The structure of Zhu's algebras for certain W-algebras

We introduce a new approach that allows us to determine the structure of Zhu's algebra for certain vertex operator (super)algebras which admit horizontal Z-grading. By using this method and an earlier description of Zhu's algebra for the singlet W-algebra, we completely describe the structure of Zhu's algebra for the triplet vertex algebra W(p). As a consequence, we prove that Zhu's algebra A(W(p)) and the related Poisson algebra P(W(p)) have the same dimension. We also completely describe Zhu's algebras for the N = I triplet vertex operator superalgebra SW(m). Moreover, we obtain similar results for the c = 0 triplet vertex algebra W-2,W-3, important in logarithmic conformal field theory. Because our approach is "internal" we had to employ several constant term identities for purposes of getting right upper bounds on dim(A(V)). This work is, in a way, a continuation of the results published in Adamovic and Milas (2008) [4]. (C) 2011 Elsevier Inc. All rights reserved.