Abstract
We introduce a new approach that allows us to determine the structure of Zhuʼs algebra for certain vertex operator (super)algebras which admit horizontal Z -grading. By using this method and an earlier description of Zhuʼs algebra for the singlet W -algebra, we completely describe the structure of Zhuʼs algebra for the triplet vertex algebra W ( p ) . As a consequence, we prove that Zhuʼs algebra A ( W ( p ) ) and the related Poisson algebra P ( W ( p ) ) have the same dimension. We also completely describe Zhuʼs algebras for the N = 1 triplet vertex operator superalgebra SW ( m ) . Moreover, we obtain similar results for the c = 0 triplet vertex algebra W 2 , 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 Adamović and Milas (2008) [4].
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