Finite-dimensional representations of minimal nilpotent W-algebras and zigzag algebras

Abstract

Let g be a simple finite-dimensional Lie algebra over an algebraically closed field F of characteristic 0. We denote by U(g) the universal enveloping algebra of g. To any nilpotent element e ∈ g one can attach an associative (and noncommutative as a general rule) algebra U(g, e) which is in a proper sense a "tensor factor" of U(g). In this article we consider the case in which g is simple and e belongs of the minimal nonzero nilpotent orbit of g. Under these assumptions U(g, e) was described explicitly in terms of generators and relations. One can expect that the representation theory of U(g, e) would be very similar to the representation theory of U(g). For example one can guess that the category of finite-dimensional U(g, e)-modules is semisimple.The goal of this article is to show that this is the case if g is not simply-laced. We also show that, if g is simply-laced and is not of type An, then the regular block of finite-dimensional U(g, e)-modules is equivalent to the category of finite-dimensional modules of a zigzag algebra.