  In this article, the two-parameter quantum Heisenberg enveloping algebra,
which serves as a model for certain quantum generalized Heisenberg algebras,
have been studied at roots of unity. In this context, the quantum Heisenberg
enveloping algebra becomes a polynomial identity algebra, and the dimension of
simple modules is bounded by its PI degree. The PI degree, center, and complete
classification of simple modules up to isomorphism are explicitly presented. We
work over a field of arbitrary characteristic, although our results concerning
the representations require that it is algebraically closed.
