  Let $B$ be a quantum algebra possessing a semiclassical limit $A$. We show
that under certain hypotheses $B^e$ can be thought of as a deformation of the
Poisson enveloping algebra of $A$, and we give a criterion for the Hochschild
cohomology of $B$ to be a deformation of the Poisson cohomology of $A$ in the
case that $B$ is Koszul. We verify that condition for the algebra of $2\times
2$ quantum matrices and calculate its Hochschild cohomology and the Poisson
cohomology of its semiclassical limit.
