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Book DetailsGraduate Studies in MathematicsVolume: 131; 2012; 488 ppMSC: Primary 17; Secondary 16;
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 PoincaréBirkhoffWitt 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 HarishChandra homomorphism, the Šapovalov determinant, supersymmetric polynomials, and SchurWeyl 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 selfcontained as possible, some background material is included on Lie theory, ring theory, Hopf algebras, and combinatorics.ReadershipGraduate students interested in Lie algebras, Lie superalgebras, quantum groups, string theory, and mathematical physics.

Table of Contents

Chapters

Chapter 1. Introduction

Chapter 2. The classical simple Lie superalgebras. I

Chapter 3. Borel subalgebras and DynkinKac diagrams

Chapter 4. The classical simple Lie superalgebras. II

Chapter 5. Contragredient Lie superalgebras

Chapter 6. The PBW Theorem and filtrations on enveloping algebras

Chapter 7. Methods from ring theory

Chapter 8. Enveloping algebras of classical simple Lie superalgebras

Chapter 9. Verma modules. I

Chapter 10. Verma modules. II

Chapter 11. SchurWeyl duality

Chapter 12. Supersymmetric polynomials

Chapter 13. The center and related topics

Chapter 14. Finite dimensional representations of classical Lie superalgebras

Chapter 15. Prime and primitive ideals in enveloping algebras

Chapter 16. Cohomology of Lie superalgebras

Chapter 17. Zero divisors in enveloping algebras

Chapter 18. Affine Lie superalgebras and number theory

Appendix A

Appendix B


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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 PoincaréBirkhoffWitt 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 HarishChandra homomorphism, the Šapovalov determinant, supersymmetric polynomials, and SchurWeyl 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 selfcontained as possible, some background material is included on Lie theory, ring theory, Hopf algebras, and combinatorics.
Graduate students interested in Lie algebras, Lie superalgebras, quantum groups, string theory, and mathematical physics.

Chapters

Chapter 1. Introduction

Chapter 2. The classical simple Lie superalgebras. I

Chapter 3. Borel subalgebras and DynkinKac diagrams

Chapter 4. The classical simple Lie superalgebras. II

Chapter 5. Contragredient Lie superalgebras

Chapter 6. The PBW Theorem and filtrations on enveloping algebras

Chapter 7. Methods from ring theory

Chapter 8. Enveloping algebras of classical simple Lie superalgebras

Chapter 9. Verma modules. I

Chapter 10. Verma modules. II

Chapter 11. SchurWeyl duality

Chapter 12. Supersymmetric polynomials

Chapter 13. The center and related topics

Chapter 14. Finite dimensional representations of classical Lie superalgebras

Chapter 15. Prime and primitive ideals in enveloping algebras

Chapter 16. Cohomology of Lie superalgebras

Chapter 17. Zero divisors in enveloping algebras

Chapter 18. Affine Lie superalgebras and number theory

Appendix A

Appendix B