Finite-dimensional simple modules over generalized Heisenberg algebras

TL;DR

This paper classifies all finite-dimensional simple modules over generalized Heisenberg algebras, providing conditions for isomorphism and explicit descriptions for certain polynomial cases, enriching the understanding of their module structure.

Contribution

It determines the center of generalized Heisenberg algebras and classifies all finite-dimensional simple modules for any polynomial f(h), including explicit cases involving roots of unity.

Findings

Complete classification of finite-dimensional simple modules for certain polynomials.

Conditions for isomorphism between generalized Heisenberg algebras.

Existence of infinitely many ideals leading to matrix algebra quotients.

Abstract

Generalized Heisenberg algebras $\H(f)$ for any polynomial $f(h)\in\C[h]$ have been used to explain various physical systems and many physical phenomena for the last 20 years. In this paper, we first obtain the center of $\H(f)$, and the necessary and sufficient conditions on $f$ for two $\H(f)$ to be isomorphic. Then we determine all finite dimensional simple modules over $\H(f)$ for any polynomial $f(h)\in\C[h]$. If $f=wh+c$ for any $c\in \C$ and $n$-th ($n>1$) primitive root $w$ of unity we actually obtain a complete classification of all irreducible modules over $\mathcal{H}(f)$. For many $f\in\C[h]$, we also prove that, for any $n\in\N$, $\mathcal{H}(f)$ has infinitely many ideals $I_n$ such that $\mathcal{H}(f)/I_n\cong M_n(\C)$, the matrix algebra.