On Hopf algebras of dimension \(4p\). (Q536189)

Let \(k\) be an algebraically closed field of characteristic zero and let \(p\) and \(q\) be distinct prime numbers. The classification of Hopf algebras of dimension \(pq^2\) over \(k\) is an open problem. It is known however in the semisimple case, after results of \textit{P. Etingof, D. Nikshych} and \textit{V. Ostrik} [Adv. Math. 226, No. 1, 176-205 (2011; Zbl 1210.18009)] (see also the reviewer's [Algebr. Represent. Theory 7, No. 2, 173-188 (2004; Zbl 1053.16030)] and references therein). It is also known in the pointed non-semisimple case, where the classification was given in the paper [Tsukuba J. Math. 25, No. 1, 187-201 (2001; Zbl 0998.16026)], by \textit{N. Andruskiewitsch} and the reviewer; in particular, for some of the Hopf algebras appearing in this list, the dual Hopf algebra is not pointed and not semisimple. In the paper under review the authors show that a non-semisimple Hopf algebra of dimension \(4p\) over \(k\), where \(p\) is an odd prime, containing more than two group-like elements is necessarily pointed, that is, every simple comodule is one-dimensional. This is used to prove that in dimensions 20, 28 and 44 every non-semisimple Hopf algebra is either pointed or its dual is pointed, thus completing the classification in these dimensions. Their proofs are based on a series of results on the dimensions of projective modules over a finite-dimensional Hopf algebra and its connections with the powers of the antipode, and also on results on braided Hopf algebras over the (non-semisimple) four-dimensional Sweedler Hopf algebra \(H_4\), which are related to the classification problem considered in the paper through the Radford-Majid biproduct construction. In the last context, it is shown that a \(p\)-dimensional braided Hopf algebra over \(H_4\) is semisimple and cosemisimple, and necessary and sufficient conditions for commutativity are given. Based on these results and the classification of Hopf algebras of dimension \(2p^2\) obtained in the paper [J. Lond. Math. Soc., II. Ser. 80, No. 2, 295-310 (2009; Zbl 1187.16029)] by \textit{M. Hilgemann} and the second-named author, the authors conjecture that any non-semisimple Hopf algebra of dimension \(pq^2\) over \(k\) is either pointed or its dual is pointed.