Integrals for Finite Tensor Categories

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Abstract

We introduce the notions of categorical integrals and categorical cointegrals of a finite tensor category (Formula presented.) by using a certain adjunction between (Formula presented.) and its Drinfeld center (Formula presented.). These notions can be identified with integrals and cointegrals of a finite-dimensional Hopf algebra H if (Formula presented.) is the representation category of H. We generalize basic results on integrals and cointegrals of a finite-dimensional Hopf algebra (such as the existence, the uniqueness, and the Maschke theorem) to finite tensor categories. Motivated by results of Lorenz, we also investigate relations between categorical integrals and morphisms factoring through projective objects. Finally, we extend the n-th indicator of a finite-dimensional Hopf algebra introduced by Kashina, Montgomery and Ng to finite tensor categories.

Original languageEnglish
Pages (from-to)1-35
Number of pages35
JournalAlgebras and Representation Theory
DOIs
Publication statusAccepted/In press - 2018 Mar 13

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Keywords

  • Drinfeld center
  • Finite tensor category
  • Integrals of Hopf algebras

ASJC Scopus subject areas

  • Mathematics(all)

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