Various structures associated to the representation categories of eight-dimensional nonsemisimple Hopf algebras.
type: Item
instance of: scholarly article
title: Various structures associated to the representation categories of eight-dimensional nonsemisimple Hopf algebras.
zbMATH Open document ID: 1088.16028
author: Michihisa Wakui
published in: Algebras and Representation Theory
publication date: 2005-04-07T00:00:00Z
review text: 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 (1988; unpublished)], and later by \textit{D. Ştefan} [J. Algebra 211, No. 1, 343-361 (1999; Zbl 0918.16031)] with more conceptual methods.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)].
reviewed by: Sonia Natale
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 2153665
zbMATH Keywords: Hopf algebras
zbMATH Keywords: Green rings
zbMATH Keywords: tensor categories
zbMATH Keywords: quasitriangular structures
zbMATH Keywords: ribbon structures
zbMATH Keywords: monoidal equivalences
zbMATH Keywords: categories of representations
MaRDI profile type: Publication
DOI: 10.1023/B:ALGE.0000048318.64013.DB
description: scientific article; zbMATH DE number 2153665