The distinguished invertible object as ribbon dualizing object in the Drinfeld center

We prove that the Drinfeld center Z ( C ) of a pivotal finite tensor category C comes with the structure of a ribbon Grothendieck–Verdier category in the sense of Boyarchenko–Drinfeld. Phrased operadically, this makes Z ( C ) into a cyclic algebra over the framed E 2 -operad. The underlying object o...

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Bibliographic Details
Published in:Selecta mathematica (Basel, Switzerland) Vol. 30; no. 5
Main Authors: Müller, Lukas, Woike, Lukas
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01-11-2024
Springer Nature B.V
Springer Verlag
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Summary:We prove that the Drinfeld center Z ( C ) of a pivotal finite tensor category C comes with the structure of a ribbon Grothendieck–Verdier category in the sense of Boyarchenko–Drinfeld. Phrased operadically, this makes Z ( C ) into a cyclic algebra over the framed E 2 -operad. The underlying object of the dualizing object is the distinguished invertible object of C appearing in the well-known Radford isomorphism of Etingof–Nikshych–Ostrik. Up to equivalence, this is the unique ribbon Grothendieck–Verdier structure on Z ( C ) extending the canonical balanced braided structure that Z ( C ) already comes equipped with. The duality functor of this ribbon Grothendieck–Verdier structure coincides with the rigid duality if and only if C is spherical in the sense of Douglas–Schommer-Pries–Snyder. The main topological consequence of our algebraic result is that Z ( C ) gives rise to an ansular functor, in fact even a modular functor regardless of whether C is spherical or not. In order to prove the aforementioned uniqueness statement for the ribbon Grothendieck–Verdier structure, we derive a seven-term exact sequence characterizing the space of ribbon Grothendieck–Verdier structures on a balanced braided category. This sequence features the Picard group of the balanced version of the Müger center of the balanced braided category.
ISSN:1022-1824
1420-9020
DOI:10.1007/s00029-024-00975-x