Multiparameter Weyl Algebras

TL;DR

This paper introduces multiparameter Weyl algebras, explores their module structures, and classifies simple weight and Whittaker modules, extending classical Weyl algebra theory with new algebraic parameters.

Contribution

It defines a new family of multiparameter Weyl algebras and classifies their simple modules, extending the understanding of Weyl algebra representations.

Findings

Classification of simple weight modules

Description of Whittaker modules as polynomial spaces

Extension to multiparameter twisted Weyl algebras

Abstract

We introduce a family of unital associative algebras A which are multiparameter analogues of the Weyl algebras and determine the simple weight modules and the Whittaker modules for them. All these modules can be regarded as spaces of (Laurent) polynomials with certain A-actions on them. This paper was written in February 2008 and some copies of it were distributed, but it has never been posted or published. We thank Bryan Bischof and Jason Gaddis for their interest in the work and for encouraging us to make the paper more widely available. The references in this posted version have been updated, and some cosmetic changes made; in particular, some typos have been corrected. The investigations have been generalized by V. Futorny and J. Hartwig to the context of multiparameter twisted Weyl algebras (see Journal of Algebra 357 (2012), 69-93).