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Lie Superalgebras and Enveloping Algebras

Ian M. Musson University of Wisconsin, Milwaukee, Milwaukee, WI
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Hardcover ISBN: 978-0-8218-6867-6
Product Code: GSM/131
List Price: $93.00 MAA Member Price:$83.70
AMS Member Price: $74.40 Electronic ISBN: 978-0-8218-8504-8 Product Code: GSM/131.E List Price:$87.00
MAA Member Price: $78.30 AMS Member Price:$69.60
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List Price: $139.50 MAA Member Price:$125.55
AMS Member Price: $111.60 Click above image for expanded view Lie Superalgebras and Enveloping Algebras Ian M. Musson University of Wisconsin, Milwaukee, Milwaukee, WI Available Formats:  Hardcover ISBN: 978-0-8218-6867-6 Product Code: GSM/131  List Price:$93.00 MAA Member Price: $83.70 AMS Member Price:$74.40
 Electronic ISBN: 978-0-8218-8504-8 Product Code: GSM/131.E
 List Price: $87.00 MAA Member Price:$78.30 AMS Member Price: $69.60 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$139.50 MAA Member Price: $125.55 AMS Member Price:$111.60
• Book Details

Volume: 1312012; 488 pp
MSC: 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 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.

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 Dynkin-Kac 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. Schur-Weyl 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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Volume: 1312012; 488 pp
MSC: 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 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.

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 Dynkin-Kac 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. Schur-Weyl 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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