**Graduate Studies in Mathematics**

Volume: 131;
2012;
488 pp;
Hardcover

MSC: Primary 17;
Secondary 16

Print ISBN: 978-0-8218-6867-6

Product Code: GSM/131

List Price: $87.00

Individual Member Price: $69.60

**Electronic ISBN: 978-0-8218-8504-8
Product Code: GSM/131.E**

List Price: $87.00

Individual Member Price: $69.60

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#### Supplemental Materials

# Lie Superalgebras and Enveloping Algebras

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*Ian M. Musson*

Lie superalgebras are a natural generalization of Lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. This book develops the theory of Lie superalgebras, their enveloping algebras, and their representations.

The book begins with five 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 unified approach to several results, in particular to the existence of an invariant bilinear form on \(\mathfrak{g}\).

The enveloping algebra of a finite 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 fields. Two important classes of representations are the Verma modules and the finite dimensional representations. The fundamental results here include the Jantzen filtration, the Harish-Chandra homomorphism, the Šapovalov 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.

#### Table of Contents

# Table of Contents

## Lie Superalgebras and Enveloping Algebras

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface xv16 free
- Chapter 1. Introduction 122 free
- Chapter 2. The classical simple Lie superalgebras. I 1132 free
- Chapter 3. Borel subalgebras and Dynkin-Kac diagrams 2546
- Chapter 4. The classical simple Lie superalgebras. II 6990
- Chapter 5. Contragredient Lie superalgebras 95116
- Chapter 6. The PBW Theorem and filtrations on enveloping algebras 131152
- Chapter 7. Methods from ring theory 147168
- Chapter 8. Enveloping algebras of classical simple Lie superalgebras 181202
- Chapter 9. Verma modules. I 207228
- Chapter 10. Verma modules. II 223244
- Chapter 11. Schur-Weyl duality 239260
- Chapter 12. Supersymmetric polynomials 263284
- Chapter 13. The center and related topics 281302
- Chapter 14. Finite dimensional representations of classical Lie superalgebras 307328
- Chapter 15. Prime and primitive ideals in enveloping algebras 319340
- Chapter 16. Cohomology of Lie superalgebras 355376
- Chapter 17. Zero divisors in enveloping algebras 381402
- Chapter 18. Affine Lie superalgebras and number theory 403424
- Appendix A 433454
- Appendix B 463484
- Bibliography 471492
- Index 485506 free
- Back Cover Back Cover1512

#### Readership

Graduate students interested in Lie algebras, Lie superalgebras, quantum groups, string theory, and mathematical physics.