### 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 language | English |
---|---|

Pages (from-to) | 1-35 |

Number of pages | 35 |

Journal | Algebras and Representation Theory |

DOIs | |

Publication status | Accepted/In press - 2018 Mar 13 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Algebras and Representation Theory*, 1-35. https://doi.org/10.1007/s10468-018-9777-5

**Integrals for Finite Tensor Categories.** / Shimizu, Kenichi.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Integrals for Finite Tensor Categories

AU - Shimizu, Kenichi

PY - 2018/3/13

Y1 - 2018/3/13

N2 - 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.

AB - 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.

KW - Drinfeld center

KW - Finite tensor category

KW - Integrals of Hopf algebras

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

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

U2 - 10.1007/s10468-018-9777-5

DO - 10.1007/s10468-018-9777-5

M3 - Article

SP - 1

EP - 35

JO - Algebras and Representation Theory

JF - Algebras and Representation Theory

SN - 1386-923X

ER -