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 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.