Lie Algebras with Triangular Decompositions

710 Pages·1995·23.184 MB·English

by  Robert V. Moody, Arturo Pianzola

Lie Algebras With
Triangular Decompositions
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*Indicates an out-of-print title
Lie Algebras With
Triangular
Decompositions
ROBERT V. MOODY
ARTURO PIANZOLA
University of Alberta
Edmonton, Canada
A Wiley-Interscience Publication
JOHN WILEY & SONS
New York Chichester • Brisbane • Toronto • Singapore
Photograph on title page by M. Goretz
This text is printed on acid-free paper.
Copyright © 1995 by John Wiley & Sons, Inc.
All rights reserved. Published simultaneously in Canada.
Reproduction or translation of any part of this work beyond
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Library of Congress Cataloging in Publication Data:
Moody, R. V., 1941-
Lie algebras with triangular decompositions / by Robert V. Moody,
Arturo Pianzola.
p. cm. — (Canadian Mathematical Society series of monographs
and advanced texts)
Includes bibliographical references.
ISBN 0-471-63304-6 (alk. paper)
1. Lie algebras. 2. Decomposition (Mathematics) I. Pianzola,
Arturo, 1955- . II. Title. III. Series.
QA252.3.M66 1995
512'.55—dc20 92-46890
Printed in the United States of America
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Contents
Introduction XI
How to Read This Book XV
Course Outlines xix
Chapter 1 Lie Algebras 1
1.1 Basic Definitions
1.2 Tensor, Symmetric, and Exterior Algebras 9
1.3 Gradings 15
1.4 Virasoro and Heisenberg Algebras 20
1.5 Derivations 23
1.6 Representations 28
1.7 Invariant Bilinear Forms 35
1.8 Universal Enveloping Algebras 38
1.9 Central Extensions 49
1.10 Free Lie Algebras 59
1.11 The Campbell-Baker-Hausdorff Formula 65
1.12 Extensions of Modules 71
Exercises 80
Chapter 2 Lie Algebras Admitting Triangular Decompositions 90
2.1 Triangular and Weight Space Decompositions 91
2.2 Highest Weight Modules 103
2.3 Verma Modules 106
2.4 §l2(IK)-Theorem 116
2.5 Characters 128
2.6 The Category & 141
2.7 The Radical 160
2.8 The Shapovalov form 168
2.9 Jantzen Filtrations 175
2.10 Bernstein-Gel’fand-Gerfand Duality 184
2.11 Embeddings of Verma Modules 200
2.12 Decomposition of Modules in Category Û 204
Exercises 209
vüi Contents
Chapter 3 Lattices and Root Systems 216
3.1 Lattices 217
3.2 Finite Root Systems 229
3.3 Bases for Finite Root Systems 239
3.4 Graphs and Coxèter-Dynkin Diagrams 247
3.5 Classification of Cartan Matrices and Finite Root Systems 258
3.6 The Perron-Frobenius Theorem and Its Consequences 272
3.7 Constructing Lie Algebras from Lattices 280
3.8 Central Extensions of Lattices 289
Exercises 302
Chapter 4 Contragredient Lie Algebras 309
4.1 Contragredient Lie algebras 310
4.2 Realizations of Contragredient Lie Algebras 330
4.3 Embeddings, Field Extensions, and Decomposability 342
4.4 Invariant Bilinear Forms 355
4.5 Casimir-Kac Operators 367
4.6 The Radical Theorem 375
4.7 Hermitian Contragredient Forms 382
Exercises 385
Chapter 5 The Weyl Group and Its Geometry 395
5.1 Root Data 396
5.2 The Length Function 408
5.3 Coxeter Groups and the Exchange Condition 418
5.4 The Bruhat Ordering 425
5.5 Morphisms of Root Data: Subroot Systems 430
5.6 The Geometry of a Set of Root Data 436
5.7 Subroot Systems 457
5.8 Imaginary Roots 464
5.9 Conjugacy of Bases 472
Exercises 476
Chapter 6 Category 0 for Kac-Moody Algebras 483
6.1 Integrable Modules 484
6.2 Weight Systems 508
6.3 The Triangular Decomposition of G 516
6.4 The Formulas of Weyl-Macdonald-Kac 531
6.5 Complete Reducibility 542
6.6 Shapóvalov Determinant Formula for Kac-Moody Algebras 545
6.7 The BGG Theorem and Generalization 556