Hopf algebras of dimension 12 (Q1862139)

In this article, the author concludes the classification of Hopf algebras of dimension \(12\) over \(k\), an algebraically closed field of characteristic zero. Semisimple Hopf algebras of dimension \(12\) were classified by \textit{N. Fukuda} [in Tsukuba J. Math. 21, No. 1, 43-54 (1997; Zbl 0897.16021)]. The classification of pointed nonsemisimple Hopf algebras of dimension \(12\) follows from \textit{N. Andruskiewitsch} and \textit{S. Natale} [Tsukuba J. Math. 25, No. 1, 187-201 (2001; Zbl 0998.16026)]. The main result in this paper is that either \(H\) or \(H^*\) is pointed for a nonsemsimple Hopf algebra \(H\) of dimension \(12\) over \(k\). In view of this result and Andruskiewitsch and Natale's and Fukuda's conclusions, the full classification of Hopf algebras of dimension \(12\) over \(k\) follows.