Techniques for classifying Hopf algebras and applications to dimension p^3

TL;DR

This paper reviews techniques for classifying finite-dimensional Hopf algebras, presents new results, and completes the classification for dimension 27, advancing understanding in nonsemisimple cases.

Contribution

It introduces new methods and completes the classification of Hopf algebras of dimension 27, extending previous partial results.

Findings

Classification of Hopf algebras of dimension 27 completed.

New techniques for analyzing Hopf algebra structures introduced.

Partial classification results for dimension p^3 discussed.

Abstract

The classification of all Hopf algebras of a given finite dimension over an algebraically closed field of characteristic 0 is a difficult problem. If the dimension is a prime, then the Hopf algebra is a group algebra. If the dimension is the square of a prime then the Hopf algebra is a group algebra or a Taft Hopf algebra. The classification is also complete for dimension 2p or 2p^2, p a prime. Partial results for some other cases are available. For example, for dimension p^3 the classification of the semisimple Hopf algebras was done by Masuoka, and the pointed Hopf algebras were classified by Andruskiewitsch and Schneider, Caenepeel and Dascalescu, and Stefan and van Oystaeyen independently. Many classification results for the nonsemisimple, nonpointed, non-copointed case have been proved by the second author but the classification in general for dimension p^3 is still incomplete, up to now even for dimension 27. In this paper 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.