On pointed Hopf algebras and Kaplansky's 10th conjecture

Abstract

In this paper we construct and study two new families of finite dimensional pointed Hopf algebras which generalize Radford&#39;s families. We show that over any infinite field which contains a primitive nth root of unity, one of the families contains infinitely many non-isomorphic Hopf algebras of any dimension of the form Nn^2, where 2<n<N are integers so that n divides N. We thus answer in the negative Kaplansky&#39;s 10th conjecture from 1975 on the finite number of types of Hopf algebras of a given dimension.