Table Of Contentde Gruyter 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. Meirmanov
4 Finite Soluble Groups, K. Doerk, T. O. Hawkes
5 The Riemann Zeta-Function, A. A. Karatsuba, S.M. Voronin
6 Contact Geometry and Linear Differential Equations, V. R. Nazaikinskii,
V. E. Shatalov, B. Yu. Sternin
7 Infinite Dimensional Lie Superalgebras, Yu. A. Bahturin, A. A. Mikhalev,
V. M. Petrogradsky, Μ. V. Zaicev
8 Nilpotent Groups and their Automorphisms, Ε. I. Khukhro
9 Invariant Distances and Metrics in Complex Analysis, M. Jarnicki, P. Pflug
10 The Link Invariants of the Chern-Simons Field Theory, E.Guadagnini
11 Global Affine Differential Geometry of Hypersurfaces, A.-M. Li, U. Simon,
G. Zhao
12 Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and
Theta Functions, K. Hulek, C. Kahn, S. H. Weintraub
13 Elliptic Problems in Domains with Piecewise Smooth Boundaries, S. A. Na-
zarov, B. A. Plamenevsky
14 Subgroup Lattices of Groups, R.Schmidt
Orthogonal Decompositions and
Integral Lattices
by
Alexei I. Kostrikin
Pham Huu Tiep
W
DE
Walter de Gruyter · Berlin · New York 1994
Authors
Pham Huu Tiep Alexei I. Kostrikin
Hanoi Institute of Mathematics Department of Mathematics
P.O. Box 631 MEHMAT
10000 Hanoi, Vietnam Moscow State University
Present address: 119899 Moscow GSP-1, Russia
Institute for Experimental Mathematics
University of Essen
Ellernstraße 29
D-45326 Essen, Germany
1991 Mathematics Subject Classification: 11H31, 17Bxx, 20Bxx, 20Cxx, 20Dxx, 94Bxx
Keywords: Orthogonal decompositions, Euclidean lattices, finite groups, Lie algebras
© Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.
Library of Congress Cataloging-in-Publication Data
Kostrikin, A. I. (AlekscT Ivanovich)
Orthogonal decompositions and integral lattices / by
A. I. Kostrikin, Pham Huu Tiep.
p. cm. — (De Gruytcr expositions in mathematics ; 15)
Includes bibliographical references and index.
ISBN 3-11-013783-6
1. Lie algebras. 2. Orthogonal decomposition. 3. Lattice theory.
I. Pham Huu Tiep, 1963- . II. Title. III. Series.
QA252.3.K67 1994
512'.55-dc20 94-16850
CIP
Die Deutsche Bibliothek — Cataloging-in-Publication Data
Kostrikin, Alekscj I.:
Orthogonal decompositions and integral lattices / by
Aleksei I. Kostrikin, Pham Huu Tiep. — Berlin ; New York :
de Gruytcr, 1994
(De Gruyter expositions in mathematics ; 15)
ISBN 3-11-013783-6
NE: Tiep, Pham Huu:; GT
(Ο Copyright 1994 by Walter de Gruytcr & Co., D-10785 Berlin.
All rights reserved, including those of translation into foreign languages. No part of this book
may be reproduced in any form or by any means, electronic or mechanical, including photocopy,
recording, or any information storage and retrieval system, without permission in writing from
the publisher.
Printed in Germany.
Typeset with LATp.X: D. L. Lewis, Berlin. Printing: Gerikc GmbH, Berlin.
Binding: Lüderitz & Bauer GmbH, Berlin. Cover design: Thomas Bonnie, Hamburg.
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, automor-
phism 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 Com-
mittee'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 com-
pleted 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 ν
Introduction 1
Part I: Orthogonal decompositions of complex simple Lie algebras 11
Chapter 1
Type A 13
n
1.1 Standard construction of ODs for Lie algebras of type Ap»>- \ 13
1.2 Symplectic spreads and ./-decompositions 17
1.3 Automorphism groups of /-decompositions 25
1.4 The uniqueness problem for ODs of Lie algebras of type A,„ η < 4.... 31
1.5 The uniqueness problem for orthogonal pairs of subalgebras 37
1.6 A connection with Hecke algebras 49
Commentary 51
Chapter 2
The types B , C and D 56
n n n
2.1 Type C„ 56
2.2 Partitions of complete graphs and ^-decompositions of Lie algebras of
types B and D„: automorphism groups 64
n
2.3 Partitions of complete graphs and Ε-decompositions of Lie algebras of
types B and D : admissible partitions 68
n n
2.4 Classification of the irreducible ^-decompositions of the Lie algebra of
type D„ 76
Commentary 81
Chapter 3
Jordan subgroups and orthogonal decompositions 84
3.1 General construction of TODs 84
3.2 Root orthogonal decompositions (RODs) 88
3.3 Multiplicative orthogonal decompositions (MODs) 93
Commentary 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 C and D 136
p p
Commentary 139
Chapter 5
Classification of irreducible orthogonal decompositions of complex simple
Lie algebras of type A 141
n
5.1 The S-case 142
5.2 The P-case. I. Generic position 144
5.3 The P-case. II. Affine obstruction 152
5.4 The P-case. III. Completion of the proof 158
Commentary 160
Chapter 6
Classification of irreducible orthogonal decompositions of complex simple
Lie algebras of type B 162
n
6.1 The monomiality of G = Aut(£>) 163
6.2 Every IOD is an ^-decomposition 168
6.3 Study of f-decompositions 176
Commentary 186
Chapter 7
Orthogonal decompositions of semisimple associative algebras 187
7.1 Definitions and examples 187
7.2 The divisibility conjecture 190
7.3 A construction of ODs 195
Commentary 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 203
8.2 Invariant lattices in £ 208
8.3 Automorphism groups 216
Commentary 229
Chapter 9
Invariant lattices, the Leech lattice and even unimodular analogues of it
in Lie algebras of type A _i 231
p
9.1 Preliminary results 232
9.2 Classification of indivisible invariant lattices 241
9.3 Metric properties of invariant lattices 248
9.4 The duality picture 253
9.5 Study of unimodular invariant lattices 259
9.6 Automorphism groups of projections of invariant lattices 264
9.7 On the automorphism groups of invariant lattices of type 273
Commentary 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 (</)-module VR/pVR 301
2
10.4 The structure of the SL {q)-module YR/q2VR 309
2
10.5 A series of unimodular invariant lattices 320
10.6 Reduction theorem: statement of results 326
10.7 Invariant lattices of type A„: the imprimitive case 329
10.8 Invariant lattices of type A„: the primitive case 339
Commentary 353
Chapter 11
The types /?2"·-ι and D^n 355
11.1 Preliminaries 356
11.2 Possible configurations of root systems 361
11.3 Automorphism groups of lattices: the classes 1Z\ and 7ΖΊ 366
11.4 Lattices of nonroot-type: the case Bj 383
11.5 Lattices of nonroot-type: the case £> 391
4
χ Table of Contents
11.6 Z-forms of Lie algebras of types G2, #3 and D4 403
Commentary 408
Chapter 12
Invariant lattices of types F4 and E(>, and the finite simple groups 1-4(3),
Ω (3), Fi 409
7 21
12.1 On invariant lattices of type F4 410
12.2 Invariant lattices of type E(,: the imprimitive case 411
12.3 Character computation 419
12.4 Invariant lattices of type E^. the primitive case 432
Commentary 439
Chapter 13
Invariant lattices of type Εχ and the finite simple groups F3, ^(5) 440
13.1 The Thompson-Smith lattice 440
13.2 Statement of results 443
13.3 The imprimitive case 445
13.4 The primitive case 452
13.5 The representations of SL^(q) of degree (q — 1 ){qi — l)/2 455
Commentary 469
Chapter 14
Other lattice constructions 471
14.1 A Moufang loop, the Dickson form, and a lattice related to Ωγ(3) 471
14.2 The Steinberg module for SL2(q) and related lattices 479
14.3 The Weil representations of finite symplectic groups and the Gow-Gross
lattices 483
14.4 The basic spin representations of the alternating groups, the Barnes-Wall
lattices, and the Gow lattices 487
14.5 Globally irreducible representations and some Mordell-Weil lattices.... 493
Commentary 504
Appendix 505
Bibliography 511
Notation 526
Author Index 529
Subject Index 531
