Abelianizing vertex algebras

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.