We study isomorphisms between generalized Weyl algebras, giving a complete answer to the quantum case of this problem for R = k[h].

We exhibit a Poisson module restoring a twisted Poincaré duality between Poisson homology and cohomology for the polynomial algebra R = C[X 1 , . . . , X n ] endowed with Poisson bracket arising from a uniparametrised quantum affine space. This Poisson module is obtained as the semiclassical limit of the dualising bimodule for Hochschild homology of the corresponding quantum affine space. As a corollary we compute the Poisson cohomology of R, and so retrieve a result obtained by direct methods (so completely different from ours) by Monnier.

Let k be a ®eld, and P L k a n-dimensional quantum torus parametrized by a n  n matrix L with entries in k à . The main object of this paper is to prove that, if P L k is simple, then any k-algebra endomorphism of P L k is surjective (and therefore is an automorphism). We use this result to prove that some quantum polynomial algebras are isomorphic if and only if they are rationally equivalent. In the case where P L k is not simple, we give a condition for a k-algebra endomorphism of P L k to be an automorphism.