Homologie et cohomologie de Hochschild de certaines algebres polynomiales classiques et quantiques

TL;DR

This paper investigates Hochschild homology and cohomology for polynomial algebras with classical and quantum relations, showing that quantum Weyl algebras share homological properties with classical Weyl algebras.

Contribution

It demonstrates that quantum Weyl algebras have identical Hochschild homology and duality properties as classical Weyl algebras, extending known results to quantum settings.

Findings

Quantum Weyl algebras share Hochschild homology with classical Weyl algebras.

Hochschild duality relations hold for quantum affine space algebras.

The results unify classical and quantum algebra homological properties.

Abstract

We study Hochschild homology and cohomology for some polynomial algebras mixing both ``classical'' relations ($XY-YX=1$) and ``quantum'' relations ($XY={\l}YX$). More specifically, we prove that the algebra of differential operators on any quantum affine space (quantum Weyl algebra) have the same Hochschild homology, and satisfy the same duality relation, as the classical Weyl algebra does.