Classification of irreducible modules for Bershadsky-Polyakov algebra at certain levels

We study the representation theory of the Bershadsky-Polyakov algebra W-k = W-k(sl(3), f(theta)). In particular, Zhu algebra of W-k is isomorphic to a certain quotient of the Smith algebra, after changing the Virasoro vector. We classify all modules in the category O for the Bershadsky-Polyakov algebra W-k for k = - 5/3,- 9/4,- 1, 0. In the case k = 0, we show that the Zhu algebra A(W-k) has two-dimensional indecomposable modules.