Hochschild homology and cohomology of some classical and quantum noncommutative polynomial algebras

Abstract

We study Hochschild homology and cohomology for a class of noncommutative polynomial algebras which are both quantum (in the sense that they contain some copies of Manin's quantum plane as subalgebras) and classical (in the sense that they also contain some copies of the Weyl algebra A 1). We obtain explicit computations for some significant families of such algebras. In particular, we prove that the algebra of twisted differential operators on a quantum affine space (simple quantum Weyl algebra) has the same Hochschild homology, and satisfies the same duality relation, as the classical Weyl algebra does.