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Lie Superalgebras and Enveloping Algebras Lie Superalgebras and Enveloping Algebras Ian M. Musson Graduate Studies in Mathematics Volume 131 American Mathematical Society Lie Superalgebras and Enveloping Algebras http://dx.doi.org/10.1090/gsm/131 Lie Superalgebras and Enveloping Algebras Ian M. Musson Graduate Studies in Mathematics Volume 131 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Rafe Mazzeo Martin Scharlemann Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 17B35; Secondary 17B10, 17B20, 17B22, 17B25, 17B30, 17B40, 17B55, 17B56, 16S30. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-131 Library of Congress Cataloging-in-Publication Data Musson, Ian M. (Ian Malcolm), 1953– Lie superalgebras and enveloping algebras / Ian M. Musson. p. cm. — (Graduate studies in mathematics ; v. 131) Includes bibliographical references and index. ISBN 978-0-8218-6867-6 (alk. paper) 1. Lie superalgebras. 2. Universal enveloping algebras. I. Title. QA252.3.M87 2012 510—dc23 2011044064 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2012 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 171615141312 Dedicated to my parents, Eric and Jessie Contents Preface xv Chapter 1. Introduction 1 §1.1. Basic Definitions 1 §1.2. Simple Lie Superalgebras 3 §1.3. Classification of Classical Simple Lie Superalgebras 8 §1.4. Exercises 9 Chapter 2. The Classical Simple Lie Superalgebras. I 11 §2.1. Introduction 11 §2.2. Lie Superalgebras of Type A(m,n) 12 §2.3. The Orthosymplectic Lie Superalgebras 14 2.3.1. The Lie Superalgebras osp(2m +1, 2n)15 2.3.2. The Lie Superalgebras osp(2m, 2n)16 2.3.3. The Lie Superalgebras osp(2, 2n − 2) 16 §2.4. The Strange Lie Superalgebras p(n) and q(n)17 2.4.1. The Lie Superalgebras p(n)17 2.4.2. The Lie Superalgebras q(n)17 §2.5. Rationality Issues 19 §2.6. The Killing Form 19 §2.7. Exercises 20 Chapter 3. Borel Subalgebras and Dynkin-Kac Diagrams 25 §3.1. Introduction 25 §3.2. Cartan Subalgebras and Borel-Penkov-Serganova Subalgebras 28 vii viii Contents §3.3. Flags, Shuffles, and Borel Subalgebras 30 §3.4. Simple Roots and Dynkin-Kac Diagrams 38 3.4.1. Definitions and Low Rank Cases 38 3.4.2. From Borel Subalgebras and Shuffles to Simple Roots 39 3.4.3. From Simple Roots to Diagrams 41 3.4.4. Back from Diagrams to Shuffles and Simple Roots 44 3.4.5. Distinguished Simple Roots and Diagrams 47 3.4.6. Cartan Matrices 48 3.4.7. Connections with Representation Theory 50 §3.5. Odd Reflections 51 §3.6. Borel Subalgebras in Types A(1, 1), p(n), and q(n)55 3.6.1. Lie Superalgebras of Type A(1,1) 55 3.6.2. The Lie Superalgebra p(n)58 3.6.3. The Lie Superalgebra q(n)63 §3.7. Exercises 64 Chapter 4. The Classical Simple Lie Superalgebras. II 69 §4.1. Introduction and Preliminaries 69 §4.2. The Lie Superalgebras D(2, 1; α)71 §4.3. Alternative Algebras 75 §4.4. Octonions and the Exceptional Lie Superalgebra G(3) 78 §4.5. Fierz Identities and the Exceptional Lie Superalgebra F (4) 82 §4.6. Borel Subalgebras versus BPS-subalgebras 88 §4.7. Exercises 88 Chapter 5. Contragredient Lie Superalgebras 95 §5.1. Realizations and the Algebras g(A, τ)96 §5.2. Contragredient Lie Superalgebras: First Results 101 5.2.1. The Center, Root Space Decomposition, and Antiautomorphism 101 5.2.2. Equivalent Matrices 104 5.2.3. Integrability and Kac-Moody Superalgebras 107 5.2.4. Serre Relations 108 §5.3. Identifying Contragredient Lie Superalgebras 109 5.3.1. The Exceptional Lie Superalgebras 110 5.3.2. The Nonexceptional Lie Superalgebras 111 Contents ix §5.4. Invariant Bilinear Forms on Contragredient Lie Superalgebras 112 5.4.1. The Invariant Form 112 §5.5. Automorphisms of Contragredient Lie Superalgebras 115 5.5.1. Semisimple Lie Algebras 115 5.5.2. Automorphisms Preserving Cartan and Borel Subalgebras 116 5.5.3. Diagram and Diagonal Automorphisms 119 5.5.4. The Structure of H and (Aut g)0 120 5.5.5. More on Diagram Automorphisms 121 5.5.6. Outer Automorphisms 124 5.5.7. Automorphisms of Type A Lie Superalgebras 125 §5.6. Exercises 127 Chapter 6. The PBW Theorem and Filtrations on Enveloping Algebras 131 §6.1. The Poincar´e-Birkhoff-Witt Theorem 131 §6.2. Free Lie Superalgebras and Witt’s Theorem 135 §6.3. Filtered and Graded Rings 136 §6.4. Supersymmetrization 138 §6.5. The Clifford Filtration 142 §6.6. The Rees Ring and Homogenized Enveloping Algebras 143 §6.7. Exercises 145 Chapter 7. Methods from Ring Theory 147 §7.1. Introduction and Review of Basic Concepts 147 7.1.1. Motivation and Hypothesis 147 7.1.2. Bimodules 148 7.1.3. Prime and Primitive Ideals 149 7.1.4. Localization 150 §7.2. Torsion-Free Bimodules, Composition Series, and Bonds 152 §7.3. Gelfand-Kirillov Dimension 154 §7.4. Restricted Extensions 161 7.4.1. Main Results 161 7.4.2. Applications 165 §7.5. Passing Properties over Bonds 166 §7.6. Prime Ideals in Z2-graded Rings and Finite Ring Extensions 170 7.6.1. Z2-graded Rings 170 x Contents 7.6.2. Lying Over and Direct Lying Over 172 7.6.3. Further Results 177 §7.7. Exercises 178 Chapter 8. Enveloping Algebras of Classical Simple Lie Superalgebras 181 §8.1. Root Space and Triangular Decompositions 181 §8.2. Verma Modules and the Category O 184 8.2.1. Verma Modules 184 8.2.2. Highest Weight Modules in the Type I Case 187 8.2.3. The Category O 188 8.2.4. Central Characters and Blocks 189 8.2.5. Contravariant Forms 190 8.2.6. Base Change 192 8.2.7. Further Properties of the Category O 193 §8.3. Basic Classical Simple Lie Superalgebras and a Hypothesis 195 8.3.1. Basic Lie Superalgebras 195 8.3.2. A Hypothesis 197 §8.4. Partitions and Characters 199 §8.5. The Casimir Element 201 §8.6. Changing the Borel Subalgebra 204 §8.7. Exercises 205 Chapter 9. Verma Modules. I 207 §9.1. Introduction 207 §9.2. Universal Verma Modules and Sapovalovˇ Elements 208 9.2.1. Basic Results and Hypotheses 208 9.2.2. Universal Verma Modules 210 9.2.3. Sapovalovˇ Elements for Nonisotropic Roots 210 9.2.4. Sapovalovˇ Elements for Isotropic Roots 212 §9.3. Verma Module Embeddings 213 9.3.1. Reductive Lie Algebras 213 9.3.2. Contragredient Lie Superalgebras 214 9.3.3. Typical Verma Modules 217 §9.4. Construction of Sapovalovˇ Elements 218 §9.5. Exercises 221 Chapter 10. Verma Modules. II 223 §10.1. The Sapovalovˇ Determinant 223 Contents xi §10.2. The Jantzen Filtration 227 10.2.1. The p-adic Valuation of a Certain Determinant 227 10.2.2. The Jantzen Filtration 228 10.2.3. Evaluation of the Sapovalovˇ Determinant 229 §10.3. The Jantzen Sum Formula 232 §10.4. Further Results 233 10.4.1. The Typical Case 233 10.4.2. Reductive Lie Algebras 233 10.4.3. Restriction of Verma Modules to g0 234 §10.5. Exercises 235 Chapter 11. Schur-Weyl Duality 239 §11.1. The Double Commutant Theorem 239 §11.2. Schur’s Double Centralizer Theorem 240 §11.3. Diagrams, Tableaux, and Representations of Symmetric Groups 246 §11.4. The Robinson-Schensted-Knuth Correspondence 249 §11.5. The Decomposition of W and a Basis for U λ 253 §11.6. The Module U λ as a Highest Weight Module 258 §11.7. The Robinson-Schensted Correspondence 259 §11.8. Exercises 260 Chapter 12. Supersymmetric Polynomials 263 §12.1. Introduction 263 §12.2. The Sergeev-Pragacz Formula 265 §12.3. Super Schur Polynomials and Semistandard Tableaux 272 §12.4. Some Consequences 278 §12.5. Exercises 278 Chapter 13. The Center and Related Topics 281 §13.1. The Harish-Chandra Homomorphism: Introduction 281 §13.2. The Harish-Chandra Homomorphism: Details of the Proof 284 §13.3. The Chevalley Restriction Theorem 293 §13.4. Supersymmetric Polynomials and Generators for I(h) 298 §13.5. Central Characters 299 13.5.1. Equivalence Relations for Central Characters 299 13.5.2. More on Central Characters 302 §13.6. The Ghost Center 304 xii Contents §13.7. Duality in the Category O 304 §13.8. Exercises 306 Chapter 14. Finite Dimensional Representations of Classical Lie Superalgebras 307 §14.1. Introduction 307 §14.2. Conditions for Finite Dimensionality 308 §14.3. The Orthosymplectic Case 310 14.3.1. Statements of the Results 310 14.3.2. A Special Case 311 14.3.3.
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