Non-degeneracy conditions for braided finite tensor categories

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Abstract

For a braided finite tensor category C with unit object 1∈C, Lyubashenko considered a certain Hopf algebra F∈C endowed with a Hopf pairing ω:F⊗F→1 to define the notion of a ‘non-semisimple’ modular tensor category. We say that C is non-degenerate if the Hopf pairing ω is non-degenerate. In this paper, we show that C is non-degenerate if and only if it is factorizable in the sense of Etingof, Nikshych and Ostrik, if and only if its Müger center is trivial, if and only if the linear map HomC(1,F)→HomC(F,1) induced by the pairing ω is invertible. As an application, we prove that the category of Yetter-Drinfeld modules over a Hopf algebra in C is non-degenerate if and only if C is.

Original languageEnglish
Article number106778
JournalAdvances in Mathematics
Volume355
DOIs
Publication statusPublished - 2019 Oct 15

Keywords

  • Briaded tensor category
  • Finite tensor category
  • Hopf algebra
  • Modular tensor category

ASJC Scopus subject areas

  • Mathematics(all)

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