Finite-dimensional simple modules over generalized Heisenberg algebras

Generalized Heisenberg algebras H(f) for any polynomial f(h)∈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)∈C[h]. More precisely, there are three classes of them, AH(f)′(λ,z˙,a), BH(f)′(λ,z˙,a), and CH(f)(z˙,n). If f=wh+c for any c∈C and n-th (n>1) primitive root w of unity we actually obtain a complete classification of all irreducible modules over H(f).