### Abstract

Let C be a finite tensor category with simple unit object, let Z(C) denote its monoidal center, and let L and R be a left adjoint and a right adjoint of the forgetful functor U: Z(C) → C. We show that the following conditions are equivalent: (1) C is unimodular, (2) U is a Frobenius functor, (3) L preserves the duality, (4) R preserves the duality, (5) L(1) is self-dual, and (6) R(1) is self-dual, where 1 ∈ C is the unit object. We also give some other equivalent conditions. As an application, we give a categorical understanding of some topological invariants arising from finite-dimensional unimodular Hopf algebras.

Original language | English |
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Pages (from-to) | 277-322 |

Number of pages | 46 |

Journal | International Mathematics Research Notices |

Volume | 2017 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2017 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**On unimodular finite tensor categories.** / Shimizu, Kenichi.

Research output: Contribution to journal › Article

*International Mathematics Research Notices*, vol. 2017, no. 1, pp. 277-322. https://doi.org/10.1093/imrn/rnv394

}

TY - JOUR

T1 - On unimodular finite tensor categories

AU - Shimizu, Kenichi

PY - 2017

Y1 - 2017

N2 - Let C be a finite tensor category with simple unit object, let Z(C) denote its monoidal center, and let L and R be a left adjoint and a right adjoint of the forgetful functor U: Z(C) → C. We show that the following conditions are equivalent: (1) C is unimodular, (2) U is a Frobenius functor, (3) L preserves the duality, (4) R preserves the duality, (5) L(1) is self-dual, and (6) R(1) is self-dual, where 1 ∈ C is the unit object. We also give some other equivalent conditions. As an application, we give a categorical understanding of some topological invariants arising from finite-dimensional unimodular Hopf algebras.

AB - Let C be a finite tensor category with simple unit object, let Z(C) denote its monoidal center, and let L and R be a left adjoint and a right adjoint of the forgetful functor U: Z(C) → C. We show that the following conditions are equivalent: (1) C is unimodular, (2) U is a Frobenius functor, (3) L preserves the duality, (4) R preserves the duality, (5) L(1) is self-dual, and (6) R(1) is self-dual, where 1 ∈ C is the unit object. We also give some other equivalent conditions. As an application, we give a categorical understanding of some topological invariants arising from finite-dimensional unimodular Hopf algebras.

UR - http://www.scopus.com/inward/record.url?scp=85014471090&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85014471090&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnv394

DO - 10.1093/imrn/rnv394

M3 - Article

VL - 2017

SP - 277

EP - 322

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 1

ER -