Invariant Bilinear Forms on a Vertex Algebra

Abstract

Invariant bilinear forms on vertex algebras have been around for quite some time now. They were mentioned by Borcherds in [1] and were used in many early works on vertex algebras, especially in relation with the vertex algebras associated with lattices [2, 3, 8]. The first systematic study of invariant forms on vertex algebras is due to Frenkel, Huang and Lepowsky [7]. This theory was developed further by Li [13]. However, these authors imposed certain assumptions on their vertex algebras which are too restrictive for the applications we have in mind. Specifically, this paper is motivated by the study of vertex algebras of OZ type generated by their Griess subalgebras [10, 19]. We show that all the results of Li [13] hold in the greatest possible generality, basically, as long as the definitions make sense. We construct a linear space that parameterizes all bilinear forms on a given vertex algebra and also prove that every invariant bilinear form on a vertex algebra is symmetric. Our methods, however, are very different from the methods used in [7] and [13]. As an application of invariant bilinear forms, we introduce a very useful notion of radical of a vertex algebra. We also use these techniques to obtain a result on simple vertex algebras, which can be viewed as a first step in their classification theory. We prove that a radical-free simple vertex algebra must be non-negatively graded and the component of degree 0 must have dimension 1 after a suitable change of constants.

Cite this paper

@inproceedings{Roitman2002InvariantBF, title={Invariant Bilinear Forms on a Vertex Algebra}, author={Michael Roitman}, year={2002} }