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

• Lukas Müller ^[1] ; Lukas Woike ^[2]
  1. [1] Perimeter Institute

     Perimeter Institute

     Canadá

     
  2. [2] Institut de Mathèmatiques de Bourgogne,Universitè de Bourgogne, Dijon, France
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 5, 2024
• Idioma: inglés
• DOI: 10.1007/s00029-024-00975-x
• Enlaces
  • Texto completo
• Resumen

  • 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 E2-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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