Generalized Vertex Algebras and Relative Vertex Operators

✍ Scribed by Chongying Dong, James Lepowsky (auth.)

• Publisher: Birkhäuser Basel
• Year: 1993
• Tongue: English
• Leaves: 206
• Series: Progress in Mathematics 112
• Edition: 1
• Category: Library

✦ Synopsis

The rapidly-evolving theory of vertex operator algebras provides deep insight into many important algebraic structures. Vertex operator algebras can be viewed as "complex analogues" of both Lie algebras and associative algebras. They are mathematically precise counterparts of what are known in physics as chiral algebras, and in particular, they are intimately related to string theory and conformal field theory.

Dong and Lepowsky have generalized the theory of vertex operator algebras in a systematic way at three successively more general levels, all of which incorporate one-dimensional braid groups representations intrinsically into the algebraic structure: First, the notion of "generalized vertex operator algebra" incorporates such structures as Z-algebras, parafermion algebras, and vertex operator superalgebras. Next, what they term "generalized vertex algebras" further encompass the algebras of vertex operators associated with rational lattices. Finally, the most general of the three notions, that of "abelian intertwining algebra," also illuminates the theory of intertwining operator for certain classes of vertex operator algebras.

The monograph is written in a n accessible and self-contained manner, with detailed proofs and with many examples interwoven through the axiomatic treatment as motivation and applications. It will be useful for research mathematicians and theoretical physicists working the such fields as representation theory and algebraic structure sand will provide the basis for a number of graduate courses and seminars on these and related topics.

✦ Table of Contents

Front Matter....Pages i-ix
Introduction....Pages 1-14
The setting....Pages 15-17
Relative untwisted vertex operators....Pages 19-25
Quotient vertex operators....Pages 27-31
A Jacobi identity for relative untwisted vertex operators....Pages 33-47
Generalized vertex operator algebras and their modules....Pages 49-58
Duality for generalized vertex operator algebras....Pages 59-75
Monodromy representations of braid groups....Pages 77-81
Generalized vertex algebras and duality....Pages 83-94
Tensor products....Pages 95-96
Intertwining operators....Pages 97-104
Abelian intertwining algebras, third cohomology and duality....Pages 105-140
Affine Lie algebras and vertex operator algebras....Pages 141-160
Z-algebras and parafermion algebras....Pages 161-189
Back Matter....Pages 191-206

✦ Subjects

Algebra;Associative Rings and Algebras;Operator Theory;Group Theory and Generalizations;Topological Groups, Lie Groups;Theoretical, Mathematical and Computational Physics