Yoneda algebras of the triplet vertex operator algebra

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