  We study reflexivity and structure properties of operator algebras generated
by representations of the discrete Heisenberg semi-group. We show that the left
regular representation of this semi-group gives rise to a semi-simple reflexive
algebra. We exhibit an example of a representation which gives rise to a
non-reflexive algebra. En route, we establish reflexivity results for subspaces
of $H^{\infty}(\bb{T})\otimes\cl B(\cl H)$.
