x = (x1 , x2 , x3 ) is identified with .x1 i + x2 j + x3 k, .y = (y1 , y2 , y3 ) is identified with 3 .y1 i + y2 j + y3 k, and .x × y = (x2 y3 − x3 y2 , x3 y1 − x1 y3 , x1 y2 − x2 y1 ) ∈ K is identified with .(x2 y3 − x3 y2 )i + (x3 y1 − x1 y3 )j + (x1 y2 − x2 y1 )k. Observe that this identification is a ring isomorphism .(K 3 , ×) ≈ (H(K)Re=0 , ×) and a K-linear isometry, once we endow .K 3 with the inner product .−, − and .H(K)Re=0 with the inner product .(−, −), where .(x, y) = Re(xy) ¯ = −Re(xy) for .x, y ∈ H(K)Re=0 .

2 Exceptional Groups .G2 (K) and .F4 (K)

54

Therefore, we may define the cross product .× : K 3 × K 3 → K 3 by x × y = xy − Re(xy) = xy + (x, y)

.

for .x, y ∈ H(K)Re=0 .  (K) = V (K), and the map .f : V (K) → In this particular case, .V3,2 3,2 3,2 SO3 (K) of property (6) above is now given by ⎛

⎞ x1 y1 x2 y3 − x3 y2 .f (x|y) = ⎝x2 y2 x3 y1 − x1 y3 ⎠ x3 y3 x1 y2 − x2 y1 ⎛

⎞ x1 y1 for .x = x1 i +x2 j +x3 k, y = y1 i +y2 j +y3 k ∈ H(K)Re=0 and .(x|y) = ⎝x2 y2 ⎠ ∈ x3 y3 V3,2 (K).

Cross Product in C(K)n Suppose now that .L = C(K). The following property is then clear w1 × · · · × wn−1 = w¯ 1 × · · · × w¯ n−1

.

for .wα ∈ C(K)n , .α = 1, . . . , n − 1. Therefore, .

det(w1 | · · · |wn−1 , |w1 × · · · × wn−1 ) = {w1 × · · · × wn−1 , w1 × · · · × wn−1 } = w1 × · · · × wn−1 , w1 × · · · × wn−1 

.

which is the square of some element of .K. Write wn = w1 × · · · × wn−1 ,

.

W = (w1 | · · · |wn−1 |w¯ n ) ∈ Mn (C(K))

.

2.1 The Exceptional Group G2 (K)

55

and ⎞ w1 , w1  · · · wn−1 , wn−1  w1 , w¯ n  ⎟ ⎜ .. .. .. ⎟ ⎜ . . . .G = ⎜ ⎟= ⎝wn−1 , w1  · · · wn−1 , wn−1  wn−1 , w¯ n ⎠ w¯ n , w1  · · · w¯ n , wn−1  w¯ n , w¯ n  ⎛

⎞ w1 , w1  · · · wn−1 , wn−1  0 ⎟ ⎜ .. .. .. ⎟ ⎜ . . . .⎜ ⎟. ⎠ ⎝wn−1 , w1  · · · wn−1 , wn−1  0 0 ··· 0 wn , wn  ⎛

But, W¯ t W = G

.

and so, det W det W = det G = (det1≤α,β≤n−1 (wα , wβ )wn , wn  =

.

det(w1 | · · · |wn−1 |w¯ n ) det(w1 , · · · |wn−1 |w¯ n ) = wn , wn 2 .

.

Therefore, wn , wn  = det1≤α,β≤n−1 (wα , wβ ).

.

Thus, we have a polynomial bijection g : Vn,n−1 (C(K)) → SUn (C(K))

.

given by g(w1 | · · · |wn−1 ) = (w1 | · · · |wn−1 |w1 × · · · × wn−1 )

.

for .(w1 | · · · |wn−1 ) ∈ Vn,n−1 (C(K)).

2 Exceptional Groups .G2 (K) and .F4 (K)

56

Cross Product in C(K)3 Any element .x = λ + μi + x1 j + x2 k + x3 l + x4 il + x5 j l + x6 kl ∈ O(K) can be written uniquely as x = u0 + u1 j + u2 l + u3 (j l)

.

with .u0 = λ + μi, u1 = x1 + x2 i, u2 = x3 + x4 i, u3 = x5 − x6 i ∈ C(K). Observe that for any .a ∈ C(K) we have ax = au0 + (au1 )j + (au2 )l + (au3 )(j l)

.

and xa = ax ¯ + 2(Im(a))iu0 .

.

Let now .x = u0 + u1 j + u2 l + u3 (j l), y = v0 + v1 j + v2 l + v3 (j l) be elements of .O(K) with .uα , vα ∈ C(K), for .α = 1, 2, 3 and write .u = u1 j + u2 l + u3 (j l), .v = v1 j + v2 l + v3 (j l). Then, we have xy = u0 v0 + u0 v + v¯0 u − v, u + (u2 v3 − u3 v2 )j + (u3 v1 − u1 v3 )l

.

+ (u1 v2 − u2 v1 )(j l). In particular, if x and y belong to the orthogonal complement .{1, i}⊥ of the subspace .{1, i} in .O(K) with respect to .Re−, −, where .x, y = xy ¯ for .x, y ∈ O(K). Then, x = u = u1 j + u2 l + u3 (j l), y = v = v1 j + v2 l + v3 (j l)

.

and xy = uv = −v, u + (u2 v3 − u3 v2 )j + (u3 v1 − u1 v3 )l + (u1 v2 − u2 v1 )(j l).

.

Therefore, if we identify .C(K)3 with .{1, i}⊥ by .f : C(K)3 → {1, i}⊥ , given by .f (u1 , u2 , u3 ) = u1 j + u2 l + u3 (j l), then, we define the cross product u × v = (u2 v3 − u3 v2 )j + (u3 v1 − u1 v3 )l + (u1 v2 − u2 v1 )(j l),

.

and so f ((u1 , u2 , u3 ) × (v1 , v2 , v3 )) = f (u1 , u2 , u3 ) × f (v1 , v2 , v3 ).

.

2.1 The Exceptional Group G2 (K)

57

For any .x = u0 + u1 j + u2 l + u3 (j l) ∈ O(K), define x˜ = u¯ 0 + u¯ 1 j + u¯ 2 l + u¯ 3 (j l)

.

and observe that .x˜ does not coincide, in general, with x¯ = u¯ 0 − u1 j − u2 l − u3 (j l),

.

and u × v = uv + u, v.

.

In particular, we have x y = x˜ y, ˜

.

for .x, y ∈ O(K) and u × v = u˜ × v˜

.

for .u, v ∈ {1, i}⊥ . This shows that the map .

∼: O(K) → O(K)

belongs to the group .Aut(O(K)) of K-linear automorphisms of .O(K). Its restriction .

∼ |{1,i}⊥ : {1, i}⊥ → {1, i}⊥

belongs to the group .Aut({1, i}⊥ , ×) and, in view of Proposition 2.3, we have Aut({1, i}⊥ , ×) = SO3 (C(K)).

.

Notice that the conjugation map .− : O(K) → O(K) does not belong to the group .Aut(O(K)) because xy = y¯ x¯

.

for any .x, y ∈ O(K).

2 Exceptional Groups .G2 (K) and .F4 (K)

58

Cross Product in K 7 Making use of B. Eckmann [8], similarly to the case .K 3 and .H(K), we identify .K 7 with .O(K)Re=0 and define the map .× : K 7 × K 7 → K 7 by x × y = xy + (x, y)

.

¯ = −Re(xy) for .x, y ∈ K 7 , where we identify .x = x1 i + with .(x, y) = Re(xy) x2 j +x3 k+x4 l +x5 il +x6 j l +x7 kl, y = y1 i +y2 j +y3 k+y4 l +y5 il +y6 j l +y7 kl ∈ O(K)Re=0 . Thus, x × y = (x2 y3 − x3 y2 + x4 y5 − x5 y4 + x7 y6 − x6 y7 , x3 y1 − x1 y3 + x4 y6 − x6 y4

.

+ x5 y7 − x7 y5 , x1 y2 − x2 y1 + x4 y7 − x7 y4 + x6 y5 − x5 y6 , x5 y1 − x1 y5 + x6 y2 − x2 y6 + x7 y3 − x3 y7 ,

.

x1 y4 − x4 y1 + x7 y2 − x2 y7 + x3 y6 − x6 y3 , x2 y4 − x4 y2 + x5 y3 − x3 y5 + x1 y7 − x7 y1 ,

.

x2 y5 − x5 y2 + x3 y4 − x4 y3 + x6 y1 − x1 y6 ).

.

Observe that this identification leads to a (non-associative) ring isomorphism (K 7 , ×) ∼ = (O(K)Re=0 , ×)

.

and a K-linear isometry, once we endow .K 7 with the inner product .−, − and .O(K)Re=0 with the inner product .(−, −). The map .× : K 7 × K 7 → K 7 is clearly K-bilinear and skew-symmetric, and observe that inclusion .ι : K 3 → K 7 , given by .ι(x1 , x2 , x3 ) = (x1 , x2 , x3 , 0, 0, 0, 0), satisfies .ι(x × y) = ι(x) × ι(y). Furthermore, observe that x, x × y = y, x × y = 0

.

for any .x, y ∈ K 7 ≡ O(K))Re=0 and, if x, x = y, y = z, z = 1, x, y = x, z = y, z = x × y, z = 0,

.

then .

det(x|y|x × y|z|x × z|y × z|(x × y) × z) = 1

(2.5)

and so the matrix .(x|y|x × y|z|x × z|y × z|((x × y) × z) ∈ M7 (K) belongs to SO7 (K).

.

2.1 The Exceptional Group G2 (K)

59

The Group G2 (K) Define .G2 (K) = Aut(O(K)), i.e., .ϕ ∈ G2 (K) if and only if .ϕ is a K-linear isomorphism of .O(K) such that .ϕ(xy) = ϕ(x)ϕ(y) for any .x, y ∈ O(K). Identify now .O8 (K) with the orthogonal group of .O(K) with respect to the inner product .(−, −). Next, identify the group .O7 (K) with the orthogonal subgroup of .O(K)Re=0 = {1}⊥ . Then, it is a subgroup of .O8 (K) by extending a K-orthogonal isomorphism ϕ : O(K)Re=0 → O(K)Re=0

.

to ϕ  : O(K) → O(K)

.

given by ϕ  (λ + x) = λ + ϕ(x)

.

for .λ ∈ K and .x ∈ O(K)Re=0 . Furthermore, observe that ϕ(x × y) = ϕ(x) × ϕ(y)

.

for .ϕ ∈ G2 (K) and .x, y ∈ O(K)Re=0 . Proposition 2.6 There is a canonical inclusion .G2 (K) ⊆ SO7 (K). Proof If .ϕ ∈ G2 (K) then .ϕ(1) = 1. Next, take .x = λ + x  with .Re(x  ) = 0 and .λ ∈ K, and suppose .ϕ(x  ) = α + y with .α ∈ K, .Re(y) = 0. Then, ϕ(x  )2 = ϕ(x  ) = −ϕ(x¯  x  ) = −x¯  x  . 2

.

But, ϕ(x  ) = (α + y)2 = α 2 − yy ¯ + 2αy.

.

2

In particular, .αy = 0 and so, either .α = 0 and .ϕ(x  ) = −ϕ(x  ), or .y = 0 and = −x¯  x  = x  2 . This implies .(x  + α)(x  − α) = 0 → x  ∈ K and so .x  = 0. Thus, .ϕ(x  ) = 0 yields .ϕ(x  ) = −ϕ(x  ). Therefore, we always have

2 .α

ϕ(x  ) = −ϕ(x  ).

.

2 Exceptional Groups .G2 (K) and .F4 (K)

60

In particular, ϕ(x) ¯ = ϕ(λ − x  ) = λ − ϕ(x  ) = λ + ϕ(x  ) = ϕ(x).

.

Then, ϕ(x)ϕ(x) = (λ + ϕ(x  ))(λ + ϕ(x  )) = (λ − ϕ(x  ))(λ + ϕ(x  )) =

.

λ2 − ϕ(x  ) = λ2 + ϕ(x¯  x  ) = λ2 + x¯  x  = xx. ¯ 2

.

This implies that .ϕ ∈ O7 (K). It is clear that the map .f : G2 (K) → V7,3 (K) given by f (ϕ) = (ϕ(i)|ϕ(j )|ϕ(l))

.

for .ϕ ∈ G2 (K) is a K-polynomial injective map, and its image is giving by f (G2 (K)) = {(x|y|z) ∈ V7,3 (K); x × y, z = 0}.

.

(2.7)

In fact, ϕ(i) × ϕ(j ), ϕ(l) = ϕ(i × j ), ϕ(l) = ϕ(k), ϕ(l) = k, l = 0.

.

On the other hand, given .(x|y|z) ∈ V7,3 (K) such that .x × y, z = 0, define ϕ : O(K) → O(K) by .ϕ(i) = x, .ϕ(j ) = y, .ϕ(l) = z. Then, it is clear that .ϕ ∈ G2 (K), where .ϕ is given by .

ϕ(λ1 + λ2 i + λ3 j + λ4 k + λ5 l + λ6 il + λ7 j l + λ8 kl) =

.

λ1 + λ2 x + λ3 y + λ4 x × y + λ5 z + λ6 x × z + λ7 y × z + λ8 (x × y) × z.

.

In particular, f identifies .G2 (K) with a K-affine algebraic subvariety of .V7,3 (K) and .dim G2 (K) = dim V7,3 (K) − 1 = 15 − 1 = 14. Finally, observe that the inclusion .G2 (K) ⊆ O7 (K) is given by the composition .ιf, where .ι : V7,3 (K) → SO7 (K) is given by ι(x|y|z) = (x|y|x × y|z|x × z|y × z|(x × y) × z) =

.

(x|y|xy|z|xz|yz|(xy)z + xy, z).

.

Then, by (2.5) the proof is complete.  It the sequel, we write .ϕx,y,z instead of .ι(ϕ(i)|ϕ(j )|ϕ(l)) for .x = ϕ(i), y = ϕ(j ), z = ϕ(l).

2.1 The Exceptional Group G2 (K)

61

Furthermore, observe that since multiplication map for .G2 (K) is polynomial, G2 (K) is a closed algebraic subgroup of .SO7 (K), i.e., a compact Lie subgroup for .K = R. Then, the following conclusion is clear. .

Corollary 2.8 There is an isomorphism groups .G2 (K) ≈ Aut(K 7 , ×). Remark 2.9 Be careful of not confusing Aut(K 7 , ×) ≈ G2 (K) = Aut(O(K))

.

with Aut(K 7 , ×6 ) = SO7 (K).

.

Automorphisms of A For .A = K, C(K), H(K) or .O(K), we also define the commutative multiplication ◦ : A × A −→ A

.

given by .a ◦ b = 12 (ab + ba) for .a, b ∈ A. Of course, it coincides with the standard multiplication for .A = K or C(K). Instead, for .A = H(K) or O(K), we have (λ + x) ◦ (μ + y) = λμ − x, y + μx + λy,

.

for .λ, μ in .K, .x, y in .H(K) (respectively .O(K)) with .Re(x) = Re(y) = 0. Further, we define .Aut(A) (respectively .Aut(A, ◦)) as the group of K-linear automorphisms of the algebra .A with it usual multiplication (respectively .(A, ◦), i.e., .ϕ ∈ Aut(A) (respectively .ϕ ∈ Aut(A, ◦)) if and only if .ϕ is a K-linear isomorphism of .A and .ϕ(AB) = ϕ(A)ϕ(B) (respectively .ϕ(AB) = ϕ(A) ◦ ϕ(B)) for any .A, B ∈ A. Proposition 2.10 (1) .Aut(K) = Aut(K, ◦) is the trivial group;  1 0 (2) .Aut(C(K)) = Aut(C(K)), ◦) = I2 , = τ (O1 (K)); 0 −1 (3) .Aut(H(K)) = τ (SO3 (K)), Aut(H(K), ◦) = τ (O3 (K)); (4) .Aut(O(K)) = G2 (K) ⊆ SO7 (K), Aut(O(K), ◦) = τ (O7 (K)). Proof (1) and (2) are clear.

2 Exceptional Groups .G2 (K) and .F4 (K)

62

(3): Consider the K-basis .1, i, j, k of .H(K) = K 4 , and so, we regard both .Aut(H(K)) and .Aut(H(K), ◦) as subgroups of .GL4 (K). Let ⎛ a00 ⎜a10 .⎜ ⎝a20 a30

a01 a11 a21 a31

a02 a12 a22 a32

⎞ a03 a13 ⎟ ⎟ a23 ⎠ a33

be the matrix representing .ϕ in .Aut(H(K)) (respectively .Aut(H(K), ◦)) in the above basis. Since .ϕ(1) = 1, we have .a00 = 1 and .a10 = a20 = a30 = 0. 2 + a 2 + a 2 = a 2 + 1 and .a a Also, .−1 = ϕ(i)2 yields .a11 01 11 = a01 a21 = 21 31 01 2 + a 2 + a 2 = 1. a01 a31 = 0 and, being K formally real, we have .a01 = 0 and .a11 21 31 2 + a2 + Similarly, .ϕ(j )2 = −1 and .ϕ(k)2 = −1 yields .a02 = a03 = 0 and .a12 22 2 = a 2 + a 2 + a 2 = 1. On the other hand, if .ϕ ∈ Aut(H(K)) we deduce from a32 13 23 33 the relations .ϕ(i)ϕ(j ) = ϕ(k), .ϕ(j )ϕ(k) = ϕ(i), .ϕ(k)ϕ(i) = ϕ(j ) that the matrix representing .ϕ is ⎛ 1 0 ⎜0 a11 .⎜ ⎝0 a21 0 a31

0 a12 a22 a32

⎞ 0 a13 ⎟ ⎟, a23 ⎠ a33

i.e it belongs to .τ (SO3 (K)). Finally, it is clear that any matrix as above represents an element of .Aut(H(K)). Thus, .Aut(H(K)) = τ (SO3 (K)). Suppose now that .ϕ ∈ Aut(H(K), ◦), then ϕ(i)ϕ(j ) = −ϕ(j )ϕ(i), ϕ(j )ϕ(k) = −ϕ(k)ϕ(j ), ϕ(k)ϕ(i) = −ϕ(i)ϕ(k).

.

This gives that the matrix representing .ϕ belongs to .τ (O3 (K)). It is clear that any matrix as above represents an element of .Aut(H(K), ◦). Thus, .Aut(H(K), ◦) = τ (O3 (K)). (4): We already know (Proposition 2.6) that G2 (K) = Aut(O(K)) ⊆ SO7 (K).

.

Similarly to (3), from the relations .ϕ(q)2 = −1, for .q ∈ {i, j, k, l, il, j l, kl} and ϕ(q)ϕ(q  ) = −ϕ(q  )ϕ(q) for .q = q  in .{i, j, k, l, il, j l, kl}, we get

.

Aut(O(K), ◦) = τ (O7 (K)).

.

2.1 The Exceptional Group G2 (K)

63

On the other hand, no other relations is needed for the case .Aut(O(K), ◦), and so it is clear that .Aut(O(K), ◦) = τ (O7 (K)) and this completes the proof.  Observe that both multiplication and the inverse map in .G2 (K) are given by polynomial maps in .K[x1 , . . . , x7 , y1 , . . . , y7 , z1 , . . . , z7 ]. Furthermore, the following result is an easy checking. Lemma 2.11 (1) There is a monomorphism of groups .f : S1 (K) × S1 (K) → G2 (K) given by ⎛ 1 ⎜0 ⎜ ⎜0 ⎜ ⎜ .f (a, b, c, d) = ⎜0 ⎜ ⎜0 ⎜ ⎝0 0

0 a b 0 0 0 0

0 −b a 0 0 0 0

0 0 0 c d 0 0

⎞ 0 0 0 ⎟ 0 0 0 ⎟ ⎟ 0 0 0 ⎟ ⎟ −d 0 0 ⎟, ⎟ ⎟ c 0 0 ⎟ 0 ac − bd −(ad + bc)⎠ 0 ad + bc ac − bd

where .a, b, c, d ∈ K, and .a 2 + b2 = c2 + d 2 = 1. (2) Write .T2 (K) = f (S1 (K) × S1 (K)) and let .N(T2 (K)) be the normalizer of .T2 (K) in .G2 (K). Then, the Weyl group .N(T2 (K))/T2 (K) has only two elements: the class of the identity and the class of the diagonal matrix in .M7 (K) having .−1 at places .(1, 1), (3, 3), (5, 5), (7, 7) and 1 at .(2, 2), (4, 4), (6, 6). Remark 2.12 (1) Notice that .T2 (K) in Lemma 2.11 is the maximal torus contained in .G2 (K). For a torus .T (K) ⊆ G2 (K) with .T2 (K) ⊆ T (K) we have .N(T (K)) ⊆ N (T2 (K)) and so .T (K)/T2 (K) ⊆ N(T2 (K))/T2 (K). This is contrary to the finiteness of the Weyl group .N(T2 (K))/T2 (K). (2) It is important to insist that we are considering K-linear maps, if not, wild situations as the following may happen: for a formally real closed field K and transcendence bases .B, B  of .C(K) over the field of rationals .Q and any bijection .ϕ0 : B → B  there is an automorphism (without hypothesis of Klinearity) .ϕ : C(K) → C(K) such that the restriction .ϕ|B = ϕ0 . In particular, if B is nonempty, then one can take .B0 to be the set of all negatives of elements in B and define .ϕ0 (b) = −b. Similarly, if we split B into two nonempty disjoint subsets .B0 ∪ B1 , take .B1 to be the set of all negatives of elements in .B1 , write    .B = B1 ∪ B , and take .ϕ0 : B → B to be the identity on .B0 and multiplication 1 by .(−1) on .B1 , then we obtain an automorphism that is different from the preceding one. In particular, if the cardinality of B is .β, this yields a family of .2β (wild) automorphisms of .C(K). In particular, for .K = R the cardinality ℵ would be .22 0 .

2 Exceptional Groups .G2 (K) and .F4 (K)

64

Action of G2 (K) on S6 (K) Notice that .S6 (K) = {x ∈ O(K)Re=0 ; xx ¯ = 1} and so we have the natural action of 6 6 .SO7 (K) on .S (K) and then the restriction action of .G2 (K) on .S (K). 6 It is clear that the action of .G2 (K) on .S (K) is transitive. In fact, any .ϕ ∈ G2 (K) is completely determined by giving .ϕ(i) = x ∈ S6 (K) and then .ϕ(j ) = y, .ϕ(l) = z such that .y, z ∈ S6 (K) with .x, y = x, z = y, z = 0 and .xy, z = 0. This shows that there exists .ϕ ∈ G2 (K) such that .ϕ(i) = x, for any .x ∈ S6 (K). If now we have .x, x  ∈ S6 (K) then there exist .ϕ, ϕ  ∈ G2 (K) such that .ϕ(i) = x,    −1 )(x) = x  . .ϕ (i) = x and so .(ϕ ϕ The isotropy subgroup at i of this action of .G2 (K) on .S6 (K), is the subgroup .G2 (K)i ⊆ G2 (K), where .ϕ ∈ G2 (K)i if and only if .ϕ(i) = i, ϕ(j ) = y, ϕ(l) = z with .y, z ∈ O(K)Re=0, Re(ix)=0 for x∈O(K) and .Re(y 2 ) = Re(z2 ) = −1, Re(yz) = Re(iyz) = 0. Write .y = v1 j + v2 l + v3 (j l), .z = w1 j + w2 l + w3 (j l), with .v1 , v2 , v3 , w1 , w2 , w3 in .C(K). Then, .ϕi,y,z ∈ G2 (K)i if and only if ⎛

⎞ v1 w1 . ⎝v2 w2 ⎠ ∈ V3,2 (C(K)). v3 w3 Then, the polynomial bijection f : V3,2 (C(K)) → SU3 (C(K)),

.

given by f (v|w) = (v|w|v × w),

.

leads to a canonical group isomorphism G2 (K)i ≈ SU3 (C(K)).

.

This gives .SU3 (C(K)) as a subgroup of .G2 (K). Then, the sphere .S6 (K) is homeomorphic (diffeomorphic for .K = R) to the orbit space .G2 (K)/SU3 (C(K)) and we have the principal bundle SU3 (C(K)) → G2 (K) → S6 (K).

.

This shows, in particular, that the group .G2 (R) is connected.

2.1 The Exceptional Group G2 (K)

65

Jordan Multiplication We define the commutative multiplication ◦ : M(A) × M(A) −→ M(A)

.

given A1 ◦ A2 =

.

1 (A1 A2 + A2 A1 ) 2

for .A1 , A2 ∈ M(A). It is called the Jordan multiplication in .M(A). This multiplication restricts to: .Herm(A), .Herm(A), .M(A), .Mn (A) and .Hermn (A) with .n ≥ 1 and .A = K, C(K), H(K), O(K). Recall that a Jordan algebra is a non-associative K-algebra .A whose multiplication satisfies the following axioms: • .xy = yx (commutative law), • .(xy)x 2 = x(yx 2 ) (Jordan identity) for .x, y ∈ A. The axioms imply that a Jordan algebra is power-associative, meaning that .x n = x · · · x is independent of how we parenthesize this expression. They also imply (see e.g., [29, pp. 35–36]) that a Jordan algebra is power-associative and satisfies the following generalization of the Jordan identity (x m y)x n = x m (yx n )

.

(2.13)

for all .x, y ∈ A and integers .m, n > 0. For further studies of Jordan algebras, the book by A. Fernández López [9] is recommended. One may check that .(M(A), ◦) is a Jordan algebra, for .A = K, C(K) or H(K). In particular, all subalgebras .(M(A), ◦), .(Mn (A), ◦), .(Herm(A), ◦), .(Herm(A), ◦) and .(Hermn (A), ◦) are also Jordan algebras. We have to exclude .M1 (K) = K and .M1 (C(K)) = C(K) since they are associative. On the other hand, .(M1 (O(K)), ◦) = (O(K), ◦) is a Jordan algebra, but .(M2 (O(K)), ◦) is not. In particular, none of the algebras .(M(O(K)), ◦), .(M(A), ◦) or .(Mn (A), ◦) for .n ≥ 2, are Jordan. The algebra .(Herm4 (O(K)), ◦) is not Jordan. Therefore, for .n ≥ 4, .(Hermn (O(K)), ◦) is not Jordan and so .(Herm(O(K)), ◦) and .(Herm(O(K)), ◦) are not Jordan algebras either. Instead, .(Herm3 (O(K)), ◦) is a Jordan algebra and, in particular .(Herm2 (O(K)), ◦) is a Jordan algebra. Of course, .Herm1 (O(K)) = K is associative and so it is not Jordan.

2 Exceptional Groups .G2 (K) and .F4 (K)

66

The Jordan algebra .(Herm3 (O(K)), ◦) is also called the exceptional Jordan algebra or the Albert algebra.

Automorphisms of M(A) We wish to study the following three groups of automorphisms: (1) .Aut(Mn (A)), consisting of the K-linear isomorphisms .ϕ : Mn (A) → Mn (A) such that .ϕ(AB) = ϕ(A)ϕ(B) for any .A, B ∈ Mn (A); (2) .Aut(Mn (A), ◦) consisting of the K-linear isomorphisms .ϕ : Mn (A) → Mn (A) such that .ϕ(A ◦ B) = ϕ(A) ◦ ϕ(B) for any .A, B ∈ Mn (A); (3) .Aut(Hermn (A), ◦) consisting of K-linear automorphisms .ϕ : Hermn (A) → Hermn (A) such that .ϕ(A ◦ B) = ϕ(A) ◦ ϕ(B) for any .A, B ∈ Hermn (A). then, we have the chains of subgroups Aut(Mn (A)) ⊆ Aut(Mn (A), ◦) ⊆ GLK (Mn (A)), Aut(Hermn (A), ◦)

.

⊆ GLK (Hermn (A)). In case .K = R, it is clear that all groups above are closed and, in particular, they are Lie groups. Remark 2.14 It is clear that any additive map .ϕ : M(A) → M(A) satisfies .ϕ(A ◦ B) = ϕ(A) ◦ ϕ(B) if and only if .ϕ(A2 ) = (ϕ(A))2 for any .A ∈ M(A). In fact, if .ϕ(A ◦ B) = ϕ(A) ◦ ϕ(B), then ϕ(A2 ) = ϕ(A ◦ A) = ϕ(A) ◦ ϕ(A) = (ϕ(A))2 .

.

Now, if .ϕ(A2 ) = (ϕ(A))2 , then ϕ(A + B)2 = ϕ(A2 + B 2 + AB + BA) = (ϕ(A))2 + (ϕ(B))2 + ϕ(AB) + ϕ(BA)

.

and (ϕ(A + B))2 = (ϕ(A))2 + (ϕ(B))2 + ϕ(A)ϕ(B) + ϕ(B)ϕ(A).

.

Consequently, we derive that .ϕ(A ◦ B) = ϕ(A) ◦ ϕ(B).

2.1 The Exceptional Group G2 (K)

67

Furthermore, .χA (X ◦ Y ) = χA (X) ◦ χA (Y ) for .A = C(K), H(K). Consider now the K-basis of .Mn (A) given by .{Eαβ q}1≤α,β≤n, q∈IA , where

IA =

.

⎧ ⎪ {1} for A = K, ⎪ ⎪ ⎪ ⎨{1, i} for A = C(K), ⎪ {1, i, j, k} for A = H(K), ⎪ ⎪ ⎪ ⎩ {1, i, j, k, l, il, j l, kl} for A = O(K).

Then, we have: Lemma 2.15 .(Eαβ q)(Eα  β  q  ) = δβα  Eαβ  qq  . Lemma 2.16 .{Eαβ q}1≤α,β≤n,q∈IA is an orthonormal basis with respect to the inner product .Re−.−. Proof Using the definition of the inner product and Lemma 2.15, we have  ReEαβ q, Eα  β  q   = Re tr(Eβα q)(E ¯ α β  q ) =

.

Re(tr(Eβα Eα  β  )qq ¯  ) = Re(tr(δαα  Eββ  )qq ¯  ) = δαα  δββ  δqq 

.

and the proof is complete.  The following two lemmas are derived from Lemmas 2.15 and 2.16: Lemma 2.17

    (1) .{Eαα q}1≤α≤n,q∈IA ; . √1 (Eαβ + Eβα )q 1≤α 0}. This is an open covering of .G3,1 (O) and we have homeomorphisms D = {(q, q  ) ∈ O2 ; |q|2 + |q  |2 < 1} → Ui

.

for .i = 1, 2, 3 given, respectively by: ⎛

⎞ 1 − |q|2 − |q  |2    ⎠ .f1 (q, q ) = ⎝ 1 − |q|2 − |q  |2 q¯ q  , q q¯  ⎛ ⎞ q      .f2 (q, q ) = ⎝ 1 − |q|2 − |q  |2 ⎠ q ¯ 1 − |q|2 − |q  |2 q  q¯  and ⎞ q      ⎠ q¯ q  1 − |q|2 − |q  |2 . .f3 (q, q ) = ⎝ q ¯  1 − |q|2 − |q  |2 ⎛

The inverses are given, respectively by: ⎛ ⎞ ⎛ ⎞  γ a¯ b   γ a¯ b    a  a b c  −1 ⎝ .f a β c¯ ⎠ = √ , √ , f2−1 ⎝ a β c¯ ⎠ = √ , √ 1 γ γ β β b¯ c α b¯ c α

144

3 Stiefel, Grassmann Manifolds

and −1 .f 3

⎛ ⎞  γ a¯ b    ⎝ a β c¯ ⎠ = √b , √c . α α b¯ c α

Proposition 3.33 The space .G3,1 (O) is a smooth submanifold of .Herm3 (O) of real dimension 16 and the maps .π : S16 p → G3,1 (O) for .p = 1, 2, 3 are smooth. Proof It is clear that the atlas above endows .G3,1 (O) with a differential structure and .π : S16 i → G3,1 (O) for .i = 1, 2, 3 are smooth surjective maps. The inclusion map .i : G3,1 (O) → Herm3 (O) is smooth. In fact, its restriction to .U1 is the composite .π σ, where .σ : U1 → V3,1 (O) is given by ⎛ ⎞ ⎛ ⎞  γ a¯ b  γ 1 ⎝ ⎝ ⎠ .σ (A) = σ =√ a⎠ a β c¯ γ ¯ ¯b c α b for .A ∈ U1 and .π : V3,1 (O) → Herm3 (O) is the Stiefel map. Analogously, .i : U2 → Herm3 (O) and .i : U3 → Herm3 (O) are smooth. The inclusion .i : G3,1 (O) → Herm3 (O) is an immersion because if1 : D → Herm3 (O), if2 : D → Herm3 (O), and if3 : D → Herm3 (O)

.

are immersions, and the proof follows. Notice that Proposition 3.33 implies: Corollary 3.34 The space .G3,1 (O) = {AA¯ t ; At = (abc) ∈ M1,3 (O), a a¯ + bb¯ + cc¯ = 1, a(bc) = (ab)c}. Therefore, the map .π : V3,1 (O) = S23 → Herm3 (O) ⎛ ⎞  a  restricts to a surjective map .π| : ⎝b ⎠ ∈ V3,1 (O; (ab)c = a(bc) → G3,1 (O). c Recall that the space .G3,1 (O) = G3,2 (O) is called the Cayley plane or octonionic projective plane and denoted by .OP 2 . Theorem 3.35 The group .F4 acts transitively on the Cayley plane .G3,1 (O). Proof Consider .A ∈ G3,1 (O). Then, certainly .ϕ(A) ∈ G3,1 (O) for any .ϕ ∈ F4 . Furthermore, by Theorem 2.48, there exists .ϕ ∈ F4 such that .ϕ(A) is diagonal. This completes the proof because it is clear there are only three diagonal matrices in .G3,1 (O): .E11 , E22 , E33 and we can go from one to any other by .ϕσ for a convenient permutation .σ. Corollary 3.36 If .X, Y ∈ G3,1 (O) then there exist .x, y ∈ R such that .x 2 + y 2 = 1, 2 2 .ReX, Y  = Re tr(XY ) = x and .2 − ReX − Y, X − Y  = 2y . In particular, the √ diameter of .G3,1 (O) is . 2 as a metric subspace of .Herm3 (O).

3.2 Grassmannians

145

Proof In fact, given .X, Y ∈ G3,1 (O). Use now Theorem 3.35 to choose .ϕ ∈ F4 such that .ϕ(X) = E11 and write ⎛ ⎞ γ a¯ b .ϕ(Y ) = ⎝ a β c ¯⎠ . b¯ c α Then, ¯ + γ 2 = x2 Re tr(XY ) = Re tr(E11 ϕ(Y )) = γ = aa ¯ + bb

.

for some .x ∈ K. In particular, .d(X, Y )2 = ReX − Y, X − Y √= 2(1 − y 2 ) ≤ 2. But, .d(E11 , E22 )2 = 2. Therefore, the diameter of .G3,1 (O) = 2 and the proof is complete. Furthermore, we have an analog of Proposition 3.28: Proposition 3.37 (1) The tangent space to .G2,1 (O) at .A ∈ G2,1 (O) is given by TA G2,1 (O) = {B ∈ Herm2 (O); AB + BA = B}

.

and the normal space to .G2,1 (O) at .A ∈ G2,1 (O) is given by NA G2,1 (O) = {B ∈ Herm2 (O); AB = BA} = A ⊕ I2 − A,

.

where .A and .I2 − A are the one dimensional subspaces generated by A and I2 − A, respectively; (2) The tangent space to .G3,1 (O) at .A ∈ G3,1 (O) is given by .

TA G3,1 (O) = {B ∈ Herm3 (O); AB + BA = B}.

.

(3) The normal space to .G3,1 (O) at .A ∈ G3,1 (O) is given by the orthogonal direct sum NA G3,1 (O) = A ⊕ I3 − A ⊕ NA G3,1 (O),

.

where .A and .I3 − A are the one dimensional subspaces generated by A and I3 − A, respectively and .NA G3,1 (O) is given by

.

NA G3,1 (O) = {B ∈ Herm3 (O); AB + BA = 0 and tr B = 0}.

.

146

3 Stiefel, Grassmann Manifolds

Proof (1): Suppose .A ∈ G2,1 (O). Then, as in Proposition 3.28 we have TA G2,1 (O) ⊆ {B ∈ Herm2 (O); AB + BA = B}.

.

It is easy to check that .{B ∈ Herm2 (O); AB +BA = B} has real dimension 8. Thus,

.

TA G2,1 (O) = {B ∈ Herm2 (O); AB + BA = B}.

.

Therefore, the normal space .NA G2,1 (O) = TA G2,1 (O)⊥ has dimension 2 with the orthonormal basis .A, I2 − A. It is also an easy checking that for any .A ∈ G2,1 (O) the real vector space .{B ∈ Herm2 (O); AB = BA} has dimension 2 and so it coincides with .NA G2,1 (O). (2): First, notice that .TA G3,1 (O) ⊆ {B ∈ Herm3 (A); AB +BA = B}. This implies .dimR TA G3,1 (O) = 16 ≤ dimR {B ∈ Herm3 (A); AB + BA = B}. ⎛ ⎞ ⎛ ⎞ γ a¯ b ν x¯ y Now let .A = ⎝ a β c¯ ⎠ ∈ G3,1 (O) with .α + β + γ = 1. Taking .B = ⎝x μ z¯ ⎠ b¯ c α y¯ z λ with .AB + BA = B, we get the system of six linear equations: ¯ (1 − 2γ )ν = 2Re(ax ¯ + by); .(1 − 2β)μ = 2Re(az ¯ + cz); ¯ ¯ + cz); .(1 − 2λ = 2Re(by ¯ ¯ .αx = (μ + ν)a + c ¯y¯ + z¯ b; .βy = (λ + ν)b + a ¯ z¯ + x¯ c; ¯ ¯ x¯ + y¯ a. .γ z = (λ + μ)c + b ¯

• • • • • •

.

From that we deduce that .dimR {B ∈ Herm3 (A); AB + BA = B} ≤ 16 and so dimR {B ∈ Herm3 (A); AB + BA = B} = 16. Therefore,

.

TA G3,1 (O) = {B ∈ Herm3 (O); AB + BA = B}.

.

(3) First observe that if .AB +BA = B, then .Re tr(AB) = Re(A(AB)+A(BA)) = 2Re tr(AB) and so .Re tr(AB) = 0. Therefore, .ReA, B = Re tr(AB) = 0. Next, if .B ∈ NA G3,1 (O), then .AB + BA = 0 yields .ReA, B = Re tr(AB) = 0. Furthermore, if .B ∈ TA G3,1 (O) and .B  ∈ NA G3,1 (O), then ReB, B   = Re tr(BB  ) = Re tr((AB + BA)B  ) = Re tr((B  A + AB  )B) = 0.

.

3.2 Grassmannians

147

Finally, .

9.

.

dimR NA G3,1 (O) = 9.

In fact, by Theorem 3.35, it is enough to show that .NE 11 G3,1 (O) has dimension But this is clear, because .B ∈ NE 11 G3,1 (O) if and only if ⎛

0 0 ⎝ .B = 0 −α 0 c

⎞ 0 c¯ ⎠ α

with .α ∈ R and .c ∈ O. Therefore, we have the orthogonal direct sum decomposition Herm3 (O) = TA G3,1 (O) ⊕ A ⊕ I3 − A ⊕ NA G3,1 (O)

.

and the proof is complete. Remark 3.38 Observe that if .A = E11 and .B ∈ Herm3 (O), the orthogonal direct sum decomposition above is given by ⎞ ⎞ ⎛ ⎛ 0 0 0 a¯ b γ a¯ b α + β ⎠ ⎝ ⎠ ⎝ ⎝ .B = (I3 − E11 ) + 0 β−α a β c¯ = a 0 0 + γ E11 + 2 2 ¯b 0 0 ¯b c α 0 c ⎛

⎞ 0 c¯ ⎠ .

α−β 2

Denote by .ISO(G3,1 (O)) the group of .R-linear isometries .ϕ of .Herm3 (O) such that .ϕ(G3,1 (O)) = G3,1 (O). In particular, .F4 ⊆ ISO(G3,1 (O)) and .(F4 )E11 ⊆ (ISO(G3,1 (O))E11 . Theorem 3.35 yields that .ISO(G3,1 (O)) acts transitively on .G3,1 (O). As for the group .F4 we have: Lemma 3.39 .ϕ(I3 ) = I3 for any .ϕ ∈ ISO(G3,1 (O)) Proof Since Reϕ(I3 ), A = ReI3 , ϕ −1 (A) = Re tr(ϕ −1 (A)) = 1

.

for any .A ∈ G3,1 (O(K)) and so Re tr(ϕ(I3 )A) = 1

.

for all .A ∈ G3,1 (O).

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3 Stiefel, Grassmann Manifolds

Write ⎛

⎞ γ a¯ b .ϕ(I3 ) = ⎝ a β c ¯⎠ . ¯b c α Then, .trϕ(I3 )E11 = γ implies .γ = 1, .trϕ(I3 )E22 = β implies .β = 1 and trϕ(I3 )E33 = α implies .α = 1. If

.

⎛ ⎞ 00 0 1 ⎠, .A = ⎝0 2 q¯ 0 q 12 with .q ∈ O and .qq ¯ =

1 4

then .A ∈ G3,1 (O) and we have

1 = trA = Re trϕ(I3 )A = 1 + Re(cq). ¯

.

Thus, .Re(cq) ¯ = 0 for all .q ∈ O, and so .c = 0. Similarly, we show that .a = 0 and .b = 0. Therefore, .ϕ(I3 ) = I3 and this completes the proof. A Canonical Decomposition for Matrices in .Herm3 (O) Consider ⎛

⎞ γ a¯ b .A = ⎝ a β c ¯⎠ , ¯b c α where .a, b, c ∈ O and .α, β, γ ∈ R. Then, ⎛

⎞ ¯ (β + γ )a¯ + bc (γ + α)b + a¯ c¯ γ 2 + aa ¯ + bb 2 .A = ⎝(β + γ )a + c ¯ + cc ¯ (α + β)c¯ + ab ⎠ , ¯b¯ β 2 + aa ¯ + cc ¯ ¯ (γ + α)b + ca (α + β)c + b¯ a¯ α 2 + bb and we have ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ γ  a¯  b     2 .A = ⎝ a ⎠ γ a ¯ b + ⎝β ⎠ a β c¯ + ⎝ c¯ ⎠ b¯ c α . b¯ c α

3.2 Grassmannians

149

Next, choose .λ1 , λ2 , λ3 ∈ K such that ¯ + γ 2 , λ22 = aa ¯ + cc λ21 = aa ¯ + bb ¯ + cc ¯ + β 2 , λ23 = bb ¯ + α2.

.

Write ⎧ ⎛ ⎞ ⎪ γ ⎪ ⎪ ⎪ ⎜ ⎟ ⎪ 1 ⎜ ⎟ ⎪ ⎪ ⎪ λ1 ⎝ a ⎠ , if λ1 = 0 ⎪ ⎪ ⎨ b¯ ⎛ ⎞ .v1 = ⎪ ⎪ ⎪⎜0⎟ ⎪ ⎪ ⎪ ⎜0⎟ , if λ1 = 0; ⎪ ⎪ ⎝ ⎠ ⎪ ⎪ ⎩ 0

⎧ ⎛ ⎞ ⎪ a¯ ⎪ ⎪ ⎪ ⎜ ⎟ ⎪ 1 ⎜ ⎟ ⎪ ⎪ ⎪ λ2 ⎝β ⎠ , if λ2 = 0 ⎪ ⎪ ⎨ c v2 = ⎛ ⎞ ⎪ ⎪ ⎪⎜0⎟ ⎪ ⎪ ⎪ ⎜0⎟ , if λ2 = 0; ⎪ ⎪ ⎝ ⎠ ⎪ ⎪ ⎩ 0

⎧ ⎛ ⎞ ⎪ b ⎪ ⎪ ⎪ ⎜ ⎟ ⎪ 1 ⎜ ⎟ ⎪ ⎪ ⎪ λ3 ⎝ c¯ ⎠ , if λ3 = 0 ⎪ ⎪ ⎨ α v3 = ⎛ ⎞ ⎪ 0 ⎪ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪⎜0⎟ , if λ3 = 0. ⎪ ⎪ ⎝ ⎠ ⎪ ⎪ ⎩ 0 Therefore, we have A2 = λ21 v1 v¯1t + λ22 v2 v¯2t + λ23 v3 v¯3t

.

with .vα v¯αt ∈ G3,1 (O) or .vα v¯αt = 0, .α = 1, 2, 3. Lemma 3.40 The space .Herm3 (O) is .R-linearly generated by elements of .G3,1 (O). Proof Any .A ∈ Herm3 (O) can be written as A=

.

1 ((A + I3 )2 − A2 − I3 ), 2

and we just have seen above that any square of a matrix in .Herm3 (O) is a linear combination of elements of .G3,1 (O). This finishes the proof. Proposition 3.41 .ISO(G3,1 (O)) = F4 . Proof We already know that .F4 ⊆ ISO(G3,1 (O)). Thus, we consider that .ϕ ∈ ISO(G3,1 (O)). Let .A, B ∈ Herm3 (O). To show that .ϕ(A ◦ B) = ϕ(A) ◦ ϕ(B), by using Lemma 3.40, we certainly may assume that .A ∈ G3,1 (O).

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3 Stiefel, Grassmann Manifolds

Write B = B1 + λA + μ(I3 − A) + B2

.

with .B1 ∈ TA G3,1 (O), .B2 ∈ NA G3,1 (O). Since .ϕ is an isometry of .Herm3 (O) restricting to .G3,1 (O), we have ϕ(B) = ϕ(B1 ) + λϕ(A) + μ(I3 − ϕ(A)) + ϕ(B2 )

.

 with .ϕ(B1 ) ∈ Tϕ(A) G3,1 (O), .ϕ(B2 ) ∈ Nϕ(A) G3,1 (O). Then,

A◦B =

.

1 B1 + λA 2

because .A ◦ B1 = 12 B1 and .A ◦ B2 = 0. Similarly, ϕ(A) ◦ ϕ(B) =

.

1 ϕ(B1 ) + λϕ(A) 2

and so, 1 ϕ(A) ◦ ϕ(B) = ϕ( B1 + λA) = ϕ(A ◦ B) 2

.

and the proof is complete. Observe that any isometry .ϕ of .Herm3 (O) restricting to .G3,1 (O) and fixing .E11 , in particular any .ϕ ∈ (F4 )E11 , is completely determined by .(ϕ1 , ϕ2 ), where .ϕ1 is the restriction of .ϕ to .TE11 G3,1 (O) and .ϕ2 is the restriction of .ϕ to .NE 11 G3,1 (O). Now, define .h : S1 × S15 → G3,1 (O) as follows: ⎛

⎞ x 2 xy a¯ xyb 2 ¯ y 2 ab⎠ .h(x, y, a, b) = ⎝xya y aa ¯ xy b¯ y 2 ab y 2 bb for .x, y ∈ S1 and .a, b ∈ S15 . We also write .hx,y : S15 → G3,1 (O), given by 1 .hx,y (a, b) = h(x, y, a, b), and .ha,b : S (K) → G3,1 (O) given by .ha,b (x, y) = h(x, y, a, b). Proposition 3.42 Any element of .ISO(G3,1 (O)) .(in particular, any element of .F4 ) is determined by its restriction to .G3,1 (O). Proof Suppose .ϕ is a .R-linear isometry of .Herm3 (O) restricting to the identity on G3,1 (O). We have to show that this implies that .ϕ is the identity.

.

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151

If .(x, y) ∈ S1 and .(a, b) ∈ S15 we have ⎛

⎛ ⎞ ⎛ ⎞ ⎞ 0 a¯ b x 2 xy a¯ xyb 0 0 0 2 ¯ y 2 ab⎠ = x 2 E + xy ⎝a 0 0⎠ + y 2 ⎝0 aa .h(x, y, a, b) = ⎝xya y aa ¯ ab⎠ , 11 2 2 ¯ ¯ ¯ ¯ b00 0 ab bb xy b y ab y bb and applying .ϕ we get ⎛

⎞ ⎛ ⎞ 0 a¯ b 0 0 0 2 2 .h(x, y, a, b) = ϕ(h(x, y, a, b)) = x E11 + xyϕ(⎝a 0 0⎠) + y ϕ(⎝0 aa ¯ ab⎠). ¯ b¯ 0 0 0 ab bb Therefore, ⎛

⎞ ⎛ ⎛ ⎞ 0 a¯ b 0 a¯ b 0 0 2 .xy(ϕ(⎝a 0 0⎠) − ⎝a 0 0⎠) + y (ϕ(⎝0 aa ¯ ¯b 0 0 ¯b 0 0 0 ab

⎞ ⎛ ⎞ 0 0 0 0 ab⎠) − ⎝0 aa ¯ ab⎠) = 0 ¯bb ¯ 0 ab bb

for all .(x, y) ∈ S1 . If we take .x = 0, y = 1 we get ⎛

⎞ ⎛ ⎞ 0 0 0 0 0 0 .ϕ(⎝0 aa ¯ ab⎠) = ⎝0 aa ¯ ab⎠ . ¯ ¯ 0 ab bb 0 ab bb Then, we have ⎛

⎞ ⎛ ⎞ 0 a¯ b 0 a¯ b .xy(ϕ(⎝a 0 0⎠) − ⎝a 0 0⎠) = 0 b¯ 0 0 b¯ 0 0 for all .(x, y) ∈ S1 . This implies, (take for instance .x = 35 , y = 45 ), ⎛

⎞ ⎛ ⎞ 0 a¯ b 0 a¯ b .ϕ(⎝a 0 0⎠) = ⎝a 0 0⎠ . b¯ 0 0 b¯ 0 0 Finally, the orthogonal direct sum decomposition Herm3 (O) = TE11 G3,1 (O) ⊕ NE11 G3,1 (O)

.

yields that .ϕ is the identity on .Herm3 (O) and the proof is complete.

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3 Stiefel, Grassmann Manifolds

Now, define .N  : S15 → NE 11 G3,1 (O) by h0,1 (a, b) =

.

1 (I3 − E11 ) + N  (a, b). 2

Properties 1. h(x, y, a, b) = h(x  , y  , a, b) if and only if (x  , y  ) = ±(x, y).

.

2.   a h(0, 1, a, b) = π( ¯ ), b

.

where .π : V2,1 (O) → G2,1 (O) is the Stiefel map. 3. By using the diffeomorphism .g : S8 → G2,1 (O) given by   1 1+λ p .g(λ, p) = , p¯ 1 − λ 2 g −1 ◦ h(0, 1, −, −) : S15 → S8 is the Hopf map sending .(a, b) ∈ S15 to .(2aa ¯ − 1, 2ab). 4. Define .Pa,b as the .R-plane through .E11 and director 2-dimensional .R-vector space with basis .

⎛

⎞ ⎛ ⎞ 0 a¯ b −1 0 0 . ⎝a 0 0⎠ , ⎝ 0 aa ¯ ab⎠ . ¯ b¯ 0 0 0 ab bb

.

Observe that this basis is orthonormal if we consider the inner product on .Herm3 (O). Then,

1 2 Re−, −

⎛ ⎞ ⎞ −1 0 0 0 a¯ b 2 .h(x, y, a, b) = E11 + xy ⎝a 0 0⎠ + y ⎝ 0 aa ¯ ab⎠ ∈ Pa,b ¯ 0 ab bb b¯ 0 0 ⎛

for all .(x, y) ∈ S1 and clearly, the image of .h(−, −, a, b) is the intersection .Pa,b ∩ G3,1 (O).

3.2 Grassmannians

153

Observe that d(h(x, y, a, b), h(x  , y  , a, b))2 =

.

1 tr(h(x, y, a, b) − h(x  , y  , a, b))2 2

= 1 − (xx  + yy  )2 . Therefore, d(h(x, y, a, b), E11 )2 = y 2

.

and d(h(x, y, a, b), h(0, 1, a, b))2 = 1 − y 2 = x 2 .

.

Thus, .h(0, 1, a, b) is the point in .h(−, −, a, b) at the maximum distance 1 to .E11 . Instead, d((x, y), (x  , y  ))2 = 2(1 − (xx  + yy  ))

.

and d((x, y), (1, 0))2 = 2(1 − x).

.

(5) Let .ϕ be an isometry of .Herm3 (O) such that .ϕ(E11 ) = E11 and .ϕ(G3,1 (O)) = G3,1 (O). Then, .ϕ restricts to .R-linear isometries .ϕ1 : TE11 G3,1 (O) → TE11 G3,1 (O) and .ϕ2 : NE 11 G3,1 (O) → NE 11 G3,1 (O). In fact, any such a .ϕ restricts to an isometry of the “cut locus” of .E11 , .G2,1 (O), i.e., the points of .G3,1 (O) at the maximum distance 1 of .E11 : matrices of the form ⎛

00 ⎝ . 0β 0 p¯

⎞ 0 p⎠ α

with .α + β = 1 and .pp ¯ = αβ. Observe that .G2,1 (O) is not contained in .NE 11 G3,1 (O), what we have is G2,1 (O) =

.

1 (I3 − E11 ) + S, 2

where .S is the sphere of radius . 12 in .NE 11 G3,1 (O), i.e., ⎛ ⎞   00 0 1 2 ⎝ ⎠ . .S = ¯ = 0 ρ x ; ρ ∈ R, x ∈ O, ρ + xx 4 0 x¯ −ρ

154

3 Stiefel, Grassmann Manifolds

Therefore, ⎛ ⎞ ⎛ ⎞ 0 0 00 0 0  p⎠ = ⎝0 12 0 ⎠ + ϕ(⎝0 β−α 2 0 0 12 α 0 p¯

⎛  00 .ϕ ⎝0 β 0 p¯

⎞ 0 p ⎠. α−β 2

Thus, .ϕ restricts to a .R-linear isometry .ϕ2 of .NE 11 G3,1 (O) and so to a .R-linear isometry .ϕ1 of .TE11 G3,1 (O). (6) We consider the inner product . 12 Re−, − in .Herm3 (O), i.e., for .X, Y ∈ Herm3 (O) we have .

1 1 ReX, Y  = Re tr(XY ) ∈ R. 2 2

In this way, the map .f : O2 → TE11 G3,1 (O) ⊆ Herm3 (O) given by ⎛

⎞ 0 a¯ b .f (a, b) = ⎝a 0 0⎠ , b¯ 0 0 is an .R-linear isometry, once we consider .O2 with its standard inner product: ¯  ). (a, b), (a  , b ) = Re(aa ¯  + bb

.

In particular, f identifies the unit sphere .S15 in .O2 with the unit sphere .S15 E11 in the tangent space .TE11 G3,1 (O). Consider now .ϕ ∈ ISO(G3,1 (O))E11 and denote by .ρ1 (ϕ) and .ρ2 (ϕ) the restriction isometries of .ϕ to .TE11 G3,1 (O) and .NE 11 G3,1 (O). Let .θ (ϕ) = (θ1 (ϕ), θ2 (ϕ)) the corresponding isometry of .O2 : ρ1 (ϕ) ◦ f = f ◦ θ (ϕ).

.

Thus, ⎛ ⎞ ⎛ ⎞  0 a¯ b  0 θ1 (ϕ)(a, b) θ2 (ϕ)(a, b) ⎠ .ρ1 (ϕ) ⎝a 0 0⎠ = ⎝θ1 (ϕ)(a, b) 0 0 ¯b 0 0 θ2 (ϕ)(a, b) 0 0 for .a, b ∈ O and we have

3.2 Grassmannians

155

Lemma 3.43 If .ϕ ∈ (F4 )E11 then ⎛  0 0 .ϕ ⎝0 aa ¯ 0 ab

⎞ ⎛ ⎞ 0  0 0 0 ab⎠ = ⎝0 θ1 (ϕ)(a, b)θ1 (ϕ)(a, b) θ1 (ϕ)(a, b)θ2 (ϕ)(a, b)⎠ . ¯ bb 0 θ1 (ϕ)(a, b)θ2 (ϕ)(a, b) θ2 (ϕ)(a, b)θ2 (ϕ)(a, b)

Proof In fact, ⎛ ⎞2 ⎞ 0 0 0 0 a¯ b ¯ . ⎝a 0 0⎠ = (aa ¯ + bb)E ¯ ab⎠ 11 + ⎝0 aa ¯ 0 ab bb b¯ 0 0 ⎛

and so ⎛ ⎛ ⎞2  0 a¯ b   0 0 ¯ = (aa ¯ + bb)E .ϕ ⎝a 0 0⎠ ¯ 11 + ϕ ⎝0 aa ¯b 0 0 0 ab

⎞ 0  ab⎠ . ¯ bb

But ⎛

⎞2 ⎛ ⎞2 0 a¯ b 0 θ1 (ϕ)(a, b) θ2 (ϕ)(a, b) ⎠ = .ϕ(⎝a 0 0⎠ ) = ⎝θ1 (ϕ)(a, b) 0 0 b¯ 0 0 θ2 (ϕ)(a, b) 0 0 (θ1 (ϕ)(a, b)θ1 (ϕ)(a, b) + θ2 (ϕ)(a, b)θ2 (ϕ)(a, b))E11 +

.

⎛

⎞ 0 0 0 . ⎝0 θ1 (ϕ)(a, b)θ1 (ϕ)(a, b) θ1 (ϕ)(a, b)θ2 (ϕ)(a, b)⎠ . 0 θ1 (ϕ)(a, b)θ2 (ϕ)(a, b) θ2 (ϕ)(a, b)θ2 (ϕ)(a, b) This ends the proof. Then, we have: Proposition 3.44 If .ϕ is an .R-linear isometry of .Herm3 (O) which fixes .E11 then the following conditions are equivalent: (1) .ϕ ∈ (F4 )E11 = ISO(G3,1 (O))E11 ; (2) .ϕ(h(x, y, a, b)) = h(x, y, θ1 (ϕ)(a, b), θ2 (ϕ)(a, b)), for all .(x, y) ∈ S1 , 15 and any isometry .θ = (θ , θ ) of .O2 such that .(a, b) ∈ S 1 2 ⎛

⎞ ⎛ ⎞ 0 a¯ b 0 θ1 (a, b) θ2 (a, b) .ϕ(⎝a 0 0⎠) = ⎝θ1 (a, b) 0 0 ⎠; b¯ 0 0 θ2 (a, b) 0 0

156

3 Stiefel, Grassmann Manifolds

15  −1 : (3) .ρ1 (ϕ) : S15 E11 → SE11 is a bundle isometry with respect to the bundle .N ◦ f S15 E11 → S.

Proof .(1) ⇒ (2) : Suppose .ϕ ∈ (F4 )E11 , ⎛ ⎞ 0 0 0 a¯ b 2 y ⎜ ¯ bb ¯ 2 .h(x, y, a, b) = xy ⎝a 0 0⎠ + x E11 + (I3 − E11 ) + y 2 ⎝0 aa− 2 2 b¯ 0 0 0 ab ⎛

⎞ 0 ⎟ ab ⎠ .

¯ aa bb− ¯ 2

Observe that ⎛ ⎞ ⎛ 0 0 0 0 0 1 ⎜ aa− ¯ ⎠ ⎝ . 0 aa ¯ ab = (I3 − E11 ) + ⎝0 ¯ 2 bb 2 ¯ 0 ab bb 0 ab

⎞ 0 ⎟ ab ⎠

¯ aa bb− ¯ 2

is the point in the image of .S1 by .h(−, −, a, b) at the maximum distance of .E11 , i.e., “the cut point of .E11 following the geodesic .h(−, −, a, b) . In view of Lemma 3.43, we have ⎛ ⎞ ⎛ ⎞ 0 0 0 0 0 0 1 ⎜ aa− ⎟ ¯ ¯ bb .ρ2 (ϕ)(⎝0 ¯ ab⎠) = ab ⎠) = − (I3 − E11 ) + ϕ(⎝0 aa 2 2 ¯ aa bb− ¯ ¯ 0 ab bb 0 ab 2 ⎛ 0 ⎜ . ⎝0 0

0

0

θ1 (ϕ)(a,b)θ1 (ϕ)(a,b)−θ2 (ϕ)(a,b)θ2 (ϕ)(a,b) 2

θ1 (ϕ)(a, b)θ2 (ϕ)(a, b)

θ1 (ϕ)(a, b)θ2 (ϕ)(a, b)

θ2 (ϕ)(a,b)θ2 (ϕ)(a,b)−θ1 (ϕ)(a,b)θ1 (ϕ)(a,b) 2

Thus, ⎛

⎞ 0 θ1 (ϕ)(a, b) θ2 (ϕ)(a, b) ⎠ .ϕ(h(x, y, a, b)) = xy ⎝θ1 (ϕ)(a, b) 0 0 θ2 (ϕ)(a, b) 0 0 + x 2 E11 + ⎛

0 0 ⎜ aa− ¯ 2 ¯ bb .y ρ2 (ϕ)(⎝0 2 0 ab

y2 (I3 − E11 )+ 2

⎞ 0 ⎟ ab ⎠) = h(x, y, θ1 (ϕ)(a, b), θ2 (ϕ)(a, b)).

¯ aa bb− ¯ 2

(2) ⇒ (3) : This is clear because if we take .(x, y) = (0, 1), we have

.

ρ2 (ϕ) ◦ N  ◦ f −1 = N  ◦ f −1 ◦ ρ1 (ϕ).

.

⎞ ⎟ ⎠.

3.2 Grassmannians

157

(3) ⇒ (1) : Consider now isometries .ϕ1 , ϕ2 of .TE11 G3,1 (O) and .NE 11 G3,1 (O), respectively and define an isometry .ϕ : Herm3 (O) → Herm3 (O) by .ϕ = ϕ1 on  .TE11 G3,1 (O), .ϕ = ϕ2 on .N E11 G3,1 (O), .ϕ(E11 ) = E11 and .ϕ(I3 −E11 ) = I3 −E11 . 15 15 Suppose that .ϕ1 : S15 E11 → SE11 is an isometry preserving the fibres of .SE11 → S and such that the induced map .ϕ2 : S → S is again an isometry. Let .ϕ1 be given by .

⎛

⎞ ⎛ ⎞ 0 a¯ b 0 θ1 (a, b) θ2 (a, b) .ϕ1 (⎝a 0 0⎠) = ⎝θ1 (a, b) 0 0 ⎠ b¯ 0 0 θ2 (a, b) 0 0 for some isometry .θ = (θ1 , θ2 ) : O2 → O2 , In particular, we have a well defined map .ϕ2 : S → S given by ⎛

0 0 .ϕ2 (⎝0 aa ¯ − 0 ab

1 2

⎞ ⎛ 0 0 0 ab ⎠) = ⎝0 θ1 (a, b)θ1 (a, b) − ¯ −1 bb 0 θ1 (a, b)θ2 (a, b) 2

1 2

⎞ 0 θ1 (a, b)θ2 (a, b) ⎠ θ2 (a, b)θ2 (a, b) − 12

and .ϕ2 N  f −1 = N  f −1 ϕ1 . This shows that the isometry .ϕ = ϕ1 on .TE11 G3,1 (O), .ϕ = ϕ2 on .NE 11 G3,1 (O) and .ϕ(E11 ) = E11 , .ϕ(I3 − E11 ) = I3 − E11 , satisfies ϕ(h(x, y, a, b)) = h(x, y, θ1 (a, b), θ2 (a, b))

.

for all .(x, y) ∈ S1 and all .(a, b) ∈ S15 . Therefore, .ϕ ∈ ISO(G3,1 (O))E11 and this completes the proof. Let .Iso(G3,1 (O)) be the group of isometries of .G3,1 (O). Then, we may state: Proposition 3.45 The restriction map ISO(G3,1 (O)) −→ Iso(G3,1 (O))

.

is a group isomorphism. Proof First, notice that by Proposition 3.42, the restriction map ISO(G3,1 (O)) −→ Iso(G3,1 (O))

.

is a monomorphism. Now, we show that the restriction map above is an epimorphism. To aim that, for a given .ϕ ∈ Iso(G3,1 (O)), we construct its extension on .Herm3 (O). Recall that Herm3 (O) = TE11 G3,1 (O) ⊕ NE11 G3,1 (O)

.

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3 Stiefel, Grassmann Manifolds

and, by Proposition 3.28(3), the normal space NE11 G3,1 (O) = E11  ⊕ I3 − E11  ⊕ NE 11 G3,1 (O).

.

We may assume that .ϕ ∈ Iso(G3,1 (O)) fixes the matrix .E11 , just composing with a convenient element of .F4 . Then, we have the induced linear isometry (dϕ)E11 : TE11 G3,1 (O) −→ TE11 G3,1 (O).

.

But, we do not have, in general, an induced linear isometry of the normal space NE11 G3,1 (O). Now, we extend the isometry .(dϕ)E11 on the subspace .E11  with the basis .E11 and the subspace .I3 − E11  with the basis .I3 − E11 by the identity map. To extend  .(dϕ)E11 on .N E11 G3,1 (O), we observe that .ϕ satisfies .ϕ(G2,1 (O)) = G2,1 (O) which is included in .G3,1 (O) by adding a first row and column of zeros, because .G2,1 (O) are points in .G3,1 (O) at the maximum possible distance of .E11 , i.e., 1 with respect to the induced distance from the inner product . 12 −, − on .Herm3 (O) or . π2 with respect to the Riemannian metric induced by the inner product above. In particular, we have that .ϕ restricts to an isometry on .G2,1 (O). Next, by Property (5), we have .G2,1 (O) = 12 (I3 − E11 ) + S, where .S is the sphere of radius . 12 in .NE 11 G3,1 (O). Therefore, .ϕ restricts to an isometry on the sphere .S. Thus, it is the restriction to .S of a linear isometry of .NE 11 G3,1 (O). This isometry is an extension of .(dϕ)E11 on .NE11 (O) we are looking for. Then, it remains to check that in view of Proposition 3.44, the linear isometry we got on .Herm3 (O) restricts to .ϕ on .G3,1 (O) and the proof is complete. .

A Polynomial Inclusion .S8 → (F4 )E11 The idea for this subsection comes from [26, Part II, 14]. Take now .S8 = {(r, u) ∈ R × O; r 2 + uu ¯ = 1} and define a map g1 : S8 → GLR (O2 )

.

by ¯ au g1 (r, u)(a, b) = (ra + ub, ¯ − rb)

.

for .(r, u) ∈ S8 and .(a, b) ∈ O2 . Observe that .g1 (r, u) is an involution, i.e., 2 8 .g1 (r, u) = id for any .(r, u) ∈ S . Then, we have: Lemma 3.46 The map .g1 is an injective isometry .g1 : S8 → SO(O2 ) Proof It is clear that .g1 (r, u) is .R-linear. Suppose now that (a, b) ∈ Ker(g1 (r, u)).

.

3.2 Grassmannians

159

Then,  .

ra + ub¯ = 0, au ¯ − rb = 0

and so  r 2 a + rub¯ = 0, . ¯ = 0. (uu)a ¯ − r(ub)

.

Thus, .a = 0 and then .b = 0. The map .g1 (r, u) is an .R-linear isometry: .

    ra + ub¯ ¯ = aa ¯ ¯ aa ¯ + bb) ¯ + bb. = (r 2 + uu)( r a¯ + bu¯ ua ¯ − r b¯ au ¯ − rb

Therefore, .det ◦g1 : S8 → {−1, 1} and so .det(g1 (r, u)) = 1 for any .(r, u) ∈ S8 . Next, we show that .g1 is injective: In fact, suppose .g1 (r, u) = g1 (r  , u ) for .(r, u), (r  , u ) ∈ S8 . Then,  .

(r − r  )a + (u − u )b = 0 u − u a − (r − r  )b = 0

for all .a, b ∈ O. Therefore, ((r − r  )2 + (u − u )(u − u ))a = 0, ((r − r  )2 + (u − u )(u − u ))b = 0.

.

Thus, (r − r  )2 + (u − u )(u − u ) = 0,

.

and so .r = r  , u = u . This completes the proof. Define then an .R-linear map : g˜ 1 (r, u) : TE11 G3,1 (O) → TE11 G3,1 (O)

.

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3 Stiefel, Grassmann Manifolds

by ⎛ ⎞ ⎛ ⎞ ¯ au 0 r a¯ + bu ¯ − rb  0 a¯ b  .g ˜ 1 (r, u) ⎝a 0 0⎠ = ⎝ra + ub¯ 0 0 ⎠. b¯ 0 0 ua ¯ − r b¯ 0 0 Lemma 3.46 says that .g˜ 1 (r, u) is an involutive isometry for any .(r, u) ∈ S8 and we have an injective map g˜ 1 : S8 → SO(TE11 G3,1 (O)).

.

Then, we have the map N  f −1 g˜ 1 (r, u) : S15 E11 → S

.

given by ⎛

⎞ ⎛ ⎞ 0 a¯ b 000 1  −1 .(N f g˜ 1 (r, u))(⎝a 0 0⎠) = − ⎝0 1 0⎠ + 2 b¯ 0 0 001 ⎛

⎞ 0 0 0 ¯ uu ¯ . ⎝0 r 2 aa ¯ + 2rRe(uab) ¯ + bb ¯ −r 2 ab + r(aa ¯ − bb)u + uabu⎠ . 2 2 ¯ u¯ + u(ab) ¯ − 2rRe(uab) ¯ − bb) ¯ u¯ r bb ¯ + aa ¯ uu ¯ 0 −r ab + r(aa 15 Proposition 3.47 The map .g˜ 1 (r, u) : S15 E11 → SE11 is an isometry preserving the 15  −1 : SE11 → S and the induced map .g˜ 2 (r, u) : S → S is fibres of the bundle .N f again an isometry.

Proof In fact, .g˜ 2 (r, u) is given by ⎛ ⎞ ⎛ 000 00 1⎝ .g ˜ 2 (r, u) − 0 1 0⎠ + ⎝0 β 2 001 0 x¯ 

⎞ 0  x⎠ = α

⎞ ⎛ ⎞ ⎛ 000 0 0 0 1⎝ .− ¯ + α uu ¯ −r 2 x + r(β − α)u + uxu ¯ ⎠, 0 1 0⎠ + ⎝0 r 2 β + 2rRe(ux) 2 2 2 ¯ u¯ r α − 2rRe(ux) ¯ + β uu ¯ 0 −r x¯ + r(β − α)u¯ + ux 001 where .α+β = 1 and .xx ¯ = αβ. It is easy to check that .g˜ 2 (r, u) ∈ SO(NE 11 G3,1 (O)) and N  f −1 g˜ 1 (r, u) = g˜ 2 (r, u)N  f −1 .

.

3.2 Grassmannians

161

This completes the proof. Proposition 3.48 (1) .g˜ 2 (r, u) is a symmetry for all .(r, u) ∈ S8 . In particular, .det g˜ 2 (r, u) = 1. (2) If .g˜ 2 (r, u) = g˜ 2 (r  , u ) for .(r, u), (r  , u ) ∈ S8 , then .(r  , u ) = ±(r, u). (3) .g˜ 2 (S8 ) generates .SO(NE 11 G3,1 (O)) Proof (1): We have ⎛ ⎞ 00 0 .g ˜ 2 (r, u)(⎝0 ρ x ⎠) 0 x¯ −ρ ⎛ ⎞ 0 0 0 ⎠. = ⎝0 ρ(2r 2 − 1) + 2rRe(ux) ¯ −r 2 x + 2rρu + uxu ¯ 2 2 ¯ u¯ −ρ(2r − 1) − 2rRe(ux) ¯ 0 −r x¯ + 2rρ u¯ + ux It is easy to check that the eigenvector space of the eigenvalue 1 has ⎛ ⎞ 00 0 . ⎝0 r u ⎠ 0 u¯ −r as the basis. Then, it is again easily checked that the orthogonal complement of ⎛

⎞ 00 0 . ⎝0 r u ⎠ , 0 u¯ −r i.e., ⎛ ⎞ ⎛ ⎞⊥ 00 0  00 0  . ⎝0 r u ⎠ = ⎝0 ρ x ⎠ ; ρr + Re(ux) ¯ =0 0 x¯ −ρ 0 u¯ −r is the eigenvector space with the eigenvalue .−1. In particular, since the dimension of .NE 11 G3,1 (O) is .9, an odd number, we conclude that .det g˜ 2 (r, u) = 1 for all .(r, u) ∈ S8 . (2): It is just a simple exercise.

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3 Stiefel, Grassmann Manifolds

(3): This is a corollary of the classical Cartan-Dieudonné Theorem (see, e.g., [3, Theorem 3.20]) saying that any element of .OK (n) is a product of at most n symmetries. Then, the proof follows. Proposition 3.49 (1) The map .ρ1 is a monomorphism. (2) The kernel of .ρ2 has two elements, I and .κ, where I is the identity and .κ is given by ⎞ ⎞ ⎛ ⎛ γ −a¯ −b γ a¯ b .κ(⎝ a β c ¯ ⎠) = ⎝−a β c¯ ⎠ . −b¯ c α b¯ c α (3) The image of .ρ2 coincides with .SO(NE 11 G3,1 (O)). (4) .(Iso(G3,1 (O))E11 = (F4 )E11 = Spin(9). Proof (1): It is clear, because if .ρ1 (ϕ) is the identity, then by using Proposition 3.44 we see that .ρ2 (ϕ) is the identity, and so .ϕ is also the identity. To show (2) suppose .ρ2 (ϕ) = id. Then, for all .a, b ∈ O we have ⎧ ⎪ ⎪ ⎨θ1 (ϕ)(a, b)θ2 (ϕ)(a, b) = ab, ¯ . θ1 (ϕ)(a, b)θ1 (ϕ)(a, b) − θ2 (ϕ)(a, b)θ2 (ϕ)(a, b) = aa ¯ − bb, ⎪ ⎪ ⎩θ (ϕ)(a, b)θ (ϕ)(a, b) + θ (ϕ)(a, b)θ (ϕ)(a, b) = aa ¯ ¯ + bb 1 1 2 2 In particular, ¯ for all a, b ∈ O. θ1 (ϕ)(a, b)θ1 (ϕ)(a, b) = aa, ¯ θ2 (ϕ)(a, b)θ2 (ϕ)(a, b) = bb

.

Therefore, .θ1 (ϕ)(0, b)θ1 (ϕ)(0, b) = 0 for all .b ∈ O, and so .θ1 (ϕ)(0, b) = 0 for all .b ∈ O and we have .θ1 (ϕ)(a, b) = θ1 (ϕ)(a, 0) + θ1 (ϕ)(0, b) = θ1 (ϕ)(a, 0). Similarly, we have .θ2 (ϕ)(a, b) = θ2 (ϕ)(0, b) for all .b ∈ O. Write .θ1 (a) = θ1 (ϕ)(a, b) and .θ2 (b) = θ2 (ϕ)(a, b), so we have ¯ θ1 (a)θ2 (b) = ab, θ1 (a)θ1 (a) = aa, ¯ θ2 (b)θ2 (b) = bb

.

for all .a, b ∈ O. Write .θ1 (1) = q, and so .qq ¯ = 1. But .θ1 (1)θ2 (1) = 1 gives .θ2 (1) = q, ¯ and so we have θ1 (a) = aq, θ2 (b) = qb ¯

.

3.3 Flag Varieties

163

with ab = (aq)(qb) ¯

.

for all .a, b ∈ O. Then, Lemma 2.41 gives .q = ±1 and this proves (2). (3): We know by Proposition 3.48 that .SO(NE 11 G3,1 (O)) ⊆ Im(ρ2 ), and we show then that SO(NE 11 G3,1 (O)) = Im(ρ2 ).

.

In fact, suppose .det(ρ2 (ϕ)) = −1 for some .ϕ ∈ (Iso(G3,1 (O))E11 , then −ρ2 (ϕ −1 ) = −ρ2 (ϕ)−1 ∈ SO(NE 11 G3,1 (O)) ⊆ Im(ρ2 ). Thus, .−ρ2 (ϕ −1 ) = ρ2 (ϕ  ) for some .ϕ  ∈ (I so(G3,1 (O))E11 . Therefore, .ρ2 (ϕϕ  ) = −id. But, this is a contradiction because then .(ϕϕ  )(E22 + E33 ) = −E22 − E33 . (4): We have an isomorphism .ρ1 : F4 → Im(ρ1 ) = ρ1 (F4 ) and an epimorphism  .ρ2 : F4 → SO(N E11 G3,1 (O)) with kernel .{I, κ}. Observe that .ρ1 (κ) = g˜ 1 (1, 0)g˜ 1 (−1, 0) = g˜ 1 (−1, 0)g˜ 1 (1, 0) and .ρ2 (κ) = g˜ 2 (1, 0) = g˜ 2 (−1, 0). Therefore, we have an epimorphism .Im(ρ1 ) → SO(NE 11 G3,1 (O)) that in particular, is a 2-sheeted covering map. Since .NE 11 G3,1 (O) is a vector space of dimension 9, there are only two possibilities: either .Im(ρ1 ) = Spin(9), i.e., the nontrivial double covering, or .Im(ρ1 ) has two connected components both diffeomorphic to  .SO(N E11 G3,1 (O)) and the connected component that contains the identity, being an isomorphism. But, this second case does not happen because both I and .κ are in that component of the identity. This completes the proof. .

Corollary 3.50 The map .F4 → G3,1 (O) given by .ϕ → ϕ(E11 ) for .ϕ ∈ F4 yields a diffeomorphism ≈

F4 /Spin(9) −→ G3,1 (O).

.

Problem 3.51 We do not know answers to the following questions: (1) what are the possible values of .tr : Gn (O) → R for .n ≥ 4?; (2) what are the connected components of .Gn (O) for .n ≥ 4?; (3) is .Gn (O) a smooth manifold for .n ≥ 4?

3.3 Flag Varieties In this section .A denotes one of the following fields: .R, .C or .H. A right flag over .A is a finite ordered strictly increasing sequence of finite dimensional right subspaces of .M−,1 (A) = A∞ : .W1 ⊆ W2 ⊆ · · · ⊆ Ws . It

164

3 Stiefel, Grassmann Manifolds

is clear that given a right flag .W1 ⊆ W2 ⊆ · · · ⊆ Ws we may consider the sequence .V1 , . . . , Vs of non-zero mutually orthogonal right .A-linear spaces given by .V1 = W1 , and .Vi+1 = Wi⊥ ∩ Wi+1 for .1 ≤ i < s, with respect to the inner product .−, −, given by .A, B = tr(A¯ t B). Thus, we better prefer taking as the definition of the right flag in .A, any finite sequence of finite dimensional, distinct, non-zero, mutually orthogonal subspaces .V1 , . . . , Vs . Given a flag .(V1 , . . . , Vs ), we say that it has length s and type .(r1 , . . . , rs ), where .rα = dimA Vα . s,right s,right We denote by .Fn;r1 ,...,rs (A) (respectively .F−;r1 ,...,rs (A)) the flag variety as the n ∞ set of all flags in .A (respectively .A ) of length s and type .(r1 , . . . , rs ). Then, we write s,right

Fn

.

s,right

s,right

(A) = ∪r1 +···+rs ≤n Fn;r1 ,...,rs (A), F s,right (A) = ∪n≥1 Fn right

Fn

.

(A),

(A) = ∪ns=1 Fns (A), F right (A) = ∪s≥1 F s,right (A).

Similarly, all can be done replacing right by left, and we have the natural bijection F right (A) → F left (A) given by .(V1 , . . . , Vs ) → (V¯1 , . . . , V¯s ), and restricting to bijections

.

right

s,right

(A) → Fnleft (A), F s,right (A) → F s,left (A), Fn

Fn

.

s,right

(A) → Fns,left (A),

s,right

s,left s,left F−;r1 ,...,rs (A) → F−;r (A), Fn;r1 ,...,rs (A) → Fn;r (A). 1 ,...,rs 1 ,...,rs

.

Observe that 1,right

F 1,right (A) = Gright (A), Fn

.

1,right

F−;r

.

right

1,right

(A) = G−,r (A), Fn;r

right

(A) = Gn

(A),

right

(A) = Gn,r (A)

and similarly replacing right by left. s,right As for the Grassmannians (see Proposition 3.25), we identify .Fn;r1 ,...,rs (A) with s .F n;r1 ,...,rs (A) defined by s Fn;r (A) = {(A1 , . . . , As ) ∈ Gn,r1 (A)×· · ·×Gn,rs (A); Aα Aβ = 0 for α = β}, 1 ,...,rs

.

s,right

s and we identify .F−;r1 ,...,rs (A) with .F−;r (A) defined by 1 ,...,rs s F−;r (A) = {(A1 , . . . , As ) ∈ G−,r1 (A) × · · · × G−,rs (A); 1 ,...,rs

.

Aα Aβ = 0 for α = β}.

3.3 Flag Varieties

165 s,right

Similarly, we identify .Fn

(A) with .Fns (A) = right .Fn (A)



s r1 +···+rs ≤n Fn;r1 ,...,rs (A), n s .Fn (A) = s=1 Fn (A),

F s,right (A) with .F s (A) =  ∪n≥1 Fns (A), with right and .F (A) with .F (A) = s≥1 F s (A). s Observe that .F (A) and .F−,r (A) are closed metric subspaces of the .R-vector 1 ,...,rs ∞ space .Herm(A) and .Herm(A)s respectively, where .Herm(A)∞ is the direct sum s of countable copies of .Herm(A), while .Fn (A) and .Fn,r (A) are closed metric 1 ,...,rs s subspaces of .Hermn (A) . Further, we define .BU s (A) as the quotient of .F s (A) by the equivalence relation given by .(A1 , . . . , As ) ∼ (B1 , . . . , Bs ) provided there exist natural numbers m n .m1 , n1 , . . . , ms , ns such that .τ α Aα = τ α Bα , .α = 1, . . . , s. s It is clear that .F−;r1 ,...,rs (A) → BU s (A), given by .(A1 , . . . , As ) → [Aa , . . . , , As ]∼ , is injective. .

Stiefel Maps Over Flags Varieties We have the natural surjective map right

s,right

πr1 ,...,rs : V−,r (A) → F−,r1 ,...,rs (A)

.

with .r = r1 + · · · + rs , called the Stiefel map given by .πr1 ,...,rs (A1 | · · · |As ) = right

(V1 , . . . , Vs ) for .(A1 | · · · |As ) ∈ V−,r (A), where .Vα , .1 ≤ α ≤ s, is the subspace with the orthonormal basis determined by .Aα . Thus, we have .A¯ tα Aα = Irα for .1 ≤ α ≤ s, and .A¯ tα Aβ = 0 for .α = β, .1 ≤ α, β ≤ s. Using the identification at the beginning of this section, we have canonical surjective maps s πr1 ,...,rs : V−,r (A) → F−,r (A) with r = r1 + · · · + rs 1 ,...,rs

.

given by .πr1 ,...,rs (A1 | · · · |As ) = (A1 A¯ t1 , . . . , As A¯ ts ) for .(A1 | · · · |As ) ∈ V−,r (A). The maps above restrict to surjective maps s πr1 ,...,rs : Vn,r (A) → Fn,r (A) with r = r1 + · · · + rs ≤ n. 1 ,...,rs

.

s Observe that those maps, being polynomial, are continuous and so .F−,r (A) 1 ,...,rs s s are connected and closed subspaces of .Herm(A) and .Fn,r1 ,...,rs (A) are compact connected subspaces of .Hermn (A)s . s Furthermore, .F−,r (A) are the connected components of .F s (A) and 1 ,...,rs s s .Fn,r ,...,r (A) are those of .Fn (A). s 1

Flags Varieties as Homogeneous Spaces We have a right action of .Ur1 (A) × · · · × Urs (A) on .V−,r (A) with .r = r1 + · · · + rs , given by (A1 | · · · |As )(B1 , . . . , Bs ) = (A1 B1 | · · · |As Bs )

.

166

3 Stiefel, Grassmann Manifolds

for .(A1 | · · · |As ) ∈ V−,r (A) and .(B1 , . . . , Bs ) ∈ Ur1 (A) × · · · × Urs (A). This action restricts to an action on .Vn,r (A). Further, observe that for A = (A1 | · · · |As ) ∈ Vn,r1 ,...,rs (A)

.

and (B1 , . . . , Bs ) ∈ Ur1 (A) × · · · × Urs (A),

.

we have πr1 ,...,rs ((A1 | · · · |As )(B1 , . . . , Bs )) = πr1 ,...,rs (A1 | · · · |As ).

.

We also have that for .πr1 ,...,rs (A) = πr1 ,...,rs (A ) there exists .B = (B1 , . . . , Bs ) ∈ Ur1 (A) × · · · × Urs (A) such that .A = AB. In fact, we can take .Bα = A¯ tα Aα for .α = 1, . . . , s. Therefore, we have a bijection s πr1 ,...,rs : V−,r (A)/Ur1 (A) × · · · × Urs (A), → F−,r (A) 1 ,...,rs

.

restricting to a bijection s πr1 ,...,rs : Vn,r (A)/Ur1 (A) × · · · × Urs (A), → Fn,r (A). 1 ,...,rs

.

Since .πr1 ,...,rs is continuous, it is a homeomorphism. s s Hence, we may identify .F−,r (A) (resp. .Fn,r (A)) with the homoge1 ,...,rs 1 ,...,rs neous space U (A)/Ur1 (A) × · · · × Urs (A) × τ r U (A),

.

(resp. Un (A)/Ur1 (A) × · · · × Urs (A) × τ r Un−r (A)).

.

s We use this identification to endow .Fn,r (A) with a smooth structure so that 1 ,...,rs s .πr1 ,...,rs : Vn,r (A) → Fn,r ,...,r (A) is a smooth principal bundle with structure s 1 group .Ur1 (A) × · · · × Urs (A).

3.3 Flag Varieties

167

In particular, .

s dimR Fn,r (A) = dimR Vn,r (A) − dimR Skr1 (A) × · · · × Skrs (A) = 1 ,...,rs

 rα (rα + 1) 1 (d dr(2n − r + 1) − r − − rα ) = 2 2 s

.

α=1

 1 rα2 ). d(2nr − r 2 − 2 s

.

α=1

Following the proof of Lemma 3.26, we get: Lemma 3.52 If .A = (A1 | · · · |As ) ∈ Vn,r (A) with .r = r1 + · · · + rs ≤ n then .

Ker(dπr1 ,...,rs )A = TA (πr−1 (A1 A¯ t1 , . . . , As A¯ ts )) = 1 ,...,rs

TA (A(Ur1 (A) × · · · × Urs (A))) = (A1 Skr1 (A)| · · · |As Skrs (A)).

.

Then, an analog to the proof of Proposition 3.27, yields: s Proposition 3.53 The space .Fn,r (A) is a compact smooth submanifold of 1 ,...,rs Hermn (A))s .

.

Furthermore, an analog to the proof of Proposition 3.28, leads to: Proposition 3.54 s s (1) The tangent space to .Fn,r (A) at .A = (A1 , . . . , As ) ∈ Fn,r (A) is 1 ,...,rs 1 ,...,rs given by s TA Fn,r (A) = 1 ,...,rs

.

{(B1 , . . . , Bs ) ∈ (Hermn (A))s ; Aα Bβ + Bα Aβ = δαβ Bα , 1 ≤ α, β ≤ s};

.

s s (2) for the normal space to .Fn,r (A) at .A = (A1 , . . . , As ) ∈ Fn,r (A), we 1 ,...,rs 1 ,...,rs have the inclusion

{(B1 , . . . , Bs ) ∈ (Hermn (A))s ; Aα Bα = Bα Aα , α = 1, . . . , s}

.

s ⊆ NA Fn,r (A). 1 ,...,rs

168

3 Stiefel, Grassmann Manifolds

s s We denote by .T Fn,r (A) the total tangent space of .Fn,r (A). Namely, 1 ,...,rs 1 ,...,rs s s T Fn,r (A) = {((A1 , . . . , As ), (B1 , . . . , Bs )) ∈ Fn,r (A) × (Hermn (A))s ; 1 ,...,rs 1 ,...,rs

.

Aα Bβ + Bα Aβ = δα,β Bα , 1 ≤ α, β ≤ n}

.

with .rα ≥ 1 for .α = 1, . . . , s, .r1 + · · · + rs ≤ n and any .A. s Furthermore, we denote by .NFn,r (A) the total normal space of 1 ,...,rs s .Fn,r ,...,r (A). Namely, s 1 s s N Fn,r (A) = {((A1 , . . . , As ), (B1 , . . . , Bs )) ∈ Fn,r (A) × (Hermn (A))s ; 1 ,...,rs 1 ,...,rs

.

Aα Bα = Bα Aα , α = 1, . . . , n}

.

with .rα ≥ 1 for .α = 1, . . . , s, .r1 + · · · + rs ≤ n and any .A. Remark 3.55 Nothing new is obtained if we replace the standard one by the Jordan multiplication of matrices: s Fn,r (A) = {(A1 , . . . , As ) ∈ Gn,r1 (A) × · · · × Gn,rs (A); 1 ,...,rs

.

Aα ◦ Aβ = 0 for α = β}. t

In fact, if A and B belongs to .Hermn (A) and .AB = 0, then .BA = AB = 0, and so .A ◦ B = 0. On the other hand, if A and B belong to .Gn (A) and .A◦B = 0, then .AB = −BA, and so AB = AB 2 = (AB)B = −(BA)B = −B(AB) = B(BA) = B 2 A = BA = −AB.

.

Therefore, .AB = BA = 0. Remark 3.56 We can define flag varieties over the octonions by s Fn,r (O) = {(A1 , . . . , As ) ∈ Gn,r1 (O) × · · · × Gn,rs (O); 1 ,...,rs

.

Aα Aβ = 0 for all α = β} or s (O) = {(A1 , . . . , As ) ∈ Gn,r1 (O) × · · · × Gn,rs (O); F˙n,r 1 ,...,rs

.

Aα ◦ Aβ = 0 for all α = β} But, in general, due to lack of associativity, they do not coincide and they are only real affine algebraic varieties.

Chapter 4

More Classical Matrix Varieties

In this chapter we generalize Stiefel, Grassmann and flag manifolds, defined in Chap. 3, to what we call here i-Stiefel, i-Grassmann and i-flag manifolds. This “i” comes from idempotent. Those manifolds do not seem to have being enough studied in the literature. In particular, they do not have even a name. As in Chap. 2, .A denotes the field of reals, .R, the field of complex numbers, .C, the skew field of quaternions, .H and, occasionally, the octonion division algebra .O.

4.1 i-Grassmannians and i-Stiefel Varieties In this section .A = R, C or H. We have identified at the beginning of Sect. 2.2, a right .A-linear subspace V of n .A with the orthogonal projection determined by considering the decomposition n ⊥ .A = V ⊕ V . This yields the definition of Grassmannian .G(A) as the set of all idempotent and hermitian matrices, check Proposition 3.25 and the natural bijection ≈ right between .Gright (A) and .G(A), restricting to bijections .G−,r (A) −→ G−,r (A) and right

≈

Gn,r (A) −→ Gn,r (A). It is natural to consider the set of all projections, and not only orthogonal projections. Given a right .A-linear subspace of .An , and a complement .W, i.e., n n → V . This suggests to .V ⊕ W = A , we have the associated projection .A define the “classical variety of idempotents” over .A, Idemright (A), as the set of all ordered couples .(V , W ), where V is a right .A-finite dimensional vector subspace of ∞ =M ∞ .A −,1 (A), W is a complement, so that .V + W = A and the sum is direct. right right We also define .Idem−,r (A) = {(V , W ) ∈ Idem (A); dimA V = r}, and .

right

Idemn,r (A) as the subset of all couples .(V , W ) in .Idem−,r (A), where both V and W are right .A-subspaces of .A∞ , respectively .An .

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Golasi´nski, F. Gómez Ruiz, Grassmann and Stiefel Varieties over Composition Algebras, RSME Springer Series 9, https://doi.org/10.1007/978-3-031-36405-1_4

169

170

4 More Classical Matrix Varieties

The analog to the Grassmannian .G(A) is the i-Grassmannian over .A, which is defined as the set .Idem(A) of all idempotent matrices over .A. Similarly, we define .Idem−,r (A) and .Idemn,r (A). ¯ (A) as the quotient of .Idem(A) by the equivalence Furthermore, we define .BU relation, where .A ∼ B provided .τ m (A) = τ n (B), for some natural numbers m and .n. We clearly have a natural bijection .Idemright (A) → Idem(A) given by sending the couple .(V , W ) to the idempotent matrix .A ∈ M(A) being the identity on V and ≈ right vanishing on .W. This bijection restricts to bijections .Idem−,r (A) → Idem−,r (A) right

≈

and .Idemn,r (A) → Idemn,r (A). Observe that .Idem(A) and .Idem−,r (A) are closed metric subspaces of the .Rvector space .M(A), .Idemn (A) = Idem(A) ∩ Mn (A) and .Idemn,r (A) are closed metric subspaces of .Mn (A). Similarly, we define the i-Stiefel variety over .A V¯−,r (A) = {(A, B) ∈ (M−,r (A))2 |B t A = Ir }, V¯n,r (A) = V¯−,r (A) ∩ (Mn,r (A))2

.

for .r ≥ 1.  Therefore, .V¯−,r (A) = n≥r V¯n,r (A). Define also .V¯ (A) as the subset of .(M(A))2 given by V¯ (A) =



.

V¯−,r (A).

r≥1

¯ Observe that there is a canonical inclusion .V (A) → V¯ (A) given by .A → (A, A) for .A ∈ V (A) which restricts to inclusions .V−,r (A) → V¯−,r (A) and .Vn,r (A) → V¯n,r (A). The map .τ : M(A) → M(A), as defined in Chap. 1, leads to inclusions V¯−,r (A) → V¯−,r+1 (A).

.

Then, we define ¯ (A) = colimr V¯−,r (A). EU

.

¯ (A) is the quotient of .V¯ (A) by the equivalence relation .∼, where Therefore, .EU we say that .(A, B) and .(A , B ) in .V¯ (A) are equivalent if there exist natural numbers .m, n such that τ m (A) = τ n (A ), τ m (B) = τ n (B ).

.

4.1 i-Grassmannians and i-Stiefel Varieties

171

¯ (A) by As for the case of .EU (A), we define a distance in .EU ˜ d([A, B], [A , B ]) =

.



d(A, A )2 + d(B, B )2

and as in Proposition 3.1 distance and quotient topology coincide. ¯ (r, A) = V¯−,r (A) and so we have .EU ¯ (r K) ⊆ We also use the notation .EU ¯ EU (A). ¯ yields an inclusion .EU (A) → EU ¯ (A). Further, observe that .A → (A, A) Smooth Structures on i-Stiefel Varieties Consider the real polynomial map .f¯ : (Mn,r (A))2 → Mr (A) given by f¯(A, B) = B t A

.

for .(A, B) ∈ (Mn,r (A))2 , where .n ≥ r ≥ 1 and .A = R, C, H or O. Its differential .(d f¯)A,B : (Mn,r (A))2 → Mr (A) at .(A, B) ∈ (Mn,r (A))2 is given by (d f¯)A,B (X, Y ) = B t X + Y t A

.

for .(X, Y ) ∈ (Mn,r (A))2 . Notice that given .C ∈ Mr (A), for .A = R, C or H, we have 1 1 (d f¯)A,B ( AC, ( CB t )t ) = C 2 2

.

for .(A, B) ∈ V¯n,r (A). Hence, we may state: Proposition 4.1 The differential .(d f¯)A,B is an .R-epimorphism for .(A, B) ∈ V¯n,r (A), .A = R, C or H and arbitrary .n ≥ r ≥ 1. Then, Proposition 4.1 yields: Proposition 4.2 Suppose that .A = R, C or H and .1 ≤ r ≤ n. Then, .Ir is a regular value for .f¯. In particular, .V¯n,r (A) is a closed regular smooth submanifold of .(Mn,r (A))2 and .

dimR V¯n,r (A) = dimR Mn,r (A)2 − dimR Mr (A) = 2dnr − dr 2 = dr(2n − r)

for .d = 1, 2, 4. The tangent space at .(A, B) ∈ V¯n,r (A) is given by TA,B V¯n,r (A) = Ker(d f¯)A,B = {(X, Y ) ∈ (Mn,r (A))2 ; B t X + Y t A = 0}.

.

172

4 More Classical Matrix Varieties

Observe that we have an involutive diffeomorphism ≈ V¯n,r (A) −→ V¯n,r (A)

.

¯ A) ¯ for .(A, B) ∈ V¯n,r (A) whose the fixed point set is the given by .(A, B) → (B, image of .Vn,r (A) by the canonical inclusion .Vn,r (A) → V¯n,r (A). This diffeomorphism clearly extends to a homeomorphism .V¯ (A) → V¯ (A), ¯ A) ¯ for .(A, B) ∈ V¯ (A) which restricts to homeomorphisms given by .(A, B) → (B, ≈ ≈ ¯ (A). ¯−,r (A) → V¯−,r (A) inducing a homeomorphism .EU ¯ (A) → .V EU The following result gives an interesting relation between .V¯n,1 (A) and .T Vn,1 (A). Proposition 4.3 There is a diffeomorphism f :

.

T Vn,1 (A) × (R − {0}) → V¯n,1 (A) × {q ∈ A; Re(q) = 0}, ∼

defined by 1 f ([X, Y, λ]) = (λX, (1 − λY¯ t X)X¯ + Y¯ , λX¯ t Y ) λ

.

for .[X, Y, λ] ∈

T Vn,1 (A)×(R−{0}) , ∼

where .∼ is the equivalence relation given by

(X, Y, λ) ∼ (−X, Y, −λ).

.

Its inverse map g is given by 1 1 g(A, B, q) = [ A, B¯ − 2 A(1 − q), λ] λ λ

.

for .(A, B, q) ∈ V¯n,1 (A) × {q ∈ A; Re(q) = 0}, where .λ2 = A¯ t A. Proof The definition of f is correct. In fact, 1 1 ( (1 − λY¯ t X)X¯ + Y¯ )t λX = ( (1 − λY¯ t X)X¯ t + Y¯ t )λX = λ λ

.

X¯ t X − λ(Y¯ t X)(X¯ t X) + λY¯ t X = 1 + λ(−Y¯ t X + Y¯ t X) = 1.

.

Further, .X¯ t Y = −X¯ t Y implies .Re(X¯ t Y ) = 0. Certainly, it is clear that .f [X, Y, λ] = f [−X, Y, −λ].

4.1 i-Grassmannians and i-Stiefel Varieties

173

The definition of g is correct. In fact, t

1 1 1 ( A) A = 2 A¯ t A = 1; λ λ λ

.

t

t

1 1 1 1 ( A) (B¯ − 2 A(1 − q)) + B¯ − 2 A(1 − q) A = λ λ λ λ

.

.

1 ¯t ¯ 1 1 1 A B − 3 A¯ t A(1 − q) + B t A − 3 (1 + q)A¯ t A = λ λ λ λ .

1 1 1 1 − (1 − q) + − (1 + q) = 0. λ λ λ λ

It is easy to check that .(gf )[X, Y, λ] = [X, Y, λ] and .(f g)(A, B, q) = (A, B, q) and the proof follows.   Lemma 4.4 Given .(A, B) ∈ V¯n,r (A) with .1 ≤ r < n, there exists .(a, b) ∈ V¯n,1 (A), such that .(A|a, B|b) ∈ V¯n,r+1 (A). Proof This is equivalent of saying that both .R-vector spaces KerB t = {x ∈ Mn,1 (A); B t x = 0}

.

and KerA = {x ∈ M1,n (A); xA = 0}

.

are non-zero. But, this is clear because otherwise, either .B t : An → Ar given by .x → B t x or the map .A : An → A given by .x → x t A, should be injective with .n > r, which is impossible. This completes the proof.   We have natural polynomial, and so smooth maps, ρ : V¯n,s (A) → V¯n,r (A)

.

for .1 ≤ r < s ≤ n given by .(A, B) = (A1 |A2 , B1 |B2 ) → (A1 , B1 ) for .A, B) ∈ V¯n,s (A). Observe that we have a natural diffeomorphism GLn (A) → V¯n,n (A)

.

given by .A → (A, (A−1 )t ) for .A ∈ GLn (A).

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Corollary 4.5 The map .ρ : GLn (A) = V¯n,n (A) → V¯n,r (A) is surjective for n > r. In particular, all maps .ρ : V¯n,s (A) → V¯n,r (A) are surjective for .1 ≤ r < s ≤ n.

.

Proposition 4.6 The map .ρ : GLn (A) → V¯n,r (A) is a submersion. Proof We already know, by Corollary 4.5, that the map .ρ is surjective and smooth. For the proof, it is enough to show that the map .ρ : V¯n,r+1 (A) → V¯n,r (A) given by ¯n,r (A), .(a, b) ∈ V¯n,1 (A) .(A|a, B|b) → (A, B), is a submersion, where .(A, B) ∈ V and .B t a = 0, .bt A = 0. Observe that .(X|x, Y |y), with .(X, Y ) ∈ Mn,r (A)2 and .(x, y) ∈ Mn,1 (A)2 , belongs to .T(A|a,B|b) V¯n,r+1 (A) if and only if (X, Y ) ∈ TA,B V¯n,r (A), bt x + y t a = 0, B t x + Y t a = 0, bt X + y t A = 0.

.

But, we have then .(dρ)(A|a,B|b) (X|x, Y |y) = (X, Y ). To show the surjectivity of .(dρ)(A|a,B|b) , given .(A|a, B|b) and .(X, Y ), we take t t t t .x = −AY a, .y = −(b XB ) and the proof is complete.   Proposition 4.7 The map .ρ : GLn (A) → V¯n,r (A is a principal bundle with group .GLn−r (A) for .A = R, C or H. r r Proof We have the subgroup  .τ (GLn−r (A) ⊆ GLn (A), where .τ (GLn−r (A)) I 0 consists of all matrices . r with .C ∈ GLn−r (A). 0 C The action of .GLn−r (A) on .GLn (A), by right multiplication of .τ r (GLn−r (A)), is obviously free and restricts to the fibres of .ρ. On the other hand, if .ρ(A1 |A2 , B1 |B2 ) = ρ(A1 |A 2 , B1 |B2 ) = (A1 , B1 ), then there exists .C ∈ GLn−r (A) such that .(A 2 , B2 ) = (A2 , B2 )C and the proof follows.  

We derive now from Propositions 4.6 and 4.7: Corollary 4.8 The map .ρ : V¯n,s (A) → V¯n,r (A) is a smooth fibre bundle with the fibre .V¯n−r,n−s (A) for .1 ≤ r < s ≤ n. Proposition 4.9 (1) The spaces .V¯n,r (C) and .V¯n,r (H) are connected for .1 ≤ r ≤ n; (2) the space .V¯n,r (R) is connected for .1 ≤ r < n and .V¯n,n (R) = GL(n; R) has two connected components. Proof (1): Let .A = C or .H, then .GL1 (A) = A − {0} is connected, by Corollary 4.8 and the induction, we have that .GLn (A) is connected for all .n ≥ 1. Therefore ¯n,r (A) is connected, being the image of a connected space. .V

4.1 i-Grassmannians and i-Stiefel Varieties

175

(2): .GL1 (R) = R−{0} has two connected components. Then, by Corollary 4.8 and the induction any .GLn (R) has two connected components, and any .V¯n,r (R) is connected for .1 ≤ r < n, because one always may complete a .(n − 1)-linearly independent vectors so the determinant is positive. This completes the proof.   Observe that the Proposition 4.7 shows in particular that .V¯n,r (A) is diffeomorphic to the homogeneous space .GLn (A)/τ r GLn−r (A), and by the definition of the colimit we have a principal bundle .ρ : GL(A) → V¯−,r (A) with the structure group .GL(A) acting by right multiplication of .τ r GL(A). Thus, .V¯−,r (A) is homeomorphic to the homogeneous space .GL(A)/τ r GL(A). Consider now the .R-bilinear, symmetric and non-degenerate inner product on 2 .M(A) given by (X, Y ), (X , Y ) = Re tr(XY t + X Y t ),

.

where .X, Y, X , Y ∈ M(A). Then, the following holds: Proposition 4.10 The normal space at .(A, B) ∈ V¯n,r (A) with respect to the above inner product in .Mn,r (A) is given by NA,B V¯n,r (A) = {(AX, (XB t )t )}X∈Mr (A) .

.

Proof We check now that if .(A, B) ∈ V¯n,r (A) and .(AX, (XB t )t ) ∈ TA,B V¯n,r (A) with .X ∈ Mr (A), then .X = 0. In fact, we have 0 = XB t A + B t AX = 2X,

.

and so .X = 0. Furthermore, the map .Mr (A) → NA,B V¯n,r (A) which sends .X → (AX, (XB t )t ) for .X ∈ Mr (A) is an isomorphism of vector spaces, because if .(AX, (XB t )t ) = 0, then .AX = 0 and so .X = B t AX = 0. It is clear that for .(A, B) ∈ V¯n,r (A) we have the canonical decomposition (Mn,r (A))2 = TA,B V¯n,r (A) ⊕ NA,B V¯n,r (A)

.

given by (M, N ) → (M − A

.

(A N

.

t A+B t M

2

, (N

t A+B t M

2

N t A + Bt M t t N t A + Bt M ,N − ( B ) )⊕ 2 2

B t )t ) for .(M, N ) ∈ (Mn,r (A))2 .

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4 More Classical Matrix Varieties

Observe that the canonical decomposition above is orthogonal. In fact, let .X ∈ Mr (A) and .(C, D) ∈ TA,B V¯−,r (A), then (AX, (XB t )t ), (C, D) = Re tr(AXD t + CXB t ) = Re tr(XD t A + XB t C) = 0,

.

because .D t A + B t C = 0. This completes the proof.

 

Remark 4.11 We also have the canonical orthogonal decomposition NA,B V¯n,r (A) = R(A, B) ⊕ {(AX, (XB t )t )}X∈Mr (A),Re tr(X)=0

.

given by (AX, (XB t )t ) =

.

Re tr(X) Re tr(X) Re tr(X) (A, B) + (A(X − Ir ), ((X − Ir )B t )t ). r r r

i-Stiefel Maps Define the i-Stiefel map ¯ r (A) = V¯−,r (A) → Idem−,r (A) = BU ¯ r (A) ρ : EU

.

¯ r (A) = V¯−,r (A) which restrict to by .(A, B) → AB t for .(A, B) ∈ EU ρ : V¯n,r (A) → Idemn,r (A)

.

and leads to ρ : V¯ (A) → Idem(A),

.

and finally to ¯ (A) → BU ¯ (A) ρ : EU

.

¯ (A). given by .[A, B] → [AB t ] for .[A, B] ∈ EU All maps above are surjective. In fact, given .C ∈ Idemn,r (A), choose  X=

.

X11 X12 X21 X22

 ∈ GLn (A)

with the inverse X−1 =

.

  Y11 Y12 Y21 Y22

4.1 i-Grassmannians and i-Stiefel Varieties

177

such that X

.

−1

  Ir 0 CX = . 0 0

Then,    t  X11 Y , 11 ∈ V¯n,r (A) . t X21 Y12 and   t   Y X11 , 11 = C. .ρ t X21 Y12 i-Grassmannians as Homogeneous Spaces The free right action .V¯−,r (A) × GL(r, A) → V¯−,r (A) given by (A, B) · X = (AX, (X−1 B t )t )

.

for .(A, B) ∈ V¯−,r (A) and .X ∈ GL(r, A) induces a bijection from the orbit space of .V¯−,r (A) to .Idem−,r (A) and so we have also bijections from the orbit spaces of ¯n,r (A) to .Idemn,r (A). .V In fact, if .ρ(A, B) = ρ(A , B ), for .(A, B), (A , B ) ∈ V¯−,r (A) then (B A)(B t A ) = B (AB t )A = B A B A = Ir Ir = Ir t

.

t

t

t

and analogously (B t A )(B A) = Ir . t

.

Therefore, B t A ∈ GL(r, K)

.

and we have (A, B) · (B t A ) = (AB t A , (B AB t )t ) = (A , B ).

.

t

≈ Thus, we have a bijection .ρ : V¯−,r (A)/GLr (A) → Idem−,r (A) restricting to a bijection .ρ : V¯n,r (A)/GLr (A) → Idemn,r (A). Since .ρ is continuous, this map is a homeomorphism.

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4 More Classical Matrix Varieties

Thus, we may identify .Idem−,r (A) (respectively .Idemn,r (A)) with the homogeneous space .GL(A)/GLr (A)τ r (GL(A)) (respectively .GLn (A)/GLr (A)τ r (GLn−r (A))). We use this identification to endow .Idemn,r (A) with a smooth structure so that ¯n,r (A) → Idemn,r (A) is a smooth principal bundle with the structure group .ρ : V .GLr (A). In particular, .

dimR Idemn,r (A) = dimR V¯n,r (A) − dimR Mr (A) = 2dr(n − r).

Similarly, ρ

E U¯ r (A) = V¯−,r (A) → Idem−,r (A) = B U¯ r (A)

.

is a principal bundle with the structure group .GLr (A). Furthermore, observe that .GLn (A) acts on .Idemn,r (A) by .A → U AU −1 .   I 0 The isotropy subgroup at . r is .GLr (A)τ r GLn−r (A) and so .GLn (A) acts 0 0 transitively on .Idemn,r (A). The following is an analog of Lemma 3.26: Lemma 4.12 If .(A, B) ∈ V¯n,r (A) then Ker(dρ)A,B = TA,B ρ −1 (AB t ) = TA,B ((A, B)GLr (A)) =

.

{(AX, −(XB t ))t ); X ∈ Mr (A)}.

.

Proof It is similar to the proof of Lemma 3.26. In fact, consider a smooth path .ω : (−ε, ε) → GLr (A) with .ω(0) = Ir , and so ω (0) = X ∈ Mr (A) = TIr GLr (A).

.

Then, the map .(−ε, ε) → ρ −1 (AB t ) given by .s → γA,B (s) = (A, B)ω(s) for −1 (AB t ) with .s ∈ (−ε, ε) is a smooth path in .ρ γA,B (0) = (A B) and γA,B (0) = (AX, −(XB t )t ).

.

Therefore, TA,B (ρ −1 (AB t )) = {(AX, −(XB t )t ); X ∈ Mr (A)}.

.

Observe that the map .(−ε, ε) → Idemn,r (A) given by .s → ρ(γA,B (s)) = AB t for .s ∈ (−ε, ε) so it is constant. Thus, .TA,B (ρ −1 (AB t )) ⊆ Ker(dρ)A,B and so they coincide because both have same dimension and the proof follows.  

4.1 i-Grassmannians and i-Stiefel Varieties

179

The following result and its proof is analogous to that of Proposition 3.27: Proposition 4.13 The space .Idemn,r (A) is a closed regular submanifold of .Mn (A). Proof The inclusion .i : Idemn,r (A) → Mn (A) is smooth, being locally the composition of a smooth cross section of the bundle .ρ : V¯n,r (A) → Idemn,r (A) followed by the map .ρ˜ : V¯n,r (A) → Mn (A) given by .ρ(A, ˜ B) = AB t . Observe that .ρ˜ is polynomial and so smooth, and we have .ρ˜ = iρ. To show that i is an immersion, suppose that .(di)AB t (C) = 0 with .C ∈ TAB t Idemn,r (A) and choose .(X, Y ) ∈ T(A,B) V¯n,r (A) such that .(dρ)(A,B) (X, Y ) = C. Therefore, 0 = (di)AB t (dρ)A,B (X, Y ) = (d ρ) ˜ A,B (X, Y ) = AY t + XB t .

.

On the other hand, since .(X, Y ) ∈ T(A.B) V¯n,r (A), we have .B t X + Y t A = 0. Thus, .0 = AY t A + XB t A = −AB t X + XB t A = −AB t X + X implies .X = AB t X and so .0 = B t AY t + B t XB t = Y t + B t XB t . Hence, by Lemma 4.12, we have (X, Y ) = (A(B t X), −((B t X)B t )t ) ∈ Ker(dρ)A,B

.

and so .C = 0. Finally, i being an injective immersion of a smooth manifold yields .Idemn,r (A) that is a regular smooth submanifold of .Mn (A) and the proof is complete.   Proposition 4.14 (1) The tangent space to .Idemn,r (A) at .A ∈ Idemn,r (A) is given by TA Idemn,r (A) = {B ∈ Mn (A); AB + BA = B}.

.

(2) The normal space to .Idemn,r (A) at .A ∈ Idemn,r (A) is given by NA Idemn,r (A) = {B ∈ Mn (A); AB = BA}.

.

Proof (1): Let .σ : (−ε, ε) → Idemn,r (A) be a smooth path with .σ (0) = A and write 2 .B = σ (0). Then, .σ (s) = σ (s) implies .AB + BA = B. Therefore, TA Idemn,r (A) ⊆ {B ∈ Mn (A); AB + BA = B}.

.

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4 More Classical Matrix Varieties

  Ir 0 In particular, if we take .A = then we have 0 0 

 0 C .TA Idemn,r (A) ⊆ { ; C ∈ Mr,n−r (A), D ∈ Mn−r,r (A)} D 0 

 0 C ; C ∈ Mr,n−r (A), D ∈ Mn−r,r (A)}, D 0 because both spaces have same dimension .2dr(n − r). Finally, we use the transitivity of the action of .GLn (A) on .Idemn,r (A) to conclude the proof of (1). (2): We show first that if .C ∈ Mn (A) and .AC = CA then .C ∈ (TA Idemn,r (A))⊥ . In fact, let .B ∈ TA Idemn,r (A). In particular, .B ∈ Mn (A) and .AB + BA = B then and so .TA Idemn,r (A) = {

Re tr(BC) = Re tr(ABC + BCA) = 2Re tr(ABC) =

.

2Re tr(A2 BC) = 2Re tr(ABCA) = 2Re tr(ABAC) = 0.

.

Since .AB + BA = B, the last equality yields .ABA = 0. Therefore, {C ∈ Mn (A); AC = CA} ⊆ (TA Idemn,r (A))⊥ .

.

We have .{C ∈ Mn (A); AC = CA} ∩ TA Idemn,r (A) = 0. In fact, .AB = BA, AB + BA = B imply .2AB = B and .0 = 2ABA = BA = AB, and so .B = 0. Finally, observe that we have the orthogonal sum Mn (A) = TA Idemn,r (A) ⊕ {C ∈ Mn (A); AC = CA}.

.

In fact, given any .B ∈ Mn (A) and any .A ∈ Idemn,r (A) we have .B = B1 +B2 , where .B1 = AB + BA − 2ABA ∈ TA Idemn,r (A) and .B2 = B + 2ABA − AB − BA satisfies .AB2 = B2 A.d This implies .NA Idemn,r (A) = {C ∈ Mn (A); AC = CA} and the proof is complete.   The following results give an important relation between Grassmannians and iGrassmannians (see [11] and [12]). Lemma 4.15 If .B ∈ Idemn (A) then: (1) .B + B¯ t − In is invertible; (2) .B(B + B¯ t − In )−1 = (B + B¯ t − In )−1 B¯ t ;

4.1 i-Grassmannians and i-Stiefel Varieties

181

(3) .B(B + B¯ t − In )−1 ∈ Gn (A), and it is the unique element A of .Gn (A) such that .AB = B and .BA = A; (4) .Re trA = Re trB. (5) Suppose that .B, .B belong to .Idemn (A) and .A, .A are the unique element of .Gn (A) such that .AB = B, .BA = A, .A B = B , .B A = B , then .AA = 0 if t and only if .B¯ B = 0. Proof (1): It is clear that .B + B¯ t − In ∈ Hermn (A). Next we show that the matrix .B + B¯ t − In is invertible. Let .v ∈ Mn,1 (A) and suppose that we have .(B + B¯ t − In )v = 0, or equivalently .Bv + B¯ t v = v. Then, .B(Bv) + B(B¯ t v) = Bv, and so, ¯ t v) = 0. Thus, .B(B v¯ t (B(B¯ t v)) = 0.

.

Therefore, t

B¯ t v B¯ t v = 0,

.

and so B¯ t v = 0.

.

t Similarly, .B¯ t (Bv) + B¯ t (B¯ t v) = B¯ t v gives .B¯ t (Bv) = 0 and .Bv Bv = 0, and so .Bv = 0. Therefore,

v = Bv + B¯ t v = 0 + 0 = 0.

.

This shows that Ker(B + B¯ t − In ) = 0

.

and so .B + B¯ t − In is invertible. (2): This is clear. (3): It is clear that .A ∈ Hermn (A) and A2 = B(B+B¯ t −In )−1 B(B+B¯ t −In )−1 = B B¯ t (B+B¯ t −In )−1 (B+B¯ t −In )−1 .

.

Therefore, .A2 = A if and only if B B¯ t (B + B¯ t − In )−1 (B + B¯ t − In )−1 = B(B + B¯ t − In )−1 ,

.

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4 More Classical Matrix Varieties

or equivalently, B B¯ t = B(B + B¯ t − In ).

.

Whence, A2 = A.

.

To show the uniqueness, suppose .A and A belong to .Gn (A) such that A B = AB = B, BA = A and BA = A.

.

Then, (A − A)B = 0 and B(A − A) = A − A

.

and so (A − A)B¯ t = A − A.

.

Therefore, (A − A)(A − A)t = ((A − A)B¯ t )(A − A) =

.

t (A − A)(B¯ t (A − A)t ) = (A − A)(A − A)B = 0.

.

Thus, .A = A. Next, we show that .A = B(B + B¯ t − In )−1 satisfies the required properties: 2 ¯ t − In )−1 = A. .B = B implies .BA = B(B + B On the other hand, B B¯ t = B(B + B¯ t − In )

.

yields B B¯ t (B + B¯ t − In )−1 = B,

.

and so, AB = B(B + B¯ t − In )−1 B = B B¯ t (B + B¯ t − In )−1 = B.

.

(4): .Re trA = Re tr(BA) ≤ Re trB = Re tr(AB) ≤ Re tr(A). Thus, .Re trA = Re trB.

4.1 i-Grassmannians and i-Stiefel Varieties

183

(5): Notice that .AA = 0 if and only if .B(B + B¯ t − In )−1 B (B + B¯ − In )−1 = 0. Thus, .B(B + B¯ t − In )−1 B = 0 .(B + B¯ t − In )−1 B¯ t B = 0. Equivalently, ¯ t B = 0 and this completes the proof. .B   t

Then, the following result is a consequence of Lemma 4.15: Theorem 4.16 The map .T Gn,r (A) → Idemn,r (A) given by .(A, B) → AB + A, is a diffeomorphism with the inverse .Idemn,r (A) → T Gn,r (A) given by .C → (A, B), where .B = C + C¯ t − 2A and A is the unique element in .Gn,r (A) such that .AC = C and .CA = A. Explicitly, .A = C(C + C¯ t − In )−1 . Remark 4.17 The diffeomorphism above extends to homeomorphisms T G(A) = {(A, B) ∈ G(A) × Herm(A); AB + BA = B} → Idem(A)

.

given by .(A, B) → AB + B, restricting to homeomorphisms T G−,r (A) = {(A, B) ∈ G−,r (A) × Herm(A); AB + BA = B} → Idem−,r (A)

.

and inducing the homeomorphisms ¯ (A) T BU (A) → BU

.

given by .[(A, B)] → [AB + B]. Here, we define .T BU (A) = colimr T G−,r (A), i.e., .T BU (A) is the quotient of .T G(A) by the equivalent relation .(A, B) ∼ (A , B ) provided there exist .m, n such     0 0 0 0 that .τ m = A = τ n A and . m = n . 0 B 0 B Corollary 4.18 The inclusion maps .Gn,r (A) → Idemn,r (A) are strong deformation retracts, extending to strong deformation retracts .G−,r (A) → Idem−,r (A) ¯ (A). and inducing a strong deformation retract .BU (A) → BU Proof In fact, just use Lemma 4.15 to show that H : Idemn,r (A) × R → Idemn,r (A),

.

given by H (B, λ) = (1 − λ)B + λB(B + B¯ t − In )−1

.

for .(B, λ) ∈ Idemn,r (A) × R is the required homotopy and the proof follows.

 

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4 More Classical Matrix Varieties

4.2 i-Flag Varieties In this section .A denotes one of the following: .R, .C, or .H. By analogy with the flag varieties defined in Sect. 3.3, we have two possible candidates for the definition of an i-flag variety: (1). .F¯ s (A) is the set of all finite sequences (A1 , . . . , As ) ∈ Idem(A)s

.

such that .A¯ tα Aβ = 0 for all .α = β; (2). .F˜ s (A) is the set of all finite sequences (A1 , . . . , As ) ∈ Idem(A)s

.

such that .Aα Aβ = 0 for all .α = β. Observe that if we replace, in definition (1), the conditions .A¯ tα Aβ = 0, for all ¯ t = 0, for all .α = β, we obtain a homeomorphic space, where the .α = β, by .Aα A β homeomorphism is given by .(A1 , . . . , As ) → (A¯ t1 , . . . , A¯ ts ). Similarly to Remarks 3.55 and 3.56, nothing new is obtained if we replace standard by Jordan multiplication in the definitions above, because of the following remark: Remark 4.19 Suppose that A and B belong to .M(A), with either A or B in Idem(A), and .A ◦ B = 0. Then .AB = BA = 0.

.

In fact, if .AB = −BA and .A2 = A then we have AB = A2 B = A(AB) = −A(BA) = −(AB)A = (BA)A = BA2 = BA = −AB

.



and the claim follows. Then, we set

.

F¯ (A) =



.

F¯ s (A), F˜ (A) = ∪s≥1 F˜ s (A)

s≥1

Similarly, we define s (A) = {(A1 , . . . , As ) ∈ Idem−,r1 (A) × · · · × Idem−,rs (A); F¯−,r 1 ,...,rs

.

A¯ tα Aβ = 0 for all α = β}, s (A) = {(A1 , . . . , As ) ∈ Idem−,r1 (A) × · · · × Idem−,rs (A); F˜−,r 1 ,...,rs

.

Aα Aβ = 0 for all α = β},

4.2 i-Flag Varieties

185

s (A) = {(A1 , . . . , As ) ∈ Idemn,r1 (A) × · · · × Idemn,rs (A); F¯n;r 1 ,...,rs

.

A¯ tα Aβ = 0 for all α = β}, s (A) = {(A1 , . . . , As ) ∈ Idemn,r1 (A) × · · · × Idemn,rs (A); F˜n;r 1 ,...,rs

.

Aα Aβ = 0 for all α = β}. Furthermore, we define F¯ns (A) = {(A1 , . . . , As ) ∈ Idemn (A); A¯ tα Aβ = 0 for all α = β},

.

F˜ns (A) = {(A1 , . . . , As ) ∈ Idemn (A); Aα Aβ = 0 for , all α = β}

.

and F¯n (A) =

n 

.

s=1

F¯ns (A), F˜n (A) =

n 

F˜ns (A).

s=1

s s Observe that .F¯ (A), .F˜ (A), .F¯−,r (A) and .F˜−,r (A) are closed metric 1 ,...,rs 1 ,...,rs s s s s ¯ ˜ subspaces of the .R-vector space .M(A) , while .Fn (A), .Fn (A), .F¯n;r (A) and 1 ,...,rs s s ˜ .F (A) are closed metric subspaces of .Mn (A) . n;r1 ,...,rs

¯ s (A) (respectively .BU ˜ s (A)) as the quotient of .F¯ s (A) Finally, we also define .BU (respectively .F˜ s (A)) by the equivalence relation .(C1 , . . . , Cs ) ∼ (C1 , . . . , Cs ), provided there exist .m, n such that .τ m Cα = τ n Cα , .α = 1, . . . , s, and .T BU s (A) as the quotient of .T F s (A) by the equivalent relation .((A1 , . . . , As ), (B1 , . . . , Bs )) ∼ ((A 1 , . . . , A s ), (B1 , . . . , Bs )) provided there exist .m, n ≥ 1 such that .τ m Aα =     0 0 0 0 = n for .α = 1, . . . , s. τ n A α and . m 0 Bα 0 Bα We clearly have 1 1 1 1 (A) = F˜n;r (A) = Idemn,r (A), F¯−;r (A) = F˜−;r (A) = Idem−,r (A), F¯n;r

.

¯ 1 (A) = BU ˜ 1 (A) = BU ¯ (A). F¯n1 (A) = F˜n1 (A) = Idemn (A), BU

.

Lemma 4.15 yields the following generalization of Theorem 4.16: s s Theorem 4.20 The map .T Fn;r (A) → F¯n;r (A), given by 1 ,...,rs 1 ,...,rs

((A1 , . . . , As ), (B1 , . . . , Bs )) → (A1 B1 + A1 , . . . , As Bs + As ),

.

s s is a homeomorphism with inverse .F¯n;r (A) → T Fn;r (A), given by 1 ,...,rs 1 ,...,rs

(C1 , . . . , Cs ) → ((A1 , . . . , As ), (B1 , . . . , Bs )),

.

186

4 More Classical Matrix Varieties

where .Bα = Cα + C¯ αt − 2Aα , .Aα = Cα (Cα + C¯ αt − In )−1 is the unique element in .Gn,rα (A) such that .Aα Cα = Cα and .Cα Aα = Aα , .α = 1, . . . , s. Remark 4.21 The homeomorphisms above extend to homeomorphisms .T F (A) → s s F¯ (A), restricting to homeomorphisms .T F−;r (A) → F¯−;r (A) and induce 1 ...,rs 1 ,...,rs s s ¯ (A). the homeomorphisms .T BU (A) → BU s We endow now .F¯n;r (A) with the smooth structure such that the homeo1 ,...,rs s s (A) → F¯n;r (A), is a diffeomorphism. morphism of Theorem 4.20, .T Fn;r 1 ,...,rs 1 ,...,rs s Then, it is clear that .F¯n;r1 ,...,rs (A) is a regular smooth submanifold of .Idemn,r1 (A)× · · · × Idemn,rs (A) and

.

s s dim F¯n;r (A) = 2 dim Fn;r (A) = d(2nr − r 2 − 1 ,...,rs 1 ,...,rs

s 

rα2 ),

α=1

where .r =

s

α=1 rα

and .d = 1, 2, 4.

s s Corollary 4.22 The inclusions .Fn;r (A) → F¯n;r (A) are strong 1 ,...,rs 1 ,...,rs s (A) → deformation retracts, extending to strong deformation retracts .F−;r 1 ,...,rs s s s ¯ ¯ F−;r1 ,...,rs (A) and inducing a strong deformation retract .BU (A) → BU (A).

Proof In fact, s s H : F¯n;r (A) × R → F¯n;r (A), 1 ,...,rs 1 ,...,rs

.

given by H ((B1 , . . . , Bs ), λ) = (1 − λ)(B1 , . . . , Bs )

.

+ λ(B1 (B1 + B¯ 1t − In )−1 , . . . , Bs (Bs + B¯ st − In )−1 ), is the required homotopy. Then, the proof follows.   i-Stiefel Maps We have the natural surjective map s ρr1 ,...,rs : V¯−,r (A) → F˜−;r (A) 1 ,...,rs

.

with .r = r1 + · · · + rs , given by .ρr1 ,...,rs (A1 | · · · |As , B1 | · · · |Bs ) (A1 B1t , . . . , As Bst ). These maps restrict to surjective maps s ρr1 ,...,rs : V¯n,r (A) → F˜n;r (A) 1 ,...,rs

.

with .r = r1 + · · · + rs ≤ n.

=

4.2 i-Flag Varieties

187

Observe that the maps above, being polynomial, are continuous and so s s (A) are connected and closed subspaces of .M(A)s , and .F˜n;r (A) F˜−;r 1 ,...,rs 1 ,...,rs s are closed connected subspaces of .Mn (A) . s Furthermore, .F˜−;r (A) are the connected components of .F˜ (A) and 1 ,...,rs ˜s .F (A) are those of .F˜n (A). .

n;r1 ,...,rs

i-Flags Varieties as Homogeneous Spaces Given a matrix A and .r = r1 + · · · + rs , we write .A = (A1 | · · · |As ), where .Aα is given by appropriate .rα -columns of A for .α = 1, . . . , s. We have a right action of .GLr1 (A) × · · · × GLrs (A) on .V¯−,r (A) with .r = r1 + · · · + rs given by ((A1 | · · · |As ), (B1 | . . . |Bs ))(X1 , . . . , Xs )

.

= ((A1 X1 | · · · |As Xs ), ((X1−1 B1t )t | · · · |(Xs−1 Bst )t )). This action restricts to an action on .V¯n,r (A). Furthermore, observe that for (A, B) = (A1 | · · · |As , B1 | · · · |Bs ) ∈ V¯n,r (A)

.

and (X1 , . . . , Xs ) ∈ GLr1 (A) × · · · × GLrs (A),

.

we have ρr1 ,...,rs ((A1 | · · · |As , B1 | · · · |Bs )(X1 , . . . , Xs )) = ρr1 ,...,rs (A, B).

.

We also have that if .ρr1 ,...,rs (A, B) = ρr1 ,...,rs (A , B ) then there exists .X = (X1 , . . . , Xs ) ∈ GLr1 (A) × · · · × GLrs (A) such that .(A , B ) = (A, B)X. In fact, take .Xα = Bαt A α , for .α = 1, . . . , s. Therefore, we have a bijection s ρr1 ,...,rs : V¯−,r (A)/GLr1 (A) × · · · × GLrs (A) → F˜−;r (A) 1 ,...,rs

.

restricting to a bijection s ρr1 ,...,rs : V¯n,r (A)/GLr1 (A) × · · · × GLrs (A) → F˜n;r (A). 1 ,...,rs

.

Since .ρr1 ,...,rs is continuous, this map is a homeomorphism. s s (A) (respectivly .F˜n;r (A)) with the homoThus, we may identify .F˜−;r 1 ,...,rs 1 ,...,rs geneous space GL(A)/GLr1 (A) × · · · × GLrs (A) × τ r GL(A),

.

188

4 More Classical Matrix Varieties

(respectively GLn (A)/GLr1 (A) × · · · × GLrs (A) × τ r GLn−r (A).)

.

s We use this identification to endow .F˜n;r (A) with a smooth structure so that 1 ,...,rs s ¯ ˜ .ρr1 ,...,rs : Vn,r (A) → F n;r1 ,...,rs (A) is a smooth principal bundle with the structure group .GLr1 (A) × · · · × GLrs (A). In particular, .

s dimR F˜n;r (A) = dimR V¯n,r (A) − dimR Mr1 (A) × · · · × Mrs (A) = 1 ,...,rs

dr(2n − r) − d(

s 

.

rα2 ) = d(2nr − r 2 −

α=1

s 

rα2 ).

α=1

The proof of the fact below, is completely analog to that of Lemma 3.26: Lemma 4.23 If .(A, B) = (A1 | · · · |As , B1 | · · · |Bs ) ∈ V¯n,r (A), .r = r1 + · · · + rs ≤ n then Ker(dρr1 ,...,rs )(A,B) = T(A.B) (ρr−1 (A1 B1t , . . . , As Bst )) = 1 ,...,rs

.

T(A,B) ((A, B)GLr1 (A)×· · ·×GLrs (A)) = ((A1 , B1 )Mr1 (A)| · · · |(As , Bs )Mrs (A)).

.

The proof of the result below is completely analog to that of Proposition 3.27. s Proposition 4.24 The space .F˜n;r (A) is a closed regular smooth submanifold 1 ,...,rs s of .Mn (A) .

Next, the proof of Proposition 3.28 yields: Proposition 4.25 s s (1) The tangent space to .F˜n;r (A) at .A = (A1 , . . . , As ) ∈ F˜n;r (A) is 1 ,...,rs 1 ,...,rs given by s TA F˜n;r (A) = {(B1 , . . . , Bs ) ∈ Mn (A)s ; Aα Bβ + Bα Aβ 1 ,...,rs

.

= δαβ Bα , 1 ≤ α, β ≤ s}. s s (2) For the normal space to .F˜n;r (A) at .A = (A1 , . . . , As ∈ F˜n;r (A) we 1 ,...,rs 1 ,...,rs have the inclusion s {(B1 , . . . , Bs ) ∈ Mn (A)s ; Aα Bα = Bα Aα , α = 1, . . . , s} ⊆ NA F˜n;r (A). 1 ,...,rs

.

4.2 i-Flag Varieties

189

i-Stiefel Varieties Over the Octonions It is natural to define i-Stiefel varieties over the octonions in the same way that for .A = R, C or .H. But again, things are more complicated due to lack of associativity. We define V¯n,r (O) = {(A, B) ∈ Mn,r (O); B t A = Ir },

.

which is a real algebraic affine variety. For the particular case of .r = 1, we generalizes Proposition 4.2 as follows: Proposition 4.26 The space .V¯n,1 (O) is a closed regular smooth submanifold of 2 .Mn,1 (O) and .

dimR V¯n,1 (O) = 8(2n − 1).

The tangent space at .(A, B) ∈ V¯n,1 (O) is given by TA,B V¯n,1 (O) = {(X, Y ) ∈ Mn,1 (O)2 ; B t X + Y t A = 0}.

.

Proof Define Uβ = {(A, B) ∈ V¯n,1 (O); aβ = 0},

.

⎛ ⎞ ⎛ ⎞ b1 a1 ⎜ .. ⎟ ⎜ .. ⎟ where .A = ⎝ . ⎠ , B = ⎝ . ⎠ . an bn Notice that the sets .U1 , . . . , Un forms an open covering of .V¯n,1 (O). Define now ϕβ : Uβ → O × · · · × (O − {0}) × · · · × O × On−1

.

by .(A, B) → (A, Bβ ) for .(A, B) ∈ Uβ , where .Bβ ∈ Mn−1 (O) is obtained by deleting the column .bβ from .B. The inverse .ϕβ−1 : O × · · · × (O − {0}) × · · · × O × On−1 → Uβ sends .(A, B ) → (A, B) for .(A , B) ∈ O × · · · × (O − {0}) × · · · × O × On−1 , where B is obtained from .B by inserting bβ =

.

 1 (a¯ β − (bα aα )a¯ β ). 2 |aβ | α=β

It is clear that transition functions are smooth and so we have a smooth structure on V¯n,1 (O) and .dimR V¯n,1 (O) = 8(2n − 1).

.

190

4 More Classical Matrix Varieties

Finally, as Proposition 4.2, the real vector space .{(X, Y ) ∈ Mn,1 (O)2 ; B t X + = 0} ⊆ T(A,B) V¯n,1 (O), and since it is clear that both spaces have same real dimension, they coincide and this completes the proof.   YtA

The following result presents an interesting relation between .V¯n,1 (O) and .T Vn,1 (O) and its proof is the same as that of Proposition 4.3: Proposition 4.27 There is a diffeomorphism T Vn,1 (O) × (R − {0}) → V¯n,1 (O) × {q ∈ K; Re(q) = 0}, ∼

f :

.

where .∼ is the equivalence relation given by .(X, Y, λ) ∼ (−X, Y, −λ) and 1 f ([X, Y, λ]) = (λX, (1 − λY¯ t X)X¯ + Y¯ , λX¯ t Y ) λ

.

T Vn,1 (O)×(R−{0}) . ∼

for .[X, Y, λ] ∈

Its inverse g is given by

1 1 g(A, B, q) = [ A, B¯ − 2 A(1 − q), λ] λ λ

.

for .(A, B, q) ∈ V¯n,1 (O) × {q ∈ K; Re(q) = 0}, where .λ2 = A¯ t A. Then, it is easily to check the following: Proposition 4.28 The normal space at .(A, B) ∈ V¯n,1 (O), with respect to the inner product in .Mn,1 (O), is given by NA,B V¯n,1 (O) = {(Aq, qB)}q∈O .

.

Remark 4.29 We also have the canonical orthogonal decomposition NA,B V¯n,1 (O) = R(A, B) ⊕ {(Aq, qB)}q∈O, Re(q)=0 }.

.

Proposition 4.30 The space .V¯n,2 (O) contains an open and dense subset U which is a regular smooth submanifold of .(Mn,2 (O))2 and .

dimR U = 32(n − 1).

The tangent space at .(A, B) ∈ U is given by TA,B V¯n,2 (O) = {(X, Y ) ∈ Mn,2 (O)2 ; B t X + Y t A = 0}.

.

4.2 i-Flag Varieties

191

Proof If .σ : (−ε, ε) → Mn,2 (O) is smooth path such that .σ (s) ∈ V¯n,2 (O) for all s ∈ (−ε, ε) with .σ (0) = (A, B) and .σ (0) = (X, Y ), we clearly have .σ2 (s)t σ1 (s)+ σ2 (s)t σ1 (s) = 0 for .σ (s) = (σ1 (s), σ2 (s)) ∈ V¯n,2 (O) ⊆ Mn,2 (O). In particular, taking .s = 0 we get .B t X +Y t A = 0. I Take now .A = B = 2 ∈ Mn,2 (O). Then, .B t X + Y t A = 0 for arbitrary .Y ∈ 0   X1 Mn,2 (O) and .X = with .X1 = −Y t I2 ∈ M2 (O) and any .X2 ∈ Mn−2,2 (O). X2 Therefore, once we prove that .V¯n,2 (O) is a smooth regular submanifold of .Mn,2 (O), we conclude that .

TA,B V¯n,2 (O) = {(X, Y ) ∈ Mn,2 (O)2 ; B t X + Y t A = 0}.

.

Next, define .Uαβ for .1 ≤ α, β ≤ n, as follows: ¯n,2 (O) belongs to .Uαβ provided .aα2 bβ1 = 0 and the maps .fαβ : O → .(A, B) ∈ V O, .gαβ : O → O given by −1 −1 fαβ (yβ1 ) = yβ1 − (((yβ1 aβ2 )aα2 )aα1 )aβ1 ,

.

−1 −1 gαβ (yα2 ) = yα2 − (((yα2 aα1 )aβ1 )aβ2 )aα2

.

for .yβ1 , yα2 ∈ O are isomorphisms of real vector spaces. It is clearly an open subset of .V¯n,2 (O). Then, define ϕαβ : Uαβ → (Mn,2 (O)\{X ∈ Mn,2 (O); xα1 xβ2 = 0}) × Mn−1,2 (O)

.

by .ϕαβ (A, B) = (A, B ), where .B is obtained by deleting .bα1 and .bβ2 from .B. The map .ϕαβ is clearly injective, because .bα1 and .bβ2 are given by the formulas bα1 = −(



.

−1 bγ 1 aγ 2 )aα2 , bβ2 = −(

γ =α



−1 bγ 2 aγ 1 )aβ1 .

γ =β

The image of .ϕαβ is given by {(A, B ) ∈ Mn,2 (O) × Mn−1,2 (O);

.

aα2 aβ1 = 0,



.

 γ =β

bγ 1 aγ 1 − ((

γ =α

bγ 2 aγ 2 − ((

 γ =β



−1 bγ 1 aγ 2 )aα2 )aα1 = 1,

γ =α −1 bγ 2 aγ 1 )aβ1 )aβ2 = 1}.

(*)

192

4 More Classical Matrix Varieties

−1 −1 The inverse, .ϕαβ : Im(ϕαβ ) → V¯n,2 (O), is given by .ϕαβ (A, B ) = (A, B), where B is obtained by inserting .bα1 and .bβ2 at places .α and .β respectively of the first and second column of .B , with .bα1 and .bβ2 given by the formulas .(∗). Thus, we have a homeomorphism .ϕαβ : Uαβ → Im(ϕαβ ). In order to show that .Im(ϕαβ ) is a regular submanifold of .Mn,2 (O)×Mn−1,2 (O), define

f : Mn,2 (O) × Mn−1,2 (O) → O2

.

as f (A, B ) = (



.

bγ 1 aγ 1 − 1 − ((

γ =α





−1 bγ 1 aγ 2 )aα2 )aα1 ,

γ =α

bγ 2 aγ 2 − 1 − ((

γ =β



−1 bγ 2 aγ 1 )aβ1 )aβ2 ).

γ =β

Then, we have then   −1 .(df )A,B (X, Y ) = ( (yγ 1 aγ 1 + bγ 1 xγ 1 ) − (( (yγ 1 aγ 2 + bγ 1 xγ 2 ))aα2 )aα1 + γ =α



((

.

γ =α

−1 −1 bγ 1 aγ 2 )(aα2 (xα2 aα2 )))aα1 − ((

γ =α

 .

−1 bγ 1 aγ 2 )aα2 )xα1 ,

γ =α

(yγ 2 aγ 2 + bγ 2 xγ 2 ) − ((

γ =β

((





−1 (yγ 2 aγ 1 + bγ 2 xγ 1 ))aβ1 )aβ2 +

γ =β



.

−1 −1 bγ 2 aγ 1 )(aβ1 (xβ1 aβ1 )))aβ2 − ((

γ =β



−1 bγ 2 aγ 1 )aβ1 )xβ2 ).

γ =β

Take xγ 1 = xγ 2 = yγ 1 = yγ 2 = 0

.

for .γ = α and .γ = β and xα1 = aα1 , xα2 = aα2 , xβ1 = aβ1 , xβ2 = aβ2 .

.

We have then −1 (df )A,B (X, Y ) = (yβ1 aβ1 + bβ1 aβ1 − ((yβ1 aβ2 + bβ1 aβ2 )aα2 )aα1 ,

.

−1 yα2 aα2 + bα2 aα2 − ((yα2 aα1 + bα2 aα1 )aβ1 )aβ2 ) =

.

4.2 i-Flag Varieties

193

−1 (fαβ (yβ1 )aβ1 + bβ1 aβ1 − ((bβ1 aβ2 )aα2 )aα1 , gαβ (yα2 )aα2 + bα2 aα2

.

−1 − ((bα2 aα1 )aβ1 )aβ2 ).

Now, using the definitions of .fαβ and .gαβ , we conclude that .(df )A,B is surjective, and so .(0, 0) is a regular value of f and therefore, .Im(ϕαβ ) is a regular submanifold of .Mn,2 (O) × Mn−1,2  (O). Finally, observe that .U = αβ Uαβ satisfies the required hypothesis and the proof follows.   ⎞ ⎛ |a11 | |a12 | ⎜ .. ⎟ has Remark 4.31 If .(A, B) ∈ V¯n,2 (O) and A is such that the matrix .⎝ ... . ⎠ rank two then .(A, B) ∈ U.

|an1 | |an2 |

It is easy to check the following: Proposition 4.32 The normal space at .(A, B) ∈ V¯n,2 (O), with respect to the inner product on .Mn,2 (O), is given by NA,B V¯n,2 (O) = {(AX, (XB t )t }X∈M2 (O) .

.

Remark 4.33 There is also the canonical orthogonal decomposition NA,B V¯n,2 (O) = R(A, B) ⊕ {(AX, (XB t )t }X∈M2 (O),Re tr(X)=0 .

.

We may imitate definitions (1) and (2) at the beginning of Sect. 3.2 to define octonionic i-flag varieties: s (O) = {(A1 , . . . , As ) F˜n;r 1 ,...,rs

.

∈ Idemn,r1 (O) × · · · × Idemn,rs (O); Aα Aβ = 0 for α = β} and s (O) = {(A1 , . . . , As ) F¯n;r 1 ,...,rs

.

∈ Idemn,r1 (O) × · · · × Idemn,rs (O); A¯ tα Aβ = 0 for α = β}. They are real affine algebraic varieties.

194

4 More Classical Matrix Varieties

Remark 4.34 If we replace standard by Jordan multiplication of matrices in the above definitions of i-flag varieties over the octonions, we get spaces bigger than the corresponding standard i-flag spaces: s (O) = {(A1 , . . . , As ) F¯n;r 1 ,...,rs

.

∈ Idemn,r1 (O) × · · · × Idemn,rs (O); A¯ tα Aβ = 0 for α = β} .

⊆ {(A1 , . . . , As ) ∈ Idemn,r1 (O) × · · · × Idemn,rs (O); A¯ tα ◦ Aβ = 0 for α = β}

and s (O) = {(A1 , . . . , As ) F˜n;r 1 ,...,rs

.

∈ Idemn,r1 (O) × · · · × Idemn,rs (O); Aα Aβ = 0 for α = β} .

⊆ {(A1 , . . . , As ) ∈ Idemn,r1 (O) × · · · × Idemn,rs (O); Aα ◦ Aβ = 0 for α = β}.

Chapter 5

Algebraic Generalizations of Matrix Varieties

We use Chaps. 1 and 2 to define and extend results of Chaps. 3 and 4 to matrix varieties over more general division algebras. That includes extending the classical definitions of Riemannian, Hermitian and symplectic manifolds. All along this chapter K is a formally real Pythagorean field and .A denotes either K, the complex K-algebra .C(K), the quaternion K-algebra .H(K) or the octonion K-algebra .O(K).

5.1 Varieties of Idempotent Matrices The following shows the natural algebraic generalization of all classical manifolds and spaces introduced in Chaps. 3 and 4:

Stiefel Varieties Here, we generalize the classical definitions given in Sect. 3.1. ¯ t A = Ir }. .Vn,r (A) = {A ∈ Mn,r (A); A If .A = K, then .Vn,r (K) is the zero set of the .r 2 polynomials  .

xαp xαq − δpq , for 1 ≤ p, q ≤ r.

α

If .A = C(K), then Vn,r (C(K)) = {A + Bi ∈ Mn,r (C(K); At A + B t B = Ir , At B − B t A = 0}

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Golasi´nski, F. Gómez Ruiz, Grassmann and Stiefel Varieties over Composition Algebras, RSME Springer Series 9, https://doi.org/10.1007/978-3-031-36405-1_5

195

196

5 Generalizations of Matrix Varieties

is the zero set of the .r 3r−1 2 polynomials   . (xαp xαq +yαp yαq )−δpq for 1 ≤ p, q ≤ r, (xαp yαq −yαq xαp ), 1 ≤ p < q ≤ r. α

α

If .A = H(K), then Vn,r (H(K)) = {A0 + A1 i + A2 j + A3 k ∈ Mn,r (H(K));

.

At0 A0 + At1 A1 + At2 A2 + At3 A3 = Ir , At0 A1 − At1 A0 + At3 A2 − At2 A3 =

.

At0 A2 − At2 A0 + At1 A3 − At3 A1 = At0 A3 − At3 A0 + At2 A1 − At1 A2 = 0}

.

and it is clearly a K-zero set. If .A = O(K), then Vn,r (O(K)) = {A0 +A1 i +A2 j +A3 k +A4 l +A5 il +A6 j l +A7 kl ∈ Mn,r (O(K));

.

(At0 − At1 i − At2 j − At3 k − At4 l − At5 il − At6 j l − At7 kl)

.

(A0 + A1 i + A2 j + A3 k + A4 l + A5 il + A6 j l + A7 kl) = Ir }

.

and it is again clear that it is a K-zero set. We also define .Vn (A) = Vn,1 (A) ∪ · · · ∪ Vn,n (A). Therefore, .Vn,r (A) (respectively .Vn (A)) are affine K-algebraic varieties and they have the Zariski, the natural and (if K has been ordered) the ordering topology, induced from .Mn,r (A) (respectively .Mn (A)).  .V−,r (A) = n≥r Vn,r (A), .V (A) = r≥1 V−,r (A) with the corresponding induced topology and .EU (A) = colimr V−,r (A), i.e., .EU (A) is the quotient of .V (A) by the equivalent relation .∼ given by .A ∼ B provided there exist integers m n .m, n ≥ 1 such that .τ A = τ B. We endow .EU (A) with the corresponding quotient topology. Suppose K has been given an order and so we have the corresponding d-distance and ordering topology, then the analog of Proposition 3.1 reads now: Proposition 5.1 The d-distance and quotient topologies on .EU (A) coincide. Furthermore, the analog of Lemma 3.2 and Corollary 3.3 hold also here. The map .f : Mn,r (A) → Hermr (A) given by .f (A) = A¯ t A is polynomial and it makes sense to consider its differential .(df )A : Mn,r (A) → Hermr (A) given by (df )A (X) = A¯ t X + X¯ t A,

.

and we have the analog of Proposition 3.4:

5.1 Varieties of Idempotent Matrices

197

Proposition 5.2 The differential .(df )A is a K-epimorphism if .A ∈ Vn,r (A), for A = K, C(K) or H(K) and arbitrary .n, r > 0; or for .A = O(K), but .r = 1 or .r = 2 and .n ≥ r. .

Tangent to Stiefel Varieties We define then the tangent space of .Vn,r (A) at A, as the K-vector space TA Vn,r (A) = Ker(df )A = {X ∈ Mn,r (A); A¯ t X + X¯ t A = 0}. Its dimension is . 12 dr(2n − r + 1) − r, .d = 1, 2, 4 or 8. We also define the total tangent space to .Vn,r (A) by

.

T Vn,r (A) = {(A, X) ∈ Vn,r (A) × Mn,r (A); A¯ t X + X¯ t A = 0}.

.

It is clearly an affine K-algebraic variety. Finally, we define for .A ∈ V−,r (A), TA V−,r (A) = {X ∈ M−,r (A); A¯ t X + X¯ t A = 0},

.

T V−,r (A) = {(A, X) ∈ V−,r (A) × M−,r (A); A¯ t X + X¯ t A = 0},

.

T V (A) = ∪r≥1 T V−,r (A),

.

T EU (A) = colimr T V−,r (A),

.

i.e., .T EU (A) is the quotient of .T V (A) by the equivalent relation  .(A,X) ∼  (B, Y) 0m 0 0 0 m n provided there exist integers .m, n ≥ 1 such that .τ A = τ B, . , = n 0 X 0 Y where .0m (respectively .0n ) represents the matrix 0 in .Mm (A) (respectively in .Mn (A)). We endow .T EU (A) with the corresponding quotient topology.

Normal to Stiefel varieties Consider now the inner product in .Mn,r (A) given by .X, Y = Re tr(X¯ t Y ) for .X, Y ∈ Mn,r (A). If .A = K, C(K) or H(K), then the analog of Proposition 3.8 holds: Proposition 5.3 The normal space at .A ∈ Vn,r (A), with respect to the above inner product is given by .NA Vn,r (A) = AHermr (A). We also define the total normal space of .Vn,r (A) by NVn,r (A) = {(A, AX) ∈ Vn,r (A) × Mn,r (A); X ∈ Hermr (A)}.

.

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5 Generalizations of Matrix Varieties

Then, as in Proposition 3.8 we have the direct orthogonal sum Mn,r (A) = TA Vn,r (A) ⊕ NA Vn,r (A).

.

(5.4)

Finally, we define for .A ∈ V−,r (A), NA V−,r (A) = A Hermr (A),

.

N V−,r (A) = {(A, AX) ∈ V−,r (A) × M−,r (A); X ∈ Hermr (A)},

.

NV (A) = ∪r≥1 NV−,r (A),

.

NEU (A) = colimr NV−,r (A),

.

i.e., .N EU (A) is the quotient of .NV (A) by the equivalent relation .(A, AX) ∼ m n (B, BY ) provided   there exist positive integers .m, n such that .τ A = τ B, 0m 0 0 0 . , where .0m (respectively .0n ) represents the matrix 0 in .Mm (A) = n 0 X 0 Y (respectively in .Mn (A)). We endow .N EU (A) with the corresponding quotient topology. Write now .Un (A) = Vn,n (A) and then we get the analog of Proposition 3.9: Proposition 5.5 The map .π : Un (A) → Vn,r (A), sending .(A|B) ∈ Un (A) to .A ∈ Vn,r (A) is a submersion, except possibly for the cases .A = O(K) and .3 ≤ r ≤ n. Here the word submersion means that .π is surjective and .(dπ )(A|B) is Kepimorphism for any .(A|B) ∈ Un (A). Corollary 5.6 The map .π : Vn,s (A) → Vn,r (A) is a fibre bundle with the fibre Vn−r,s−r (A) for .1 ≤ r < s ≤ n and .A = K, C(K) or H(K). Furthermore, the map .π : Vn,2 (O(K)) → Vn,1 (O(K)) = S8n−1 (K) is a fibre bundle with the fibre .Vn−1,1 (O(K)) = S8n−9 (K).

.

Remark 5.7 It is interesting to observe that, in particular, if we take .r = 1 and s = n ≥ 2, in the above corollary we get fibre bundles

.

S0 (K) → U2 (K) → S1 (K); S1 (K) → U2 (C(K)) → S3 (K);

.

S3 (K) → U2 (H(K)) → S7 (K) and S7 (K) → U2 (O(K)) → S15 (K).

.

Furthermore, notice that the first three bundles are principal, but the last one is not, because of .S7 (K) not being a group. We have analog Examples 3.16 and 3.17 giving explicit trivializations for the π π bundles .U2 (A) → V2,1 (A) = S2d−1 (K) and .Un (A) → Vn,1 (A) = Sdn−1 (K). Observe that the open subsets .Uα appearing there are Zariski open subsets.

5.1 Varieties of Idempotent Matrices

199

Then, Proposition 3.18 reads now as follows: Proposition 5.8 The map .π : Un (A) → Vn,r (A) given by .π(A|B) = A, for (A|B) ∈ Un (A) is a principal bundle with the structure group .Un−r (A), .A = K, C(K) or H(K).

.

r r Proof We have  thesubgroup .τ (Un−r (A)) ⊆ Un (A), where .τ (Un−r (A)) consists I 0 with .C ∈ Un−r (A). of matrices . r 0 C The action of .Un−r (A) on .Un (A), by right multiplication of .τ r (Un−r (A)), is obviously free and restricts to the fibres of .π :

  Ir 0 ) = π(A|BC) = π(A). .π((A|B) 0 C On the other hand, if .π(A|B) = π(A|B ) = A, then B and .B are orthonormal basis of .A⊥ and so, there exists .C ∈ Un−r (A) such that .B = BC and the proof is complete.   In particular .Vn,r (A) is homeomorphic to the homogeneous space Un (A)/τ r Un−r (A), and by definition of the colimit we have a principal bundle .π : U (A) → V−,r (A) with structure group .U (A) acting by right multiplication of .τ r (U (A)). Thus, .V−,r (A) is homeomorphic to the homogeneous space r .U (A)/τ U (A). .

Grassmann Varieties We generalize here the classical definitions given in Sect. 3.2 The Grassmannian over .A is the set .G(A) of all idempotent and hermitian matrices .X ∈ M(A). ¯ t = X, X2 = X, tr(X) = r}, and We also define .G−,r (A) = {X ∈ M−,r (A); X  so .G(A) = r≥1 G−,r (A), ¯ t = X, X2 = X}, .Gn (A) = {X ∈ Mn (A); X X¯ t = X, X2 = X, tr(X) = r}, and so .Gn (A) = n.Gn,r (A) = {X ∈ Mn,r (A);  r=0 Gn,r (A) and .G−,r (A) = n≥r Gn,r (A). Next, .BU (A) = colimr G−,r (A), i.e., .BU (A) is the quotient of .G(A) by the equivalent relation .∼ given by .A ∼ B provided there exist positive integers .m, n such that .τ m A = τ n B. It is clear that .Gn,r (A) and .Gn (A) are affine K-algebraic varieties as can be easily deduce from the definition of Grassmannian. Thus, they have Zariski topology as well as the natural and ordering topology (in case K is an ordered field). Those topologies extend as we know to .G−,r (A) and .G(A). We endow .BU (A) with the quotient topology.

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5 Generalizations of Matrix Varieties

The Stiefel Map For .A = K, C(K) or H(K) we define the Stiefel map .π : V (A) → G(A), by .π(A) = AA¯ t for .A ∈ V (A). It restricts to polynomial surjective maps V−,r (A) → G−,r (A), Vn (A) → Gn (A), Vn,r (A) → Gn,r (A).

.

Next, .π being polynomial, is continuous and so .Gn (A) and .Gn,r (A) are clearly closed subspaces of .Hermn (A).

Grassmannians as Homogeneous Spaces First, we continue with hypothesis of .A = O(K). We have a right action of .Ur (A) on .V−,r (A) given by .(A, B) → AB, for .A ∈ V−,r (A) and .B ∈ Ur (A). t In fact, .AB AB = B¯ t A¯ t AB = B¯ t B = Ir . This action restricts to an action on .Vn,r (A). Furthermore, observe that for .A ∈ t Vn,r (A), .B ∈ Ur (A) we have .π(AB) = π(A). In fact, .ABAB = AB B¯ t A¯ t = AA¯ t . We have also that for .π(A) = π(A ) there exists .B ∈ Ur (A) such that .A = AB. t t In fact, if .AA¯ t = A A¯ then .A = A A¯ A = A(A¯ t A ) and t t t A¯ t A A¯ t A = A¯ AA¯ t A = A¯ A A¯ A = Ir Ir = Ir .

.

Thus, .B = A¯ t A ∈ Ur (A) and .A = AB. Therefore, we have a bijection .π : V−,r (A)/Ur (A) → G−,r (A) restricting to a bijection .π : Vn,r (A)/Ur (A) → Gn,r (A). Since .π is continuous, it is easy to check that it is a homeomorphism. Thus, we may identify .G−,r (A), resp. .Gn,r (A), with the homogeneous space .U (A)/Ur (A)τ r (U (A)), resp. .Un (A)/Ur (A)τ r (Un−r (A)). and then .π : Vn,r (A) → Gn,r (A) is a principal bundle with structure group .Ur (A). Similarly, π : V−,r (A) −→ G−,r (A)

.

is a principal bundle with structure group .Ur (A). Furthermore, there is the action .Un (A) × Gn,r (A) → Gn,r (A) given by   ¯ t with .Ur (A)τ r Un−r (A) as the isotropy subgroup at . Ir 0 . .(U, A) → U AU 0 0 Since, we identify .Gn,r (A) with the homogeneous space .U (n (A)/Ur (A)τ r (Un−r (A)), it follows that .Un (A) acts transitively on .Gn,r (A).

5.1 Varieties of Idempotent Matrices

201

Tangent and Normal to Grassmannian Varieties By analogy to the classical case (check Proposition 3.28) we define the tangent space to .Gn,r (A) at .A ∈ Gn,r (A) by TA Gn,r (A) = {B ∈ Hermn (A); AB + BA = B}.

.

Its dimension is .dr(n − r) where .d = 1, 2, 4 or 8 depending on .A = K, C(K), H(K) or O(K). The normal space to .Gn,r (A) at .A ∈ Gn,r (A) is then NA Gn,r (A) = TA Gn,r (A)⊥

.

and so we have the orthogonal direct sum decomposition Hermn (A) = TA Gn,r (A) ⊕ NA Gn,r (A).

.

It is clear that .A ∈ NA Gn,r (A). In fact, AB + BA = B implies A(AB) + A(BA) = AB.

.

Hence Re tr(A(AB)) + Re tr(A(BA)) = Re tr(AB) yields

.

2Re tr(AB) = Re tr(AB) and so Re tr(AB) = 0.

.

Also we have .In − A ∈ NA Gn,r (A). In fact, if .B ∈ TA Gn,r (A), then (In − A)B = B − AB = BA

.

and so Re tr((In − A)B) = Re tr(BA) = 0.

.

Define then NA Gn,r (A) = {X ∈ NA Gn,r (A); tr(X) = 0 and Re tr(AX) = 0}

.

and we have the orthogonal direct sum decomposition Hermn (A) = TA Gn,r (A) ⊕ A ⊕ In − A ⊕ NA Gn,r (A).

.

202

5 Generalizations of Matrix Varieties

For .A = O(K) we get, as in Proposition 3.28, that the normal space to .Gn,r (A) at .A ∈ Gn,r (A) is given by NA Gn,r (A) = {B ∈ Hermn (A); AB = BA}.

.

Now, we define the total tangent space to .Gn,r (A) by T Gn,r (A) = {(A, B) ∈ Gn,r (A) × Hermn (A); AB + BA = B}.

.

It is clearly an affine K-algebraic variety. Also we define the total normal space to .Gn,r (A) by N Gn,r (A) = {(A, B) ∈ Gn,r (A) × Hermr (A); Re tr(BX)

.

= 0 for all X ∈ TA Gn,r (A)}. It is an affine K-algebraic variety. Next, we define for .A ∈ G−,r (A), TA G−,r (A) = {B ∈ Herm(A); AB + BA = B},

.

NA G−,r (A) = {B ∈ Herm(A); Re tr(BX) = 0 for all X ∈ TA G−,r (A)}

.

and the corresponding total spaces T G−,r (A) = {(A, B) ∈ G−,r (A) × Herm(A); AB + BA = B},

.

NG−,r (A) = {(A, B) ∈ G−,r (A) × Herm(A); Re tr(BX)

.

= 0 for all X ∈ TA G−,r (A)} and then T G(A) = ∪r≥1 T G−,r (A) and NG(A) = ∪r≥1 NG−,r (A).

.

Then, we have the direct orthogonal sum Herm(A) = T G(A) ⊕ NG(A).

.

Furthermore, we define .BU (A) = colimr G−,r (A), i.e., .BU (A) is the quotient of .G(A) = ∪r≥1 G−,r (A) by the equivalent relation .∼ given by .A ∼ B if there exist .m, n ≥ 1 such that .τ m A = τ n B. Observe that we have, as in the classical case, a well defined Stiefel map .π : EU (A) → BU (A) given by .π([A]) = [AA¯ t ] for .A ∈ EU (A).

5.1 Varieties of Idempotent Matrices

203

Then, we define .T BU (A) = colimr T G−,r (A), i.e., .T BU (A) is the quotient of T G(A) by the equivalent relation ∼ (B, Y ) provided there exist .m, n ≥ 1  .(A,X)  0 0 0 0 m n such that .τ m = A = τ n B and . . = 0 X 0 Y Finally, .N BU (A) = colimr NG−,r (A), i.e., .NBU (A) is the quotient of .NG(A) by the equivalent relation  .(A, X)  ∼(B, Y) provided there exist integers .m, n ≥ 1 0 0 0 0 such that .τ m A = τ n B, . m . = n 0 X 0 Y We endow .BU (A) and .NBU (A) with the corresponding quotient topologies.

.

Grassmann Varieties over K-Octonions This section generalizes the analog section of Chap. 3 and so we only recall the results and definitions except when something special is needed. The discussion above for .A, C(K), H(K) allows to define the Grassmannian G(O(K)) = {X ∈ M(O(K)); X¯ t = X, X2 = X}

.

over K-octonions .O(K) and Gn (O(K)) = {X ∈ Mn (O(K)); X¯ t = X, X2 = X},

.

G−,r (O(K)) = {X ∈ M(A); X¯ t = X, X2 = X, tr(X) = r},

.

Gn,r (O(K)) = {X ∈ Mn,r (A); X¯ t = X, X2 = X, tr(X) = r}.

.

It is clear (as for .A = K, C(K) or H(K)) that .G(O(K)) and .G−,r (O(K)) are closed subspaces of .Herm(O(K)) and .Gn (O(K)), and .Gn,r (O(K)) are closed subspaces of .Hermn (O(K)). We also define .BU (O(K)) as the quotient of .G(O(K)) by the equivalent.∼ relation given by .A ∼ B if and only .τ m A = τ n B for some .m, n ≥ 1. We have then a distance given by ˜ d([A], [B]) = d(A, B) =

.



A − B, A − B ,

where A and B are taken in the same .G−,r (O(K)) ⊆ Herm(O(K)). Furthermore, observe that distance and quotient topology coincide on .BU (O(K)). We have a polynomial map .Vn,r (O(K)) → Hermn (O(K)), given by .A → AA¯ t for .A ∈ Vn,r (O(K)) but we do not have, in general, a Stiefel map .Vn,r (O(K)) → Gn,r (O(K)) for lack of associativity. It is clear that .Gn,r (O(K)) are K-affine algebraic varieties.

204

5 Generalizations of Matrix Varieties

The following is clear for any .A = K, C(K), H(K) or O(K): Proposition 5.9 The map .Gn,r (A) → Gn,n−r (A) given by .A → In − A, for .A ∈ Gn,r (A) is an isometry .(in particular, a homeomorphism.) for all .A. 

.

Particular Cases (1) G1 (O(K)) = G1,0 (O(K)) ∪ G1,1 (O(K)), and obviously Herm1 (O(K)) = K, G1,0 (O(K)) = {0}, and G1,1 (O(K)) = {1}.  λ q   (2) Herm2 (O(K)) = ∈ M2 (O(K)); λ, μ ∈ K, q ∈ O(K) . q¯ μ Thus, it is a K-vector space of dimension  10.  λq Then, G2 (O(K)) consists of the matrices in Herm2 (O(K)) such that q¯ μ λ2 + |q|2 = λ, μ2 + |q|2 = μ, (λ + μ)q = q.

.

In particular, G2 (O(K)) = G2,0 (O(K)) ∪ G2,1 (O(K)) ∪ G2,2 (O(K)),

.

where  

 

00 10 .G2,0 (O(K)) = , G2,2 (O(K)) = 00 01 and G2,1 (O(K)) =

.

  1 1 + α p ; α 2 + |p|2 = 1, α ∈ K, p ∈ O(K }. p¯ 1 − α 2

Therefore, G2,1 (O(K)) is homeomorphic to the K-sphere S8 (K) = {(α, p) ∈ K × O(K); α 2 + |p|2 = 1}.

.

Observe that the linear affine injective map g : K × O(K) → Herm2 (O(K)), given by g(α, p) =

.

      1 1+α p 1 10 1 α p = + , p¯ 1 − α 2 2 01 2 p¯ −α

5.1 Varieties of Idempotent Matrices

205

has the image  as the hyperplane H of Herm2 (O(K)) going through the 1 10 point 2 and having the director subspace Herm2 (O(K))tr=0 = {A ∈ 01 Herm2 (O(K)); tr(A) = 0}. Consider the standard inner product in K × O(K) : (λ, p), (μ, q) = λμ + Re(pq) ¯

.

and the inner product 12 −, − in Herm2 (O(K))tr=0 : .

       1 1 λ p λ p μ q μ q Re , = Re tr = λμ + Re(pq). ¯ p¯ −λ p¯ −λ q¯ −μ q¯ −μ 2 2

8 Then, the map g : K ×O(K) → H is a K-linear affine isometry and g(S  (K))  = 1 0 G2,1 (O(K)) is the K 8-sphere in the hyperplane above with the center 12 and 01 of radius 12 . In particular, g : S8 (K) → G2,1 (O(K)) is a homeomorphism. Furthermore, in this case,

π : V2,1 (O(K)) = S15 (K) → G2,1 (O(K)) = S8 (K)

.

is a well defined bundle with fibre S7 (K). (3) ⎛ ⎞ γ a¯ b

⎝ ⎠ .Herm3 (O(K)) = a β c¯ ∈ M3 (O(K)); α, β, γ ∈ K, , a, b, c ∈ O(K) . b¯ c α Thus, it is a K-vector space of dimension 27, and then G3 (O(K)) consists of ⎛ ⎞ γ a¯ b the hermitian matrices A = ⎝ a β c¯ ⎠ such that the following relations hold: b¯ c α

.

⎧ ¯ = γ (1 − γ ), ⎪ aa ¯ + bb ⎪ ⎪ ⎪ ⎪ ⎪ aa ¯ + cc ¯ = β(1 − β), ⎪ ⎪ ⎪ ⎨bb ¯ + cc ¯ = α(1 − α), ⎪ ab = (1 − α − β)c, ¯ ⎪ ⎪ ⎪ ⎪ ⎪ bc = (1 − β − γ )a, ¯ ⎪ ⎪ ⎪ ⎩ ¯ ca = (1 − γ − α)b.

(*)

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5 Generalizations of Matrix Varieties

Writing ρ(A) = (ab)c, we get: Lemma 5.10 If A ∈ G3 (O(K)) then ρ(A) ∈ K and ρ(A) = (ab)c = a(bc) = (bc)a = b(ca) = (ca)b = c(ba) =

.

¯ ((1 − α − β)cc ¯ = (1 − β − γ )aa ¯ = (1 − γ − α)bb,

.

ρ(A) = (1 − α − β)(1 − β − γ )(1 − γ − α), except for A = 0 or A = I3 .

.

Lemma 5.11 The trace of any element A ∈ G3 (O(K)) is 0, 1, 2 or 3. If tr(A) = 0 then A = 0; if tr(A) = 3, then A = I3 ; ⎛ ⎞ γ a¯ b if tr(A) = 1 then A = ⎝ a β c¯ ⎠ such that b¯ c α ¯ = γ α, cc ¯ α + β + γ = 1, aa ¯ = βγ , bb ¯ = αβ, ab = γ c, ¯ bc = α a, ¯ ca = β b;

.

if tr(A) = 2, then tr(I3 − A) = 1 and so I3 − A is a matrix as before. Define ⎛ ⎞ q1

16 ⎝q2 ⎠ ∈ V3,1 (O(K)); qp ∈ K for p = 1, 2, 3. .Sp (K) = q3 Proposition 5.12 The spaces G3,0 (O(K)), G3,1 (O(K)), G3,2 (O(K)), G3,3 (O(K)) are the connected components of G3 (O(K)). The Stiefel map π : V3,1 (O(K)) = S 23 (K) → Herm3 (O(K)), given by π(A) = ¯ AAt for A ∈ V3,1 (O(K)) restricts to surjective maps π| : S16 p (K) → G3,1 (O(K)) for p = 1, 2, 3. The space G3,1 (O(K)) = G3,2 (O(K)) is called the K-Cayley plane or the Koctonionic projective plane. Theorem 5.13 If K is a Pythagorean formally real field then the group F4 (K) acts transitively on the Cayley plane G3,1 (O(K)). Proof Consider A ∈ G3,1 (O(K)). If we furthermore assume K being real closed, then by Theorem 2.48 there exists ϕ ∈ F4 (K) such that ϕ(A) is diagonal. This completes the proof because it is clear there are only three diagonal matrices in G3,1 (O(K)) : E11 , E22 , E33 and we can go from one to any other by ϕσ for a convenient permutation σ .

5.1 Varieties of Idempotent Matrices

207

In case, we do not assume that K is real closed we prove this result by using Lemma 2.53 as follows: by Proposition 2.47 we may suppose ⎛

⎞ γ z¯ λ .A = ⎝ z β 0 ⎠ . λ 0α with α, β, γ , λ ∈ K, and z ∈ C(K). Consider A ∈ G3,1 (O(K)). Since A2 = A and tr(X) = 1, we have

.

⎧ ⎪ α + β + γ = 1, ⎪ ⎪ ⎪ ⎪ ⎪γ 2 + z¯ z + λ2 = γ , ⎪ ⎪ ⎪ ⎨(β + γ )z = z, ⎪ β 2 + z¯ z = β, ⎪ ⎪ ⎪ ⎪ ⎪ λz = 0, ⎪ ⎪ ⎪ ⎩ 2 α + λ2 = α.

(5.14)

Certainly, we consider the cases λ = 0 or z = 0. Notice that (5.14) yields that α, β, γ are sums of squares. Furthermore, the fifth equation of (5.14) yields λ = 0 or z = 0. Now, we examine the following cases. (1): z = 0 and λ = 0. The fourth equation of (5.14) gives β = 0 or β = 1. If β = 1 then α+γ = 0. Since α, γ are sums of squares, we get that α = γ = 0. Then, the sixth equation of (5.14) gives λ = 0 which is a contradiction. Therefore, β = 0 and so ⎛

⎞ 1−α 0 λ .A = ⎝ 0 0 0⎠ λ 0α Next, consider ⎛

⎞ x 0 −y .P = ⎝ 0 1 0 ⎠ , y0 x with x, y ∈ K such that x 2 + y 2 = 1. Thus, P ∈ O3 (K) and, by (2) in the subsection "Some observations” of section "The exceptional group F4 (K)” of Chap. 1, the map Herm3 (O(K)) → Herm3 (O(K)) defined by X → P XP t for X ∈ Herm3 (O(K)) belongs to F4 (K).

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5 Generalizations of Matrix Varieties

We have ⎛

⎞ (1 − α)x 2 − 2λxy + αy 2 0 (1 − 2α)xy − λy 2 + λx 2 t ⎠. .P AP = ⎝ 0 0 0 2 2 2 2 (1 − 2α)xy − λy + λx 0 (1 − α)y + 2λxy + αx But, x 2 + y 2 = 1, so x = 0 or y = 0. Choose x = 0 and x, y such that y y λy 2 − (1 − 2α)xy − λx 2 = 0 or equivalently λ( )2 − (1 − 2α) − λ = 0. x x

.

Taking t=

.

1 − 2α ±

 (1 − 2α)2 + 4λ2 , 2λ

we have y = tx and x such that 1 . x=√ 1 + t2

.

Since the field K is Pythagorean, t, x ∈ K. (2): λ = 0 and z = 0. The first and third equations of (5.14) yield β + γ = 1 and α = 0. Thus, ⎛ ⎞ 1 − β z¯ 0 .A = ⎝ z β 0⎠ . 0 00 We show that there are ρ ∈ K and q ∈ C(K) with ρ 2 + qq ¯ = 1 and such that the matrix (U X)U¯ t = U (XU¯ t ) is diagonal for ⎛

⎞ ρ q¯ 0 ¯ t. .U = ⎝q −ρ 0⎠ = U 0 0 1 In fact, (U X)U ⎛ 2 ρ (1 − β ) + ρ(qz ¯ + z¯ q) + β qq ¯ ρ(1 − 2β )q¯ + qz ¯ q¯ − ρ 2 z¯ 2 ⎝ = (1 − β )qq ¯ − ρ(zq¯ + q z¯ ) + ρ 2 β (1 − 2β )ρq − ρ z + q z¯ q 0 0

.

⎞ 0 0⎠ 0

5.1 Varieties of Idempotent Matrices

209

Choose now ρ ∈ K and q ∈ C(K) such that z¯ q 2 + ρ(1 − 2β)q − ρ 2 z = 0.

.

In fact, take q=

.

ρ (−(1 − 2β) ± 2¯z



(1 − 2β)2 + 4¯zz.

Observe that (1 − 2β)2 + 4¯zz is square of some element of K, because K is Pythagorean. Then, take ρ such that 1 = qq ¯ + ρ2 =

.

ρ2 (−(1 − 2β) ± 4¯zz



(1 − 2β)2 + 4¯zz)2 + ρ 2 .

Again, because K is Pythagorean, there exists ρ ∈ K such that ρ2 =

.

4¯zz  . (−(1 − 2β) ± (1 − 2β)2 + 4¯zz)2 + 4¯zz

This completes the proof because it is clear there are only three diagonal matrices in G3,1 (O(K)): E11 , E22 , E33 and we can go from one to any other by ϕσ for a convenient permutation σ .   The following generalization of Corollary 3.36 is now obvious: Corollary 5.15 If K is a Pythagorean formally real field and X, Y ∈ G3,1 (O(K)), there exist then x, y ∈ K such that x 2 + y 2 = 1, ReX, Y = Re tr(XY ) = x 2 and 2 − ReX − Y, X − Y = 2y 2 . In particular, the square of the diameter of G3,1 (O(K)) is 2. Furthermore, we have the analog of Proposition 3.28: Proposition 5.16 (1) The tangent space to G2,1 (O(K)) at A ∈ G2,1 (O(K)) is given by TA G2,1 (O(K)) = {B ∈ Herm2 (O(K)); AB + BA = B}

.

and the normal space to G2,1 (O(K)) at A ∈ G2,1 (O(K)) is given by NA G2,1 (O(K)) = {B ∈ Herm2 (O(K)); AB = BA} = A ⊕ I2 − A ,

.

where A and I2 − A are respectively the one dimensional subspaces generated by A and I2 − A;

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5 Generalizations of Matrix Varieties

(2) The tangent space to G3,1 (O(K)) at A ∈ G3,1 (O(K)) is given by TA G3,1 (O(K)) = {B ∈ Herm3 (O(K)); AB + BA = B}.

.

(3) The normal space to G3,1 (O(K)) at A ∈ G3,1 (O(K)) is given by the orthogonal direct sum NA G3,1 (O(K)) = A ⊕ I3 − A ⊕ NA G3,1 (O(K)),

.

where A and I3 − A are respectively the one dimensional subspaces generated by A and I3 − A, and NA G3,1 (O(K)) is given by NA G3,1 (O(K)) = {B ∈ Herm3 (O(K)); AB + BA = 0 and trB = 0}.

.

Remark 5.17 Observe that if A = E11 and B ∈ Herm3 (O(K)), the orthogonal direct sum decomposition above is given by ⎛ ⎛ ⎞ ⎛ ⎞ 0 0 γ a¯ b 0 a¯ b α + β ⎝ ⎝ ⎠ ⎝ ⎠ (I3 − E11 ) + 0 β−α .B = a β c¯ = a 0 0 + γ E11 + 2 2 ¯b c α ¯b 0 0 0 c

⎞ 0 c¯ ⎠ .

α−β 2

Let ISO(G3,1 (O(K))) denote the group of K-linear isometries ϕ of Herm3 (O(K)) such that ϕ(G3,1 (O(K))) = G3,1 (O(K)). In particular, F4 (K) ⊆ ISO(G3,1 (O(K))) and (F4 (K))E11 ⊆ (ISO(G3,1 (O(K)))E11 . Theorem 5.13 yields that if K is a Pythagorean formally real field, ISO(G3,1 (O(K))) acts transitively on G3,1 (O(K)). As for the group F4 (see Lemma 3.39) we have: Lemma 5.18 ϕ(I3 ) = I3 for any ϕ ∈ ISO(G3,1 (O(K)))

A Canonical Decomposition for Matrices in Herm3 (O(K)) Consider ⎛

⎞ γ a¯ b .A = ⎝ a β c ¯⎠ , b¯ c α where .a, b, c ∈ O(K) and .α, β, γ ∈ K.

5.1 Varieties of Idempotent Matrices

211

Then, ⎛

⎞ ¯ (β + γ )a¯ + bc (γ + α)b + a¯ c¯ γ 2 + aa ¯ + bb 2 .A = ⎝(β + γ )a + c ¯b¯ β 2 + aa ¯ + cc ¯ (α + β)c¯ + ab ⎠ , ¯ + cc (γ + α)b¯ + ca (α + β)c + b¯ a¯ α 2 + bb ¯ and we have ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ b  a¯  γ  2 .A = ⎝ a ⎠ γ a ¯ b + ⎝β ⎠ a β c¯ + ⎝ c¯ ⎠ b¯ c α . α c b¯ Next, since K is Pythagorean, we may choose .λ1 , λ2 , λ3 ∈ K such that ¯ + γ 2 , λ22 = aa ¯ + cc λ21 = aa ¯ + bb ¯ + cc ¯ + β 2 , λ23 = bb ¯ + α2.

.

Then, as in Chap. 3, we have A2 = λ21 v1 v¯1t + λ22 v2 v¯2t + λ23 v3 v¯3t

.

with .vα v¯αt ∈ G3 (O(K)), α = 1, 2, 3. The analog of Lemma 3.40 holds: Lemma 5.19 The space .Herm3 (O(K)) is K-linearly generated by elements of G3,1 (O(K)).

.

We also have the analog of Proposition 3.41: Proposition 5.20 .ISO(G3,1 (O(K))) = F4 (K). Proof We already know that .F4 (K) ⊆ ISO(G3,1 (O(K))). Thus, we consider now that .ϕ ∈ ISO(G3,1 (O(K)). Let .A, B in .Herm3 (O(K)). To show that .ϕ(A ◦ B) = ϕ(A) ◦ ϕ(B), using Lemma 5.19, we certainly may assume that .A ∈ G3,1 (O(K)). Write B = B1 + λA + μ(I3 − A) + B2

.

with .B1 ∈ TA G3,1 (O(K)), .B2 ∈ NA G3,1 (O(K)). Then, since .ϕ is an isometry of .Herm3 (O(K)) restricting to .G3,1 (O(K)), we also have ϕ(B) = ϕ(B1 ) + λϕ(A) + μ(I3 − ϕ(A)) + ϕ(B2 )

.

G3,1 (O(K)). with .ϕ(B1 ) ∈ Tϕ(A) G3,1 (O(K)), .ϕ(B2 ) ∈ Nϕ(A)

212

5 Generalizations of Matrix Varieties

Then, A◦B =

.

1 B1 + λA 2

because .A ◦ B1 = 12 B1 and .A ◦ B2 = 0. Similarly, ϕ(A) ◦ ϕ(B) =

.

1 ϕ(B1 ) + λϕ(A) 2

and so, 1 ϕ(A) ◦ ϕ(B) = ϕ( B1 + λA) = ϕ(A ◦ B) 2

.

 

and the proof is complete.

Observe that any isometry .ϕ of .Herm3 (O(K)) restricting to .G3,1 (O(K)) and fixing .E11 , in particular any .ϕ ∈ (F4 (K))E11 , is completely determined by a pair .(ϕ1 , ϕ2 ), where .ϕ1 is the restriction of .ϕ to .TE11 G3,1 (O(K)) and .ϕ2 is the restriction of .ϕ to .NE 11 G3,1 (O(K)). We consider the inner product . 12 Re−, − in .Herm3 (O(K)), i.e., for .X, Y ∈ Herm3 (O(K)) we have .

1 1 ReX, Y = Re tr(XY ) ∈ K. 2 2

In this way, the map .f : O(K)2 → TE11 G3,1 (O(K)) ⊆ Herm3 (O(K)), given by ⎛

⎞ 0 a¯ b .f (a, b) = ⎝a 0 0⎠ , b¯ 0 0 is a K-linear isometry, once we consider .O(K)2 with its standard inner product: ¯ ). (a, b), (a , b ) = Re(aa ¯ + bb

.

In particular, we identify the unit sphere .S15 (K) in .O(K)2 with the unit sphere in the tangent space .TE11 G3,1 (O(K)). Observe that the diameter of .G3,1 (O(K)) as a metric subspace of .Herm3 (O(K)) with the inner product . 12 Re−, − is now 1 (check Corollary 5.15). The matrices

5.1 Varieties of Idempotent Matrices

213

A ∈ G3,1 (O(K)) at the maximum distance 1 of .E11 are clearly those matrices

.

⎛

⎞ 00 0 .A = ⎝0 β c ¯⎠ 0c α with .α + β = 1 and .cc ¯ = αβ. They can be identify with .G2,1 (O(K)) by sending B=

.

  β c¯ ∈ G2,1 (O(K)) c α

to ⎛ ⎞ 00 0 . ⎝0 β c ¯⎠ . 0c α Thus, with the matrices ⎛ ⎞ 00 0 1 1⎝ . (I3 − E11 ) + 0ρ x ⎠ 2 2 0 x¯ −ρ with .ρ ∈ K, .x ∈ O(K) and .ρ 2 + xx ¯ = 1. In particular, .G2,1 (O(K)) is contained in the hyperplane of .I3 −E11 ⊕NE 11 G3,1 (O(K)) going through . 12 (I3 −E11 ) and director subspace .NE 11 G3,1 (O(K)). It is the sphere with center . 12 (I3 − E11 ) and of radius . 12 contained in that hyperplane. Define now .h : S1 (K) × S15 (K) → G3,1 (O(K)) as follows: ⎛

⎞ x 2 xy a¯ xyb 2 ¯ y 2 ab⎠ .h(x, y, a, b) = ⎝xya y aa ¯ xy b¯ y 2 ab y 2 bb for .(x, y) ∈ S1 (K) and .(a, b) ∈ S15 (K). We also write .hx,y : S15 (K) → G3,1 (O(K)) given by .hx,y (a, b) = h(x, y, a, b), and .ha,b : S1 (K) → G3,1 (O(K)) given by .ha,b (x, y) = h(x, y, a, b). Define .N : S15 (K) → NE 11 G3,1 (O(K)) by h0,1 (a, b) =

.

1 (I3 − E11 ) + N (a, b). 2

214

5 Generalizations of Matrix Varieties

Properties (1) h(x, y, a, b) = h(x , y , a, b) if and only if (x , y ) = ±(x, y).

.

(2)   a .h(0, 1, a, b) = π( ), b¯ where π : V2,1 (O(K)) → G2,1 (O(K)) is the Stiefel map. (3) By using the homeomorphism g : S8 (K) → G2,1 (O(K)) given by   1 1+λ p .g(λ, p) = , p¯ 1 − λ 2 g −1 ◦ h(0, 1, −, −) : S15 (K) → S8 (K) is the Hopf map sending (a, b) ∈ S15 (K) to (2aa ¯ − 1, 2ab) ∈ S8 (K). (4) Define Pa,b as the K-plane through E11 and the director 2-dimensional Kvector space with basis ⎛

⎞ ⎛ ⎞ 0 a¯ b −1 0 0 . ⎝a 0 0⎠ , ⎝ 0 aa ¯ ab⎠ . ¯b 0 0 ¯ 0 ab bb Observe that this basis is orthonormal if we consider the inner product 1 2 Re−, − on Herm3 (O(K)). Then, ⎛ ⎞ ⎞ −1 0 0 0 a¯ b 2 .h(x, y, a, b) = E11 +xy ⎝a 0 0⎠+y ⎝ 0 aa ¯ ab⎠ ∈ Pa,b for all (x, y) ∈ S1 (K) ¯ 0 ab bb b¯ 0 0 ⎛

and clearly, the image of h(−, −, a, b) is the intersection Pa,b ∩ G3,1 (O(K)). We have the analog of Proposition 3.42: Proposition 5.21 Any element of ISO(G3,1 (O(K))) (in particular, any element of F4 (K)) is determined by its restriction to G3,1 (O(K)).

5.1 Varieties of Idempotent Matrices

215

Observe that d(h(x, y, a, b), h(x , y , a, b))2 =

.

1 tr(h(x, y, a, b) − h(x , y , a, b))2 2

= 1 − (xx + yy )2 . Therefore, d(h(x, y, a, b), E11 )2 = y 2

.

and d(h(x, y, a, b), h(0, 1, a, b))2 = 1 − y 2 = x 2 .

.

Thus, h(0, 1, a, b) is the point in h(−, −, a, b) at maximum distance 1 to E11 . Instead, d((x, y), (x , y ))2 = 2(1 − (xx + yy ))

.

and d((x, y), (1, 0))2 = 2(1 − x).

.

(5) Let ϕ be an isometry of Herm3 (O(K)) such that ϕ(E11 ) = E11 and ϕ(G3,1 (O(K))) = G3,1 (O(K)). Then, ϕ restricts to K-linear isometries ϕ1 : TE11 G3,1 (O(K)) → TE11 G3,1 (O(K)) and ϕ2 : NE 11 G3,1 (O(K)) → NE 11 G3,1 (O(K)). In fact, any such a ϕ restricts to an isometry of the “cut locus” of E11 , G2,1 (O(K)), i.e., the points of G3,1 (O(K)) at maximum distance 1 of E11 : matrices of the form ⎛

00 . ⎝0 β 0 p¯

⎞ 0 p⎠ α

with α + β = 1 and pp ¯ = αβ. Observe that G2,1 (O(K)) is not contained in NE 11 G3,1 (O(K)) what we have is G2,1 (O(K)) =

.

1 (I3 − E11 ) + S(K), 2

216

5 Generalizations of Matrix Varieties

where S(K) is the sphere of radius

1 2

in NE 11 G3,1 (O(K)), i.e.,

⎛ ⎞ 00 0 1 2 .S(K) = {⎝0 ρ x ⎠ ; ρ ∈ K, x ∈ O(K), ρ + xx ¯ = }. 4 0 x¯ −ρ Therefore, ⎛  00 .ϕ ⎝0 β 0 p¯

⎛ ⎞ ⎛ ⎞ 0 0 00 0 0  1 ⎝ ⎠ ⎝ ⎠ = 0 2 0 + ϕ( 0 β−α p 2 0 0 12 α 0 p¯

⎞ 0 p ⎠. α−β 2

Thus, ϕ restricts to a K-linear isometry ϕ2 on NE 11 G3,1 (O(K)) and so to a K-linear isometry ϕ1 on TE11 G3,1 (O(K)). (6) We consider the inner product 12 Re−, − in Herm3 (O(K)), i.e., for X, Y ∈ Herm3 (O(K)) we have .

1 1 ReX, Y = Re tr(XY ) ∈ K. 2 2

In this way, the map f : O(K)2 → TE11 G3,1 (O(K)) ⊆ Herm3 (O(K)), given by ⎛

⎞ 0 a¯ b .f (a, b) = ⎝a 0 0⎠ , b¯ 0 0 is an K-linear isometry, once we consider O(K)2 with its standard inner product: ¯ ). (a, b), (a , b ) = Re(aa ¯ + bb

.

In particular, f identifies the unit sphere S15 (K) in O(K)2 with the unit sphere S15 E11 (K) in the tangent space TE11 G3,1 (O(K)). Consider now ϕ ∈ ISO(G3,1 (O(K)))E11 and denote by ρ1 (ϕ) and ρ2 (ϕ) the restriction isometries of ϕ to TE11 G3,1 (O(K)) and NE 11 G3,1 (O(K)). Let θ (ϕ) = (θ1 (ϕ), θ2 (ϕ)) be the corresponding isometry of O(K)2 : ρ1 (ϕ) ◦ f = f ◦ θ (ϕ).

.

5.1 Varieties of Idempotent Matrices

217

Thus, ⎛

⎛ ⎞ ⎞ 0 a¯ b 0 θ1 (ϕ)(a, b) θ2 (ϕ)(a, b) ⎠ .ρ1 (ϕ)(⎝a 0 0⎠) = ⎝θ1 (ϕ)(a, b) 0 0 b¯ 0 0 θ2 (ϕ)(a, b) 0 0 for a, b ∈ O(K) and we have Lemma 5.22 If ϕ ∈ (F4 (K))E11 then ⎛

⎞ ⎛ ⎞ 0 0 0 0 0 0 .ϕ(⎝0 aa ¯ ab⎠) = ⎝0 θ1 (ϕ)(a, b)θ1 (ϕ)(a, b) θ1 (ϕ)(a, b)θ2 (ϕ)(a, b)⎠ . ¯ 0 ab bb 0 θ1 (ϕ)(a, b)θ2 (ϕ)(a, b) θ2 (ϕ)(a, b)θ2 (ϕ)(a, b) Then, we have: Proposition 5.23 Let ϕ be an K-linear isometry of Herm3 (O(K)) which fixes E11 . Then, the following conditions are equivalent: (1) ϕ ∈ (F4 (K))E11 = ISO(G3,1 (O(K))E11 ; (2) ϕ(h(x, y, a, b)) = h(x, y, θ1 (ϕ)(a, b), θ2 (ϕ)(a, b)), for all (x, y) ∈ S1 (K), (a, b) ∈ S 15 (K) and any isometry θ = (θ1 , θ2 ) of O(K)2 such that ⎞ ⎛ ⎞ 0 0 a¯ b θ1 (a, b) θ2 (a, b) .ϕ(⎝a 0 0⎠) = ⎝θ1 (a, b) 0 0 ⎠; ¯b 0 0 θ2 (a, b) 0 0 ⎛

15 (3) ρ1 (ϕ) : S15 E11 (K) → SE11 (K) is a bundle isometry with respect to the bundle N ◦ f −1 : S15 E11 (K) → S(K).

Let Iso(G3,1 (O(K))) be the group of isometries of G3,1 (O(K)). Then, we may state the analog of Proposition 3.45: Proposition 5.24 The restriction map ISO(G3,1 (O)) −→ Iso(G3,1 (O))

.

is a group isomorphism. As in Chap. 3, we define:

A Polynomial Inclusion S8 (K) → (F4 (K))E11 The idea for this subsection comes from [26, Part II, 14].

218

5 Generalizations of Matrix Varieties

Take now .S8 (K) = {(r, u) ∈ K × O(K); r 2 + uu ¯ = 1} and define a map g1 : S8 (K) → GLK (O(K)2 )

.

by ¯ au g1 (r, u)(a, b) = (ra + ub, ¯ − rb).

.

Observe that .g1 (r, u) is an involution, i.e., .g1 (r, u)2 = id for any .(r, u) ∈ S8 (K). Then, we have: Lemma 5.25 The map .g1 is an injective isometry .g1 : S8 (K) → SO(O(K)2 ) Define then a K-linear map: g˜ 1 (r, u) : TE11 G3,1 (O(K)) → TE11 G3,1 (O(K))

.

by ⎛ ⎞ ⎛ ⎞ ¯ au  0 a¯ b  0 r a¯ + bu ¯ − rb .g ˜ 1 (r, u) ⎝a 0 0⎠ = ⎝ra + ub¯ 0 0 ⎠. ¯ ¯b 0 0 ua ¯ − rb 0 0 Lemma 5.25 says that .g˜ 1 (r, u) is an involutive isometry for any .(r, u) ∈ S8 (K) and we have an injective map g˜ 1 : S8 (K) → SO(TE11 G3,1 (O(K))).

.

Then, we have N f −1 g˜ 1 (r, u) : S15 E11 (K) → S(K)

.

is given by ⎛ ⎞ ⎛ ⎞  0 a¯ b  000 1 −1 .(N f g˜ 1 (r, u)) ⎝a 0 0⎠ = − ⎝0 1 0⎠ + 2 001 b¯ 0 0 ⎛

⎞ 0 0 0 ¯ uu ¯ . ⎝0 r 2 aa ¯ + 2rRe(uab) ¯ + bb ¯ −r 2 ab + r(aa ¯ − bb)u + uabu⎠ . 2 2 ¯ ¯ ¯ − bb)u¯ + u(ab) ¯ u¯ r bb − 2rRe(uab) ¯ + aa ¯ uu ¯ 0 −r ab + r(aa

5.1 Varieties of Idempotent Matrices

219

15 Proposition 5.26 .g˜ 1 (r, u) : S15 E11 (K) → SE11 (K) is an isometry preserving the fibres of the bundle .N f −1 : S15 E11 (K) → S(K) and the induced map .g˜ 2 (r, u) : S(K) → S(K) is again an isometry.

We also have the generalization of Proposition 3.48: Proposition 5.27 (1) .g˜ 2 (r, u) is a symmetry for all .(r, u) ∈ S 8 (K). In particular, .det g˜ 2 (r, u) = 1. (2) If .g˜ 2 (r, u) = g˜ 2 (r , u ) for .(r, u), (r , u ) ∈ S8 (K) then .(r , u ) = ±(r, u). (3) .g˜ 2 (S8 (K)) generates .SO(NE 11 G3,1 (O(K))) Then, the following generalization of Proposition 3.49 holds with K being a Pythagorean formally real field: Proposition 5.28 (1) .ρ1 is a monomorphism. (2) The kernel of .ρ2 has two elements, I and .κ, where I is the identity and .κ is given by ⎞ ⎛ ⎞ ⎛  γ a¯ b  γ −a¯ −b ⎝ a β c¯ ⎠ = ⎝−a β c¯ ⎠ . .κ −b¯ c α b¯ c α (3) The image of .ρ2 coincides with .SO(NE 11 G3,1 (O(K))). (4) .(Iso(G3,1 (O(K)))E11 = (F4 (K))E11 = SpinK (9). Then, we get: Corollary 5.29 If K is a Pythagorean formally real field, the map .F4 (K) → G3,1 (O(K)) given by .ϕ → ϕ(E11 ) for .ϕ ∈ F4 (K) yields a bijection ≈

F4 (K)/SpinK (9) −→ G3,1 (O(K)).

.

Problem 5.30 As in Chap. 3, we do not know answers to the following questions: (1) what are the possible values of .tr : Gn (O(K)) → K for .n ≥ 4?; (2) what are the connected components of .Gn (O(K)) for .n ≥ 4?

220

5 Generalizations of Matrix Varieties

Flag Varieties As for the Grassmannians (see Proposition 3.25), we write .F s (A) for all finite sequences (A1 , . . . , As ) ∈ G(A)s

.

such that .Aα Aβ = 0 for .α = β. Similarly, we define the flag variety s F−,r (A) = {(A1 , . . . , As ) ∈ G−,r1 (A) × · · · × G−,rs (A); 1 ,...,rs

.

Aα Aβ = 0 for all α = β} and s Fn,r (A) = {(A1 , . . . , As ) ∈ Gn,r1 (A) × · · · × Gn,rs (A); 1 ,...,rs

.

Aα Aβ = 0 for all α = β}. s Observe that .F s (A) and .F−,r (A) are Zariski closed subspaces of the K1 ,...,rs ∞ vector space .Herm(A) , where .Herm(A)∞ is the direct sum of countable copies s of .Herm(A), while .Fn,r (A) are Zariski closed subspaces of .Hermn (A). Also 1 ,...,rs they could be taken with the natural or the ordering topology (in case of K being an ordered field).

Stiefel Maps over Flag Varieties We have canonical surjective maps ( Stiefel maps) s πr1 ,...,rs : V−,r (A) → F−,r (A) with r = r1 + · · · + rs 1 ,...,rs

.

given by .πr1 ,...,rs (A1 | · · · |As ) = (A1 A¯ t1 , . . . , As A¯ ts ) for .(A1 | · · · |As ) ∈ V−,r (A). The maps above restrict to surjective maps s πr1 ,...,rs : Vn,r (A) → Fn,r (A) with r = r1 + · · · + rs ≤ n. 1 ,...,rs

.

Observe that those maps, being polynomial, are continuous for Zariski and ordering topology.

5.1 Varieties of Idempotent Matrices

221

Flag Varieties as Homogeneous Spaces As in Sect. 3.3, we have a right action of .Ur1 (A) × · · · × Urs (A) on .V−,r (A) with r = r1 + · · · + rs given by

.

(A1 | · · · |As )(B1 , . . . , Bs ) = (A1 B1 | · · · |As Bs )

.

for .(A1 | · · · |As ) ∈ V−,r (A) and .(B1 , . . . , Bs ) ∈ Ur1 (A) × · · · × Urs (A). Therefore we have, a bijection s πr1 ,...,rs : V−,r (A)/Ur1 (A) × · · · × Urs (A), → F−,r (A) 1 ,...,rs

.

restricting to a homeomorphism s πr1 ,...,rs : Vn,r (A)/Ur1 (A) × · · · × Urs (A), → Fn,r (A). 1 ,...,rs

.

s s Hence, we may identify .F−,r (A) (respectively .Fn,r (A)) with the 1 ,...,rs 1 ,...,rs homogeneous space

U (A)/Ur1 (A) × · · · × Urs (A) × τ r U (A)

.

(respectively Un (A)/Ur1 (A) × · · · × Urs (A) × τ r Unr (A)).

.

s Therefore, .πr1 ,...,rs : Vn,r (A) → Fn,r (A) is a principal bundle with structure 1 ,··· ,rs group .Ur1 (A) × · · · × Urs (A). Then, we have the analog of Lemma 3.26:

Lemma 5.31 If .A = (A1 | · · · |As ) ∈ Vn,r (A) with .r = r1 + · · · + rs ≤ n then Ker(dπr1 ,...,rs )A = TA (πr−1 (A1 A¯ t1 , . . . , As A¯ ts )) = 1 ,...,rs

.

TA (AUr1 (A) × · · · × Urs (A)) = (A1 Skr1 (A)| · · · |As Skrs (A)).

.

Further, an analog to the proof of Proposition 3.28, leads to: Proposition 5.32 s s (1) The tangent space to .Fn,r (A) at .A = (A1 , . . . , As ) ∈ Fn,r (A) is 1 ,...,rs 1 ,...,rs given by s TA Fn,r (A) = {(B1 , . . . , Bs ) ∈ (Hermn (A))s ; Aα Bα + Bα Aα 1 ,...,rs

.

= Bα , α = 1, . . . , s};

222

5 Generalizations of Matrix Varieties

s s (2) the normal space to .Fn,r (A) at .A = (A1 , . . . , As ∈ Fn,r (A) is given 1 ,...,rs 1 ,...,rs by

NA Fn,r1 ,...,rs (A) = {(B1 , . . . , Bs ) ∈ (Hermn (A))s ;

.

Aα Bα = Bα Aα , α = 1, . . . , s}.

i-Grassmann and i-Stiefel Varieties Here .A = K, C(K), H(K) or O(K). We generalize now definitions of i-Stiefel varieties of Sect. 4.1 as follows: .

V¯−,r (A) = {(A, B) ∈ M−,r (A)2 ; B t A = Ir }, V¯ (A) =



V¯−,r (A),

r≥1 .

V¯n,r (A) = V¯−,r (A) ∩ Mn,r (A)2 , ¯ (A) = colimr V¯−,r (A), EU

.

¯ (A) is the quotient of .V¯ (A) by the equivalent relation .∼, where we say i.e., .EU that .(A, B) ∼ (A , B ) for .(A, B), (A , B ) ∈ V¯ (A) are equivalent if there exist integers .m, n ≥ 1 such that .τ m A = τ n A , .τ m B = τ m B . Also, we generalize the definition of i-Grassmann varieties of Sect. 4.1 as follows: .Idem(A) denotes the subspace of .M(A) given by the idempotent matrices, i.e., Idem(A) = {A ∈ M(A)| A2 = A}.

.

Observe then that the Grassmannian over .A, .G(A), is given by G(A) = Idem(A) ∩ Herm(A) = {A ∈ M(A)| A2 = A, A¯ t = A}.

.

Idem−,r (A) = {A ∈ Idem(A); Re tr(A) = r}

.

and so Idem(A) =



.

Idem−,r (A),

r≥1

Idemn,r (A) = Idem−,r (A) ∩ Mn (A)

.

5.1 Varieties of Idempotent Matrices

223

and ¯ (A) = colimr Idem−,r (A), BU

.

¯ (A) is the quotient of .Idem(A) by the equivalent relation .A ∼ B if there i.e., .BU exist integers .m, n ≥ 1 such that .τ m A = τ n B. We also consider .Idemn (A) = Idem(A) ∩ Mn (A) and we have Idemn (A) =

n 

.

Idemn,r (A)

r=0

It is clear that .Idemn,r (A) and .Idemn (A) are affine K-algebraic varieties as can be easily deduce from the definition of i-Grassmannian. But now, .Idemn,r (C(K)) and .Idemn (C(K)) are also affine .C(K)-algebraic varieties, i.e., zero set of polynomials in .C(K)[xαβ ]. Thus, they have Zariski topology as well as the natural and ordering topology (in case K is an ordered field). Those topologies extend, as we know, to .Idem−,r (A) and .Idem(A). ¯ (A) with the quotient topology. We endow .BU Furthermore, observe that the map .Mn (A) → Mn (A) given by .A → In − A for .A ∈ Mn (A) yields homeomorphisms of varieties ≈

≈

Idemn,r (A) → Idemn,n−r (A) and Gn,r (A) → Gn,n−r (A).

.

Now, consider the map .f¯ : (Mn,r (A))2 → Mr (A) given by f¯(A, B) = B t A

.

for .(A, B) ∈ (Mn,r (A))2 , where .n ≥ r ≥ 1 and .A = K, C(K), H(K) or O(K). Since .f¯ is polynomial, it makes sense to consider in general its differential 2 .(d f¯)A,B : (Mn,r (A)) → Mr (A). It is given then by (d f¯)A,B (X, Y ) = B t X + Y t A

.

for .(X, Y ) ∈ (Mn,r (A))2 . Notice that given .C ∈ Mr (A), for .A = K, C(K) or H(K), we have 1 1 (d f¯)A,B ( AC, ( CB t )t ) = C. 2 2

.

Hence, the analog of Proposition 4.1 holds: Proposition 5.33 The differential .(d f¯)A,B is an K-epimorphism if .(A, B) ∈ V¯n,r (A), for .A = K, C(K) or H(K) and arbitrary .n ≥ r ≥ 1.

224

5 Generalizations of Matrix Varieties

We define, by analogy to the classical case, the tangent of .V¯n,r (A) at .(A, B) ∈ V¯n,r (A) by T(A,B) V¯n,r (A) = Ker(d f¯)A,B = {(X, Y ) ∈ Mn,r (A)2 ; B t X + Y t A = 0}.

.

It is clear that we have a homeomorphism ≈ V¯n,r (A) −→ V¯n,r (A)

.

¯ A) ¯ for .(A, B) ∈ V¯n,r (A). given by .(A, B) → (B, We also define T V¯n,r (A) = {((A, B), (X, Y )) ∈ V¯n,r (A) × Mn,r (A)2 ; B t X + Y t A = 0},

.

T V¯−,r (A) = {((A, B), (X, Y )) ∈ V¯−,r (A) × M−,r (A)2 ; B t X + Y t A = 0},

.

T V¯ (A) =



.

T V¯−,r (A)

r≥1

and ¯ (A) = colimr T V¯−,r (A), T EU

.

¯ (A) is the quotient of .T V¯ (A) by the equivalent relation .∼ given by i.e., .T EU ((A, B), (X, Y )) ∼ ((A , B ), (X , Y ))

.

m n m n if  thereexist  numbers  that .τ A = τ A , .τ B = τ B ,  natural  .m,n such 0m 0 0 0 0 0 0 0 . = n ,. m = n . 0 X 0 X 0 Y 0 Y The following relation between .V¯n,1 (A) and .T Vn,1 (A), as in Proposition 4.3, holds also in this general situation:

Proposition 5.34 There is a homeomorphism f :

.

T Vn,1 (A) × (K\{0}) → V¯n,1 (A) × {q ∈ K; Re(q) = 0}, ∼

where .∼ is the equivalence relation given by .(X, Y, λ) ∼ (−X, Y, −λ), and 1 f ([X, Y, λ]) = (λX, (1 − λY¯ t X)X¯ + Y¯ , λX¯ t Y ) λ

.

for .[X, Y, λ] ∈

T Vn,1 (A)×(K\{0}) . ∼

5.1 Varieties of Idempotent Matrices

225

Its inverse map g is given by 1 1 g(A, B, q) = [ A, B¯ − 2 A(1 − q), λ], λ λ

.

where .λ2 = A¯ t A for .(A, B, q) ∈ V¯n,1 (A) × {q ∈ K; Re(q) = 0}. Observe that we have a natural homeomorphism Gn (A) → V¯n,n (A)

.

given by .A → (A, (A−1 )t ) for .A ∈ Gn (A). Furthermore, Corollary 4.5 gives now: Corollary 5.35 The map .ρ : GLn (A) = V¯n,n (A) → V¯n,r (A) is surjective for n > r. In particular, all maps .ρ : V¯n,s (A) → V¯n,r (A), .1 ≤ r < s ≤ n, are surjective.

.

In the following result is analog to Proposition 4.26: Proposition 5.36 The map .ρ : GLn (A) → V¯n,r (A) given by .(A1 |A2 , B1 |B2 ) → (A1 , B1 ) is a submersion for .(A1 |A2 , B1 |B2 ) ∈ GLn (A) = V¯n,n (A). Then we have, as in Corollary 4.8: Corollary 5.37 The map .ρ : V¯n,s (A) → V¯n,r (A) is a fibre bundle with fibre V¯n−r,n−s (A) for .1 ≤ r < s ≤ n.

.

Furthermore: Proposition 5.38 The map .ρ : GLn (A) → V¯n,r (A sending .(A1 |A2 , B1 |B2 ) → (A1 , B1 ) for .(A1 |A2 , B1 |B2 ) ∈: GLn (A) = V¯n,n (A) is a principal bundle with group .GLn−r (A) for .A = K, C(K) or H(K). In particular, .V¯n,r (A) is homeomorphic to the homogeneous space .GLn (A)/ n−r (A), and we have a principal bundle .ρ : GL(A) → V¯−,r (A) with the structure group .GL(A) acting by right multiplication of .τ r GL(A). Thus, .V¯−,r (A) is homeomorphic to the homogeneous space .GL(A)/τ r GL(A). Recall now the K-bilinear, symmetric and non-degenerate inner product on 2 .(M(A)) is given by τ r GL

(X, Y ), (X , Y ) = Re tr(XY t + X Y t ),

.

where .X, Y, X , Y ∈ M(A). Then, the following holds: Proposition 5.39 The normal space at .(A, B) ∈ V¯n,r (A) with respect to the above inner product in .(Mn,r (A))2 is given by NA,B V¯n,r (A) = {(AX, (XB t )t )}X∈Mr (A) .

.

226

5 Generalizations of Matrix Varieties

Furthermore, the map .Mr (A) → NA,B V¯n,r (A) which sends .X → (AX, (XB t )t ) for .X ∈ Mr (A) is an isomorphism of vector spaces, because if .(AX, (XB t )t ) = 0, then .AX = 0 and so .X = B t AX = 0. It is clear that for .(A, B) ∈ V¯n,r (A) we have the canonical orthogonal decomposition (Mn,r (A))2 = TA,B V¯n,r (A) ⊕ NA,B V¯n,r (A)

.

(5.40)

given by (M, N ) → (M − A

.

(A N

.

t A+B t M

2

, (N

t A+B t M

2

N t A + Bt M t t N t A + Bt M ,N − ( B ) )⊕ 2 2

B t )t ) for .(M, N ) ∈ (Mn,r (A))2 .

Remark 5.41 There is also the canonical orthogonal decomposition NA,B V¯n,r (A) = K(A, B) ⊕ {(AX, (XB t )t )}X∈Mr (A),Re tr(X)=0

.

given by (AX, (XB t )t ) =

.

Re tr(X) Re tr(X) Re tr(X) (A, B) + (A(X − Ir ), ((X − Ir )B t )t ). r r r

We also define in a similar way the total normal to i-Stiefel varieties: ¯ (A). N V¯−,r (A) N V¯ (A), .N V¯n,r (A) and .N EU

.

i-Stiefel Maps Define the i-Stiefel map ¯ r (A) = V¯−,r (A) → Idem−,r (A) = BU ¯ r (A), ρ : EU

.

¯ r (A) = V¯−,r (A) which restrict to given by .(A, B) → AB t for .(A, B) ∈ EU ρ : V¯n,r (A) → Idemn,r (A)

.

and leads to ρ : V¯ (A) → Idem(A),

.

5.1 Varieties of Idempotent Matrices

227

and finally we have ¯ (A) → BU ¯ (A) ρ : EU

.

¯ (A). given by .[A, B] → [AB t ] for .[A, B] ∈ EU All maps above are surjective. In fact, check Sect. 4.1.

i-Grassmannians as Homogeneous Spaces The free right action .V¯−,r (A) × GL(r, A) → V¯−,r (A) given by (A, B).X = (AX, (X−1 B t )t )

.

induces a bijection from the orbit space of .V¯−,r (A) to .Idem−,r (A) and so we have also bijections from the orbit spaces of .V¯n,r (A) to .Idemn,r (A). Therefore, B t A ∈ GL(r, K)

.

and we have (A, B).(B t A ) = (AB t A , (B AB t )t ) = (A , B ). t

.

Thus, we have a bijection .ρ : V¯−,r (A)/GLr (A) → Idem−,r (A) restricting to a bijection .ρ : V¯n,r (A)/GLr (A) → Idemn,r (A). Since .ρ is continuous, this map is a homeomorphism. Similarly, ρ

E U¯ r (A) = V¯−,r (A) → Idem−,r (A) = B U¯ r (A)

.

is a principal bundle with the structure group .GLr (A). Further observe that .GLn (A) acts on .Idemn,r (A) by .A → U AU −1 . The isotropy   Ir 0 subgroup at . is .GLr (A)τ r GLn−r (A), and so .GLn (A) acts transitively on 0 0 .Idemn,r (A). The following is an analog of Lemmas 3.26 and 4.12: Lemma 5.42 If .(A, B) ∈ V¯n,r (A) then Ker(dρ)A,B = TA,B ρ −1 (AB t ) = TA,B ((A, B)GLr (A)) =

.

{(AX, −(XB t )t ); X ∈ Mr (A)}.

.

228

5 Generalizations of Matrix Varieties

We have the analog of Proposition 4.14: Proposition 5.43 (1) The tangent space to .Idemn,r (A) at .A ∈ Idemn,r (A) is given by TA Idemn,r (A) = {B ∈ Mn (A); AB + BA = B}.

.

(2) The normal space to .Idemn,r (A) at .A ∈ Idemn,r (A) is given by NA Idemn,r (A) = {B ∈ Mn (A); AB = BA}.

.

Then, we have the direct orthogonal decomposition Mn (A) = TA Idemn,r (A) ⊕ NA Idemn,r (A)

(5.44)

.

for .A ∈ Idemn,r (A). We extend the analogous classical definitions to define the tangent to iGrassmann varieties: ¯ (A) and the normal to i.T Idem−,r (A), .T Idem(A), .T Idemn,r (A), .T BU Grassmann varieties: ¯ (A). .NIdem−,r (A), .NIdem(A), .NIdemn,r (A) and .N BU The following results, the analog of Lemma 4.15 and Theorem 4.16, hold also here and give an important relation between Grassmannians and i-Grassmannians (see [11] and [12]). Lemma 5.45 If .B ∈ Idemn (A) then: (1) .B + B¯ t − In is invertible; (2) .B(B + B¯ t − In )−1 = (B + B¯ t − In )−1 B¯ t ; (3) .B(B + B¯ t − In )−1 ∈ Gn (A), and it is the unique element A of .Gn (A) such that .AB = B and .BA = A; (4) .Re trA = Re trB. (5) Suppose that .B, B ∈ Idemn (A) and A, .A are the unique element of .Gn (A) such that .AB = B, .BA = A, .A B = B , .B A = B , then .AA = 0 if and only if .B¯ t B = 0. Theorem 5.46 The map .T Gn,r (A) → Idemn,r (A) given by .(A, B) → AB + A for .(A, B) ∈ T Gn,r (A) is a polynomial homeomorphism .(with respect to the order and Zariski topologies.) with the inverse .Idemn,r (A) → T Gn,r (A) given by .C → (A, B) for .C ∈ Idemn,r (A) is a regular map, where .B = C + C¯ t − 2A and A is the unique element in .Gn,r (A) such that .AC = C and .CA = A. Explicitly, ¯ t − In )−1 . .A = C(C + C Remark 5.47 The homeomorphism above extend to homeomorphisms ≈

T G(A) = {(A, B) ∈ G(A) × Herm(A); AB + BA = B} → Idem(A)

.

5.1 Varieties of Idempotent Matrices

229

given by .(A, B) → AB + B for .(A, B) ∈ T G(A), restricting to homeomorphisms T G−,r (A) = {(A, B) ∈ G−,r (A) × Herm(A); AB + BA = B} → Idem−,r (A)

.

and inducing homeomorphisms ≈

¯ (A) T BU (A) → BU

.

given by .[(A, B)] → [AB + B] for .[(A, B)] ∈ T BU (A). Here, we define .T BU (A) = colimr T G−,r (A), i.e., .T BU (A) is the quotient of T G(A) by the equivalent relation .(A, B) ∼ (A , B )provided  there exist integers 0 0 0 0 m n m = A = τ n A and . .m, n ≥ 1 such that .τ . = 0 B 0 B Then, the analog of Corollary 4.18 holds: .

Corollary 5.48 The inclusions .Gn,r (A) → Idemn,r (A) are strong deformation retracts .(with respect to the order and Zariski topologies.), extending to strong deformation retracts .G−,r (A) → Idem−,r (A) and inducing a strong deformation ¯ (A). retract .BU (A) → BU

i-Flag Varieties Here .A = K, C(K), H(K) or O(K). We generalize now definitions of i-flag varieties of Sect. 4.2 as follows: (1) .F¯ s (A) is the set of all finite sequences (A1 , . . . , As ) ∈ Idem(A)s

.

such that .A¯ tα Aβ = 0 for all .α = β; (2) .F˜ s (A) is the set of all finite sequences (A1 , . . . , As ) ∈ Idem(A)s

.

such that .Aα Aβ = 0 for all .α = β. Observe that if we replace in definition (1) the conditions .A¯ tα Aβ = 0, for all .α = β, by .Aα A¯ tβ = 0, we obtain a homeomorphic space, where the homeomorphism is given by .(A1 , . . . , As ) → (A¯ t1 , . . . , A¯ ts ).

230

5 Generalizations of Matrix Varieties

Similar to Remarks 3.55, 3.56 and 4.19 nothing new is obtained, for .A = K, C(K), H(K), if we replace the standard multiplication by the Jordan multiplication in the definitions above, because of the following: Remark 5.49 Suppose that A and B belong to .M(A), with either A or B in Idem(A), and .A ◦ B = 0. Then .AB = BA = 0.

.

Then, we set F¯ (A) =



.

F¯ s (A), F˜ (A) =

s≥1



F˜ s (A)

s≥1

Similarly, we define s (A) = {(A1 , . . . , As ) ∈ Idem−,r1 (A) × · · · × Idem−,rs (A); F¯−,r 1 ,...,rs

.

A¯ tα Aβ = 0 for all α = β}, s (A) = {(A1 , . . . , As ) ∈ Idem−,r1 (A) × · · · × Idem−,rs (A); F˜−,r 1 ,...,rs

.

Aα Aβ = 0 for all α = β}, s (A) = {(A1 , . . . , As ) ∈ Idemn,r1 (A) × · · · × Idemn,rs (A); F¯n;r 1 ,...,rs

.

A¯ tα Aβ = 0 for all α = β}, s (A) = {(A1 , . . . , As ) ∈ Idemn,r1 (A) × · · · × Idemn,rs (A); F˜n;r 1 ,...,rs

.

Aα Aβ = 0 for all α = β}. Furthermore, we define F¯ns (A) = {(A1 , . . . , As ) ∈ Idemn (A); A¯ tα Aβ = 0 for all α = β},

.

F˜ns (A) = {(A1 , . . . , As ) ∈ Idemn (A); Aα Aβ = 0 for all α = β}

.

and F¯n (A) =

n 

.

F¯ns (A), F˜n (A) = ∪ns=1 F˜ns (A).

s=1 s s Observe that .F¯ (A), .F˜ (A), .F¯−,r (A) and .F˜−,r (A) are closed subspaces 1 ,...,rs 1 ,...,rs s s s s s ¯ ˜ ¯ (A) of the K-vector space .M(A) , while .Fn (A), .Fn (A), .Fn;r1 ,...,rs (A) and .F˜n;r 1 ,...,rs s are closed subspaces of .Mn (A) .

5.1 Varieties of Idempotent Matrices

231

¯ s (A) (resp. .BU ˜ s (A)) as the quotient of .F¯ s (A) Finally, we also define .BU s (respectively .F˜ (A)) by the equivalence relation .(C1 , . . . , Cs ) ∼ (C1 , . . . , Cs ), provided there exist integers .m, n ≥ 1 such that .τ m Cα = τ n Cα for .α = 1, . . . , s, and .T BU s (A) as the quotient of .T F s (A) by the equivalent relation ((A1 , . . . , As ), (B1 , . . . , Bs )) ∼ ((A 1 , . . . , A s ), (B1 , . . . , Bs ))

.

provided there exist integers .m, n ≥ 1 such that τ m Aα = τ n A α and

.

    0 0 0m 0 = n for α = 1, . . . , s. 0 Bα 0 Bα

We clearly have 1 1 1 1 (A) = F˜n;r (A) = Idemn,r (A), F¯−;r (A) = F˜−;r (A) = Idem−,r (A), F¯n,r

.

¯ 1 (A) = BU ˜ 1 (A) = BU ¯ (A). F¯n1 (A) = F˜n1 (A) = Idemn (A), BU

.

Lemma 5.45 yields the following generalization of Theorem 4.20: s s Theorem 5.50 The map .T Fn;r (A) → F¯n;r (A), given by 1 ,...,rs 1 ,...,rs

((A1 , . . . , As ), (B1 , . . . , Bs )) → (A1 B1 + A1 , . . . , As Bs + As )

.

s for .((A1 , . . . , As ), (B1 , . . . , Bs )) ∈ T Fn;r (A) is a homeomorphism with the 1 ,...,rs s s inverse .F¯n;r1 ,...,rs (A) → T Fn;r1 ,...,rs (A) given by

(C1 , . . . , Cs ) → ((A1 , . . . , As ), (B1 , . . . , Bs ))

.

s for .(C1 , . . . , Cs ) ∈ F¯n;r (A), where .Bα = Cα + C¯ αt −2Aα , .Aα = Cα (Cα + C¯ αt − 1 ,...,rs In )−1 is the unique element in .Gn,rα (A) such that .Aα Cα = Cα and .Cα Aα = Aα , .α = 1, . . . , s.

Remark 5.51 The homeomorphisms above extend to homeomorphisms .T F (A) → s s F¯ (A), restricting to homeomorphisms .T F−;r (A) → F¯−;r (A) and induce 1 ...,rs 1 ,...,rs s s ¯ the homeomorphisms .T BU (A) → BU (A). s s Corollary 5.52 The inclusions .Fn;r (A) → F¯n;r (A) are strong 1 ,...,rs 1 ,...,rs s deformation retracts, extending to strong deformation retracts .F−;r (A) → 1 ,...,rs s s s ¯ ¯ F−;r1 ,...,rs (A) and inducing a strong deformation retract .BU (A) → BU (A).

232

5 Generalizations of Matrix Varieties

i-Stiefel Maps As in Sect. 4.2, for .A = K, C(K), H(K), we have the natural surjective map (iStiefel maps) s ρr1 ,...,rs : V¯−,r (A) → F˜−;r (A) 1 ,...,rs

.

with .r = r1 + · · · + rs , given by .ρr1 ,...,rs (A1 | · · · |As , B1 | · · · |Bs ) (A1 B1t , . . . , As Bst ). These maps restrict to surjective maps

=

s ρr1 ,...,rs : V¯n,r (A) → F˜n;r (A) 1 ,...,rs

.

with .r = r1 + · · · + rs ≤ n. Observe that the maps above, being polynomial, are continuous and so s ˜s ˜s .F −;r1 ,...,rs (A) are connected and closed subspaces of .M(A) , and .Fn;r1 ,...,rs (A) s are closed connected subspaces of .Mn (A) . s Furthermore, .F˜−;r (A) are the connected components of .F˜ (A) and 1 ,...,rs s ˜ ˜ .F (A) are those of .Fn (A). n;r1 ,...,rs

i-Flags Varieties as Homogeneous Spaces Again, for .A = K, C(K), H(K), we have, as in Sect. 3.2, a right action of GLr1 (A) × · · · × GLrs (A) on .V¯−,r (A) with .r = r1 + · · · + rs given by

.

((A1 | · · · |As ), (B1 | . . . |Bs ))(X1 , . . . , Xs )

.

= ((A1 X1 | · · · |As Xs ), ((X1−1 B1t )t | · · · |(Xs−1 Bst )t )). This action restricts to an action on .V¯n,r (A). Further, observe that for (A, B) = (A1 | · · · |As , B1 | · · · |Bs ) ∈ V¯n,r (A)

.

and (X1 , . . . , Xs ) ∈ GLr1 (A) × · · · × GLrs (A),

.

we have ρr1 ,...,rs ((A1 | · · · |As , B1 | · · · |Bs )(X1 , . . . , Xs )) = ρr1 ,...,rs (A, B).

.

5.1 Varieties of Idempotent Matrices

233

We also have that if .ρr1 ,...,rs (A, B) = ρr1 ,...,rs (A , B ), there exists .X = (X1 , . . . , Xs ) ∈ GLr1 (A) × · · · × GLrs (A) such that .(A , B ) = (A, B)X. In fact, take .Xα = Bαt A α , for .α = 1, . . . , s. Therefore, we have a bijection s ρr1 ,...,rs : V¯−,r (A)/GLr1 (A) × · · · × GLrs (A), → F˜−;r (A) 1 ,...,rs

.

restricting to a bijection s ρr1 ,...,rs : V¯n,r (A)/GLr1 (A) × · · · × GLrs (A), → F˜n;r (A). 1 ,...,rs

.

Since .ρr1 ,...,rs is continuous, this map is a homeomorphism. We also have the following results, similarly to those of Sect. 4.2: Lemma 5.53 If .(A, B) = (A1 | · · · |As , B1 | · · · |Bs ) ∈ V¯n,r (A), .r = r1 + · · · + rs ≤ n then Ker(dρr1 ,...,rs )(A,B) = T(A.B) (ρr−1 (A1 B1t , . . . , As Bst )) = 1 ,...,rs

.

T(A,B) ((A, B)GLr1 (A) × · · · × GLrs (A)) = ((A1 , B1 )

.

×Mr1 (A)| · · · |(As , Bs )Mrs (A)). s Proposition 5.54 The space .F˜n;r (A) is a closed subvariety of .Mn (A)s . 1 ,...,rs

Proposition 5.55 s s (1) The tangent space to .F˜n;r (A) at .A = (A1 , . . . , As ) ∈ F˜n;r (A) is 1 ,...,rs 1 ,...,rs given by s TA F˜n;r (A) = {(B1 , . . . , Bs ) ∈ Mn (A)s ; Aα Bβ + Bα Aβ 1 ,...,rs

.

= δαβ Bα , 1 ≤ α, β ≤ s}. s s (2) For the normal space to .F˜n;r (A) at .A = (A1 , . . . , As ∈ F˜n;r (A) we 1 ,...,rs 1 ,...,rs have the inclusion s {(B1 , . . . , Bs ) ∈ Mn (A)s ; Aα Bα = Bα Aα , α = 1, . . . , s} ⊆ NA F˜n;r (A). 1 ,...,rs

.

The Shuffle Product For .A = K, C(K), H(K), O(K), define injective K-linear maps odd : M(A) → M(A), even : M(A) → M(A)

.

234

5 Generalizations of Matrix Varieties

given by odd Aodd 2p−1,2q−1 = Apq and Aαβ = 0, otherwise;

.

and even Aeven 2p,2q = Apq and Aαβ = 0, otherwise

.

for .A ∈ M(A). The following properties are easily checked: ¯ odd = Aodd , .(A) ¯ even = Aeven , .(At )odd = (Aodd )t , .(At )even = (Aeven )t . (i) .(A) In particular, we have restrictions odd : Herm(A) → Herm(A)

.

and even : Herm(A) → Herm(A).

.

(ii) tr(A) = tr(Aodd ) = tr(Aeven ).

.

(iii) .(AB)odd = Aodd B odd , .(AB)even = Aeven B even . Furthermore, we have restrictions: odd : Idem(A) → Idem(A), even : Idem(A) → Idem(A),

.

and odd : G(A) → G(A), even : G(A) → G(A).

.

(iv) Aodd B even = Aeven B odd = 0.

.

(v) {Aodd , B odd } = {A, B} = {Aeven , B even },

.

Aodd , B odd = A, B = Aeven , B even

.

5.1 Varieties of Idempotent Matrices

235

and {Aodd , B even } = Aodd , B even = 0.

.

Define then the shuffle product .μ : M(A) × M(A) → M(A) by μ(A, B) = Aodd + B even

.

for .A, B ∈ M(A). Then, we have .μ(0, 0) = 0. Observe that, by properties above, we have restrictions μ : Idem(A) × Idem(A) → Idem(A), μ : Idem−,r (A) × Idem−,s (A)

.

→ Idem−,r+s (A) and .

μ : G(A) × G(A) → G(A), μ : G−,r (A) × G−,s (A) → G−,r+s (A).

¯ (A) × BU ¯ (A) → BU ¯ (A), (resp. .BU (A) × BU (A) → Finally, define .μ : BU BU (A)) as follows: μ([A], [B]) = [μ(A, B)] = [Aodd + B even ]

.

such that .A, B belong to .Idem−,r (A) (respectively to .G−,r (A)) for some r. The definition above is correct, i.e., it does not depend on the choices, because of the following property that holds for any .A, B ∈ M(A): μ(τ (A), τ (B)) = τ (A)odd + τ (B)even = τ 2 (Aodd + B even ) = τ 2 (μ(A, B)).

.

Similarly, we have shuffle products ¯ s (A) × BU ¯ s (A) → BU ¯ s (A), μ : BU s (A) × BU s (A) → BU s (A), BU

.

and ˜ s (A) → BU ˜ s (A). ˜ s (A) × BU BU

.

¯ (A), .BU s (A), .BU ¯ s (A) Finally, one checks easily that for .K = R, .BU (A), .BU s ˜ and .BU (A) are H -spaces with multiplication .μ, i.e., the maps given by .[A] → μ([A], 0) and .[A] → μ(0, [A]) are homotopic to the identity relative to 0 (see [39, Problem 32.19, p. 688]).

236

5 Generalizations of Matrix Varieties

Idempotent Maps Representing the Tangent Bundles to V¯ (A), V (A), Idem(A) and G(A) Following the classical construction of representing vector bundles by idempotent matrices and applying (5.4), (5.40) and (5.44), we define: ⎧ 2 2 ⎪ ⎪fV¯ : V¯−,r (A) → HomK ((M−,r (A)) , (M−,r (A)) ), ⎪ ⎪ ⎨f : V (A) → Hom (M (A), M (A)), V −,r K −,r −,r . ⎪ fI : Idem(A) → HomK (M(A), M(A)), ⎪ ⎪ ⎪ ⎩ fG : G(A) → HomK (Herm(A), Herm(A))

(5.56)

respectively, by: fV¯ (A, B)(M, N ) = (M − A

.

N t A + Bt M t t N t A + Bt M ,N − ( B)) 2 2

for .(A, B) ∈ V¯−,r (A) and .(M, N ) ∈ (M−,r (A))2 , fV (A)(M) = M −

.

AM¯ t A + AA¯ t M 2

for .A ∈ V−,r (A) and .M ∈ M−,r (A), fI (A)(M) = AM + MA − 2AMA

.

for .A ∈ Idem(A) and .M ∈ M(A), and fG (A)(M) = AM + MA − 2AMA

.

for .A ∈ G(A) and .M ∈ Herm(A). The following result is clear now: Theorem 5.57 (1) For .(A, B) ∈ V¯−,r (A), .fV¯ (A, B) is idempotent and .fV¯ represents the Ktangent bundle to .V¯−,r (A); (2) for .A ∈ V−,r (A), .fV (A) is idempotent and .fV represents the K-tangent bundle to .V−,r (A); (3) for .A ∈ Idem(A), .fI (A) is idempotent and .fI represents the K-tangent bundle to .Idem(A); (4) for .A ∈ G(A), .fG (A) is idempotent and .fG represents the K-tangent bundle to .G(A). Furthermore, if .A is commutative, .fI : Idem(A) → HomA (M(A), M(A)) represents the .A-tangent bundle to .Idem(A).

5.2 Atlas on Varieties of Matrices

237

Also, .fI (A) : M(A) → M(A) (resp. .fG (A) : Herm(A) → Herm(A)) is a symmetric map with respect to the inner product .Re{−, −} for all .A ∈ Idem(A) (resp. .A ∈ G(A)). In fact, for all .A ∈ Idem(A), and for all .C, D ∈ M(A), we have Re{fI (A)(C), D} = Re tr((AC + CA − 2ACA)D) =

.

Re tr(CDA) + Re tr(CAD) − Re tr(CADA) =

.

Re tr(C(AD + DA − 2ADA)) = Re tr(C(f (A)(D)) = Re{C, fI (A)(D)}.

.

Similarly, for all .A ∈ G(A), and for all .C, D ∈ Herm(A), we have Re{fG (A)(C), D} = Re{C, fG (A)(D)}.

.

5.2 Atlas on Varieties of Matrices In this section, .A = K, C(K) or .H(K).

Some Zariski Closed Subsets of M(A) For .A ∈ M(A), if the rank .rk(A2 ) = r, we define ZA = {B ∈ M(A)|rk(AB) < r or rk(BA) < r}.

.

In particular, .A ∈ / ZA . Thus, by Proposition 1.30 (2), .B ∈ ZA if and only if all minors of order r of AB and all minors of order r of BA are non-invertible. In case .A = K or .A = C(K), that is the same as saying that determinants of all minors of order r of AB and BA vanish, and if .A = H(K) that means that all determinants of order 2r of .χH(K) (AB) and .χH(K) (BA) vanish. In fact, see Proposition 1.30 (1). Therefore, .ZA is Zariski K-closed for any .A, and it is Zariski .C(K)-closed for .A = C(K). Observe that UA = M(A)\ZA = {B ∈ M(A); rk(AB) = rk(BA) = r}

.

for all .A ∈ M(A) is a Zariski open set which contains A.

238

5 Generalizations of Matrix Varieties

It is also clear that we have ZBAB −1 = BZA B −1 and UBAB −1 = BUA B −1

.

for all .A ∈ M(A) and .B ∈ GL(A). Let .Ir = (aij ) be the matrix with .aii = 1, .i = 1, . . . , r and .aij = 0 otherwise. Then, we check immediately that .UIr is the subset of matrices of the form  .

 A A , A A

with .A ∈ GL(r, A), . ∈ M−,r (A), . ∈ Mr,− (A). Namely, .UIr consist of matrices of rank r having invertible the minor of order r of the first r rows and columns. In particular, .UIr ∩ Mn (A) is the above set of matrices in .Mn (A) with .A ∈ GL(r, A), . ∈ Mn−r,r (A) and . ∈ Mr,n−r (A). It is clear that we have a polynomial bijection GL(r, A) × M−,r (A) × Mr,− (A) → UIr ,

.

which sends .(A, ,  ) to  .

 A A . A A

Its inverse is given by  .

 AC  (A, BA−1 , A−1 C), → BD

which is K-regular for any .A, and .A-regular if .A is commutative. In particular, we have a polynomial bijection GL(r, A) × Mn−r,r (A) × Mr,n−r (A) → UIr ∩ Mn (A),

.

which sends .(A, ,  ) to  .

 A A . A A

Its inverse is K-regular for any .A, .A-regular if .A is commutative, and it is given as above. Given .I = {i1 , . . . , ir } with .1 ≤ i1 < · · · < ir . We also denote by I the matrix having 1 at the .ik ik entry, .k = 1, . . . , r, and 0 otherwise. In particular .I ∈ G−,r (A).

5.2 Atlas on Varieties of Matrices

239

.UI = M(A)\ZI are those matrices of rank r which have invertible the minor of order r consisting of elements .(aiα iβ ) with .α, β ∈ {1, . . . , r}. Let V be the matrix of the right .A-linear map of .An to itself, sending the canonical basis .e1 , . . . , er , er+1 , . . . , en to .ei1 , . . . , eir , eir+1 , . . . , ein , where .ir+1 , . . . , in is the ordered complement of .{i1 , . . . , ir }. Then, if .A ∈ UI , we have .V −1 AV ∈ UIr . Therefore

UI = UV Ir V −1 = V UIr V −1 .

.

Observe that .V ∈ Un (A), because its elements are either 1 or 0, and clearly V −1 = V t .

.

Atlas in Idem−,r (A) if .A2 = A then .ZA ∩ Idem(A) (respectively .UA ∩ Idem(A)) is a Zariski closed (respectively open) in .Idem(A), with respect to K in all cases and with respect to .A whenever .A is commutative. Observe that .UIr ∩ Idem(A) is the set of matrices of the form 

 A A , A A

.

with .A ∈ GL(r, A) such that .A−1 = Ir +  . In fact, the idempotency of  .

A A A A



is equivalent to the identity .A−1 = Ir +  . Thus, .UIr ∩ Idem(A) consists of those idempotent matrices having invertible the minor of the first r rows and columns. In the intersection with .Idem(A) we have the bijection ≈

VIr = {(,  ) ∈ M−,r (A) × Mr,− (A)|Ir +   ∈ GL(r, A)} → UIr ∩ Idem(A)

.

which sends .(,  ) to the matrix  .

 (Ir +  )−1 (Ir +  )−1  , (Ir +  )−1 (Ir +  )−1 

240

5 Generalizations of Matrix Varieties

with the inverse .ϕ given by  ϕ

.

 AC = (BA−1 , A−1 C) BD

  AC for . ∈ UIr ∩ Idem(A). BD Both maps are always K-regular and .A-regular, provided .A is commutative. If we restrict ourselves to .Mn (A) we have VIr ∩ Mn (A) = {(,  ) ∈ Mn−r,r (A) × Mr,n−r (A)|Ir +   ∈ GL(r, A)}

.

and so we get induced regular bijections VIr ∩ Mn (A) → UIr ∩ Idemn,r (A).

.

In particular, the above maps are always .R-analytic if .K = R and .C-analytic (holomorphic) if .A = C. If .A ∈ Idem−,r (A) there exists .B ∈ GL(A) such that .BAB −1 = Ir . Thus, the action by conjugation of .GL(A) in .Idem−,r (A) is transitive. Therefore, if .ϕ : UIr → VIr is given as above, we have a regular atlas in −1 ). .Idem−,r (A) given by .UA → VIr sending X to .ϕ(BXB The above atlas restricts to an atlas in .Idemn,r (A) = Idem−,r (A)∩Mn (A) which is always K-regular, and .A-regular if .A is commutative. Further, observe the induced structure in .Idemn,r (A), if .A is commutative, is that of a .A-regular submanifold of .Mn (A). In particular, the .C(K)-structure in .TA Idem(C(K)) is the one induced by that of .M(C(K)). Furthermore, if .F ∈ GL(C(K)), the .C(K)-regular bijection, Idem(C(K)) → Idem(C(K))

.

given by .A → F AF −1 for .A ∈ Idem(C(K)) induces .C(K)-linear isomorphisms ≈

TA Idem(C(K)) → TF AF −1 Idem(C(K)).

.

Another Atlas in Idem−,r (A) Let .A ∈ Idem−,r (A) and suppose .A ∈ Idemn,r (A). Let .i1 , . . . , ir with .1 ≤ i1 < · · · < ir ≤ n such that the rows .i1 , . . . , ir of A are K-linearly independents on the left, and set .ir+1 , . . . , in such that .1 ≤ ir+1 < · · · < in ≤ n and .{i1 , . . . , ir } ∪ {ir+1 , . . . , in } = {1, . . . , n}.

5.2 Atlas on Varieties of Matrices

241

Consider the following minors of A: B = (aαβ ) with .α, β ∈ {i1 , . . . , ir }. B is a minor of order r; C = (aαβ ) with .α ∈ {i1 , . . . , ir }, .β ∈ {ir+1 , . . . , in }. It is a minor of r rows and .n − r columns; .B = (aαβ ) with .α ∈ {ir+1 , . . . , in }, .β ∈ {i1 , . . . , ir }. It is a minor of .n − r rows and r columns; .C = (aαβ ) with .α, β ∈ {ir+1 , . . . , in }. It is a minor of order .n − r. . .

Using the fact that the rows .i1 , . . . , ir are K-linearly independents on the left, there exists a unique . ∈ Mn−r,r (A) such that .B = B y .C = C. On the other hand, idempotency of A is equivalent to (B + C)B = B, (B + C)C = C.

.

Thus, the matrix .B + C of order r is the identity in the subspace on the right of .Ar generated by the columns of B together with that of C. But, we know that subspace has dimension r. Therefore, .B + C is the identity in .Ar . Whence, .B + C = Ir . Therefore, we have B = Ir − C, B = (Ir − C), C = C.

.

Let .Uileft the set of matrices of .Idemn,r (A) such that have rows .i1 , . . . , ir .A1 ,...,ir linearly independents on the left. We have then a bijection ≈

Mr,n−r (A) × Mn−r,r (A) → Uileft 1 ,...,ir

.

sending .(C, ) to the matrix A with minors .B, B , C, C above. This bijection is polynomial toward the right and K-regular, .A-regular if .A is commutative, toward the left. Furthermore, observe that the .Uileft above are Zariski K-open, Zariski .A-open 1 ,...,ir if .A is commutative.  In this way, we get an atlas of .Idemn,r (A) consisting of the . nr open sets above, and by taking as coordinates the .r(n − r) elements of C together with the .(n − r)r of ., .2r(n − r) coordinates in total. right Analogously, we can define .Ui1 ,...,ir as the set of idempotent matrices with the .Alinearly columns independent .i1 , . . . , ir on the right. We get like this another atlas which is compatible with the one above. right ∩ Ui1 ,...,ir = UI , where .I = {i1 , . . . , ir } and .UI is the set of Clearly, .Uileft 1 ,...,ir idempotent matrices having invertible the minor with rows and columns .i1 , . . . , ir . It is easy to check that both atlas given above are compatible.

242

5 Generalizations of Matrix Varieties

Atlas in G−,r (A) If .A ∈ G−,r (A), we have ZA ∩ G−,r (A) = {B ∈ G−,r (A); rk(AB) < r}

.

and UA ∩ G−,r (A) = {B ∈ G−,r (A); rk(AB) = r}.

.

Also it is clear that B(ZA ∩ G−,r (A))B −1 = ZBAB −1 ∩ G−,r (A)

.

and B(UA ∩ G−,r (A))B −1 = UBAB −1 ∩ G−,r (A)

.

for all .B ∈ U (A). Observe that if .A ∈ G−,r (A) then there exists .B ∈ U (A) such that .BAB −1 = Ir . Thus, the conjugate action of .U (A) in .G−,r (A) is transitive. The intersection .UIr ∩ G−,r (A) is given by matrices of the form  .

¯t ¯ t )−1 (Ir +  ¯ t )−1  (Ir +  ¯t ¯ t )−1 (Ir +  ¯ t )−1  (Ir + 



with . ∈ M−,r (A). ¯ t  is already invertible: Observe that .Ir +  ¯ t v = −v, then .(v)t (v) = v¯ t  ¯ t v = −v¯ t v, and this if .v = 0 is such that . is impossible. ≈ In this way, we get a K-regular bijection .M−,r (A) → UIr ∩ G−,r (A) sending . to the matrix   ¯t ¯ t )−1 (Ir +  ¯ t )−1  (Ir +  . ¯t . ¯ t )−1 (Ir +  ¯ t )−1  (Ir +   Its inverse .ϕ, which is K-regular as well, is given by .ϕ   AC . ∈ UIr ∩ G−,r (A). BD By restricting to .Mn (A) we have K-regular bijection ≈

Mn−r,r (A) → UIr ∩ Gn,r (A)

.

AC BD



= BA−1 for

5.2 Atlas on Varieties of Matrices

243

and so a K-regular atlas in .Gn,r (A), UA ∩ Gn,r (A) → Mn−r,r (A)

.

given by .X → ϕ(BXB −1 ) for .X ∈ UA ∩ Gn,r (A), where .B ∈ Un (A) is such that −1 = I . .BAB r In particular, the above atlas is .R-analytic for .A = R.

Another Atlas in G−,r (A) Let .I = {i1 , . . . , ir }. Then, .UI ∩ Gn,r (A) = {B ∈ Gn,r (A); rk(B) = r} consists of those matrices of .Gn,r (A) whose minor of order r with rows and columns in .i1 , . . . , ir is invertible. On the other hand, these open sets .UI ∩ Gn,r (A) cover .Gn,r (A). Therefore, we get a subatlas of the one given above, just taking the . nr Zariski open sets above. As above, the matrices of .UI ∩ Gn,r (A) are  .

 ¯ t )−1 (Ir +  ¯ t )−1  ¯t (Ir +  ¯ t )−1 (Ir +  ¯ t )−1  ¯t , (Ir + 

after changing the basis .e1 , . . . , en by .ei1 , . . . , eir , eir+1 , . . . , ein . Proposition 5.58 The matrix .A ∈ UI ∩ Gn,r (A) if and only if the restrictions to Im(A) of .ei∗1 , . . . , ei∗r is basis for the dual space .(Im(A))∗ , which is .A-linear space on the right, because .Im(A) is a .A-linear space on the left.

.

Proof Suppose .A ∈ UI ∩ Gn,r (A). The dimension of .Im(A)∗ is r, and so it is enough to show that the restrictions of .ei∗k for .k = 1, . . . , r, to the image .Im(A) are K-linearly independent on the left.  But, the restriction to the image .Im(A) of . rk=1 λk ei∗k is zero if and only if r  .

λk ei∗k (Aeiα ) = 0, α = 1, . . . , n,

k=1

i.e., if and only if ⎛ ⎞ ai1 i1 . . . ai1 in ⎜ . . ⎟ ⎟   ⎜ ⎜ ⎟ . λ1 · · · λr ⎜ . . ⎟= 0...0 . ⎜ ⎟ ⎝ . . ⎠ air i1 . . . air in

244

5 Generalizations of Matrix Varieties

But, the matrix ⎛ ⎞ ai1 i1 . . . ai1 ir ⎜ . . ⎟ ⎜ ⎟ ⎜ ⎟ .⎜ . . ⎟ ⎜ ⎟ ⎝ . . ⎠ air i1 . . . air ir is invertible. Therefore, .λ1 = · · · = λr = 0. Conversely, if .A ∈ Gn,r (A) is such that restriction to the image .Im(A) of ∗ ∗ ∗ .e , . . . , e i1 ir is basis of the dual .(Im(A)) then, arguing again as above, yields that the matrix ⎛ ⎞ ai1 i1 . . . ai1 ir ⎜ . . ⎟ ⎜ ⎟ ⎜ ⎟ .⎜ . . ⎟ ⎜ ⎟ ⎝ . . ⎠ air i1 . . . air ir has rank r and, since A is hermitian, we get that the minor with rows and columns i1 , . . . , ir has to be invertible. Finally, we check that matrix . coincides with the one obtained by writting the restriction of .ei∗k for .k = r + 1, . . . , n, to the image .Im(A) as the linear combination on the left of the restriction to the image .Im(A) of .ei∗k , .k = 1, . . . , r :

.

ei∗k =

r 

.

λkα ei∗α

α=1

in .Im(A). In fact, by applying both members to .Aeiβ for .β = 1, . . . , r we have aik iβ =

r 

.

λkα aiα iβ ,

α=1

β = 1, . . . , r.

.

 

Observe that this is the atlas appearing in [30, p. 133]. It is now easy to check that change of coordinates is regular, if .A is commutative, and so we get the structure of complex manifold for .A = C. Proposition 5.59 If .A ∈ G(C(K)) then the structure of .C(K)-vector space on TA G(C(K)), determined by the above atlas, is given by .i ∗A B = i(BA − AB) for all .B ∈ TA G(C(K)).

.

5.2 Atlas on Varieties of Matrices

245

Proof It is obvious that the vector space .C(K)-structure on .TA G(C(K)) cannot be iB, because if .B¯ t = B, then .(iB)t = −iB. It is clear, however, that .i ∗A B = i(BA − AB), for all .B ∈ TA G(C(K)), is well defined and it is a structure of .C(K)-vector space on .TA G(C(K)), which restricts to .TA G−,r (C(K)) and to .TA Gn,r (C(K)). In order to see that the above structure is the one determined by the atlas, let .A ∈ UI ∩ Gn,r (C(K)), and suppose for simplicity that .I = {1, . . . , r}. Set     t t t (Ir + 0 0 )−1 (Ir + 0 0 )−1 0 A11 A12 = .A = t t t , A21 A22 0 (Ir + 0 0 )−1 0 (Ir + 0 0 )−1 0 with .0 ∈ Mn−r,r (C(K)). If now . ∈ Mn−r,r (C(K)), we consider the paths .σ , σi : C(K) → Gn,r (C(K)) given by σ (s) =

.

  B11 (s) B12 (s) , B21 (s) B22 (s)

where B11 (s) = (Ir + (0 + s)t (0 + s))−1 ,

.

B12 (s) = (Ir + (0 + s)t (0 + s))−1 (0 + s)t ,

.

B21 (s) = (0 + s)(Ir + (0 + s)t (0 + s))−1 ,

.

B22 (s) = (0 + s)(Ir + (0 + s)t (0 + s))−1 (0 + s)t ;

.

and σi (s) = σ (is)

.

for .s ∈ C(K). (0) and so we have to show that By definition, it has to be .i ∗A σ (0) = σi σi (0) = i(σ (0)A − Aσ (0)).

.

(0) = iσ (0)!) (Be careful of not thinking that .σi  But,

B11 (s)(Ir + (0 + s)t (0 + s)) = Ir

.

246

5 Generalizations of Matrix Varieties

and so, ¯ t A21 . B11 (0) = −A12 A11 − A11 

.

Therefore, ⎛

⎞ ¯ t A21 ¯ t A22 + A11  ¯t −A12 A11 − A11  −A12 A12 − A11  ⎠ ¯ t A21 ¯ t − A22 .σ (0) = ⎝A11 − A22 A11 − A21  A12 + A21  t ¯ A22 ×A12 − A21  and ⎛

¯ t A21 −A12 A11 + A11 

⎜ σi (0) = i ⎜ ⎝A11 − A22 A11 + A21  ¯ t A21

.

¯ t A22 ⎞ −A12 A12 + A11  ⎟ ¯t −A11  ⎟ t ¯ − A22 A12 ⎠ . A12 − A21  ¯ t A22 +A21 

On the other hand, a simple calculation, by using the idempotency of A, yields .σ (0)A



−A12 A12 −A12 A11 = A11 − A22 A11 A12 − A22 A12



and .Aσ (0)

  ¯ t A21 −A11  ¯ t A22 + A11  ¯t −A11  = ¯ t A21 −A21  ¯ t A22 + A21  ¯t . −A21 

Therefore, i(σ (0)A − Aσ (0)) = σi (0),

.

and this is what we wanted to prove.

 

It is clear that isomorphisms .TA G(C(K)) → HomC(K) (Im(A), Ker(A)) are C(K)-linear with respect to the .C(K)-structure above. Furthermore, .i ∗A B = i(BA − AB) defines also a structure of .C(K)-vector space on .TA Idem(C(K)) if .A ∈ Idem(C(K)). For the particular case of .K = R we have in the way above a quasi-complex structure on .Idem(C) (being also a complex structure on .Idem(C)) which does not coincide with the one as a submanifold of .M(C). That quasi-complex structure on .Idem(C) induces on .G(C) the complex structure given above. ≈ Also observe that the natural bijection .Gn,r (C(K)) → Gn,n−r (C(K)) given by .A → In − A for .A ∈ Gn,r (C(K)) does not preserve the complex structure because .i∗In −A (−B) = i∗A B = −i∗A B. So, for .K = R, this bijection is not holomorphic. .

5.2 Atlas on Varieties of Matrices

247 ≈

If .F ∈ U (C), i.e., .F¯ t = F −1 , the .C(K)-regular bijection .G(C(K)) → G(C(K)) given by .A → F AF −1 for .A ∈ G(C(K)) induces .C(K)-vector space isomorphisms ≈

TA G(C(K)) → TA G(C(K)).

.

In fact, just observe that we have F (i ∗A B)F −1 = i ∗F AF −1 (F BF −1 )

.

for all .B ∈ TA G(C(K)). Observe that if . ∈ Mn−r,r (C(K)) and . ∈ Mr,n−r (C(K)), we may consider the path .σ, given by σ, (s) =

.

  B11 (s) B12 (s) , B21 (s) B22 (s)

where B11 (s) = (Ir + (0 + s )(0 + s))−1 ,

.

B12 (s) = (Ir + (0 + s )(0 + s))−1 (0 + s ),

.

B21 (s) = (0 + s)(Ir + (0 + s )(0 + s))−1 ,

.

B22 (s) = (0 + s)(Ir + (0 + s )(0 + s))−1 (0 + s ).

.

If .A ∈ Gn,r (C(K)), we have then σ (s) = σ,t∗ (s)

.

and so i ∗A σ (0) = σi (0) = σi,(i) t∗ (0) = σi,−it∗ (0) = iσ,−t∗ (0).

.

Hermitian Metric on G(C(K)) For .B, B ∈ TA G(C(K)) we define B, B A = tr(BAB ) ∈ C(K).

.

248

5 Generalizations of Matrix Varieties

We get then a Hermitian metric with respect to the .C(K)-structure on TA G(C(K)):

.

i ∗A B, B A = itr(BAB − ABAB ) = iB, B A ,

.

B, i ∗A B A = itr(BAB A − BAB ) = −iB, B A ,

.

¯ = trB t At B t = tr(BAB )t = B, B A . B , B A = tr(B¯ A¯ B)

.

Also we have 2B, B A = tr(BB ) + itr(B(i ∗A B )).

.

In fact, tr(BB ) = tr((AB + BA)(AB + B A)) =

.

B, B A + B , B A = 2Re(B, B A ).

.

On the other hand, itr(B(B A − AB )) = i(B , B A − B, B A ) =

.

i(B, B A − B, B A ) = i(−2iI mB, B A ) = 2Im(B, B A ).

.

Finally, observe that tr((i ∗A B)(i ∗A B )) = tr(BB )

.

for .B, B ∈ TA G(C(K)). In fact, i ∗A B, i ∗A B = tr((i ∗A B)(i ∗A B )) = tr((BA − AB)(B A − AB )) =

.

tr(BAB + ABB ) = 2ReB, B A = tr(BB ) = B, B .

.

In particular, we extend the classical definition of the Riemannian metric on a smooth manifold and define the Riemannian metric on .G(C(K)) by .ReB, B = Re tr(BB ) for .B, B ∈ TA G(C(K)). We have shown above that this metric is Hermitian as well. The associated fundamental 2-form A : TA G(C(K)) × TA G(C(K)) → K

.

5.2 Atlas on Varieties of Matrices

249

is given by A (B, B ) = B, i ∗A B

.

for .B, B ∈ TA G(C(K)). Since the matrices .B, i ∗A B are Hermitian, .B, i ∗A B ∈ K and .A is welldefined. Then, .A (B, B ) = −i ∗A B, B = −A (B , B) and the 2-form .A is antisymmetric. Clearly, we also have A (B, B ) = 2ImB, B A .

.

In fact, A (B, B ) = B, i ∗A B = tr(B(i ∗A B )) = tr(B(i(B A − AB ))

.

= itr(BB A − BAB ) = i(tr(B AB) − tr(BAB )) = i(B , B A − B, B A ) = i(B, B A − B, B A )

.

= 2ImB, B A . Some authors, for instance Sakai [34], take the opposite. It is clear that the fundamental 2-form . is non-degenerate and, for .K = R, it defines a symplectic structure on .G(C). Therefore, we have defined above the symplectic structure on .G(C(K)). It is also clear now that . is invariant by the action by conjugation of .Un (C(K)) on .Gn,r (C(K)). We show in the next chapter that the fundamental 2-form . is closed, i.e., .d = 0, and so that .G(C) is a Kähler manifold.

Chapter 6

Curvature, Geodesics and Distance on Matrix Varieties

In this chapter we study more closely the Riemannian structure of classical matrix manifolds introduced in Chaps. 3 and 4. Here, .A = R, C, H and occasionally O. We also extend, whenever it is possible, definitions and results to the general case treated in Chap. 5, where .A = K, C(K), H(K), O(K) for K a Pythagorean formally real field. Section 6.1 extends the classical Stiefel maps .π : Vn,r (A) → Gn,r (A), resp. the not so classical Stiefel maps .ρ : V¯n,r (A) → Idemn,r (A), for .A = R, C, H, to the more general setting where .R is replaced by a Pythagorean formally real field .K. It is shown they are Riemannian, resp. semi-Riemannian, submersions. Section 6.2 greatly extend definitions of Levi-Civita connection, Kähler manifold, Riemannian curvature tensor and sectional curvature for the more general hypothesis where .R is replaced by any Pythagorean formally real field K. In particular, it is shown that .C(K)- Grassmanians are Kähler manifolds. We make use of the classifying maps given in (5.56) to find explicit formulas for the curvature tensor fields and the sectional curvatures in this general setting. Section 6.3 proves that both Grassmannians and i-Grassmannians are Einstein manifolds for the extended definitions replacing .R by any Pythagorean formally real field .K. Section 6.4 explicitly computes the second fundamental tensor field and use it to obtain the geodesics in Grassmanians .Gn,r (A) and i-Grasmannians .Idemn,r (A): the geodesic with origin A and unitary initial velocity B is given as solution of the differential equation σ  (s) = 2(In − 2σ (s))σ  (s)2

.

with initial conditions .σ (0) = A and .σ  (0) = B.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Golasi´nski, F. Gómez Ruiz, Grassmann and Stiefel Varieties over Composition Algebras, RSME Springer Series 9, https://doi.org/10.1007/978-3-031-36405-1_6

251

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6 Curvature, Geodesics and Distance

Of course, to obtain the solution σ (s) =

.

1 (In − e2s(In −2A)B (In − 2A)) 2

the field K has to be complete since we need convergency of the exponential .eX . Section 6.5 makes use of the Stiefel submersion to get the volumes of .AGrassmannians for .A = R, C, H. Sections 6.6 and 6.7, study the Riemannian geometry of the Cayley plane .G3,1 (O) and its generalization to .G3,1 (O(K)) for any Pythagorean formally real field .K. In particular, we extend and compute explicit formulas for the Riemannian curvature tensor field, the sectional curvatures and the second fundamental tensor field. We extend definition and show that .G3,1 (O(K)) is an Einstein manifold. The second fundamental tensor field is used to obtain the geodesics in .G3,1 (O(K)): the geodesic with origin A and unitary initial velocity B is given as a solution of the differential equation σ  (s) = 2(σ  (s)2 − tr(σ  (s)2 ))σ (s).

.

The solution, for which we need K being complete, is given in Proposition 6.23 by σ (s) = cos(2s)A +

.

sin(2s) B + sin2 sB 2 . 2

Finally, we discuss in Sect. 6.8 the volume of the Cayley plane .G3,1 (O) and 3! 8 Theorem 6.30 shows that .vol(G3,1 (O)) = 11! π . Certainly, this result has been already stated in the literature, e.g., [23]. But, no direct proof is known to the authors. Then, the general formula .vol(G3,1 (A)) = ( d2 −1)! d π ( 3d 2 −1)!

for .d = 2, 4, 8 and .A = C, H, O, respectively, is derived.

6.1 The Stiefel Submersion In this section we take .A = K, .C(K) or .H(K). Recall the Stiefel map given in Sect. 3.2 for the classical case or in Sect. 5.1 in the general case. It is a surjective K-polynomial map of degree two .π : V (A) → G(A) given by .π(A) = AA¯ t for .A ∈ V (A) and restricts to surjective maps V−,r (A) → G−,r (A), Vn (A) → Gn (A), Vn,r (A) → Gn,r (A).

.

TA V−,r (A) = {B ∈ M−,r (A)|B¯ t A + A¯ t B = 0}

.

and (dπ )A : TA V−,r (A) → TAA¯ t G−,r (A)

.

6.1 The Stiefel Submersion

253

is the K-epimorphism given by (dπ )A (B) = B A¯ t + AB¯ t

.

for .B ∈ TA V−,r (A). The kernel of the above epimorphism consists of the .B ∈ TA V−,r (A) of the form ¯ t = −X, which coincides with .B = AX with X anti-Hermitian, i.e., .X TA (π −1 (AA¯ t )) = TA (AU (A)).

.

Now, given .B ∈ TA Vn,r (A), consider .B = (B − AA¯ t B) + AA¯ t B. t Since .A¯ t B = −A¯ t B, we derive that .AA¯ t B ∈ Ker(dπ )A . Next, given .C ∈ Ker(dπ )A there is an anti-Hermitian X with .C = AX. Then, .ReC, B − AA¯ t B = Re tr(−XA¯ t (B − AA¯ t B) = Re tr(−XA¯ t B + XAA¯ t B) = 0. Consequently, we have the orthogonal decomposition TA Vn,r (A) = (Ker(dπ )A )⊥ ⊕ Ker(dπ )A .

.

Recall that a submersion of Riemannain manifolds .f : M → N is called a Riemannian submersion if the restriction .(df )x | : (Ker(df )x )⊥ → Tf (x) N is an isometry for all .x ∈ M. Proposition 6.1 The Stiefel map π : Vn,r (A) → Gn,r (A)

.

is a Riemannian submersion once we endow .Vn,r (A) with the Riemannian metric induced by .Re−, − and .Gn,r (A) with the Riemannian metric induced by 1 . Re−, −. 2 Proof In fact, if .B, C ∈ TA Vn,r (A) then we have (dπ )A B, (dπ )A (C) = Re tr((B A¯ t + AB¯ t )(C A¯ t + AC¯ t )) =

.

2Re tr(B C¯ t ) + 2Re tr(AB¯ t AC¯ t ) = 2ReB, C + 2ReAB¯ t A, C.

.

Therefore, (dπ )A B, (dπ )A (C) = 2ReB, C

.

if .C ∈ (Ker(dπ )A )⊥ , since .AB¯ t A ∈ Ker(dπ )A , and the proof is complete.



Remark 6.2 Some observations on the Riemannian metrics: (1) We extend the definition of Riemannian metric on .Vn,r (A) and .Gn,r (A), for .A = R, C, H, to the general case where K is a Pythagorean formally real

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6 Curvature, Geodesics and Distance

field by saying that it associates to each .A ∈ Vn,r (A), resp. .Gn,r (A), an inner product in the tangent space at A in continuous way, and where those inner products .Re−, − are K-bilinear, symmetric, such that .ReX, X = 0 if and only if .X = 0, and .ReX, X is always the square of some element of .K. (2) It is better to consider the Riemann metric . 12 Re−, − in .Gn,r (A) and .Re−, − in .Vn,r (A) to get the standard metric for the classical situation. For example, we get the canonical metric on .Vn,1 (R) = Sn−1 and .π : Sn−1 → Gn,1 (R) = RP n−1 is an isometry. Proposition 6.3 The Stiefel map ρ : V¯n,r (A) → Idemn,r (A)

.

is a semi-Riemannian submersion, once we endow .V¯n,r (A) with the semiRiemannian structure .V¯ (A) induced by the inner product .Re−, − on .M(A)2 given by (X, Y ), (X , Y  ) = Re tr(XY  + X Y t ), t

.

and the semi-Riemannian structure on .Idem(A) induced by the metric .Re{−, −} on .M(A), i.e., .Re{V , W } = Re tr(V W ) and we have an isometry .(dρ)A : ¯ Ker(dρ)⊥ A → TAA¯ t Idemn,r (A) for all .A ∈ Vn,r (A). Proof Define ρ : V¯ (A) → Idem(A)

.

given by.ρ(A, B) = AB t for .(A, B) ∈ V¯ (A). This restricts to surjective maps: V¯−,r (A) → Idemr (A), V¯n,r (A) → Idemn,r (A).

.

On the other hand, the group .GLr (A) acts on the right on .V¯−,r (A) by (A, B)X = (AX, (X−1 B t )t )

.

for .(A, B) ∈ V¯−,r (A) and .X ∈ GLr (A) and this action obviously restricts to actions of .GL(r, A) on .V¯n,r (A). ¯ Observe that if .X ∈ U (A) and .B = A¯ then we have .(A, A)X = (AX, AX). In this way, we get bijections from the orbit space by the action of .GL(r, A) on .V¯−,r (A), to .Idem−,r (A), which restrict to bijections from the orbit space of the action of .GL(r, A) on .V¯n,r (A), to .Idemn,r (A). If .(A, B) ∈ V¯−,r (A) and .(X, Y ) ∈ TA,B V¯−,r (A) then we have (dρ)A,B (X, Y ) = AY t + XB t .

.

6.1 The Stiefel Submersion

255

The kernel of the epimorphism .(dρ)A,B consists of .(AV , −(V B t )t ) with .V ∈ Mr (A) which obviously coincides with .TA,B (ρ −1 (AB t )) = TA,B ((A, B)GL(r, A)). Furthermore, (Ker(dρ)A,B )⊥ → TAB t Idemr (A)

.

is an isometry, i.e., {(dρ)A,B (X, Y ), (dρ)A,B (X , Y  )} = {(X, Y ), (X , Y  )}

.

for .(X, Y ), (X , Y  ) ∈ (Ker(dρ)A,B )⊥ . In fact, {(dρ)A,B (X, Y ), (dρ)A,B (X , Y  )} = (X, Y ), (X , Y  )

.

+ Re tr(AY t AY  + XB t X B t ). t

But, since .(X, Y ), (X , Y  ) ∈ T(A,B) V¯n,r (A), we have Y t A + B t X = 0, Y  A + B t X = 0 t

.

and so Re tr(AY t AY  +XB t X B t ) = Re tr(B t X B t X +B t X B t X) = 2Re tr(X (B t X)B t ). t

.

Taking .V = B t X ∈ Mr (K) and using the fact that (X , Y  ), (AV , −(V B t )t  = 0,

.

we get Re tr(AY t AY  + XB t X B t ) = 2Re tr(AB t XY  ) = t

.

t

2Re tr(B t XY  A) = −2Re tr(B t XB t X ) = −2Re tr(B t X B t X).

.

t

Therefore, Re tr(B t X B t X) = 0.

.

Thus, we have proved that .ρ : V¯n,r (A) → Idemn,r (A) is a semi-Riemannian submersion and the proof follows.



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6 Curvature, Geodesics and Distance

Similarly to Remark 6.2 (1), we extend the definition of semi-Riemannian metric on .V¯n,r (A) and .Idemn,r (A) for .A = R, C, H to .A = K, C(K), H(K), where K is a Pythagorean formally real field.

6.2 Curvatures In this section .A = K, C(K), H(K) and an ordering has being chosen in .K. Levi-Civita Connection on .Idem(A) (Resp. .G(A)) If .A ∈ M(A) (resp. .A ∈ Herm(A)) then we have the orthogonal projection πA : M(A) → TA Idem(A), (resp. πA : Herm(A) → TA G(A))

.

given by .πA (C) = AC + CA − 2ACA for .C ∈ M(A) (resp. .C ∈ Herm(A)). If now .B ∈ M(A) (resp. .B ∈ Sk(A) = {X ∈ M(A); X¯ t = −X}) then we define the vector field .ad(B) : Idem(A) → TA Idem(A) (respectively .ad(B) : G(A) → TA G(A)) by ad(B)(A) = [B, A]

.

for all .A ∈ Idem(A) (resp. .A ∈ G(A)), where .[B, A] = AB − BA. Observe that .[B, A] ∈ TA Idem(A) (resp. .TA G(A)) because A[B, A] + [B, A]A = [B, A],

.

and if .A ∈ G(A) and .B ∈ Sk(A), we have t

[B, A] = [B, A].

.

Thus, .ad(B) is a tangent vector field on .Idem(A) (resp. on .G(A)). On the other hand, if .∇ is the Levi-Civita connection on .Idem(A) (resp. on .G(A)), we have then (∇ad(B) ad(C))A = πA ((d(ad(C))A (ad(B)(A)))

.

for all .A ∈ Idem(A) (resp. .A ∈ G(A)) and for all .B, C ∈ M(A) (respectively in Sk(A)). Since

.

d(ad(C))A (ad(B)(A)) = lim

.

→0

[C, A + [B, A]] − [C, A] = [C, [B, A]], 

6.2 Curvatures

257

we obtain (∇ad(B) ad(C))A = πA ([C, [B, A]]) = [A, [[A, [A, B]], C]].

.

¯ (A). Observe that .∇ could be extended to .BU (A) and .BU Further, if .B, C ∈ Sk(A) then we have .[B, C] ∈ Sk(A). It is clear that [ad(B), ad(C)] = −ad([B, C]).

.

In fact, [ad(B), ad(C)](A) = (∇ad(B) ad(C) − ∇ad(C) ad(B))A = [A, [B, C]]

.

= −ad([B, C])(A). To show the next result we need the following. Given a smooth manifold M, write .X(M) for the .C ∞ (M)-module of tangent vector fields on M. Then, recall that given a 2-form ., its differential .d : X(M) × X(M) × X(M) → C ∞ (M) is defined as follows: (d)(X, Y, Z) = X(Y, Z) − Y (X, Z) + Z(X, Y ) − ([X, Y ], Z)

.

+ ([X, Z], Y ) − ([Y, Z], X) for .X, Y, Z ∈ X(M). Let . the fundamental form on .G(C(K)) defined in subsection “Hermitian metric on .G(C(K))” of Chap. 5. Then, Proposition 6.4 If K is a Pythagorean real field then the fundamental form . defined on .G(C(K)) is closed, and so .G(C(K)) is a Kähler manifold Proof In fact, if .A ∈ G(C(K)) and .B, C, D ∈ Sk(C(K)) then we have (d)A (ad(B), ad(C), ad(D)) =

.

[B, A](ad(C), ad(D)) − [C, A](ad(B), ad(D)) + [D, A](ad(B), ad(C))−

.

A ([ad(B), ad(C)], ad(D)) + A ([ad(B), ad(D)], ad(C))

.

− A ([ad(C), ad(D)], ad(B)).

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6 Curvature, Geodesics and Distance

But, A ([C, A], [D, A]) = Re tr([C, A](i([D, A]A − A[D, A]))) =

.

.

− Im tr([C, A](DA − ADA − ADA + AD) = −Im tr(DCA − CDA) = Im tr([C, D]A).

Therefore, [B, A](ad(C), ad(D)) =

.

.

Im tr([C, D](A + [B, A]) − Im tr([C, D]A) = Im tr([C, D][B, A]). →0  lim

Thus, [B, A](ad(C), ad(D)) − [C, A](ad(B), ad(D)) + [D, A](ad(B), ad(C)) =

.

Im tr([C, D][B, A] − [B, D][C, A] + [B, C][D, A]) = 0.

.

On the other hand, A ([[B, C], A], [D, A]) − A ([[B, D], A], [C, A]) + A ([[C, D], A], [B, A]) =

.

Im tr([[B, C], D]A − [[B, D], C]A + [[C, D], B]A) = 0.

.

But, the map Sk(C(K)) → TA G(C(K))

.

given by .X → [X, A] for .X ∈ Sk(C(K))is surjective if we further assume K being  Ir 0 . But, since K is a Pythagorean real real closed. In fact, this is clear for .A = 0 0 ¯t closed  field, for .A ∈ G(C(K)) there exists a matrix .U ∈ U (C(K)) with .U AU = Ir 0 . Then, .U [X, A]U¯ t = [U XU¯ t , U AU¯ t ] concludes the proof that .d = 0. 0 0 Therefore, by restriction we also have .d = 0 without the hypothesis of K being

real closed. Observe that, everything being polynomial, we have extended the definition to say that .G(C(K)) is a Kähler manifold for any Pythagorean formally real field .K.

6.2 Curvatures

259

Riemannian Curvature Tensor on .Idem(A) and .G(A) To show the next result we need the following. Given a smooth manifold M with linear connection .∇, the map R∇ : X(M) × X(M) → EndC ∞ (M) (X(M)

.

given by .R∇ (X, Y ) = ∇X ∇Y − ∇Y ∇X − ∇[X,Y ] for .X, Y ∈ X(M) is called the Riemannian curvature tensor on M associated to the connection .∇. We write for simplicity f instead of .fI or .fG defined in (5.56) and matrices are in .Idem(A), .M(A), or .G(A), .Herm(A), depending of the corresponding case. Next, let .R = R∇ be the Riemannian curvature tensor associated to the Levi-Civita connection .∇. Then, by Gómez Ruiz [21] we have .R = f (df ∧ df ). Therefore, RA (B, C) = f (A)((df ∧ df )(A))(B, C) = f (A)((Bf )(Cf ) − (Cf )(Bf )),

.

where 1 (f (A + B)(D) − f (A)(D)) = BD + DB − 2ADB − 2BDA, →0 

(Bf )(D) = lim

.

and similarly for the other cases. Then, we have ((Bf )(Cf ) − (Cf )(Bf ))(D) = BCD − CBD + DCB − DBC

.

.

− 2ACDB + 2ABDC − 2BDCA + 2CDBA − 2BADC + 2CADB − 2CDAB + 2BDAC .

+ 4ACDAB − 4ABDAC + 4BADCA − 4CADBA.

Therefore, RA (B, C)D = A((Bf )(Cf ) − (Cf )(Bf )(D)) + ((Bf )(Cf ) − (Cf )(Bf )(D))A

.

.

− 2A((Bf )(Cf ) − (Cf )(Bf )(D))A,

and since .ABA = ACA = 0, RA (B, C)D = A(BC−CB)D−D(BC−CB)A−AD(BC−CB)+(BC−CB)DA

.

.

− 2A(BC − CB)DA + 2AD(BC − CB)A − 2ACD(B − AB) .

+ 2ABD(C − AC) − 2(B − BA)DCA + 2(C − CA)DBA.

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6 Curvature, Geodesics and Distance

But then, .B = AB + BA y .C = AC + CA, yields RA (B, C)D = [[[[B, C], D], A], A].

.

Thus, .RA (B, C) : TA Idem(A) → TA Idem(A), is given by the composition RA (B, C) = ad(A)2 ◦ ad([B, C]).

.

It restricts to .RA (B, C) : TA G(A) → TA G(A) for .A ∈ G(A) and .B.C ∈ TA G(A). Therefore, for .E ∈ TA Idem(A) we have Re{RA (B, C)D, E} =

.

Re tr(A[B, C]DE − D[B, C]AE − AD[B, C]E + [B, C]DAE

.

.

− 2A[B, C]DAE + 2AD[B, C]AE),

and since .AEA = 0, we have Re{RA (B, C)D, E} = Re tr(A[B, C]DE − D[B, C]AE − AD[B, C]E

.

+ [B, C]DAE) = Re tr([B, C]D(EA + AE) − D[B, C](EA + AE)).

.

But, .E = AE + EA and so Re{RA (B, C)D, E} = Re tr([B, C]DE − D[B, C]E) =

.

(6.1)

Re tr([B, C][D, E]) = Re{[B, C], [D, E]}.

.

Sectional Curvatures on .G(A) Let M be a Riemannian manifold, R the Riemannian curvature tensor on M and .F ⊆ Tx M for .x ∈ M and .u, v ∈ F an orthonormal basis of F . Recall that the sectional curvature at .x ∈ M is defined as follows: kx (F ) = −Rx (u, v)u, v.

.

We consider the Riemannian metric on .G(A) given by .(X, Y ) → 12 Re tr(XY ) for .X, Y ∈ TA G(A) with .A ∈ G(A). Then, if .A ∈ G(A) and .B, C ∈ TA G(A) is an orthonormal basis of a 2-dimensional K-vector subspace F of .TA G(A), i.e., 2 2 .trB = trC = 2 and .Re tr(BC) = 0. Then, by relation (6.1), the sectional curvature at A with respect to the K-vector subspace F is given by 1 1 kA (F ) = − Re tr([B, C]2 ) = − tr([B, C]2 ). 2 2

.

6.2 Curvatures

261

In particular, if .A = C(K) and the 2-dimensional K-vector subspace F of TA G(C(K)) is a one-dimensional .C(K)-vector subspace of .TA G(C(K)), regarded as a K-vector space, then .kA (F ) is called the holomorphic sectional curvature at A hol (F ). with respect to the .C(K)-space F and denote by .kA Observe that if .A ∈ G(C(K)) and B is a non zero vector of .TA G(C(K)) then .B, i ∗A B = i(BA − AB) are obviously orthogonal, i.e., .Re tr(B(i ∗A B)) = 0. Further, if .tr(B 2 ) = 2 then .tr(i ∗A B)2 = 2. Therefore, we get an orthonormal basis of some one-dimensional .C(K)-vector space .F, regarded as a two-dimensional Kvector space and so the holomorphic sectional curvature .

hol (F ) = − 1 tr([B, i ∗ B]2 ). kA A 2

.

Proposition 6.6 Let K be a Pythagorean real field and choose an ordering on it. (1) If .kA (F ) is the sectional curvature at .A ∈ Gn,r (A) with respect to a 2dimensional K-vector space .F ⊆ TA Gn,r (A) then 0 ≤ kA (F ) ≤ 4.

.

hol (B) is the holomorphic sectional curvature at .A ∈ G (C(K)) with (2) If .kA n,r respect to any non zero .B ∈ TA Gn,r (C(K)) then

.

4 hol (F ) ≤ 4, ≤ kA r

where F is one dimensional .C(K)-subspace of .TA Gn,r (C(K)) and both the minimum . 4r as well as the maximum 4 are reached. In particular, the projective spaces .Gn,1 (C(K)) have constant holomorphic sectional curvature .4. Proof Observe that .[B, C] ∈ Sk(A) for .B, C ∈ Ta Gn,r (A) and so, if we write [B, C]pq for the .pq-entry of .[B, C], we have

.

1 1 kA (F ) = − tr([B, C]2 ) = [B, C]pq [B, C]pq . 2 2 p,q

.

Therefore, the sectional curvature .kA (F ) is non-negative, i.e., the square of some element of .K, and it is zero if and only if F admits an orthonormal basis .B, C ∈ TA Gn,r (A) with .BC = CB. In order to show both  (1) and (2), we clearly may take K real closed and so Ir 0 we may take .A = , because the field K being Pythagorean real closed, 0 0 for any .U ∈ Un (A), the maps .Aut(U ) : Gn,r (A) → Gn,r (A) given by ¯ t , are isometries and the action of .U (A) on .Gn,r (A) is .Aut(U )(A) = U AU transitive by Proposition 1.29.

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6 Curvature, Geodesics and Distance

  Ir 0 Thus, we take now .A = . The tangent to .Idem−,r (A) at A are clearly all 0 0   0X matrices . for any .X ∈ Mr,− (A) and any .Y ∈ M−,r (A), and the tangent to Y 0   0 X .G−,r (A) at A are the matrices . for any .X ∈ Mr,− (A). X¯ t 0 Then, using (6.1), we have        

0 B 0 C 0 D 0 E ⎛ ⎞ , , = .Re R I 0 B 0 C 0 D 0 E 0 ⎝ r ⎠ 0 0 Re tr((BC  − CB  )(DE  − ED  )) + Re tr((B  C − C  B)(D  E − E  D)).

.

In particular,          ⎞( 0 B , 0 C ) 0 B , 0 C = Re R⎛  I 0 C 0 B 0 C 0 ⎝ r ⎠ B 0 0 0

.

(6.2)

Re tr((BC  − CB  )(BC  − CB  )) + Re tr((B  C − C  B)(B  C − C  B)) =

.

Re{BC  − CB  , BC  − CB  } + Re{B  C − C  B, B  C − C  B} =

.

2Re tr((BC  )2 − BB  CC  ) + 2Re tr((B  C)2 − B  BC  C).

.

Suppose that .B  = B¯ t and .C  = C¯ t . We have then (6.2) is equal to 4Re tr((B C¯ t )2 ) − 2Re tr(B¯ t B C¯ t C) − 2Re tr(B B¯ t C C¯ t ).

.

If we assume that .Re tr(B B¯ t ) = Re tr(C C¯ t ) = 1 and .Re tr(B C¯ t ) = 0 then    

0 B 2

0 C 2 = 2 and .Re tr = 2. .Re tr B¯ t 0 C¯ t 0   ⎞ G (A) with the basis . 0 B , Therefore, for the subspace .F ⊆ T⎛ I 0 n,r B¯ t 0 ⎝ r ⎠ 0 0   0 C we get the sectional curvature C¯ t 0 ⎞ (F ) = −2Re tr((B C ¯ t )2 ) + Re tr(B¯ t B C¯ t C) + Re tr(B B¯ t C C¯ t ) = k = k⎛ I 0 ⎝ r ⎠ 0 0

.

Re(−2B C¯ t , C B¯ t  + B¯ t B, C¯ t C + B B¯ t , C C¯ t ).

.

6.2 Curvatures

263

If we choose an order of K and use Corollary1.20(1) then we get k ≤ 2B C¯ t , B C¯ t  2 C B¯ t , C B¯ t  2 + 1

1

.

B¯ t B, B¯ t B 2 C¯ t C, C¯ t C 2 + B B¯ t , B B¯ t  2 C C¯ t , C C¯ t  2 . 1

1

1

1

.

But, B C¯ t , B C¯ t  = Re tr(B C¯ t C B¯ t ) = Re tr(B¯ t B C¯ t C) = B¯ t B, C¯ t C

.

and similarly C B¯ t , C B¯ t  = B¯ t B, C¯ t C.

.

Therefore, k ≤ 4B B¯ t , B B¯ t  2 C C¯ t , C C¯ t  2 ≤ 4||B||2 ||C||2 = 4. 1

1

.

  Ir 0 On the other hand, if .A = then for .A = C(K) we have 0 0  i ∗A

.

0 B B¯ t 0





0 −B = i ¯t B 0



and so  [

.

    t  0 0 B 0 −B B B¯ , i ] = 2i . 0 −B¯ t B B¯ t 0 B¯ t 0

Hence, hol (F ) = 4tr(B B¯ t )2 kA

.

 0 B . for .F = C(K) ¯ t B 0 But, .B B¯ t is Hermitian and the field K is Pythagorean and real closed, so there exists .U ∈ Ur (C(K)) such that .U B B¯ t U¯ t is a diagonal matrix with principal diagonal given by the eigenvalues .λ1 , . . . , λr of .B B¯ t . Observe that .λp > 0, i.e., square of some non zero element of .K, for .p = 1, . . . , r. 

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6 Curvature, Geodesics and Distance

In fact, if .B B¯ t v = vλ with v and .λ both non-zero then t λ¯ v¯ t v = vλ v = v¯ t B B¯ t v = v¯ t vλ.

.

Hence, .λ ∈ K and .λ is the square of some element of K because .λ = Thus, we have

t

B¯ t v B¯ t v v¯ t v .

  hol ( 0 B ) = 4(λ2 + · · · + λ2 ), kA r 1 B¯ t 0

.

with .tr(B B¯ t ) = λ1 + · · · + λr = 1. Therefore,  hol ( .k A

 4 0 B )≥ . t ¯ B 0 r

√ In fact, write .u = r(1, . . . ,√ 1) ∈ K r and .v =√(λ1 , . . . , λr ). Then, .||u||2 = r 2 ,

r r 2 2 .||v|| = p=1 λp and .u, v = r( p=1 λp ) = r. Finally, the Cauchy-Schwarz inequality yields r = |u, v|2 ≤ ||u||2 ||v||2 = r 2

r 

.

λ2p

p=1

and so r  .

λ2p ≥

p=1

1 . r

    Ir 0 I 0 and . Then, .B B¯ t = 1r r 0 0 0 0 so the holomorphic curvature is in this case is equal to . 4r . Finally, if we take B with 1 at the entry .(1, 1) and all other entries being 0 then the holomorphic curvature is equal to .4.

Assume now .r ≤ n − r and take .B =

√1 r

Remark 6.8 (1) It is clear that the Levi-Civita connection .∇, the Riemannian curvature tensor ¯ (A)). R and the sectional curvatures .kF , extend naturally to .BU (A) (resp. .BU (2) Observe that if .B, C ∈ TA Idem(A) then .[B, C] ∈ NA Idem(A). In fact, ABC − ACB = (B − BA)C − (C − BA)B

.

.

= BC − CB − B(C − CA) + C(B − BA) = [B, C]A.

6.2 Curvatures

265

(3) Instead, if .A ∈ G(A), and .B, C ∈ TA G(A) then .[B, C]2 ∈ NA G(A), because t t 2 2 2 .([B, C]2 ) = ((([B, C]) ) = (−[B, C]) = [B, C] . Proposition 6.9 (1) If .A = C(K) or H(K), .r ≥ 2 and .n − r ≥ 2, then the maximum 4 as well as the minimum 0 of the sectional curvatures are reached. (2) If .A = C(K) or H(K), and .r = 1 or .r = n − 1, then any sectional curvature satisfies .1 ≤ k ≤ 4, and both the minimum 1 as well as the maximum 4 are reached. (3) All sectional curvatures are constant and equal to 4 for .G2,1 (C(K)) and .G2,1 (H(K)) (4) All sectional curvatures are constant and equal to 1 for .Gn,1 (K) and .Gn,n−1 (K) Proof In fact, take for instance B with all entries  zero,  except for the .(1, 1) entry Ir 0 we have .kA (F ) = 4 for the which equals .1, and .C = iB. Then, for .A = 0 0     0 B 0 C subspace F generated by matrices . ¯ t . , ¯t B 0 C 0 (1): If we take B with all entries zero except that at the entry .(1, 1) equal to 1 and C with all entries zero except that at the entry .(2, 2) which equal to 1 then, as above, we get .kA (F ) = 0. (2): If .r = 1 and we write .B = (λ1 , . . . , λn−1 ) and .C = (μ1 , . . . , μn−1 ), one row matrices, we get that (6.2) equal to 2Re(

n−1 

.

λp μ¯ p )2 − (

p=1

n−1  p=1

|λp |2 )(

n−1 

n−1 

|μp |2 ) − |

p=1

λp μ¯ p |2 ,

p=1

n−1

n−1 2 2 and if we also assume . n−1 ¯ p) p=1 |λp | = p=1 |μp | = 1, and .Re( p=1 λp μ = 0 then we get that the sectional curvature k is given by k = 1+|

n−1 

.

p=1

λp μ¯ p |2 − 2Re((

n−1 

λp μ¯ p )2 ) = 1 + 3|

p=1

and so 1 ≤ k ≤ 4.

.

n−1  p=1

λp μ¯ p |2 ≤ 1 + 3 = 4,

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6 Curvature, Geodesics and Distance

⎞ ⎞ ⎛ λ1 μ1 ⎜ . ⎟ ⎜ . ⎟ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ The same formula holds for .r = n − 1, .B = ⎜ . ⎟ and .C = ⎜ . ⎟ . ⎟ ⎟ ⎜ ⎜ ⎝ . ⎠ ⎝ . ⎠ λn−1 μn−1 (3): This follow from the proof of (2). (4): If .A = K and .r = 1, n − 1 then . n−1 ¯ p = 0 and by the proof of (2) p=1 λp μ we get that the sectional curvature k is constant and equal to 1 and the proof is complete.

⎛

Remark 6.10 For .A = C(K) or H(K) all sectional curvatures reach any value of the closed interval .[1, 4] provided K is a formally real closed field. Proposition 6.11 If the field K is Pythagorean and formally real, .r ≥ 2 and .n−r ≥ 2 then the sectional curvature 0 ≤ k = kA (F ) ≤ 2

.

with respect to a 2-dimensional K-vector space .F ⊆ TA Gn,r (K) and both the minimum 0 as well as the maximum 2 are reached. Proof We show now that if .A = K, .r > 1 and .n − r > 1 then the sectional curvature belongs to the closed interval .[0, 2]. We may assume the field K is Pythagorean real closed and .r ≤ n − r and so we may take that .B = (bpq ) is the matrix of r rows and .n − r columns with .bpp = λp , .p = 1, . . . , r and all other entries zero.

We have then .tr(BB t ) = 1, i.e., . rp=1 λ2p = 1, .tr(BC t ) = 0, i.e.,

r

r n−r 2 t . p=1 λp cpp = 0; and .tr(CC ) = 1, i.e., . p=1 q=1 cpq = 1.   Ir 0 Therefore, for .A = and the subspace .F ⊆ TA Gn,r (K) generated by 0 0     0 B 0 C matrices . ¯ t , ¯t we have B 0 C 0 

k = −4

λp λq cpq cqp − 2

.

1≤p