Various structures associated to the representation categories of eight-dimensional nonsemisimple Hopf algebras.
Hopf algebras of dimension \(\leq 11\) over an algebraically closed field of characteristic zero were classified by \textit{R. Williams} [Ph.D. Thesis, Florida State Univ. (1988; unpublished)], and later by \textit{D. Ştefan} [J. Algebra 211, No. 1, 343-361 (1999; Zbl 0918.16031)] with more conceptual methods. Among these, the only nonsemisimple examples occur in dimensions 4, 8 and 9, and in dimension 8 they are either pointed or dual to pointed. In this paper the author determines the quasitriangular and ribbon structures on the nonsemisimple Hopf algebras of dimension 8. This problem was solved in the semisimple case by \textit{S. Suzuki} [Tsukuba J. Math. 22, No. 1, 1-29 (1998; Zbl 0939.16024)] and the author [RIMS Kokyuroku 1057, 41-53 (1998; Zbl 0940.20021)]. The author also determines the Green rings and discusses monoidal equivalences between the categories of representations. In the semisimple case, this last question was first solved by the results of \textit{D. Tambara} and \textit{S. Yamagami} [J. Algebra 209, No. 2, 692-707 (1998; Zbl 0923.46052)].
scientific article; zbMATH DE number 2153665