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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Published in: | Selecta mathematica (Basel, Switzerland) Vol. 30; no. 5 |
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Main Authors: | , |
Format: | Journal Article |
Language: | English |
Published: |
Cham
Springer International Publishing
01-11-2024
Springer Nature B.V Springer Verlag |
Subjects: | |
Online Access: | Get full text |
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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. |
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ISSN: | 1022-1824 1420-9020 |
DOI: | 10.1007/s00029-024-00975-x |