Quantum generalized Heisenberg algebras and their representations

We introduce and study a new class of algebras, which we name quantum generalized Heisenberg algebras (qGHA), including both the so-called down-up but allowing more parameters freedom, so as to encompass wider range applications provide common framework for several previously studied classes algebras. In particular, our includes enveloping Lie algebra sl2 3-dimensional algebra, well its q-deformation, neither can be realized algebra. This paper focuses mostly on classification finite-dimensional irreducible representations qGHA, reveals their rich structure. Although these are not in general noetherian, still retain Lie-theoretic flavor. work over field arbitrary characteristic results presented characteristic-free fashion.