Tensor categories with fusion rules of self-duality for finite abelian groups
type: Item
instance of: scholarly article
title: Tensor categories with fusion rules of self-duality for finite abelian groups
zbMATH Open document ID: 0923.46052
author: D. Tambara
author: Shigeru Yamagami
published in: Journal of Algebra
publication date: 1999-10-15T00:00:00Z
review text: Split semisimple tensor categories are a special class of monoidal categories. They have associated an algebra which is called the fusion algebra of the given category. Let \(A\) be a finite group, \(S\) the disjoint union of \(A\) with the element \(m\) and \({\mathbb Z}[S]\) the algebra obtained by extension of the group algebra \({\mathbb Z}[G]\) whose product satisfies \(m^2 = \sum_{a\in A}a\) and \(am = ma = m\) for all \(a\in A\). The main result of this paper classifies split semisimple tensor categories over a field \(K\) whose fusion algebra is \({\mathbb Z}[S]\). As an application is obtained the existence of four classes of semisimple tensor categories over \({\mathbb C}\) when \(A = {\mathbb Z}_2 \oplus {\mathbb Z}_2 \). Three of these are realized as the class of the category of representations of Hopf algebras of dimension \(8\) (dihedral group algebra, quaternion group algebra and the Kac and Paljutkin algebra). So these Hopf algebras have the same fusion algebra but different classes of categories of representations.
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 18D10
zbMATH DE Number: 1247622
zbMATH Keywords: Hopf algebra
zbMATH Keywords: representation
zbMATH Keywords: monoidal category
zbMATH Keywords: fusion algebra
zbMATH Keywords: dihedral group algebra
zbMATH Keywords: quaternion group algebra
zbMATH Keywords: Kac and Paljutkin algebra
reviewed by: José Antonio Cuenca Mira
MaRDI profile type: Publication
full work available at URL: jabr.1998.7558
OpenAlex ID: W2018308928
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DOI: 10.1006/JABR.1998.7558
description: scientific article; zbMATH DE number 1247622