Abelianizing vertex algebras

Abstract

To every vertex algebra $V$ we associate a canonical decreasing sequence of subspaces and prove that the associated graded vector space $gr(V)$ is naturally a vertex Poisson algebra, in particular a commutative vertex algebra. We establish a relation between this sequence and the sequence $C_n$ introduced by Zhu. By using the (classical) algebra $gr(V)$, we prove that for any vertex algebra $V$, $C_2$-cofiniteness implies $C_n$-cofiniteness for all $n\ge 2$. We further use $gr(V)$ to study generating subspaces of certain types for lower truncated $Z$-graded vertex algebras.

arXiv commentLatex, 24 pages

References in corpus (1)

• Rationality, quasirationality and finite W-algebras

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