Abelianizing vertex algebras

Haisheng Li

doi:10.1007/s00220-005-1348-z

paper

Abelianizing vertex algebras

arXiv:math/0409140 · doi:10.1007/s00220-005-1348-z

Abstract

To every vertex algebra $V$ we associate a canonical decreasing sequence of subspaces and prove that the associated graded vector space $g r (V)$ 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 $g r (V)$ , we prove that for any vertex algebra $V$ , $C_{2}$ -cofiniteness implies $C_{n}$ -cofiniteness for all $n \geq 2$ . We further use $g r (V)$ to study generating subspaces of certain types for lower truncated $Z$ -graded vertex algebras.

Latex, 24 pages