The distinguished invertible object as ribbon dualizing object in the Drinfeld center
type: Item
instance of: scholarly article
title: The distinguished invertible object as ribbon dualizing object in the Drinfeld center
zbMATH Open document ID: 1548.1802
author: Lukas Müller
author: Lukas Woike
published in: Selecta Mathematica. New Series
publication date: 2024-10-28T00:00:00Z
review text: For any finite tensor category \(\mathcal{C}\), one can define the so-called Drinfeld center \(Z(\mathcal{C})\) as the category of pairs \((X,\beta)\), where \(X\) is an object in \(\mathcal{C}\) and \(\beta\) is a \textit{half braiding}, so that \ becomes a braided finite tensor category. A pivotal structure on \(\mathcal{C}\) gives rise to a pivotal structure on \(Z(\mathcal{C})\) and hence to a balancing, i.e. a natural automorphism \(\theta\) of \(\mathrm{id}_{Z(\mathcal{C})}\) such that \(\theta_{I}=\mathrm{id}_{I}\) and\N\[\N\theta_{X\otimes Y}=c_{Y,X}c_{X,Y}(\theta_{X}\otimes\theta_{Y})\N\]\Nthe for \(X,Y\in Z(\mathcal{C})\), where \(c\) denotes the braiding on \(Z(\mathcal{C})\). This is referred to as the \textit{canonical balanced braided structure} on the Drinfeld center of a pivotal finite tensor category.\N\NThis paper is concerned with the question when the Drinfeld center of a pivotal finite tensor category is ribbon [\textit{M. Müger}, Rev. Unión Mat. Argent. 51, No. 1, 95--163 (2010; Zbl 1215.18007), \S 6]. The main result is the following theorem answering the question.\N\NTheorem. Let \(\mathcal{C}\) be a pivotal finite tensor category. Then the distinguished invertible object of \(\mathcal{C}\) equipped with the half braiding induced by the Radford isomorphism of \textit{P. Etingof} et al. [Int. Math. Res. Not. 2004, No. 54, 2915--2933 (2004; Zbl 1079.16024)] and the pivotal structure of \(\mathcal{C}\) is a dualizing object that makes \(Z(\mathcal{C})\) a ribbon Grothendieck-Verdier category in the sense of [\textit{M. Boyarchenko} and \textit{V. Drinfeld}, Quantum Topol. 4, No. 4, 447--489 (2013; Zbl 1370.18007)]. Up to equivalence, this is the only ribbon Grothendieck-Verdier on \(Z(\mathcal{C})\) that extends the canonical balanced braided structure.\N\NEven though the authors change the notion of duality from the traditional rigid duality to a Grothendieck-Verdier duality, they retain control over when exactly the ribbon Grothendieck-Verdier…
reviewed by: Hirokazu Nishimura
Mathematics Subject Classification ID: 18M20
Mathematics Subject Classification ID: 18M85
zbMATH DE Number: 7932302
MaRDI profile type: Publication
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DOI: 10.1007/S00029-024-00975-X
description: scientific article; zbMATH DE number 7932302