Techniques for classifying Hopf algebras and applications to dimension p3

Abstract

Classifying Hopf algebras of a given finite dimension n over C is a challenging problem. If n is p, p^2, 2p, or 2p^2 with p prime, the classification is complete. If n = p^3, the semisimple and the pointed Hopf algebras are classified and much progress on the remaining cases was made by the second author but the general classification is still open. Here we outline some results and techniques which have been useful in approaching this problem and add a few new ones. We give some further results on Hopf algebras of dimension p^3 and finish the classification for dimension 27.