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

Abstract

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