On the Hochschild homology of smash biproducts

TL;DR

This paper introduces a spectral sequence for computing Hochschild homology of smash biproducts, enabling calculations for various quantum algebras and matrix algebras, thus advancing understanding of their homological properties.

Contribution

It develops a new spectral sequence method for Hochschild homology of smash biproducts when one algebra has low Hochschild dimension, facilitating computations for complex quantum structures.

Findings

Calculated Hochschild homology for quantum tori and quantum affine spaces.

Determined Hochschild homology for quantum complete intersections and Weyl algebras.

Extended results to quantum matrix algebra $M_q(2)$.

Abstract

We develop a new spectral sequence in order to calculate Hochschild homology of smash biproducts (also called twisted tensor products) of unital associative algebras $A\# B$ provided one of $A$ or $B$ has Hochschild dimension less than 2. We use this spectral sequence to calculate Hochschild homology of quantum tori, multiparametric quantum affine spaces, quantum complete intersections, quantum Weyl algebras, deformed completed Weyl algebras, and finally the algebra $M_q(2)$ of quantum $2\times 2$-matrices.