A remark on the Hochschild dimension of liberated quantum groups

TL;DR

This paper investigates the Hochschild dimension of certain quantum groups, showing that under specific conditions, their third Hochschild homology group is non-trivial, revealing new algebraic properties.

Contribution

It establishes a link between the structure of Hopf algebras with projections onto semisimple groups and their Hochschild homology, highlighting conditions for non-trivial third Hochschild homology.

Findings

Third Hochschild homology group is non-trivial under given conditions.

Non-trivial G-module structure induces non-trivial Hochschild homology.

Results connect quantum group structure with Hochschild homological properties.

Abstract

Let $A$ be a Hopf algebra equipped with a projection onto the coordinate Hopf algebra $\mathcal{O}(G)$ of a semisimple algebraic group $G$. It is shown that if $A$ admits a suitably non-degenerate comodule $V$ and the induced $G$-module structure of $V$ is non-trivial, then the third Hochschild homology group of $A$ is non-trivial.