Finite-dimensional simple modules over generalized Heisenberg algebras

Generalized Heisenberg algebras $\stackrel{˝}{(}f)$ for any polynomial $f(h)\in \text{C}[h]$ 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 $\stackrel{˝}{(}f)$, and the necessary and sufficient conditions on $f$ for two $\stackrel{˝}{(}f)$ to be isomorphic. Then we determine all finite dimensional simple modules over $\stackrel{˝}{(}f)$ for any polynomial $f(h)\in \text{C}[h]$. If $f=wh+c$ for any $c\in \text{C}$ 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 $f\in \text{C}[h]$, we also prove that, for any $n\in \mathbb{N}$, $\mathcal{H}(f)$ has infinitely many ideals $I_n$ such that $\mathcal{H}(f)\mathrm{/}{I}_n\cong M_n(\text{C})$, the matrix algebra.