Twisted Weyl algebras, crossed products, and representations

TL;DR

This paper introduces a new class of twisted generalized Weyl algebras called multiparameter twisted Weyl algebras, embeds them into crossed products, and classifies their simple modules and Whittaker pairs, unifying several known algebraic structures.

Contribution

It develops the theory of multiparameter twisted Weyl algebras, providing parametrization of simple quotients, embeddings into crossed products, and classification of modules and Whittaker pairs.

Findings

Embedded twisted generalized Weyl algebras into crossed products.

Parametrized all simple quotients of multiparameter twisted Weyl algebras.

Classified all simple weight modules and described Whittaker pairs.

Abstract

We prove that any twisted generalized Weyl algebra satisfying certain consistency conditions can be embedded into a crossed product. We also introduce a new family of twisted generalized Weyl algebras, called multiparameter twisted Weyl algebras, for which we parametrize all simple quotients of a certain kind. Both Jordan's simple localization of the multiparameter quantized Weyl algebra and Hayashi's $q$-analog of the Weyl algebra are special cases of this construction. We classify all simple weight modules over any multiparameter twisted Weyl algebra. Extending results by Benkart and Ondrus, we also describe all Whittaker pairs up to isomorphism over a class of twisted generalized Weyl algebras which includes the multiparameter twisted Weyl algebras.