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Lie Superalgebras and Enveloping Algebras
 
Ian M. Musson University of Wisconsin, Milwaukee, Milwaukee, WI
Lie Superalgebras and Enveloping Algebras
Hardcover ISBN:  978-0-8218-6867-6
Product Code:  GSM/131
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
eBook ISBN:  978-0-8218-8504-8
Product Code:  GSM/131.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-0-8218-6867-6
eBook: ISBN:  978-0-8218-8504-8
Product Code:  GSM/131.B
List Price: $220.00 $177.50
MAA Member Price: $198.00 $159.75
AMS Member Price: $176.00 $142.00
Lie Superalgebras and Enveloping Algebras
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Lie Superalgebras and Enveloping Algebras
Ian M. Musson University of Wisconsin, Milwaukee, Milwaukee, WI
Hardcover ISBN:  978-0-8218-6867-6
Product Code:  GSM/131
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
eBook ISBN:  978-0-8218-8504-8
Product Code:  GSM/131.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-0-8218-6867-6
eBook ISBN:  978-0-8218-8504-8
Product Code:  GSM/131.B
List Price: $220.00 $177.50
MAA Member Price: $198.00 $159.75
AMS Member Price: $176.00 $142.00
  • Book Details
     
     
    Graduate Studies in Mathematics
    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.

    Readership

    Graduate 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 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
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.

Readership

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
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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