Twisted generalized Weyl algebras and primitive quotients of enveloping algebras

TL;DR

This paper constructs and analyzes twisted generalized Weyl algebras associated with multiquivers, establishing their representations, connections to Kac-Moody algebras, and characterizing primitive quotients of enveloping algebras as TGW algebras.

Contribution

It introduces solutions to consistency equations for TGW algebras, explores their representations, and characterizes primitive quotients of enveloping algebras as TGW algebras.

Findings

Constructed canonical differential operator representations of TGW algebras.

Identified conditions for faithfulness and local surjectivity of representations.

Characterized primitive quotients of enveloping algebras as TGW algebras in terms of simple weight modules.

Abstract

To each multiquiver $\Gamma$ we attach a solution to the consistency equations associated to twisted generalized Weyl (TGW) algebras. This generalizes several previously obtained solutions in the literature. We show that the corresponding algebras $\mathcal{A}(\Gamma)$ carry a canonical representation by differential operators and that $\mathcal{A}(\Gamma)$ is universal among all TGW algebras with such a representation. We also find explicit conditions in terms of $\Gamma$ for when this representation is faithful or locally surjective. By forgetting some of the structure of $\Gamma$ one obtains a Dynkin diagram, $D(\Gamma)$. We show that the generalized Cartan matrix of $\mathcal{A}(\Gamma)$ coincides with the one corresponding to $D(\Gamma)$ and that $\mathcal{A}(\Gamma)$ contains graded homomorphic images of the enveloping algebra of the positive and negative part of the corresponding Kac-Moody algebra. Finally, we show that a primitive quotient $U/J$ of the enveloping algebra of a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero is graded isomorphic to a TGW algebra if and only if $J$ is the annihilator of a completely pointed (multiplicity-free) simple weight module. The infinite-dimensional primitive quotients in types $A$ and $C$ are closely related to $\mathcal{A}(\Gamma)$ for specific $\Gamma$. We also prove one result in the affine case.