Lie Superalgebras and Enveloping Algebras

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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.

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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 Poincaré-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. The General Case 314 §14.4. The Kac-Weyl Character Formula 316 §14.5. Exercises 317 Chapter 15. Prime and Primitive Ideals in Enveloping Algebras 319 §15.1. The Dixmier-Moeglin Equivalence 320 §15.2. Classical Simple Lie Superalgebras 322 15.2.1. A Theorem of Duﬂo and Its Superalgebra Analog 322 15.2.2. Type I Lie Superalgebras 323 §15.3. Semisimple Lie Algebras 324 15.3.1. Notation 325 15.3.2. The Characteristic Variety 326 15.3.3. Translation Functors on the Category O 329 15.3.4. Translation Maps on Primitive Ideals 332 15.3.5. Primitive Ideals for Type A Lie Algebras 337 15.3.6. The Poset of Primitive Ideals and the Kazhdan- Lusztig Conjecture 342 15.3.7. The Lie Superalgebra osp(1, 2n) 344 §15.4. More on Prime Ideals and Related Topics 346 15.4.1. Strongly Typical Representations, Annihilation, and Separation Theorems 346 15.4.2. Primeness of U(g) 347 15.4.3. The Unique Minimal Prime 348 15.4.4. The Goldie Rank of U(g) 349 15.4.5. Enveloping Algebras of Nilpotent and Solvable Lie Superalgebras 349 §15.5. Exercises 350 Chapter 16. Cohomology of Lie Superalgebras 355 §16.1. Introduction and Preliminaries 355 16.1.1. Complexes and Filtrations 355 Contents xiii

§16.2. Spectral Sequences 357 16.2.1. The Spectral Sequence Associated to a Filtered Complex 357 16.2.2. Bounded Filtrations and Convergence 359 §16.3. The Standard Resolution and the Cochain Complex 361 16.3.1. The Standard Resolution 361 16.3.2. The Cochain Complex 363 §16.4. Cohomology in Low Degrees 367 §16.5. The Cup Product 369 16.5.1. Definition and Basic Properties 369 16.5.2. Examples of Cup Products 372 §16.6. The Hochschild-Serre Spectral Sequence 374 §16.7. Exercises 379 Chapter 17. Zero Divisors in Enveloping Algebras 381 §17.1. Introduction 381 §17.2. Derived Functors and Global Dimension 383 §17.3. The Yoneda Product and the Bar Resolution 386 17.3.1. The Yoneda Product 386 17.3.2. The Bar Resolution 387 §17.4. The Löfwall Algebra 389 §17.5. Proof of the Main Results 392 §17.6. Further Homological Results 398 17.6.1. Tor and Homology of Lie Superalgebras 398 17.6.2. The Auslander and Macaulay Conditions 399 §17.7. Exercises 400 Chapter 18. Affine Lie Superalgebras and Number Theory 403 §18.1. Some Identities 403 §18.2. Affine Kac-Moody Lie Superalgebras 405 §18.3. Highest Weight Modules and the Affine Weyl Group 412 §18.4. The Casimir Operator 414 §18.5. Character Formulas 418 §18.6. The Jacobi Triple Product Identity 420 §18.7. Basic Classical Simple Lie Superalgebras 422 §18.8. The Case go = sl(2, 1) 425 §18.9. The Case go = osp(3, 2) 428 §18.10. Exercises 430 xiv Contents

Appendix A. 433 §A.1. Background from Lie Theory 433 A.1.1. Root Systems 433 A.1.2. The Weyl Group 434 A.1.3. Reductive Lie Algebras 435 A.1.4. A Theorem of Harish-Chandra 436

§A.2. Hopf Algebras and Z2-Graded Structures 437 A.2.1. Hopf Algebras 437 A.2.2. Remarks on Z2-Graded Structures: The Rule Of Signs 440 A.2.3. Some Constructions with U(g)-Modules 443 A.2.4. The Supersymmetric and Superexterior Algebras 446 A.2.5. Actions of the Symmetric Group 447 §A.3. Some Ring Theoretic Background 449 A.3.1. The Diamond Lemma 449 A.3.2. Cliﬀord Algebras 452 A.3.3. Ore Extensions 455 §A.4. Exercises 456 Appendix B. 463 Bibliography 471 Index 485 Preface

The publication of Dixmier’s book [Dix74] in 1974 led to increased interest in the structure of enveloping algebras. Considerable progress was made in both the solvable and semisimple cases. For example the primitive ideals were completely classified and much information was obtained about the structure of the primitive factor rings [BGR73], [Dix96], [Jan79], [Jan83], [Mat91]. Most of this work was complete by the early 1980s, so it was natural that attention should turn to related algebraic objects. Indeed at about this time some new noncommutative algebras appeared in the work of the Leningrad school led by L. D. Faddeev on quantum integrable systems. The term “quantum group” was used by V. G. DrinfeldandM.Jimbotodescribe particular classes of Hopf algebra that emerged in this way. This subject underwent a rapid development, spurred on in part by connections with Lie theory, low dimensional topology, special functions, and so on. The algebraic aspects of quantum groups are treated in detail in the books [CP95], [Kas95], [KS97], [Lus93], [Jos95], and [Maj95]. Against this background, Lie superalgebras seem to have been somewhat overlooked. Finite dimensional simple Lie superalgebras over algebraically closed fields of characteristic zero were classified by V. G. Kac in his seminal paper [Kac77a]. However more than thirty years after the classification, the representation theory of these algebras is still not completely understood and the structure of the enveloping algebras of these superalgebras remains rather mysterious.

xv xvi Preface

Nevertheless some fundamental progress has been made. For example the characters of finite dimensional simple representations have been de- termined for Lie superalgebras of Types A and Q and recently for the orthosymplectic Lie algebras [Bru03], [Bru04], [CLW09], [GS10], [PS97a], [PS97b], [Ser96]. Furthermore exciting connections have been uncovered between Lie superalgebra representations, Khovanov’s diagram algebra, and the parabolic category O for semisimple Lie algebras; see the series of papers [BS08], [BS10], [BS11], [BS09]. We mention here also the superduality conjectures from [CW08] and [CLW09].1 Moreover we know when the enveloping algebra U(g) is a domain, when it has finite global dimension, and some progress has been made on understanding its primitive ideals. Therefore it is timely, I hope, for a volume that brings together some of what is known about Lie superalgebras and their representations. My original motivation for writing the book was to collect results that were difficult to find. For this reason I tried to include primarily results that have only appeared in research journals. Of course it is impossible to keep to this rule consistently. No attempt was made to be comprehensive, and I needed to include a certain amount of background material that is well known. Here is a brief overview of the contents of the book. Chapter 1 contains some basic definitions and the statement of the Classification Theorem for finite dimensional classical simple Lie superalgebras over an algebraically closed field of characteristic zero. Since the proof of the Classification The- orem can be found in the book [Sch79], as well as in the paper of Kac, we do not include the proof here. However we give explicit constructions for each classical simple Lie superalgebra g. This is done in Chapter 2 if g is a close relative of gl(m, n), g is orthosymplectic, or g belongs to one of the series p or q in the Kac classification. The other classical simple Lie superalgebras, which we will call exceptional, are dealt with in Chapter 4. Now in order to construct highest weight modules for g, we need to understand its Borel subalgebras. Unlike the case of a semisimple Lie algebra, there are in general several conjugacy classes of Borel subalgebras. However, at least if g = psl(2, 2), there are only a finite number of conjugacy classes, and we give a combinatorial description of them in Chapter 3. If g = p(n), q(n), then the choice of a Borel subalgebra b of g leads, in Chapter 5, to a second construction of g as a contragredient Lie superalgebra. This approach is less explicit than the first, and some work is required to reconcile the two points of view. However contragredient Lie superalgebras give a unified approach

1Because of lack of space and time, the topics mentioned in this paragraph are not treated in this book. Preface xvii to several results, in particular to the existence of an even nondegenerate invariant bilinear form on g. An algebra that admits such a form is often called basic. In Chapter 6 we define the enveloping algebra U(k) of a Lie superalgebra k. The study of representations of k is equivalent to that of U(k), and techniques from ring theory can be utilized to investigate U(k). We prove the Poincaré-Birkhoff-Witt (PBW) Theorem using the Diamond Lemma. A crucial difference with the PBW Theorem for Lie algebras is that the basis elements for the odd part of k can only appear with exponents zero or one inaPBWbasis. Let k be a finite dimensional Lie superalgebra over a field and let R = U(k0),S = U(k) be the enveloping algebras of k0 and k respectively. By the PBW Theorem, S is finitely generated and free as a left or right R- module. Suppose that S is a ring extension of R, that is, R is a subring of S with the same 1, and that S is a finitely generated R-module. We develop some general methods for studying such finite ring extensions in Chapter 7. Particular attention is paid to the relationship between prime and primitive ideals in R and S.WhenS is commutative, we have the classical Krull relations of lying over, going up, etc. The usual definitions and proofs do not work well in the noncommutative setting, and we adopt an approach using R-S bimodules. In Chapter 8 we set up some of the notation that we will use in sub- sequent chapters to study the enveloping algebra of a classical simple Lie superalgebra g. Among the topics covered are triangular decompositions g = n− ⊕ h ⊕ n+ of g, Verma modules, and the category O. Partitions, which can be used to index a basis for U(n±), are also introduced here. For the rest of this introduction, we assume that K is an algebraically closed field of characteristic zero and all Lie superalgebras are defined over K. Chapters 9 and 10 are devoted to the study of Verma modules.Ifk is a semisimple Lie algebra, the homomorphisms between Verma modules were first described by I. N. Bern˘ste˘ın,I.M.Gelfand, and S. I. Gelfand [BGG71] and later in more explicit form by N. N. Sapovalovˇ [Sap72ˇ ]. In Chapter 9 we introduce Sapovalovˇ elements and Sapovalovˇ maps for basic simple classical Lie superalgebras. In Chapter 10 we evaluate the Sapovalovˇ determinant and study the Jantzen filtration and sum formula. Although the results of these two chapters complement each other, they can be read independently. Classical Schur-Weyl duality provides a deep connection between representations of the symmetric group and representations of the Lie algebra gl(n). This theory was extended to the Lie superalgebra g = gl(m, n)first by Sergeev in [Ser84a] and then in more detail by Berele and Regev [BR87]. In Chapter 11, we give an exposition of this work and also of a beautiful xviii Preface extension of the Robinson-Schensted-Knuth correspondence from [BR87]. This correspondence allows us to use semistandard tableaux to index a basis for the simple U(g)-modules that appear in the decomposition of the tensor powers of the defining representation of g. In the theory of symmetric polynomials a key role is played by Schur polynomials. These can be defined in three different ways: as a quotient of alternants, using the Jacobi-Trudi determinant formula in terms of elementary or complete symmetric polynomials, and by using semistandard tableaux. The first definition is directly related to the Weyl character formula. Our treatment of supersymmetric polynomials in Chapter 12 places particular emphasis on super Schur polynomials. Each of the above three definitions of the usual Schur polynomials can be extended to the super case, and the main results, due to Sergeev, Pragacz and Thorup [PT92], [Pra91] and Remmel [Rem84], demonstrate the equivalence of the extended definitions. There is a connection with Schur-Weyl duality since the characters of composition factors of tensor powers of the defining representation of gl(m, n) are given by super Schur polynomials. Chapter 13 is devoted to the center Z(g)ofU(g) and related topics. Denote the ring of invariants of S(h) under the action of the Weyl group W by S(h)W . There is an injective algebra map from Z(g)toS(h)W which we call the Harish-Chandra homomorphism. Unlike the case of semisimple Lie algebras, however, this map is not surjective, but its image can be explicitly described. This result was first formulated by Kac [Kac84], but a gap in the proof was later filled by Gorelik [Gor04] and independently by the present author (unpublished). On the other hand, Sergeev [Ser99a]proved a version of the Chevalley restriction theorem for basic classical simple Lie superalgebras, [Ser99a]. This can be used to give another proof of the theorem formulated by Kac, but we will in fact deduce Sergeev’s Theorem from the result about the center. If g = gl(m, n) or an orthosymplectic Lie superalgebra, supersymmetric polynomials can be used to give an explicit set of generators for the image of the Harish-Chandra homomorphism and to describe the central characters of g. In Chapter 14 we study finite dimensional modules for a basic classical simple Lie superalgebra. If g is a close relative of gl(m, n)org is orthosymplectic, we give necessary and sufficient conditions for a simple highest weight module to be finite dimensional in terms of the highest weight. Then we prove the Kac-Weyl character formula for finite dimensional typical modules. If k is a finite dimensional Lie algebra, then the space of primitive ideals Prim U(k) is now well understood in both the solvable and semisimple cases. By comparison much less is known about primitive ideals in the enveloping Preface xix algebra of a Lie superalgebra g, but in Chapter 15 we survey what is known, with particular emphasis on the case where g is classical simple. To do this, it is convenient to review the semisimple Lie algebra case. Unlike the Lie algebra case, the enveloping algebra U(k)ofaLiesuper- algebra k may contain zero divisors. This will be the case if k contains a nonzero odd element x such that [x, x] = 0. In the absence of such elements k is called torsion free. In Chapter 17 we prove a theorem of R. Bøgvad, stating that the enveloping algebra of a torsion-free Lie superalgebra is a domain, [Bøg84]; see also [AL85]. The proof of this result, in contrast to the simplicity of its statement, requires a considerable amount of homological algebra. In Chapter 16 we develop the necessary cohomology theory of Lie superalgebras. This includes the fact that if M is a Z2-graded k-module, then the even part of H2(k,M) parameterizes extensions of k by M,thecup product in cohomology, and the Hochschild-Serre spectral sequence. We give a self-contained account of the necessary background on spectral sequences. The cohomology of a Lie superalgebra k can be computed using the standard resolution of the trivial k-module. In contrast to the Lie algebra case, this resolution has infinitely may terms if k1 = 0. Chapter 17 deals with the more ring theoretic aspects of homological algebra needed to prove Bøgvad’s result. These include standard results on derived functors and global dimension, as well as the Yoneda product, the bar resolution, and the Löfwall algebra. In the final chapter we introduce affine Lie superalgebras and obtain some applications to number theory. These applications concern the number of ways to write an integer as the sum of a given number of squares, or as the sum of a given number of triangular numbers. The main results here are due to Kac and Wakimoto [KW94] and Gorelik [Gor09], [Gor11]. Some background material on Lie theory, Hopf algebras, and ring theory is given in Appendix A, and the Dynkin-Kac diagrams for (nonexceptional) low rank Lie superalgebras may be found in Appendix B. This book has grown to about twice the length I originally intended, and nevertheless some important results had to be left out. However some of the topics not covered here are treated in other texts. Connections with physics are dealt with in [DM99] and [Var04]. The Dictionary of Lie Algebras and Lie Superalgebras [FSS00], while not containing any proofs, is nevertheless an invaluable source for detailed information about Lie superalgebras and their representations. We also recommend the survey article of Serganova on affine Lie superalgebras and integrable representations [Ser09]. On the other hand many topics are included which I believe can be found in no other texts. These topics include, in the order they appear in the book, xx Preface the construction of the exceptional Lie superalgebras, many of the ring theoretic methods used to study enveloping algebras, material on Schur-Weyl duality, supersymmetric polynomials, the center and central characters, the question of when the enveloping algebra contains zero divisors, and applications of affine Lie superalgebras to number theory. The treatment of Borel subalgebras that we give here is probably new.

I have used parts of this book to teach courses at the University of Wisconsin- Milwaukee and elsewhere. Of course it works best for students with some background in Lie theory. Here are some suggestions about how to use this book as a textbook. Chapter 1 contains the basic deﬁnitions so is a pre- requisite for everything else. Then Chapters 2–5 form a basic course on Lie superalgebras. Chapters 6–8, possibly followed by parts of Chapters 9, 10, 13, 14, 15, could be used for a course dealing with enveloping algebras. For more combinatorics use Chapters 11 and 12, for homological topics use Chapters 16 and 17, and for applications to number theory use Chapter 18. Exercises are given at the end of each chapter, often providing examples to illustrate the theory.

Acknowledgements

I would like to thank Georgia Benkart, Jason Gaddis, Maria Gorelik, Ed Letzter, America Masaros, Ivan Penkov, Georges Pinczon, Jos´e Santos, Paul Smith, Wolfgang Soergel, Elizaveta Vishnyakova, Lauren Williams, Hiroyuki Yamane, and James Zhang for many useful comments on the presentation. In addition I thank Rikard Bøgvad for explaining the proof of his result about global dimension to me. This text has been greatly improved thanks to the many helpful suggestions of Vera Serganova and Catharina Stroppel. Thanks are also due to Hedi Benamor, Paula Carvalho, Georges Pinczon, Wolfgang Soergel, and Rosane Ushirobira for invitations to lecture on this material in Metz, Porto, Dijon, and Freiburg. For her help in typing the manuscript, I would like to thank Gail Boviall. During the time that this book was written, I was partially supported by grants from the National Science Foundation and the National Security Agency. Bibliography

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This index gives the section in which the term appears. a-basal, 15.3 Noetherian, 7.1 Admissible pair, 18.7 Bimodule composition factors, series, Ad-nilpotent, 5.5 7.2 Affine denominator, 18.7 restricted, 7.4 Affine Weyl group, 18.3 Blocks, 8.2 translation subgroup of, 18.3 Bond, 7.2 Affinization, 18.2 Borel subalgebra, 3.1 Almost commutative algebra, 6.3 adjacent, 3.1 Alternative algebra, 4.3 positive, 8.1 Ambiguity, A.3 Bounded filtration, 16.2 resolvability of, A.3 BPS-subalgebra, 3.2 Annihilator of a module, 7.1 Braid equation, A.2 Annihilator prime, 7.4 Braid group, 11.2 Anticenter, 13.6 Bruhat order, 10.4.2 Antipode, A.2 Associated graded ring, 6.3 Canonical Cartan subalgebras, 5.2 Associated Type A diagram, 3.4 Cartan matrix associated to a Associator, 4.3 basis B,3.4 Augmentation map, A.2 decomposable, symmetrizable Augmented algebra, 17.3 Cartan matrix, 5.2 Auslander condition, Cartan subalgebra, 3.2 Auslander-Gorenstein, regular element of, 3.2 Auslander regular, 17.6 Casimir element, 8.5 Casimir operator, 18.4 b-dominant, 14.2 Category O,8.2 Bar resolution, 17.3 Catenary, 15.4 Basis of simple roots, 3.2, A.1 Cayley algebra, octonion algebra, 4.4 Bernstein number, 7.3 Central character, 8.2 Bialgebra, A.2 Central simple algebra, central Bimodule, 7.1 simple graded algebra, A.3

485 486 Index

Character of a module in the Grothendieck group, 8.4 Category O,8.4 Characteristic variety, 15.3 Harish-Chandra homomorphism, Clifford algebra, A.3 13.1, A.1 Cochain complex, 16.1 Harish-Chandra projection, 8.1 filtration on, 16.1 Height, 5.1 Cochain map, 16.1 Hochschild-Serre spectral sequence, Commutant, 11.1 16.6 Comodule, A.2 Homogeneous element, 1.1 Contravariant bilinear form, 8.2 Homogenized enveloping algebra, 6.6 Coproduct, counit, A.2 Homology of Lie superalgebras, 17.6 Cup product, 16.5 Hook, 11.3 skew Schur polynomial, 12.3 δ- -diagram, 3.3 Hopf algebra, superalgebra, A.2 Diagonal automorphism, diagram coquasitriangular, A.4 automprphism, 5.5 triangular, quasitriangular, A.2 Directional derivative, 12.1 Initial part, 3.4 Distinguished simple roots, diagrams Inner derivation, 16.4 and Borel subalgebras, 3.4 Insertion and recording tableau, 11.4 Dixmier-Moeglin equivalence, 15.1 Invariant bilinear form, 1.1 Dominates, 3.6 Inversions, A.2 Duflo involutions, 15.3 Involution on an algebra, 4.4 Dynkin-Kac diagrams, 3.4 Isotropic flag, subspace, 3.3

Elementary reduction, A.3 Jacobson radical, 7.1 Equivalent matrices, 3.4 Jacobson ring, 7.5 Essential submodule, 7.1 Jacobson topology, 7.1 Extension of a Lie superalgebra by Jantzen filtration, 10.2.1 a module, 16.4 Jantzen sum formula, 10.3 Jantzen’s multiplicity matrix, 15.3 F -form, defined over F ,2.5 F -split, Q-split, 2.5 Kac module, 8.1 Faithful R-module, 7.1 Kazhdan-Lusztig polynomials, 15.3 Filtered rings and modules, 6.3 Killing form, 2.6 Flag, 3.3 Knuth equivalence, Knuth relations, 11.7 G-ideal, G-prime, 7.6 Gelfand-Kirillov Dimension, 7.3 Lattice path, 12.3 Ghost center, 13.6 Left cone representation, 15.3 GK-homogeneous, 7.4 Length lexicographic ordering, A.3 GK-Macaulay, 17.6 Length of a permutation, 3.3 Global dimension, 17.2 Lie superalgebra, 1.1 Good filtration, 6.3 abelian, nilpotent, solvable, 1.1 Graded center, graded centralizer, basic, 1.2 A.3 cohomology of, 16.3 Graded ring, 6.3 consistently graded, 1.1 Graded-prime, graded-primitive contragredient, 5.2 ideal, 7.5 derived series of, 1.1 Index 487

free, 6.2 Q-regular, 3.2 Heisenberg, 6.7 Quadratic module, A.3 ideal, left ideal of, 1.1 Quasiroot, 10.2 lower central series of, 1.1 Quasisimple, 5.2 module, 1.1 Quiver, 3.6 orthosymplectic, 2.3 Borel, 3.6 semidirect product, 17.7 representation of, 3.6 simple, classical simple, 1.1 Quotient ring, 7.1 simple superalgebras of Cartan type, 1.3 R-matrix, A.2 strange, 2.4 Rational ideal, 15.1 torsion free, absolutely torsion Realization of a matrix, 5.1 free, 17.1 isomorphic realizations, 5.1 Type I, 8.1 minimal, 5.1 Limit superior, 7.3 Reduced expression, A.1 Locally closed ideal, 7.6 Reduction system, A.3 Locally nilpotent linear operator, 5.2 Rees ring, 6.6 Löfwall algebra., 17.4 Reflection in the hyperplane Loop algebra, 18.2 orthogonal to a root, 8.3, A.1 Lying over, direct lying over, 7.6 Regular element on a module, 7.1 Ring extension, 7.1 Mate, perfect mate, 15.4 Root, 2.1 Maximal isotropic set, 13.5 decomposable, simple, 3.4 Minimal graded resolution, 17.2 isotropic, 8.3 Modular law, 1.2 positive, 8.1 Multiplicative set, 7.1 root lattice, 5.1 root space decomposition, 2.1, 8.1, Nilradical, 7.1 A.1 Norm, trace on an algebra, 4.4 root system, A.1 Numerical marks, 14.2 string, 3.4 Rule of signs, A.2 Odd reflections, 3.4 One-line notation, 3.3 Sapovalovˇ determinant, 10.1 Ore condition, 7.1 Sapovalovˇ element, 9.2 Ore domain, 7.7 Sapovalovˇ maps, 9.3 Ore extension, A.3 Semigroup ordering, A.3 Outertensorproduct,2.2 Semiprime ring, 7.1 Semistandard, standard tableau, 11.3 Parity vector, parity change functor, Separation of variables, 15.3 A.2 Serre relations, 5.2 Partition into positive roots, Set of simple roots, 3.4 partition function, 8.4 initial part of, 3.4 Partition of an integer, conjugate Shuffle, 3.3 partition, 11.3 of Type C or D,3.3 Positive root, 3.2 Signed determinant, A.3 Prime and primitive ideals, 7.1 Simple reflection, A.1 Principal Z-gradings, 5.4 Skew diagram, skew tableau, 11.3 Projective dimension, 17.2 Space of harmonics, 15.3 488 Index

Spectral sequence, 16.2 Spinors, A.3 Standard resolution, 16.3 Strongly typical, 16.5 Superbialgebra, A.2 Supercommutative, 1.4 Superderivation, 1.1 Superexterior, supersymmetric algebra, A.2 Supersymmetric polynomial, 12.1 Supertrace, 2.2 Supertranspose, A.2 Support of an h-module, 3.3 Symbol, 6.3

Tensor algebra, A.3 Torsion submodule, 7.1 Trace ideal, 7.5 Translation functors, translation maps, 15.3 Triangular decomposition, 8.1, A.1 Twist map, A.2 Twisted adjoint action, 13.6 Typical, 8.6

Universal enveloping algebra, 6.1 Clifford filtration on, 6.5 standard filtration on, 6.3 Universal Verma module, 9.1

Verma module, 8.2

Weight module, highest weight module, 8.2 Weyl chamber, A.1 Weyl denominator, 18.7 Weyl group, 8.3, A.1 longest element of, A.1 Words, A.3

Yang-Baxter equation, A.2 Yoneda product, 17.3 Young diagram, 11.3

Z-graded Lie algebra, 1.1 Z2-graded vector space, 1.1 Titles in This Series

131 Ian M. Musson, Lie superalgebras and enveloping algebras, 2012 130 Viviana Ene and Jürgen Herzog, Gröbner bases in commutative algebra, 2012 129 Stuart P. Hastings and J. Bryce McLeod, Classical methods in ordinary differential equations: With applications to boundary value problems, 2012 128 J. M. Landsberg, Tensors: Geometry and applications, 2011 127 Jeffrey Strom, Modern classical homotopy theory, 2011 126 Terence Tao, An introduction to measure theory, 2011 125 Dror Varolin, Riemann surfaces by way of complex analytic geometry, 2011 124 David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties, 2011 123 Gregory Eskin, Lectures on linear partial differential equations, 2011 122 Teresa Crespo and Zbigniew Hajto, Algebraic groups and differential Galois theory, 2011 121 Tobias Holck Colding and William P. Minicozzi II, A course in minimal surfaces, 2011 120 Qing Han, A basic course in partial differential equations, 2011 119 Alexander Korostelev and Olga Korosteleva, Mathematical statistics: asymptotic minimax theory, 2011 118 HalL.SmithandHorstR.Thieme, Dynamical systems and population persistence, 2010 117 Terence Tao, An epsilon of room, I: pages from year three of a mathematical blog. A textbook on real analysis, 2010 116 Joan Cerdà, Linear functional analysis, 2010 115 Julio González-D´ıaz, Ignacio Garc´ıa-Jurado, and M. Gloria Fiestras-Janeiro, An introductory course on mathematical game theory, 2010 114 Joseph J. Rotman, Advanced modern algebra: Second edition, 2010 113 Thomas M. Liggett, Continuous time Markov processes: An introduction, 2010 112 Fredi Tröltzsch, Optimal control of partial differential equations: Theory, methods and applications, 2010 111 Simon Brendle, Ricci flow and the sphere theorem, 2010 110 Matthias Kreck, Differential algebraic topology: From stratifolds to exotic spheres, 2010 109 John C. Neu, Training manual on transport and fluids, 2010 108 Enrique Outerelo and Jesús M. Ruiz, Mapping degree theory, 2009 107 Jeffrey M. Lee, Manifolds and differential geometry, 2009 106 Robert J. Daverman and Gerard A. Venema, Embeddings in manifolds, 2009 105 Giovanni Leoni, A first course in Sobolev spaces, 2009 104 Paolo Aluffi, Algebra: Chapter 0, 2009 103 Branko Grünbaum, Configurations of points and lines, 2009 102 Mark A. Pinsky, Introduction to Fourier analysis and wavelets, 2009 101 Ward Cheney and Will Light, A course in approximation theory, 2009 100 I. Martin Isaacs, Algebra: A graduate course, 2009 99 Gerald Teschl, Mathematical methods in quantum mechanics: With applications to Schrödinger operators, 2009 98 Alexander I. Bobenko and Yuri B. Suris, Discrete differential geometry: Integrable structure, 2008 97 David C. Ullrich, Complex made simple, 2008 96 N. V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces, 2008 95 Leon A. Takhtajan, Quantum mechanics for mathematicians, 2008 94 James E. Humphreys, Representations of semisimple Lie algebras in the BGG category O, 2008 93 Peter W. Michor, Topics in differential geometry, 2008 92 I. Martin Isaacs, Finite group theory, 2008 TITLES IN THIS SERIES

91 Louis Halle Rowen, Graduate algebra: Noncommutative view, 2008 90 Larry J. Gerstein, Basic quadratic forms, 2008 89 Anthony Bonato, A course on the web graph, 2008 88 Nathanial P. Brown and Narutaka Ozawa, C∗-algebras and finite-dimensional approximations, 2008 87 Srikanth B. Iyengar, Graham J. Leuschke, Anton Leykin, Claudia Miller, Ezra Miller, Anurag K. Singh, and Uli Walther, Twenty-four hours of local cohomology, 2007 86 Yulij Ilyashenko and Sergei Yakovenko, Lectures on analytic differential equations, 2007 85 John M. Alongi and Gail S. Nelson, Recurrence and topology, 2007 84 Charalambos D. Aliprantis and Rabee Tourky, Cones and duality, 2007 83 Wolfgang Ebeling, Functions of several complex variables and their singularities (translated by Philip G. Spain), 2007 82 Serge Alinhac and Patrick Gérard, Pseudo-differential operators and the Nash–Moser theorem (translated by Stephen S. Wilson), 2007 81 V. V. Prasolov, Elements of homology theory, 2007 80 Davar Khoshnevisan, Probability, 2007 79 William Stein, Modular forms, a computational approach (with an appendix by Paul E. Gunnells), 2007 78 Harry Dym, Linear algebra in action, 2007 77 Bennett Chow, Peng Lu, and Lei Ni, Hamilton’s Ricci flow, 2006 76 Michael E. Taylor, Measure theory and integration, 2006 75 Peter D. Miller, Applied asymptotic analysis, 2006 74 V. V. Prasolov, Elements of combinatorial and differential topology, 2006 73 Louis Halle Rowen, Graduate algebra: Commutative view, 2006 72 R. J. Williams, Introduction the the mathematics of finance, 2006 71 S. P. Novikov and I. A. Taimanov, Modern geometric structures and fields, 2006 70 Seán Dineen, Probability theory in finance, 2005 69 Sebastián Montiel and Antonio Ros, Curves and surfaces, 2005 68 Luis Caffarelli and Sandro Salsa, A geometric approach to free boundary problems, 2005 67 T.Y. Lam, Introduction to quadratic forms over fields, 2004 66 Yuli Eidelman, Vitali Milman, and Antonis Tsolomitis, Functional analysis, An introduction, 2004 65 S. Ramanan, Global calculus, 2004 64 A. A. Kirillov, Lectures on the orbit method, 2004 63 Steven Dale Cutkosky, Resolution of singularities, 2004 62 T. W. Körner, A companion to analysis: A second first and first second course in analysis, 2004 61 Thomas A. Ivey and J. M. Landsberg, Cartan for beginners: Differential geometry via moving frames and exterior differential systems, 2003 60 Alberto Candel and Lawrence Conlon, Foliations II, 2003 59 Steven H. Weintraub, Representation theory of finite groups: algebra and arithmetic, 2003 58 Cédric Villani, Topics in optimal transportation, 2003 57 Robert Plato, Concise numerical mathematics, 2003 56 E. B. Vinberg, A course in algebra, 2003

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/. Lie superalgebras are a natural generalization of Lie algebras, having applications in geometry, number theory, gauge fi eld theory, and string theory. This book develops the theory of Lie superalgebras, their enveloping algebras, and their representations. The book begins with fi ve chapters on the basic properties of Lie superalgebras, including explicit constructions for all the classical simple Lie superalgebras. Borel subalgebras, which are more subtle in this setting, are studied and described. Contragredient Lie superalgebras are introduced, allowing a unifi ed approach to several results, in particular to the existence of an invariant bilinear form on g. The enveloping algebra of a fi nite dimensional Lie superalgebra is studied as an extension of the enveloping algebra of the even part of the superalgebra. By developing general methods for studying such extensions, important information on the algebraic structure is obtained, particularly with regard to primitive ideals. Fundamental results, such as the Poincaré-Birkhoff-Witt Theorem, are established. Representations of Lie superalgebras provide valuable tools for understanding the algebras themselves, as well as being of primary interest in applications to other fi elds. Two important classes of representations are the Verma modules and the fi nite dimensional representations. The fundamental results here include the Jantzen fi ltration, the Harish-Chandra homomorphism, the Sˇapovalov determinant, supersymmetric polynomials, and Schur-Weyl duality. Using these tools, the center can be explicitly described in the general linear and orthosymplectic cases. In an effort to make the presentation as self-contained as possible, some background material is included on Lie theory, ring theory, Hopf algebras, and combinatorics.

ISBN 978-0-8218-6867-6

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