Lie Superalgebras and Enveloping Algebras

Released on by Ian Malcolm Musson
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  • Lie Superalgebras and Enveloping Algebras Book Detail

  • Author : Ian Malcolm Musson
  • Release Date : 2012-04-04
  • Publisher : American Mathematical Soc.
  • Genre : Mathematics
  • Pages : 512
  • ISBN 13 : 0821868675
  • File Size : 64,64 MB
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Lie Superalgebras and Enveloping Algebras by Ian Malcolm Musson PDF Summary

Book Description: 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 Poincare-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 Sapovalov 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.

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Lie Superalgebras and Enveloping Algebras

Lie Superalgebras and Enveloping Algebras

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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

Enveloping Algebras

Enveloping Algebras

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For the graduate student, this is a masterpiece of pedagogical writing, being succinct, wonderfully self-contained and of exceptional precision. --Mathematical