Quantum generalized Heisenberg algebras and their representations

Abstract

We introduce and study a new class of algebras, which we name \textit{quantum generalized Heisenberg algebras} and denote by ${\mathcal{H}}_q(f,g)$, related to generalized Heisenberg algebras, but allowing more parameters of freedom, so as to encompass a wider range of applications and include previously studied algebras, such as (generalized) down-up algebras. In particular, our class now includes the enveloping algebra of the $3$-dimensional Heisenberg Lie algebra and its $q$-deformation, neither of which can be realized as a generalized Heisenberg algebra. This paper focuses mostly on the classification of finite-dimensional irreducible representations of quantum generalized Heisenberg algebras, a study which reveals their rich structure. Although these algebras are not in general noetherian, their representations still retain some Lie-theoretic flavor. We work over a field of arbitrary characteristic, although our results on the representations require that it be algebraically closed.