Generalized Vertex Algebras and Relative Vertex Operators

1. Front Matter

   Pages i-ix

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2. Introduction

   • Chongying Dong, James Lepowsky

   Pages 1-14

3. The setting

   • Chongying Dong, James Lepowsky

   Pages 15-17

4. Relative untwisted vertex operators

   • Chongying Dong, James Lepowsky

   Pages 19-25

5. Quotient vertex operators

   • Chongying Dong, James Lepowsky

   Pages 27-31

6. A Jacobi identity for relative untwisted vertex operators

   • Chongying Dong, James Lepowsky

   Pages 33-47

7. Generalized vertex operator algebras and their modules

   • Chongying Dong, James Lepowsky

   Pages 49-58

8. Duality for generalized vertex operator algebras

   • Chongying Dong, James Lepowsky

   Pages 59-75

9. Monodromy representations of braid groups

   • Chongying Dong, James Lepowsky

   Pages 77-81

10. Generalized vertex algebras and duality

    • Chongying Dong, James Lepowsky

    Pages 83-94

11. Tensor products

    • Chongying Dong, James Lepowsky

    Pages 95-96

12. Intertwining operators

    • Chongying Dong, James Lepowsky

    Pages 97-104

13. Abelian intertwining algebras, third cohomology and duality

    • Chongying Dong, James Lepowsky

    Pages 105-140

14. Affine Lie algebras and vertex operator algebras

    • Chongying Dong, James Lepowsky

    Pages 141-160

15. Z-algebras and parafermion algebras

    • Chongying Dong, James Lepowsky

    Pages 161-189

16. Back Matter

    Pages 191-206

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In the past few years, vertex operator algebra theory has been growing both in intrinsic interest and in the scope of its interconnections with areas of mathematics and physics. The structure and representation theory of vertex operator algebras is deeply related to such subjects as monstrous moonshine, conformal field theory and braid group theory. Vertex operator algebras are the mathematical counterpart of chiral algebras in conformal field theory. In the Introduction which follows, we sketch some of the main themes in the historical development and mathematical and physical motivations of these ideas, and some of the current issues. Given a vertex operator algebra, it is important to consider not only its modules (representations) but also intertwining operators among the mod­ ules. Matrix coefficients of compositions of these operators, corresponding to certain kinds of correlation functions in conformal field theory, lead natu­ rally to braid group representations. In the specialbut important case when these braid group representations are one-dimensional, one can combine the modules and intertwining operators with the algebra to form a structure satisfying axioms fairly close to those for a vertex operator algebra. These are the structures which form the main theme of this monograph. Another treatment of similar structures has been given by Feingold, Frenkel and Ries (see the reference [FFR] in the Bibliography), and in fact the material de­ veloped in the present work has close connections with much work of other people, as we explain in the Introduction and throughout the text.

• Algebraic structure
• Cohomology
• Lattice
• Representation theory
• algebra
• cls
• homology
• ring theory

"The presentation is smooth, self-contained and accessible with detailed proofs. The introduction offers background and history about the generalized theory. It also uses examples to show some of the central techniques in VOA, thus offering pedagogical help to the readers. I think this book will benefit researchers in the field."

—Mathematical Reviews

• Department of Mathematics, University of California, Santa Cruz, Santa Cruz, USA

  Chongying Dong

• Department of Mathematics, Rutgers University, New Brunswick, USA

  James Lepowsky

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