The distinguished invertible object as ribbon dualizing object in the Drinfeld center
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 structure described here is a traditional ribbon structure. This is the case iff \ is \textit{spherical} in the sense of [\textit{C. L. Douglas} et al., Dualizable tensor categories. Providence, RI: American Mathematical Society (AMS) (2021; Zbl 1514.57001)], in which case the above main result reduces to an answer to the above question given by \textit{K. Shimizu} [Kodai Math. J. 46, No. 1, 75--114 (2023; Zbl 1528.18018)].\N\NIt is also shown that\N\NTheorem. For any ribbon Grothendieck-Verdier category \(\mathcal{A}\) in the symmetric monoidal bicategory \textsf{Lex}\(^{\mathsf{f}}\) of finite categories, left exact functors and natural transformations, there is an exact sequence\N\begin{multline*}\N1 \longrightarrow\mathsf{cAut}_{\otimes}(\mathrm{id}_{\mathcal{A} })\longrightarrow\mathsf{Aut}_{\otimes}(\mathrm{id} _{\mathcal{A}})\longrightarrow \mathsf{Aut}_{\mathcal{A}}(I)\longrightarrow\mathsf{cAut} _{\mathsf{f}E_{2}}(\mathcal{A})\longrightarrow\\\N\mathsf{Aut} _{\mathsf{f}E_{2}}(\mathcal{A})\longrightarrow \mathsf{Pic}(\mathsf{Z}_{2}^{\mathsf{bal}}(\mathcal{A}))\longrightarrow\mathsf{RibGV}(\mathcal{A})\longrightarrow1\N\end{multline*}\Nof groups, except for the last stage where it is an exact sequence of pointed sets. In the above, \(\mathsf{Aut}_{\mathsf{f}E_{2}}(\mathcal{A})\) is the group of isomorphism classes of balanced braided automorphisms of \(\mathcal{A}\), and \(\mathsf{Aut}_{\otimes}(\mathrm{id}_{\mathcal{A} })\) is the group of monoidal isomorphisms of the identity of \(\mathcal{A}\), while \(\mathsf{cAut}_{\mathsf{f}E_{2}}(\mathcal{A} )\) and \(\mathsf{cAut}_{\otimes}(\mathrm{id}_{\mathcal{A} })\) are cyclic analogs of these groups that are additionally compatible with the Grothendieck-Verdier duality.
scientific article; zbMATH DE number 7932302
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