Two-parameter analogs of the Heisenberg enveloping algebra

TL;DR

This paper explores two-parameter generalizations of the Heisenberg algebra, examining their structure, especially with dependent parameters, and investigates the implications for the representation theory of a two-parameter Virasoro algebra.

Contribution

It introduces and analyzes new two-parameter versions of the Heisenberg algebra, extending previous one-parameter studies and exploring their representation theory.

Findings

Two-parameter analogs of the Heisenberg algebra are constructed.

Dependent parameters lead to novel algebraic structures.

Representation theory of the two-parameter Virasoro algebra is developed.

Abstract

One-parameter analogs of the Heisenberg enveloping algebra were studied previously by Kirkman and Small. In particular, they demonstrated how one may obtain Hayashi's analog of the Weyl algebra as a primitive factor of this algebra. We consider various two-parameter versions of this problem. Of particular interest is the case when the parameters are dependent. Our study allows us to consider the representation theory of a two-parameter version of the Virasoro enveloping algebra.