Smash products for -sets, Clifford theory and duality theorems.
Let and be the Sweedler’s and Kac-Paljutkin Hopf algebras, respectively. We prove that any Hopf algebra which factorizes through and (equivalently, any bicrossed product between the Hopf algebras and ) must be isomorphic to one of the following four Hopf algebras: . The set of all matched pairs is explicitly described, and then the associated bicrossed product is given by generators and relations.
Let denote the twisted smash product of an arbitrary algebra A and a Hopf algebra H over a field. We present an analogue of the celebrated Blattner-Montgomery duality theorem for , and as an application we establish the relationship between the homological dimensions of and A if H and its dual H* are both semisimple.