Orthogonal Decompositions and Integral Lattices 9783110901757, 3110137836, 9783110137835

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de G r u y t e r Expositions in Mathematics 15

Editors

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

de Gruyter Expositions in Mathematics

1

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

2

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

3

The Stefan Problem, A. M.

4

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

5

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

6

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

7

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

8

Nilpotent Groups and their Automorphisms, Ε. I. Khukhro

9

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

Meirmanov

Voronin Nazaikinskii, Mikhalev,

10

The Link Invariants of the Chern-Simons Field Theory,

11

Global Affine Differential Geometry of Hypersurfaces, A.-M. Li, U. Simon, G. Zhao

12

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

13

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

14

Subgroup Lattices of Groups,

R.Schmidt

E.Guadagnini

Orthogonal Decompositions and Integral Lattices by Alexei I. Kostrikin Pham Huu Tiep

W DE Walter de Gruyter · Berlin · New York 1994

Authors P h a m H u u Tiep H a n o i Institute o f M a t h e m a t i c s P.O. B o x 631 10000 Hanoi, Vietnam

A l e x e i I. K o s t r i k i n Department of Mathematics MEHMAT

Present address: Institute f o r E x p e r i m e n t a l M a t h e m a t i c s University of Essen Ellernstraße 29 D - 4 5 3 2 6 Essen, G e r m a n y

119899 M o s c o w GSP-1, Russia

1991 Mathematics

Subject

Classification:

M o s c o w State University

11H31, 17Bxx, 20Bxx, 20Cxx, 2 0 D x x , 94Bxx

K e y w o r d s : O r t h o g o n a l d e c o m p o s i t i o n s , E u c l i d e a n lattices, finite g r o u p s , Lie algebras

©

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

Library of Congress Cataloging-in-Publication

Data

Kostrikin, A. I. (AlekscT Ivanovich) O r t h o g o n a l decompositions and integral lattices / by A. I. Kostrikin, P h a m Huu Tiep. p. cm. — (De G r u y t c r expositions in mathematics ; 15) Includes bibliographical references and index. ISBN 3-11-013783-6 1. Lie algebras. 2. O r t h o g o n a l decomposition. 3. Lattice theory. I. P h a m H u u Tiep, 1 9 6 3 . II. Title. III. Series. QA252.3.K67 1994 512'.55-dc20 94-16850 CIP

Die Deutsche

Bibliothek

— Cataloging-in-Publication

Data

Kostrikin, Alekscj I.: O r t h o g o n a l decompositions and integral lattices / by Aleksei I. Kostrikin, P h a m H u u Tiep. — Berlin ; New York : de Gruytcr, 1994 (De Gruyter expositions in mathematics ; 15) ISBN 3-11-013783-6 NE: Tiep, P h a m Huu:; G T

(Ο Copyright 1994 by Walter de G r u y t c r & Co., D-10785 Berlin. All rights reserved, including those of translation into foreign languages. N o part of this book may be reproduced in any form or by any means, electronic or mechanical, including p h o t o c o p y , recording, or any information storage and retrieval system, without permission in writing f r o m the publisher. Printed in G e r m a n y . Typeset with LATp.X: D. L. Lewis, Berlin. Printing: Gerikc G m b H , Berlin. Binding: Lüderitz & Bauer G m b H , Berlin. Cover design: T h o m a s Bonnie, H a m b u r g .

Preface

The present book is the result of investigations carried out by algebraists at Moscow University over the last fifteen years. It is written for mathematicians interested in Lie algebras and groups, finite groups, Euclidean integral lattices, combinatorics and finite geometries. The authors have used material available to all, and have attempted to widen as far as possible the range of familiar ideas, thus making the object of Euclidean lattices in complex simple Lie algebras even more attractive. It is worth mentioning that orthogonal decompositions of Lie algebras have not been investigated before now, for purely accidental reasons. However, automorphism groups of the integral lattices associated with them could not be investigated properly until finite group theory had reached an appropriate stage of development. No special theoretical preparation is required for reading and understanding the first two chapters of the book, though the material of these chapters enables the reader to form rather a clear notion of the subject-matter. The subsequent chapters are intended for the reader who is familiar with the basics of the theories of Lie algebras, Lie groups and finite groups, and is more-or-less acquainted with integral Euclidean lattices. As a rule, undergraduates receive such information from special courses delivered at the Faculty of Mathematics and Mechanics of Moscow University. In any case, it is essential to have in mind a small collection of classic books on the above-mentioned themes: [SSL], [SAG], [Gor 1], [Ser 1] and [CoS 7], Our teaching experience shows that the material of Part I and some chapters from Part II can be used as a basis for special courses on Lie algebras and finite groups. The material on integral lattices enrich the lecture course to a considerable extent. The interest of the audience and the readers grows, due to the great number of concrete unsolved problems on orthogonal decompositions and lattice geometry. In connection with integral lattice theory, we mention here a comprehensive book [CoS 7] and an interesting survey [Pie 3], where one can find much information contiguous with our book. It is a pleasure to acknowledge the contributions of the many people from whose insights, assistance and encouragement we have profited greatly. First of all we wish to express our thanks to Igor Kostrikin and Victor Ufnarovskii, who were among the first to investigate orthogonal decompositions, and whose enthusiasm has promoted the popularisation of this new research area. Their impetus was kept

vi

Preface

up by the concerted efforts of K. S. Abdukhalikov, A. I. Bondal, V. P. Burichenko and D. N. Ivanov, to whom the authors are sincerely grateful. We are particularly indebted to D. N. Ivanov and K. S. Abdukhalikov: Chapter 7 is based on the results of D. N. Ivanov's C. Sc. Thesis, and the first five paragraphs of Chapter 10 are taken from K. S. Abdukhalikov's C. Sc. Thesis. Some brilliant ideas came from A. I. Bondal and V. P. Burichenko. We would like to thank Α. V. Alekseevskii, Α. V. Borovik, S. V. Shpektorov, K. Tchakerian, A. D. Tchanyshev and Β. B. Venkov, who have made contributions to progress in this area of mathematics. A significant part of the book has drawn upon the Doctor of Sciences Thesis of the second author. We are indebted to our colleagues W. Hesselink, P. E. Smith and J. G. Thompson for a number of valuable ideas mentioned in the book. Our sincere thanks go to Walter de Gruyter & Co, and especially to Prof. Otto H. Kegel, for the opportunity of publishing our book. We would also like to express our gratitude to Prof. James Wiegold for his efforts in improving the English. We are grateful to Professor W. M. Kantor for many valuable comments. The authors wish to state that the writing of the book and its publication were greatly promoted by the creative atmosphere in the Faculty of Mathematics and Mechanics of Moscow University. The present work is partially supported by the Russian Federation Science Committee's Foundation Grant # 2.11.1.2 and the Russian Foundation of Fundamental Investigations Grant # 93-011-1543. The final preparation of this book was completed when the second author stayed in Germany as an Alexander von Humboldt Fellow. He wishes to express his sincere gratitude to the Alexander von Humboldt Foundation and to Prof. Dr. G. O. Michler for their generous hospitality and support. A. I. Kostrikin Pham Huu Tiep

Table of Contents

Preface Introduction

Part I: Orthogonal decompositions of complex simple Lie algebras

ν 1

11

Chapter 1

Type An

13

1.1 Standard construction of ODs for Lie algebras of type Ap»>- \ 1.2 Symplectic spreads and ./-decompositions 1.3 Automorphism groups of /-decompositions 1.4 The uniqueness problem for ODs of Lie algebras of type A,„ η < 4 . . . . 1.5 The uniqueness problem for orthogonal pairs of subalgebras 1.6 A connection with Hecke algebras Commentary

13 17 25 31 37 49 51

Chapter 2

The types Bn, Cn and Dn

56

2.1 2.2

56

Type C„ Partitions of complete graphs and ^-decompositions of Lie algebras of types Bn and D„: automorphism groups 2.3 Partitions of complete graphs and Ε-decompositions of Lie algebras of types Bn and Dn: admissible partitions 2.4 Classification of the irreducible ^-decompositions of the Lie algebra of type D„ Commentary

64 68 76 81

Chapter 3

Jordan subgroups and orthogonal decompositions 3.1 General construction of TODs 3.2 Root orthogonal decompositions (RODs) 3.3 Multiplicative orthogonal decompositions (MODs) Commentary

84 84 88 93 103

viii

Table of Contents

Chapter 4 Irreducible orthogonal decompositions of Lie algebras with special Coxeter number 107 4.1 The irreducibility condition and the finiteness theorem 107 4.2 General outline of the arguments 110 4.3 Regular automorphisms of prime order and Jordan subgroups 113 4.4 h + 1 = r. the P-case 122 4.5 h + I = r: classification of IODs 125 4.6 A characterization of the multiplicative orthogonal decompositions of the Lie algebra of type D4 131 4.7 The non-existence of IODs for Lie algebras of types Cp and Dp 136 Commentary 139

Chapter 5 Classification of irreducible orthogonal decompositions of complex simple Lie algebras of type An 141 5.1 The S-case 5.2 The P-case. I. Generic position 5.3 The P-case. II. Affine obstruction 5.4 The P-case. III. Completion of the proof Commentary

142 144 152 158 160

Chapter 6 Classification of irreducible orthogonal decompositions of complex simple Lie algebras of type Bn 162 6.1 The monomiality of G = Aut(£>) 6.2 Every IOD is an ^-decomposition 6.3 Study of f-decompositions Commentary

163 168 176 186

Chapter 7 Orthogonal decompositions of semisimple associative algebras

187

7.1 Definitions and examples 7.2 The divisibility conjecture 7.3 A construction of ODs Commentary

187 190 195 198

Table of Contents

ix

Part II: Integral lattices and their automorphism groups

199

Chapter 8 Invariant lattices of type Gi and the finite simple G2(3)

203

8.1 Preliminaries 8.2 Invariant lattices in £ 8.3 Automorphism groups Commentary

203 208 216 229

Chapter 9 Invariant lattices, the Leech lattice and even unimodular analogues of it in Lie algebras of type A p _i 231 9.1 Preliminary results 9.2 Classification of indivisible invariant lattices 9.3 Metric properties of invariant lattices 9.4 The duality picture 9.5 Study of unimodular invariant lattices 9.6 Automorphism groups of projections of invariant lattices 9.7 On the automorphism groups of invariant lattices of type Commentary

232 241 248 253 259 264 273 282

Chapter 10 Invariant lattices of type Ay-ι

285

10.1 Preliminaries 285 10.2 Classification of projections of invariant sublattices to a Cartan subalgebra 293 10.3 The structure of the SL 2 ('„ is also nilpotent, so AT(JC.V, y„) = Tr(adjt9.ad;y„) = 0. Similarly, K(ys,xn) — Kix,„yn) = 0. Therefore, we have Kix, y) = Kixs,

v,) + Kixs,

yn) + K(ys,

x„) + K(x„, >·,),

as stated. Now choose some basis { j t 1 , . . . , xm) of the complex space H, where m = dim7i = rank£. From the non-degeneracy of the form Κ it follows that the j matrix iKix', x ))\=

TrFv/F,,(M Ο u),

1 Type A,

16

where u ο u' = aß' - α'β for u = (a, ß), u' = (α', β'); α, β, α', ß' e ¥g. For this form on W there exists a symplectic basis {e\, . . . , em,f\, . . . , f m ] such that m < u Iu >=

— a'jbj)

for u = Σ?=\(a,ei + £//,), u' = ΣΤ=\(^ί + b'Ji). In view of Witt's Theorem, one can suppose that the first component F^ of W is generated (over ¥p) by e\, ..., em, and the second component by f \ , . . . , f m . In this basis set Ju -

J(ai.h,) ® JUi2,b2) ® • · · ® J(am,bm),

(1.2)

identifying a vector u with the collection of coordinates: u = («ι,

. . . , am, b\, . . . ,

bm).

In our terminology the set {Ju \ 0 ^ u e W} forms a J-basis of the Lie algebra £ of type Aq_\ as a C-space. The operations on matrices J u are performed by exactly the same rules as in the case of the matrices J(a.b) (see (1.1)): Jku =

JJ ,

[JUi J«'] =

= sB^Ju+u,

(1.3)

— 1 }Ju+u),

where

m B(u,u)

a b

= -J2 'i i·

(I·4)

1=1 By (1.3), the matrices of the form s J u form an extraspecial group Η of order p2m+\ u n £ j e r t^g u s u a i product. In particular, s

JUJVJ~]

=sJv.

(1.5)

This remark will be required later. Now it is easy to see that the subspaces

Hoc

= (A(U>

I λ e F*)c

of £ are Cartan subalgebras and the decomposition £

=

Ή-οο Θ (®aeF„η

(1.6)

is an orthogonal decomposition for C. To this end, it is sufficient to remark that i) < (λ, λα) I (λ',λ'α) > = 0 , ii) the trace TrJ(a,p)J(y,S) is not zero if and only if (y, = 0 for all u, u' e Wj·, that is, the subspace < Wi > p is totally isotropic on the one hand. On the other hand, the dimension of any totally isotropic subspace in W does not exceed m, and = pm. Hence W, coincides with < W, >, and so it is a maximal totally isotropic subspace of W. Conversely, if the subspaces Wj from (1.9) form a symplectic spread, then it is obvious that (1.7) is a ./-decomposition (again use the formulas (1.3) and Lemma 1.1.1).

Definition 1.2.3. By the kernel Κ(π) of a spread 7Γ of the form (1.8) we mean the set Κ (π) = { i and < * | * >2 on W. We shall show that we can find an element ΰ e Κ* such that < u \ ν > 2 = < üu \ ν >1 for all η, v g W. Without loss of generality, one can suppose that < u \ ν >1=

TrK/¥n(u ο υ),

where u ο υ = αδ — βγ for u = (a, β), ν = (y, 2 — < u I Xu >2 — < ν I λυ = < u \ λυ >2 — < Xu \ ν >2; in other words, < u\Xv >2=
2·

>2~

20

1 Type A,

For the fixed element w — (0, 1) e W ^ , consider the F^-linear functional f : a e Κ 0)|ιυ>2· Obviously, one can find ΰ e Κ such that f(a) = Tr^/F (#of). So, for u = (α, β), ν = (γ, δ) we have < u I ν >2=
2 =

= < (a, 0) I (0, 2 - < (y, 0) I (0, β) >2— = < (α, 0) I 8w >2 — < (y, 0) I ßw > 2 =

= < 8(a, 0) I w >2 — < β(γ, 0) | w >2= = f(aS - βγ) = TrK/Fi,(&(a8 - βγ)) = Tr K/Fp (&u ο ν) =< Du \ ν >,, as stated. 3) We return to symplectic spreads π, π' = Α(π) of the space W with the symplectic form < * | * >, and introduce a new symplectic form < u | ν >'— < Au I Αν >. Then π is a symplectic spread with respect to < * | * > and < * I * >'. Hence, in accordance with 2), there exists ΰ € K* such that < Au I Av > = < üu I ν >. In 2) we have also shown that < \ v>= < u I ϋν >. So it is easy to see that the operator Β defined on W = Κ 0 Κ by the rule B((a, β)) — (ϋα, β) preserves π, and also that < Bu \ Bv >—< §u \ υ >. In particular, < Au | Av >=< Bu \ Bv >, that is, AB X £ Sp(W). Moreover, ΑΒ~χ(π) = A{π) = π'. In other words, π and π' are S/?(W0-conjugate. 4) Finally, we compute the group Aut(7r) for a desarguesian symplectic spread π. According to the result proved above, it is enough to find this group for the spread π = {W^, Wk \ λ e K) of the space W — Κ ® Κ with the form < u I ν > = Tr^/F (μ ο v). By Lemma 1.2.5, any ψ e Aut(7r) is contained in the group YL2{q) = GL2(q).Gal(Fc//Fp) of all A'-semilinear transformations of the space W = Κ 0 Κ, Κ = ¥q. Conversely, r L 2 ( ^ ) fixes π. So, Aut(^) = ΓL 2 (q) Π Sp±(W) = (SL2(q).Ζιη).Ζυ.

Ο

We are interested firstly in symplectic spreads which correspond to groups Aut(7r) that are transitive on W\{0}. Obviously, any desarguesian spread satisfies this condition. A classification of such spreads is given in the following statement. Theorem 1.2.7. Up to Sp±(W)-conjugacy, there exists precisely one symplectic spread with group Aut(jr) transitive on W\{0). The case p"' = 27 is exceptional, when there exists one further spread, with group Aut(7r) = SLi{\ 3). Proof, i) Set A — Aut(7r). Let W be the translation group of the space W. Considering the natural permutation representation of the group Ä = W.A on W, we see that Ä acts doubly transitively on W. Such groups Ä - the so-called affine 2-transitive permutation groups - are classified by Hering [Her 2] and Liebeck [Lie 2]. In this connection, A belongs to one of the following classes.

1.2 Symplectic spreads and ./-decompositions

(a) Infinite classes. ( a l ) A c ΓΖ,,(γ) = GL\(r).Gal(¥r/¥p), ( a 2 ) A>SLa(r),

ra =p2m,

r =

21

p2'"·

a > 1;

(a3) A > S / ? 2 „ ( 0 , = / Λ α > 1; (a4) Λ > G 2 ( r ) ' , /?'" = 2'" = r \ (b) Extraspecial classes. The group A normalizes an extraspecial subgroup R, where either p'n = 5, 7, 11, 23 and R = 2 l + 2 = Qg, or pm = 9 and R = 21+ 4 = Dh * 0 8 · (c) Exceptional classes. ( c l ) pm = 9, 11, 19, 29, 59, A > 5 L 2 ( 5 ) ; (c2) = 4, Λ = A 6 or Λ = A 7 ; (c3) / / " = 21, A = SL2( 13). Since the symplectic spread is unique when m = 1, we shall assume henceforth that m > 1. Set / = dimp Κ (π). Our argument will be based on the transitivity of A on the elements W, of the spread π. ii) Here we show that π is desarguesian in case ( a l ) . Set S = GL\{r), Β = Galfßrßp) ~ Z 2 „, and Η = Α Π S. Note that ρ2'" - 1 divides \A\, so if \H\ = (p2m — 1 )/t, we have that t divides g c d ( 2 m , p 2 ' " — 1). Since A acts transitively on the elements Wj of π, the normal subgroup Η acts ^-transitively on π, that is, with orbits of the same length, say d, d \ (p'" + 1). But Η is cyclic, so that the subgroup H\ of index d in Η is unique. Hence, H\ fixes every element W, e n. Thus Η ι c Κ(π)*, which implies the divisibility of ρ' - 1 by (p2m - 1 )/td. Recall that d I (p'" + 1), t I g c d ( 2 m , p 2 m — 1), 1 < I < 2m, / | 2m. It is now easy to show that we always have I = m, with the single exception pm = 9, / = 4, d = 10, / = 1. But in this exceptional case, A cannot act transitively on W \ { 0 } . S o / = m and π is desarguesian. iii) A s s u m e now that one of cases ( a l ) , (a2) or (a3) holds. Then A \> S, where S = SLa(r), Sp2a{r) or Giir)'. Again, S acts ^-transitively on π with orbits of length d, d \ {pm + 1 ) . Note that d > 1, because the equality d = 1 means that S c K{π)* and S is soluble. Thus, S has a subgroup S | = S / s ( W i ) of index d, d > 1, d I (p'" + 1). Since d and ρ are coprime, by the Borel-Tits Theorem [BoTJ, [Sei 1], Si is contained in some maximal parabolic subgroup Ρ of 5. If S = G^ir)' we have

If S = Spjair),

ü > 1 then J k= 1

2Ui-j)+2k _ ι

22

1 Type A, 3m/2

for some j, 1 < j < a. Hence (S : P) > p a > 1 then

> pm + 1. Finally, if S =

SLa(r),

But p'n + 1 > d =

(S : S\) > (S : P), so we obtain the unique possibility S — SL2(q), q = p , d = q + 1, Si = P. In this case, we can identify the space W with the natural S-module φ F^. Because W\ is invariant under m

we must have W\ = {(m, 0) | u e F^}. Making S act on Wi, one can see that the spread π has the form | λ € F^}, where = {(w, Xu) \ u e ¥q}, = {(0, u) I u e F^}. But in that case, the kernel Κ (π) contains the multiplications by scalars from F^, that is, |ΑΓ(π)| > pm; hence π is desarguesian. iv) Direct calculations show that when pm = 9, every symplectic spread is desarguesian. In particular, cases (b) and (cl) can be eliminated. Furthermore, for pm — 4, the groups Ag and Κη have no subgroups of index 5, so case (c2) is also impossible. It remains to consider the case (c3): pm = 27, A = SL2(13). We mention some properties of A: a ) A acts transitively on W\{0}, W — F^; β) A is a maximal subgroup of Sp^O). Moreover, 3) has two conjugacy classes of maximal subgroups of type SL2( 13), which are permuted by the outer E-t. 0 \ s

(

o

)e

PtO)·,

y) every subgroup of index 28 in SL2(13) is conjugate to the subgroup

Consider the element φ — f\j of order 13 in Q. As 132 / |Sp6(3)|, we can suppose that φ lies in the maximal parabolic subgroup

(matrices are written in a symplectic basis {e\, ,/ι,/ι})· Setting 2 Fj 7 = < ϋ >, one can assume that φ\a is multiplication by ϋ , where U = I) spread of a space W. Let V be an incidence structure, where points are vectors of W and lines are affine subvarieties W, + u, u e W. Then V is a finite linear space. Moreover, transitivity of the spread π implies flag-transitivity of the space V. • Proposition 1.2.12. Suppose that W — ¥2m is a symplectic space, π — {Wt-}"_, a symplectic spread, and GQ a subgroup of Aut(N) with transitive action on Then one of the following statements holds.

25

1.3 Automorphism groups

1) Go is soluble and Go c YL \ (p2m); 2) π is desarguesian; 3) pm = 27, G 0 = 5L 2 (13); 4) = r2, r = 2 2 / + 1 (I > 1 5 z ( r ) < G 0 C Aut(5z(r)). Moreover, in cases 2) - 4), f/ze corresponding spread exists and is unique up to conjugacy. Proof. Let V be a finite linear space corresponding to π (see Lemma 1.2.11). Then G = W.GQ is a flag-transitive automorphism group of the space V, where W denotes the translation group of W. Furthermore, V has Ν = p2m points and each line of V contains k = p'" points. According to a fundamental result [BDDKLS], in this case one of the conditions 1) - 4) is satisfied. Note that here we have excluded a translation plane of order 9, which is constructed via a near-field (see [Lun] or [Hall 1]). This is because, as mentioned in the proof of Theorem 1.2.7, every symplectic spread of F3 is desarguesian. The uniqueness of η in cases 2) 4) is well-known (see [Lun]). • There are many known examples of transitive symplectic spreads with soluble automorphism groups [Kan 4, 5], These examples suggest that such spreads may be too numerous to be classified. Transitive spreads π such that d i m ^ ) W = 4 have been studied in [BaE 1,2].

1.3 Automorphism groups of /-decompositions In the preceding paragraph, with each /-decomposition V of the Lie algebra C of type Apm_x we associated a certain symplectic spread π = ττχ> of the symplectic space W = Our aim is to prove that the group Aut(D) can be expressed in terms of Η (see §1.1) and Aut(Tr). Recall that Aut(£) = Inn(£). < Τ > , where Inn(£) ~ PSLp*(C) is the group of all inner automorphisms of €., and Τ is the outer automorphim Χ μ>· —'X. (In the case of the Lie algebra of type A1, the automorphism Τ is inner). Set Ζ = Z(SIp»(C)) ~ Z ^ , Η = Η Ζ / Ζ ~ H / Z ( H ) =

A(J) = {c)

G(J) = A(J) η ΐ η η ( £ ) . Lemma 1.3.1. The following equalities hold: (i) Q(J) = H.Sp2m(p); (ii) A(J) = Q{J). < Τ >.

=< Jv >c},

1 Type A,

26

Proof. The embedding of Η in PSLpl«(C) via matrices Ju, described in §1.1, occurs in the works of many authors (see, for example, [Sup 1]). In [Sup 1] it is shown that Q{j) = Npsl^c

)(//) =

H.Sp2mip).

The extension of Η by Sp2,„(p) is split when ρ > 2. More precisely, Η consists of the maps (1.11) where f eW* mations

= Horrig (W, F ; ,), and Sp2m (p) is identified with the set of transforφ : Ju ^

(B(»Mu))-B(U.U))/2J(fiW ^

s

(1.12)

where φ e Sp(W). If ρ = 2, then the extension H.Sp2,n(p) is non-split in general. This happens if m > 2 (see below). Furthermore, equality (ii) follows from the fact that the action of Τ takes the form T{JU) = -sB{u] : £ = @"_{H\ into a ./-decomposition V2 : C = ®"=lTif. 1) Firstly, suppose that φ e Inn(£). Using the transitivity of Sp(W) on the set of maximal totally isotropic subspaces of W, we can assume that Ή\ is the standard Cartan subalgebra H () = { d i a g ( X , , . . . , λ,,™) I λ,·

> = °)·

λ

/

Suppose that φ(Ηο) = Hj for some i. As above, we can find / e Q(J) such that f(7ij) = Hq. Here, / maps V2 into a new 7-decomposition P 3 : C = θ · = 1 Η]. Consider an arbitrary element Ju φ TCq. Then fc, u G W. Finally, the factor-group G/H+ is obviously just Aut(7r). • The following assertions are immediate consequences of Proposition 1.3.3 and Lemma 1.3.2. Corollary 1.3.4. The standard orthogonal decomposition T> of the Lie algebra C of type Ap»>-\ constructed in §1.1 has automorphism group Aut(P) = The last factor

Zjm.(SL2(pm).Zm).Z2.

is absent in the case p'" = 2.

•

Corollary 1.3.5. A J-decomposition V of the Lie algebra C of type A if'·: ι is an irreducible (transitive, respectively) OD if and only if the corresponding symplectic spread π = π-ρ of the space W = Fj"' is irreducible (transitive, respectively). A classification of irreducible (transitive, respectively) J-decompositions of L up

1 Type A,

28

to Aut(C)-conjugacy is equivalent to a classification of irreducible respectively) symplectic spreads ofW up to Sp^ (W)-conjugacy.

(transitive, •

Combining Corollary 1.3.5 with Theorem 1.2.7 we obtain: Corollary 1.3.6. Every irreducible J-decomposition of the Lie algebra C of type Apm-1 is standard. The unique exception is the case pm = 27, where there exists (up to Aut(£)-conjugacy) just one more irreducible (non-standard) J-decomposition V with automorphism group Aut(£>) = Zf.SL 2 (13).

•

Note (see Chapter 4) that Corollary 1.3.6 remains valid if we exchange the words "./-decomposition" in the statement by "orthogonal decomposition" ! From Corollary 1.3.4, it follows that, for a standard decomposition V of the Lie algebra C of type Apm_b we have Inn(D) = I?pm.ΣΖ,2(ρ'"), where Inn(X>) = Aut(Z?) Π Inn(£), and ZL2(pm) =

SL2(pm).Gal(Wp„,/Fp)

consists of all special F/y»-semilinear transformations of the space W = (F,,™)2. Theorem 1.3.7. The extension Inn(X>) = Ζ2'".Σ L2(p"1) is non-split if and only if ρ — 2 and m > 2. Proof. 1) The fact that Inn(X>) is split over l?pm for ρ > 2 follows from the explicit formulas (1.11) and (1.12). Henceforth, we shall suppose that ρ = 2 and q = 2m. Then all 7 u 's are of order 4. But according to Lemma 1.3.2 every Ψ e Inn(I>) permutes the lines < Ju >c- Therefore Inn(D) consists of the elements u/ T j .ψ ·· J II /

I V

1; / ( " ) J

ι tp(u)->

where i / is a functional on W with values in Z 4 = Z/4Z, and the map φ acts by the rule φ(α + βϋ) = aa2" + bß2" + (ca2" + άβ2")ϋ

(1.13)

with 0 < η < m — 1, a, b, c, d e ¥q, ad — be = l, α, β e Fq. Here (see also §2.1), we identify the space W with F ^ = F ^ t f ) . Furthermore, for u = a + βϋ and ν = γ + δϋ we have < u I ν >= TrF(//F,(aß)j'ß^

so that g,2{ß) = g\(rxß). In particular, g^+Aß) = gi«s + t)-*ß) for all j , t € F*, s φ t. On the other hand, from (1.17) it follows that g^+Aß)

= SAß) + sAß)

= g\(s~lß)

+ g\(rlß)

= gl«*-1

rx)ß).

+

Hence, ((.5 + t)~l + s~l + t~l)ß e Kerg, for all β e Ff/ and s, t e F*, s Φ t. Since m > 2, there exist s, t e F* such that s φ t and (s + t)~l + s~l + t~] φ 0. But in this case Kergi = F^, gi = 0, and gh — 0 for all b, as claimed. Recall that we have identified W = with F ^ = ¥„(&). For the vector υ = 1 6 W choose ψ = ψα such that = 1. Suppose that ir(J\) = Μ\+αϋ, λ e C*. Then A = Ψ2(Α)

= ifaA+aü)

= mu\j'a9)

— ΙΑ J J +aü.Jaff

= ίλψν\).ψ(Αϋ)

=

— A J ,.

Hence λ 2 = — 1, on the one hand. On the other hand, Jo = f(J0)

= i(A-J\)

= Ψ iA)MA)

= λ 2 ( / , ^ ) 2 = λ 2 7ο,

which implies that λ 2 = 1. This contradiction shows that the group Inn(D) is non-split over Z 2m if m > 2. 5) Finally, we define a section S : ΣΖ^Ο?) —> Inn(D), which is in fact an embedding ΣΖ^Ο?) ^ Inn(P) if m < 2. By the Bruhat decomposition, it is

1.4 Uniqueness problem for ODs

enough to define the values φ we set \ — < 1 \Q(c/-/h)+Jraß.Trc/^h+Tr(bß+a^b) ji VbyJu) — y~l) where u — a + βϋ and the quadratic form Q : Ft/ —» F2 satisfies the condition Q(a + ß) = Q{a) + Q(ß) + Traß + Tra.Trß. (Here, the suffix F^/F 2 is omitted from the notation of the trace Trp /F2). Relations (1.15) and (1.16) show that the maps φ~, ω - , p~ and ψ^ so constructed really are contained in Inn(£>). • Corollary 1.3.8. The group Q(J) = Zj'n .Sp2m(p) is non-split over Zj'" if ρ = 2 and m > 2. • To conclude this section, we mention the following fact. Let W = F'j be the natural F^-module for SL„{q). Then the second cohomology group H2(SLn(q), W) is trivial except in the cases (n, q) = (2, 2'"), m > 2; (3, 2); (5, 2); (3, 3 m ), m > 1; (3,5) and (4,2), when it is F^ (see [Sah], [Avr]). In the first three exceptional cases a (unique) non-split extension of W by SL„{q) is realized as the automorphism group of some OD of the Lie algebra of type Α2»·_ι (as we just convinced ourselves), G2 and Ε χ (see Chapter 2), respectively.

1.4 The uniqueness problem for ODs of Lie algebras of type An, η < 4 Examples of ODs for Lie algebras of type A„, η < 4, have been constructed in §1.1. In fact the following result holds. Theorem 1.4.1. A Lie algebra C of any of the types A\, A2, A3 and A4 possesses an orthogonal decomposition, that is unique (up to conjugacy). Sketch of the proof. In principle, the problem of classifying ODs can be solved using the following scheme. As all Cartan subalgebras are conjugate, and the orthogonality property is preserved by conjugation, one can assume that one component of the OD is the subalgebra TLQ consisting of the diagonal matrices. Thus, the problem is reduced to extending 7Y0 to an orthogonal decomposition and showing that any two such decompositions are conjugate under the stabilizer of HQ in Aut(£). To this end, we first investigate the set C — {Ή.} of all Cartan subalge-

32

1 Type A,

bras orthogonal to HQ. A parametrization of C which is minimal in some sense is an essential part of this proof. Secondly, a decisive step consists of analysing the relations between the parameters that follow from the pairwise orthogonality condition for certain members of C. We make some general remarks concerning a general Ή e C. 1) Every diagonal matrix is orthogonal to H . 2) Every matrix A eTt has zero principal diagonal. 3) For any matrix A e Η and any polynomial f ( t ) , the matrix / ( A ) has equal coefficients on the principal diagonal. 4) In Ή there exists an invertible matrix X such that X" + l = En+\ (the unit matrix of degree η + 1) and Ή. = < Χ, X 2 , . . . , X n > c · 5) A nonzero matrix from H, cannot have a zero row nor a zero column. 6) The subalgebra Η admits a basis {A(k}, 2 < k < η + 1} satisfying the following conditions: (i) the first row of A(k) contains exactly one nonzero element; it is equal to 1 and stands in the £-th position; (ii) the k-th column of A{k) contains exactly one nonzero element; by (i), it is equal to 1 and stands in the first position; (iii) the y'-th row of A(k) coincides with the &-th row of A^K Only assertion 6) requires proof. From 5) it follows that there exists a basis {k) [A , 2 < k < n + 1} with property (i). To prove (iii), we use the commutativity of H: by (i), the y-th row of A{k) is equal to the first row of the product whereas A^A(k) = A(k)A^\ so this first row coincides with the £-th row of To prove (ii), it remains to note that for j > 2 the y-th element of the &-th column of A{k) is the k-th element of the j-th row of A(k\ and by (iii) the latter is equal to the /c-th element of the A>th row of and therefore it is zero by 2). Applying these elementary remarks to the case η < 3, we obtain the following parametrization. Lemma 1.4.2. Every subalgebra (i) If η = 1, then

for some α φ 0. (ii) If η = 2, then

for some a, b, ab φ 0.

7i € C has a basis of the form indicated

below.

33

1.4 Uniqueness problem for ODs

(iii) If η = 3, then Η =< A (2) , A (3) , A(4) >c, where 0 ab be 2 V cyd

/

1 0 0 0 C3 bc2/d 0 0 a 0 b 0

( cid° bc\ \ bd

\ , A(3) -

/ 0 0 d bcxja

0 1 0 0 0 a c 1 0 Cl d 0 0

( be°2 ad V be,

0 b 0 erf/a

\ /

1 \ 0 0 0

here abd φ 0 and at least two of c\, C2, (3 are equal to 0. Because of the cumbersome nature of the calculations, we omit the proof of Lemma 1.4.2 (and also of some following lemmas), referring the reader for details to [KKU 4], • If αϊ — 4, the following statements hold for Ti e C. Lemma 1.4.3. Suppose that Ή contains at least one monomial matrix. Then Ή admits a basis {A ( '\ 2 < i < 5}, where A = A(2) is monomial, Λ5 = λΕ$ for some λ φ 0, and all the matrices A(l) are proportional to powers of A. • Lemma 1.4.4. Assume that Ή = Hi does not contain monomial matrices. Then, up to simultaneous conjugation by a diagonal matrix and multiplication by a constant, its basis matrices AU) have the following form: / 0 2 1 1 V1 0 1 λ 2λ 2 ν λ

/

1 0 0 0 \ / 0 1 0 1 1 1 2 (3) 0 0 1 1 2λ , A = 0 1 0 1 λ 0 1 1 0 \ λ 0 0 λ2 λ3 λ2

0 1 0 λ2 λ

0 0 λ3 λ2 λ2

1 0 \ / 0 1 0 1 0 λ λ , Α(5) = λ 0 λ2 2 0 0 / ν 2λ

1 0 0 1 0 λ2 0 0 0 λ 0 0 λ2 λ2 λ3

0 1 0 λ λ2

0 \ 1 λ2 λ 0 / 0 1 λ 0 λ2

1 \ 0 0 0 0 /

where λ 2 + λ — 1 = 0 . Every Cartan subalgebra Ή! e C admitting no basis consisting of monomial matrices (briefly: non-monomial Cartan subalgebra) may be obtained from ΉΊ via conjugation by a diagonal matrix. •

34

1 Type A,

Having parametrized basis matrices for Η e C, we can begin the proof of Theorem 1.4.1. For example, suppose that η = 2 and let C = θ; 3 = 0 Hi be an orthogonal decomposition of the Lie algebra C of type A2. According to Lemma 1.4.2, one can suppose that

H\

=

H2

=

h3

= \ \

cd

0

0 /

\

0

d

0 ) /

c

Because of the orthogonality condition K{H\,H2) = 0, we have a = b 2 and 1 + a + a = 0. If ε is a primitive cube root of 1, either a = b = ε or a = b = ε 2 . Similarly, c = d g {ε, ε 2 }. But K(H2, Hi) = 0, so we can suppose that a = ε, c = ε 2 . Thus the decomposition is uniquely defined. The same arguments are applicable to the cases η = 1 and η = 3. Now suppose that η = 4. Firstly, by using Lemma 1.4.4, one shows (and the main difficulty of this case lies here!) that every component of the decomposition C = Θ; =0 7ί,· is a monomial subalgebra. The monomial matrix A(,) (see remark 6)) is the product of some diagonal matrix and the permutation-matrix π(Λ ( , ) ). If Η is a monomial Cartan subalgebra orthogonal to HQ, then it is easy to check that 7r(H) = {£";,+ι, π{Α(,)) | 2 < / ' < « + 1} is an abelian subgroup of order η + 1 of the symmetric group §„+|. In our case η = 4, the group π(Η) is the cyclic group Z5. For arbitrary n, the following statement holds: Lemma 1.4.5. Suppose that Η is a monomial Cartan subalgebra of the Lie algebra of type An that is orthogonal to the standard Cartan subalgebra Ho- Then π(Η) is a transitive regular abelian permutation group of degree η + 1. Proof. According to remark 3), for π e π(Η) every permutation π' is trivial or else has no fixed point. Thus, π is a product of disjoint cycles of the same length d = d(π). In particular, if 7r fixes some point, then d = 1 and π is trivial. But \π{Η)\ = η + 1, so π{Η) is a regular transitive permutation group (of degree n+1). • Lemma 1.4.6. Suppose that Η and H! are monomial Cartan subalgebras of the Lie algebra of type An that are orthogonal to the standard Cartan subalgebra HQ. If the set π(Η).π(Η') = {σ.σ \ σ e π{Η), d € π {Η')} C §„+ι

1.4 Uniqueness problem for ODs

35

contains some permutation fixing exactly one symbol, then the subalgebras Η and H' are not orthogonal. Proof. As we mentioned in the proof of Lemma 1.4.5, neither π(Η) nor π(Η') contains permutations with exactly one fixed point. Therefore, if a.c/ fixes precisely one symbol, where σ e π(Η) and d e π(Η'), then σ φ 1 and & φ 1. So a = π (A) and and π(Η') = < (1, 2, χ, >', z) >, where {jc, y, z} = {3,4,5}. The required statement follows from the following equalities: (1, 2, 4, 3, 5)(1, 2, 3,4. 5) = (1,4)(2, 5)(3); ( 1 , 2 , 3 , 5 , 4 ) 0 , 2 , 3 , 4 . 5 ) = (1, 3)(2, 5)(4); ( 1 , 2 , 5 , 4 , 3)(1, 2, 3, 4, 5) 4 = (1, 4)(3, 5)(2); (1, 2, 4, 5, 3)(1, 2, 3, 4, 5) 3 = (1, 5)(2, 3)(4); (1, 2, 5, 3, 4)(1, 2, 3, 4, 5) 3 = (2, 3)(4, 5 ) 0 ) .

•

As we have just shown, all subalgebras 7i; with i φ 0 are monomial and have the same group π(7ίί). Permuting rows and columns if necessary we can suppose that π(Τίι) = < (1, 2, 3, 4, 5) >, that is, Hi —< A, A2, A 3 , A4 >c for some A = diagO, a, b, c, d).P, where Ρ is the permutation matrix ( 1 , 2 , 3 , 4 , 5 ) . For brevity, we denote such a Cartan subalgebra by H(a, b, c, d). Direct calculations show that the subalgebras H(a, b, c, d) and H(a\ b', c', d') are orthogonal if and only if one of the following conditions holds: (i) a'/a = d'/d = 1, b'jb = c/c' = a-; (ii) a'/a = ek, b'/b - e2k, c'/c = ε Μ , d'/d = s4k\ (iii) a'/a = a.d'/d, where a + a _ l + 3 = 0, 1 < k < 4 and ε is a primitive 5-th root of 1. After establishing this fact, it is not difficult to show that (up to conjugacy) the Lie algebra of type A4 has a unique orthogonal decomposition C =

et={)H(sk,£2k,eik,e4k)®H0.

This completes the proof of Theorem 1.4.1.

•

36

1 Type A,

After proving Theorem 1.4.1, it is natural to treat the case of the Lie algebra £ of type As. Unfortunately, all attempts to prove or to disprove Conjecture 0.1 for £ undertaken so far have met with no success. In view of Theorem 1.4.1, an essential part of the argument confirming the validity of this conjecture for type A5 is contained in the following statement. Proposition 1.4.8. The Lie algebra C of type A 5 has no monomial OD. Sketch of the proof It would be boring to go into details, which often require the direct checking of a great number of special cases. Instead, we highlight the most important ideas in the proof, which are to a considerable extent typical for Lie algebras C of type A„_ 1 for arbitrary n. 1. Let C = ®"=0Hi is a monomial OD, that is, Ho consists of diagonal matrices and Hi possesses a monomial basis {Λ/, ι , . . . , A/, „_i}, I < i < n. According to Lemma 1.4.5, when η = 6 the group π {Hi) is a cyclic regular transitive permutation group of degree 6. Using Lemma 1.4.6 and checking special cases directly, one can show that K{Hi, Hj) Φ 0 if the groups π {Hi) and n{Hj) are different. 2. If all groups π{Ηι) are the same and are cyclic, then the orthogonality condition K{Hj, Hj) = 0, i φ j, leads to the following system of equations: n-l Y^

· · · Xk+s = 0, 0 / 3 ) , ( - 1 , - 1 - V 3 - V s W H , - 1 - V3 + V 3 + V I ^ ) , ( - 1 , - 1 + V3 - i\j3 + Τ Ϊ 2 , - 1 + v / 3 + /V / 3 + v / 12). Similarly, ^ = , jc'5 = and 4 = / / 3 ; < = « V / , = β"χ'{ and x" = Ϋ'χ'ί', where (α', β', / ) and (α", β", γ") belong to the same set of triples. But it follows from the condition = 1 that ota'a" = 1, β β'β" = 1 and y j / / = 1. However, no three triples from the 19 triples described above satisfy this condition. Hence we cannot even find three pairwise orthogonal Cartan subalgebras H \ , H i , Τίτ, (each of which is, by definition, orthogonal to the standard Ho). •

1.5 The uniqueness problem for orthogonal pairs of subalgebras Up to now, we have dealt only with sets consisting of η + 1 Cartan subalgebras of the Lie algebra C of type A„_i that are pairwise orthogonal. Meanwhile, orthogonal pairs of Cartan subalgebras are also worthy of attention in connection with, firstly, the uniqueness problem for OD and, secondly, the requirements of algebraic topology and geometry. The notion of a commutative square of von Neumann algebras introduced by Popa plays an essential role in recent works [GHJ], [Pop 1, 2, 3, 4], [Sun], We recall the definition in a special case. For given η >2, consider the von Neumann algebra λί = Mn(C) of complex η χ η matrices with normalized trace τ(Χ) = Definition 1.5.1. Let Λ and Β be *-subalgebras of λ ί (that is, subalgebras of the associative algebra Λί that contain the unit 1 Ar of Ν and are closed under the involutory antiisomorphism * : Χ X). We say that the diagram Β u AnB

C

G

λί u Λ

is a commutative square if τ(ΧΥ) = 0 for all X e A, Υ e Β such that τ(ΧΖ) = τ(ΥΖ) = 0 whenever Ζ e ΑΠ Β. In particular, if τ(ΧΥ) = τ(Χ)τ(Υ) for all X e A, Υ e B, then Α Π Β = < 1 ν > c and the subalgebras A, Β form a commutative square which is an orthogonal pair in the sense of [Pop 1]. Correspondingly, orthogonal pairs (A, B) and {A\ B') are said to be conjugate, if there exists an automorphism of the von Neumann algebra λί mapping A into A and Β into B'.

38

1 Type A,

In the first place we are interested in orthogonal pairs (Α, B), where Α, maximal abelian *-subalgebras of λί. It is easy to give an example of such for any n. Let ε be a primitive nth root of 1, set D = diag(l, ε, ε2,..., and let Ρ be the permutation matrix ( 1 , 2 , . . . , « ) . Then the pair (,4ο, A\), AQ =
correspond to the so-called standard UGΗ-matrix

= l/*Jn. Such ) and (Ho, H\ )

e ('-iK/-i)

Χ/1 ~

ι

V" where ε = exp(2ni/n). Roughly speaking, the task of classifying orthogonal pairs up to conjugacy in C (in Λ/*, respectively) is equivalent to that of enumerating double cosets PXP = (aXb I a, b e P) of G//-matrices (f/G//-matrices, respectively), where Ρ = 5/gl„(C)(^o) is the group of monomial matrices (P = S t u j o ( A o ) is the group of monomial unitary matrices, respectively). In fact, the pairs (A), ( a X b ) A o ( a X b ) _ 1 ) and (A), X A ^ - 1 ) are conjugate: (A0, =

(aXb)Ao(aXbΓ1)

=

( A ) , aXbA0b~lX~la~[)

(aA)a~l,aXA0X~la~])

=

a(Ao,

= XAoX~l)a~l.

The first (and also the last, as will be seen!) argument confirming the validity of Conjectures 1.5.2 and 1.5.3 is contained in the following statement: Lemma 1.5.6.

Conjectures

1.5.2

and

1.5.3

hold

if η
0 A =< E s , A, A 2 , A 3 , A4 > c , where A = ( 0 * \ a 0 b 0 , abede φ 0. Rescaling A, we may assume that abede = 1. c 0

\ d 0 / Furthermore, A contains the matrix A A* = diag(ee, aä, bb, cc, d d ) . So aä = bb = cc = dd = ee = 1. In particular, the matrix X = ] X diag(l, a, ab, abc, abed) is unitary. But XAQX~ — AQ, XAX~ = P, so the pairs (AO, A) and (Λ), - 4 i ) are conjugate. • In [HaJ] the following invariant is introduced for an orthogonal pair {A, B ) of maximal abelian *-subalgebras of λί: Ν*(Α,

Β) = {X E Β \ XX*

= E„, XAX*

=

A).

By analogy, we introduce the following invariant for an orthogonal pair (7Ί, KL) of Cartan subalgebras of C: N(H,

JC) = {X G T

I d e t X φ 0 , XUX~X

=

H).

42

1 Type A,

where /C = < fC, En > c . In fact, it is more convenient to work with the following "truncated" invariants: I*(A, B) = N*(A, B)/Z(Un(C)),

I(H, K) = N(H,

JC)/Z(GLn(C)).

Clearly, all the invariants introduced are groups (under the usual product). Note that every two conjugate (ordered) orthogonal pairs have isomorphic invariants. For pairs that are conjugate under Aut(A/) or Inn(£), this remark is obvious. Assume now that ordered orthogonal pairs (7i, /C) and (Η', Κ!) are conjugate under the automorphism Τ : X -'X. Then for X e N(H, JC), A e H, we have T(X) € Κ!, detT(X) = ( - l ) " d e t X φ 0, T(X)T(A)T(X)~{

Χ Ά ' Χ = - ' ( Χ - ' Λ Χ ) = T(X~lAX)

=

e T(X~xHX)

= T(X~l(XHX~l)X)

e

= T(H) = Hi.

Thus T(X) e N{H',K!). An isomorphism between N(H,JC) and N(H',JC') is given by the map X (->· 'X - 1 = —T(X)~l. Concerning the standard pairs (,4ο, A\) and (Ho, Hi), the following statements hold. Lemma 1.5.8. 1) The ordered pairs (Ao, Αι) and (Αι,Αο) (('Ho,H\) and (Hi, Ho), respectively) are conjugate. 2) The standard pair (Ao, A\) has invariants Ν* = S1 .Z„ and I* = Z„. The standard pair (Ho, Hi) has invariants Ν = C*.Z„ and I = Z„. Here S1 = {z € C | zz = 1}. Proof. 1) Suppose that the matrices D and Ρ are written in an (orthonormal) basis {^o, ..., e„-i}. Define a new (orthonormal) basis 1 {fj= - r Y j e ν n /= σ, where XE^X'1 = E^^k), is in fact a homomorphism N(H0,H) S„ (N*(Ao,A) S„, respectively) with kernel Z(GLn(C)) (Z(f/„(C)), respectively). Consequently, the identity πη,ρ = 1 proved above means that the exponents of 1(HQ, H) and I*(AQ, A) divide n/p. Meanwhile, the exponents of I(Ho, H\) and I*(A), A\) are equal to n. Hence, the pairs (Ho, Hi) and (Ho, Η) ((Λ), A\) and (Λ), A), respectively) are not conjugate. • In [HaJ], strongly regular graphs are used to disprove Conjecture 1.5.2 in the case η ξ 1 (mod 4), η > 5 and η a prime.

1.5 Uniqueness problem for pairs

45

Definition 1.5.11. A graph with η vertices having no loops or multiple edges and satisfying the following conditions: (i) exactly k edges issue from every vertex; (ii) any two different adjacent vertices have exactly λ adjacent vertices in common; (iii) any two non-adjacent vertices have exactly μ adjacent vertices in common; is called a strongly regular graph (s.r.g.) with parameters (η, k, λ, μ). Example 1.5.12. Let ρ be a prime, ρ = 1 (mod4). Then the Paley graph (with elements of the field F p as vertices, where a, b e Wp are adjacent if and only if a — b is a nonzero quadratic residue) is an s.r.g. with parameters (p, (p — l)/2, ip — 5)/4, (p — l)/4). Example 1.5.13. The Petersen graph

is an s.r.g. with parameters (10, 3, 0, 1). If A is the incidence matrix of an s.r.g. with parameters (n, k, λ, μ), then by [Bal], A2 = kE„ + λΑ + ~EnA), AJ = JA = kJ, where J is the matrix all of whose coefficients are equal to 1. Suppose now that μ = λ + 1 and set R = En + ßA + ß~l (J - En - A) for some β with \β\ = 1. Then for b = β + β~χ we have RR* = Ε„(2 - b - (b2 - 4)(k - μ)) + J((b2 - 4)(k - μ) + b - 2 + n). If we choose β so that (b2 — 4)(k — μ) + b — 2 + η — 0, then RR* = nEn, Ri} = 1 if i = j and Ru e [β, β~1} if i φ j, that is, i/G//-matrix.

(1.20) is a

46

1 Type A,

Set b = — j for the Petersen graph and b = for the Paley graph. Then equation (1.20) is satisfied; moreover, in both cases there exists β such that \β\ = 1, β + ß~l = b. Furthermore, if η φ 5 then β is not a root of unity. Indeed, suppose that βΝ = 1 for some N. Then β and b are algebraic integers. In particular, b φ — If b = j^jß, the element b of the field Q(^/p) has norm = an T+jji' - i ^T' so ^can algebraic integer only in the case η = ρ = 5 (if ρ = 5, then in fact β = βχρ(2τπ/5)). We have just proved the following lemma:

L e m m a 1.5.14. Suppose that η = 10 or η is a prime with η = 1 (mod 4) and η > 5. Then there exists a symmetric UGH-matrix Β = (bjj) of degree η with the following properties: sjnbij

- j

1 β or β~ι

if if

i — j, ίφ],

where \β\ = 1 but β is not a root of unity.

•

We describe one more construction of i/G//-matrices. Suppose that ρ is a prime, ρ = 3 (mod4), and η is a quadratic character of the field = {0, 1 , . . . ,p — 1}: ( 0 if λ = 0, >7 (χ) — s 1 if χ is a nonzero quadratic residue, { — 1 otherwise. Consider the matrix A = (η(ί — j))\ 4. If η is divisible by the square of some prime p, then both Conjectures 1.5.2 and 1.5.3 are false, by Lemma 1.5.10. Suppose that η = ρ\ρι.. .p,, where p\ < p2 < ... < pt are primes in ascending order, η > 6 and η φ 15. Then η is divisible by an integer q, where q = 10 or q is prime and q > 7 : η = mq. According to Lemmas 1.5.14 and 1.5.15, there exists a UGH-matrix Β = {bjj} of degree q with the following properties: (i) yjqbu = 1; (ii) if i φ j, the number qbjjbji has modulus 1 but is not a root of unity. 1 Set A = Β Xm, Η = AHqA~\ Λ = AAqA' , where Xm is a standard UGHmatrix of degree m. By Lemma 1.5.9, Λ is a UGH-matrix. Recall that every X e N(Hο, H) satisfies equations (1.18) and (1.19). We split the set Ω = {1, 2 , . . . , η] up into ρ subsets Ω,· = {(/ — l)m + 1 , . . . , im), 1 < i < p. Firstly, assume that the permutation π e §„ corresponding to some element χ e N(Ho, Ή) does not fix some component, Ω, say, of the partition Ω = ΩΙ U . . . ϋ Ω ρ , so that 7γ(Ω;) Π Qj φ 0 for some i,j, 1 < i Φ j < p. Thus, one can find k = ( / - l)m + r, 1 < r wp(An_\) > φ(η) + 1, where φ(η) is the Euler function. To engage the reader's interest, we suggest a quite elementary formulation of the Winnie-the-Pooh problem. Let C = sln(C) be the Lie algebra of type A„_i and C = Hq θ Hi Θ . . . Θ η„ an orthogonal decomposition. We know that if Ho = {diag(A,, λ 2 , . . . , λ„) \ Σ χ ί ζ

=

is the standard Cartan subalgebra and Hs = XsHqXj\ the orthogonality condition for the pair (Ho, Hs) is equivalent to the condition that Xs be a G//-matrix. Now if s, t > 0 and s φ t, then by definition of OD we have 0 = K(H.S, Ht) — K(XsHQXs

Χ,ΗοΧ, ') = K(Hq, Xs ' X/HQX, ' X.v).

So the matrices X~lX,, and Xx and X, are G//-matrices. However, if X = (xij), Y = (yij) and are G//-matrices, then in addition to the relations

k= 1

Xk

J

k= 1 XJk k=\}kJ

k= 1 yJk

which were obtained in the proof of Lemma 1.5.5, the following equations must hold:

Using the non-degeneracy of the restriction of the Killing form to an arbitrary Cartan subalgebra, it is easy to show that our arguments are reversible; that is, the following statement holds: The Lie algebra of type A„_i possesses an OD if and only if there exist η GHmatrices X\, ..., Xn of degree η such that X~1 Xj are G Η-matrices for all i,j with 1 < i j < η, ίφΐ The system of equations introduced above defines a certain affine algebraic variety. What is this variety, in fact? 8) The Garrick Club of London was founded in 1831, and would have gone out of existence long ago for reasons of bankruptcy, had not its late member A. A. Milne bequeathed it a fixed portion of the royalties from reprints of his book "Winnie-the-Pooh". The authors promise to transfer $1 to this honourable club.

Chapter 2

The types Βn, Cn and D

In this chapter, orthogonal decompositions for other classical Lie algebras are constructed. The results concerning orthogonal Lie algebras are characterized by their great variety. In this context, information about orthogonal decompositions of Lie algebras of type C„ is rather meagre in the sense that constructions available at present of ODs for this type are very limited.

2.1 Type Cn As yet, ODs for Lie algebras of type C„ are known only in the case η = 2m. It is very probable that the following conjecture is true:

Conjecture 2.1.1. The Lie algebra of type C„ possesses an OD if and only if n = 2m. This conjecture is open even in the simplest case η = 3, where we confront practically the same difficulties as in the case of the Winnie-the-Pooh problem for the Lie algebra of type A5. We realize the Lie algebra C of type C^ in the form {X e M 2m+ . (C) I X.J + J.'X = 0} ,

whereJ

ad

2 Saß

= (-l o")-"" =™· =U -UM?

then Dg = < D, Ρ > = 2+ +2 is the dihedral group of order 8 and Qg =< iD, iP >= is the quaternion group of order 8. Recall (see §1.1) that we have fixed some special basis {ei,..., em+\,f\,..., fm+\} in the space W = , and set

Ju = J{aubx) ® · · · ®

2.1 Type C, for u = («!,...,

am+1, b\,...,

/rc+I q{u) = ^ i=l

57

G W, where J(a.b) =

We write further:

+ (α, + /?ι), < u | υ >= q(u) +