Twisted Generalized Weyl Algebras and Primitive Quotients of Enveloping Algebras

Abstract

To each multiquiver Γ 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 Γ for when this representation is faithful or locally surjective. By forgetting some of the structure of Γ one obtains a Dynkin diagram, D(Γ). We show that the generalized Cartan matrix of $\mathcal {A}({\Gamma })$ coincides with the one corresponding to D(Γ) 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 Γ. We also prove one result in the affine case.