Invariant bilinear forms on a vertex algebra and the radical

2002/10/28 by Michael Roitman, Roitman, Michael
Mathematics · #17B69 #FOS: Mathematics #Quantum Algebra (math.QA) #math.QA #msc:17B69

paper · pdf · doi:10.48550/arxiv.math/0210432

arxiv created 2003/06/23 · arxiv updated 2009/11/30

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 A₀ of degree 0 is a commutative associative algebra, so that all structural maps and operations on A are A₀-linear. We also show that in this case A is simple if and only if A₀ is a field.

Related

• Symmetric Invariant Bilinear Forms on Modular Vertex Algebras
• Generalized Vertex Algebras
• Lie algebras admitting symmetric, invariant and nondegenerate bilinear forms
• The radical of a vertex operator algebra
• The Lie algebra preserving a degenerate bilinear form