Generalized Vertex Algebras

TL;DR

This paper introduces generalized vertex algebras using polylocal fields, constructing a new algebra that encompasses lattice vertex algebras and their irreducible representations, expanding the algebraic framework in vertex algebra theory.

Contribution

It presents a novel construction of generalized vertex algebras associated with a vector space and bilinear form, integrating existing lattice vertex algebras and their representations.

Findings

Constructed a generalized vertex algebra from a vector space with a symmetric bilinear form.

The algebra contains all lattice vertex algebras of matching rank.

Includes all irreducible representations of these lattice vertex algebras.

Abstract

We give a short introduction to generalized vertex algebras, using the notion of polylocal fields. We construct a generalized vertex algebra associated to a vector space h with a symmetric bilinear form. It contains as subalgebras all lattice vertex algebras of rank equal to dim h and all irreducible representations of these vertex algebras.