Extensions of the Jacobi identity for generalized vertex algebras

Abstract

In this paper the (generalized) Jacobi identity for generalized vertex algebras is extended to multi-operator identities. This work is based on the book of Dong and Lepowsky on generalized vertex algebras and relative vertex operators (Dong and Lepowsky, 1993), on the work of the author on the Jacobi identity for vertex operator algebras in Husu (1993), Section 1, and on the work in Husu (1993) and Husu (1995) on the A 1 (1)-case of the Jacobi identity for relative vertex operators. The multi-operator extensions of the (generalized) Jacobi identity (in the present paper) generalize the corresponding results on vertex operator algebras in Husu (1993), Section 1, and provide a general framework both for the vertex operator construction of generating function identities for standard A 1 (1)-modules in Husu (1995) and for further applications of the vertex operator construction to other affine Lie algebras.