Poisson and Hochschild cohomology and the semiclassical limit

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.