A remark on the Hochschild dimension of liberated quantum groups

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.