TY - JOUR

T1 - Integrals for Finite Tensor Categories

AU - Shimizu, Kenichi

N1 - Funding Information:
The author is grateful to the referee for careful reading of the manuscript. The author is supported by JSPS KAKENHI Grant Number 16K17568.

PY - 2019/4/15

Y1 - 2019/4/15

N2 - We introduce the notions of categorical integrals and categorical cointegrals of a finite tensor category C by using a certain adjunction between C and its Drinfeld center Z(C). These notions can be identified with integrals and cointegrals of a finite-dimensional Hopf algebra H if C 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.

AB - We introduce the notions of categorical integrals and categorical cointegrals of a finite tensor category C by using a certain adjunction between C and its Drinfeld center Z(C). These notions can be identified with integrals and cointegrals of a finite-dimensional Hopf algebra H if C 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.

KW - Drinfeld center

KW - Finite tensor category

KW - Integrals of Hopf algebras

UR - http://www.scopus.com/inward/record.url?scp=85043699139&partnerID=8YFLogxK

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U2 - 10.1007/s10468-018-9777-5

DO - 10.1007/s10468-018-9777-5

M3 - Article

AN - SCOPUS:85043699139

SN - 1386-923X

VL - 22

SP - 459

EP - 493

JO - Algebras and Representation Theory

JF - Algebras and Representation Theory

IS - 2

ER -