Invariant bilinear forms on a vertex algebra and the radical

Abstract

In this paper we construct a linear space that parameterizes all invariant bilinear forms on a given vertex algebra with values in a arbitrary vector space. Also we prove that every invariant bilinear form on a vertex algebra is symmetric. This is a generalization of the result of H. Li [J. Pure and Appl. Algebra, 96(3):279-297, 1994], who proved this for the case when the vertex algebra is non-negatively graded and has finite dimensional homogeneous components. As an application, we introduce a notion of a radical of a vertex algebra. We prove that a radical-free vertex algebra A is non-negatively graded, and its component A_0 of degree 0 is a commutative associative algebra, so that all structural maps and operations on A are A_0-linear. We also show that in this case A is simple if and only if A_0 is a field.