Yoneda algebras of the triplet vertex operator algebra

Abstract

Given a vertex operator algebra V, one can construct two associative algebras, the Zhu algebra A(V) and the C2-algebra R(V). This gives rise to two abelian categories A(V)-Mod and R(V)-Mod, in addition to the category of admissible modules of V. In case V is rational and C2-cofinite, the category of admissible V-modules and the category of all A(V)-modules are equivalent. However, when V is not rational, the connection between these two categories is unclear. The goal of this paper is to study the triplet vertex operator algebra W(p), as an example to compare these three categories, in terms of abelian categories. For each of these three abelian categories, we will determine the associated Ext quiver, the Morita equivalent basic algebra, i.e., the algebra End(⊕L∈IrrPL)op, and the Yoneda algebra Ext⁎(⊕L∈IrrL,⊕L∈IrrL). As a consequence, the category of admissible log-modules for the triplet VOA W(p) has infinite global dimension, as do the Zhu algebra A(W(p)), and the associated graded algebra grA(W(p)) which is isomorphic to R(W(p)). We also describe the Koszul properties of the module categories of W(p), A(W(p)) and grA(W(p)).