Extensions of a near-group category of type \((\mathbb{Z}_2;1)\)
type: Item
instance of: scholarly article
title: Extensions of a near-group category of type \((\mathbb{Z}_2;1)\)
author: Yanyan Li
published in: Acta Mathematica Hungarica
publication date: 2022-11-09T00:00:00Z
review text: The theory of fusion categories arises in many areas of mathematics and physics such as semisimple Hopf algebras, quantum groups, vertex operator algebras and topological quantum field theory. Group extensions play a significant role in the theory of fusion categories. This paper is concerned with near-group categories of type \(\left( \mathbb{Z}_{2},1\right)\).\par The principal objective in this paper is to establish the following two theorems.\par I. Let \(\mathcal{D}\)\ be a near-group fusion category of type \(\left(\mathbb{Z}_{2},1\right) \)\ and \[\mathcal{C}={\bigoplus\nolimits_{g\in G}}\mathcal{C}_{g}\] be an extension of \(\mathcal{D}\). Assume that \(\mathcal{C}\)\ is braided. Then the structure of \(\mathcal{C}\) is exactly one of the following:\par (1) If \(\mathcal{C}\) is of type \(\left( 1,2m;2,m\right) \), then \(\mathcal{C}\)\ is a \(\mathbb{Z}_{2}\)-equivariantization of a braided pointed fusion category \(\mathcal{D}\)\ of dimension \(3m\).\par (2) If \(\mathcal{C}\) is of type \(\left( 1,2m;\sqrt{3},2m;2,m\right) \), then \(\mathcal{C}\)\ is a \(\mathbb{Z}_{2}\)-equivariantization of a fusion category \(\mathcal{D}\)\ which is of type \(\left( 1,3m;\sqrt{3},m\right) \) and is a \(\mathbb{Z}_{2}\)-extension of a pointed fusion category of dimension \(3m\). In particular, \(\mathcal{D}\)\ is not braided.\par (3) If \(\mathcal{C}\) is of type \(\left( 1,2m;2,m;\sqrt{6},m\right) \), then \(\mathcal{C}\)\ is a \(\mathbb{Z}_{2}\)-equivariantization of a fusion category \(\mathcal{D}\)\ which is of type \(\left( 1,3m;\sqrt{6},m/2\right) \) and is a \(\mathbb{Z}_{2}\)-extension of a pointed fusion category of dimension \(3m\). In particular, \(\mathcal{D}\)\ is not braided.\par II. Let \(\mathcal{D}\)\ be a near-group fusion category of type \(\left(\mathbb{Z}_{2},1\right) \)\ and \[\mathcal{C}={\bigoplus\nolimits_{g\in G}}\mathcal{C}_{g}\] be an extension of \(\mathcal{D}\). Then \(\mathcal{C}\) is group-theoretical in one of the following forms:\par (…
reviewed by: Hirokazu Nishimura
Mathematics Subject Classification ID: 18M20
Mathematics Subject Classification ID: 16T05
zbMATH DE Number: 7614054
zbMATH Keywords: fusion category
zbMATH Keywords: Hopf algebra
zbMATH Keywords: extension
zbMATH Keywords: near-group category
zbMATH Keywords: group-theoretical fusion category
MaRDI profile type: Publication
full work available at URL: s10474-022-01256-9
OpenAlex ID: W4296087114
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DOI: 10.1007/S10474-022-01256-9
description: scientific article; zbMATH DE number 7614054