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

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

### Citations:

Zbl 0862.17022; Zbl 0841.17015
Full Text: