Pointed Hopf algebras of dimension 32

Abstract

We give a complete classification of the 32-dimensional pointed Hopf algebras over an algebraically closed field k with characteristic different from 2. It turns out that there are infinite families of isomorphism classes of pointed Hopf algebras of dimension 32. In [Andruskiewitsch-Schneider, J. Alg 209], [Beattie-Dascalescu-Grunenfelder, Invent. Math. 136 (1)] and [Gelaki, J. Alg 209] are given families of counterexamples for the tenth Kaplansky conjecture. Up to now, 32 is the lowest dimension where Kaplansky conjecture fails.