, Volume 259, Issue 2, pp 391-411
Date: 12 Apr 2005

Abelianizing Vertex Algebras

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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 decreasing 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≥2. We further use gr(V) to study generating subspaces of certain types for lower truncated ℤ-graded vertex algebras.

Communicated by Y. Kawahigashi
Partially supported by an NSA grant