Hopf algebras of dimension \(pq\). (Q1883066)

Let \(p\leq q\) be prime numbers, and let \(k\) be an algebraically closed field of characteristic zero. It is conjectured that a Hopf algebra of dimension \(pq\) over \(k\) is always semisimple or pointed; and thus necessarily semisimple if \(p\neq q\). The classification of semisimple Hopf algebras of dimension \(p^2\) was given by \textit{A. Masuoka} [in Proc. Am. Math. Soc. 124, No. 3, 735-737 (1996; Zbl 0848.16033)]. Later [in Lect. Notes Pure Appl. Math. 237, 193-201 (2004; Zbl 1063.16047)], the author of the paper under review proved the above mentioned conjecture for the case \(p=q\), thus giving the full classification in these cases. The conjecture remains an open problem in the general case. In this paper, the author shows that the trace of \({\mathcal S}^{2p}\), for a non-semisimple Hopf algebra of dimension \(pq\) over \(k\), equals \(p^2d\), where \(d=pq\pmod 4\). This allows to conclude the proof that every Hopf algebra of dimension \(pq\), where \(p\) and \(q\) are twin prime numbers, is necessarily semisimple, and therefore commutative or cocommutative, by results of Etingof, Gelaki and Westreich. Partial results of the general conjecture have also been obtained in papers by \textit{M. Beattie} and \textit{S. Dăscălescu} [J. Lond. Math. Soc., II. Ser. 69, No. 1, 65-78 (2004; Zbl 1060.16034)], \textit{P. Etingof} and \textit{S. Gelaki} [J. Algebra 277, No. 2, 668-674 (2004; Zbl 1068.16054)], and the reviewer [Bull. Lond. Math. Soc. 34, No. 3, 301-307 (2002; Zbl 1042.16027)].