Generalized rationality and a “Jacobi identity” for intertwining operator algebras. (English) Zbl 1013.17026

Huang, Yi-Zhi

Sel. Math., New Ser. 6, No. 3, 225-267 (2000).

Summary: We prove a generalized rationality property and a new identity that we call the “Jacobi identity” for intertwining operator algebras. Most of the main properties of genus-zero conformal field theories, including the main properties of vertex operator algebras, modules, intertwining operators, Verlinde algebras, and fusing and braiding matrices, are incorporated into this identity. Together with associativity and commutativity for intertwining operators proved by the author in [Y.-Z. Huang, J. Algebra 182, 201–234 (1996; Zbl 0862.17022) and J. Pure Appl. Algebra 100, 173–216 (1995; Zbl 0841.17015)], the results of the present paper solve completely the problem of finding a natural purely algebraic structure on the direct sum of all inequivalent irreducible modular for a suitable vertex operator algebra. Two equivalent definitions of intertwining operator algebra in terms of this Jacobi identity are given.

MSC:

17B69	Vertex operators; vertex operator algebras and related structures
81R10	Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
81T40	Two-dimensional field theories, conformal field theories, etc. in quantum mechanics

Keywords:

Jacobi identity; generalized rationality property; intertwining operator algebras; genus-zero conformal field theories; vertex operator algebras; intertwining operators; Verlinde algebras; fusing; braiding matrices

Citations:

Zbl 0862.17022; Zbl 0841.17015