Classification of Hopf algebras of dimension 18. (Q1001421)

Let \(k\) be an algebraically closed field of characteristic zero. The classification of semisimple Hopf algebras of dimension \(18\) over \(k\) was done by \textit{A. Masuoka} [Commun. Algebra 24, No. 1, 307-329 (1996; Zbl 0845.16037)]. In this paper the author proves that a non-semisimple Hopf algebra of dimension \(18\) over \(k\) is either a pointed Hopf algebra or the dual of a pointed Hopf algebra. Pointed non-semisimple Hopf algebras of dimension \(pq^2\), where \(p\neq q\) are prime numbers, were classified by \textit{N. Andruskiewitsch} and the reviewer [in Tsukuba J. Math. 25, No. 1, 187-201 (2001; Zbl 0998.16026)]. Therefore, the main result of this paper concludes the full classification of Hopf algebras of dimension \(18\).