Exact Factorizations of Rank-One Pointed Hopf Algebras in Positive Characteristic, I

Abstract

We classify exact factorizations of the third-type rank-one pointed Hopf algebras over an algebraically closed field of positive characteristic. The main step is a complete classification of matched pairs between such an algebra and a group algebra. It turns out that the group-like part must form a matched pair of finite groups, while the only possible action on the skew-primitive generator is a shift by a scalar multiple of $1-g$, where $g$ is the distinguished group-like element, encoded by a single group homomorphism. The bicrossed product is again a third-type rank-one pointed Hopf algebra, and we give a necessary and sufficient condition for two such products to be isomorphic as Hopf algebras. Consequently, up to interchanging the two factors, exact factorizations of a fixed third-type algebra correspond bijectively to exact factorizations of its underlying finite group, with the distinguished group-like element lying in the rank-one factor. As an application, all matched pairs between the Radford algebra and cyclic group algebras are classified, and the corresponding exact factorizations are determined explicitly when the cyclic group has prime order.