Twisted generalized Weyl algebras and primitive quotients of enveloping algebras

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.

38 pages

References in corpus (1)

• Locally finite simple weight modules over twisted generalized Weyl algebras

Cited by in corpus (6)

• Noncommutative fiber products and lattice models
• Cylindrical Dyck paths and the Mazorchuk-Turowska equation
• Fixed rings of twisted generalized Weyl algebras
• Classification of twisted generalized Weyl algebras over polynomial rings
• Clifford and Weyl superalgebras and spinor representations
• On rational twisted generalized Weyl algebra