### Abstract

For a certain kind of tensor functor F:C→D, we define the relative modular object χ_{F}∈D as the “difference” between a left adjoint and a right adjoint of F. Our main result claims that, if C and D are finite tensor categories, then χ_{F} can be written in terms of a categorical analogue of the modular function on a Hopf algebra. Applying this result to the restriction functor associated to an extension A/B of finite-dimensional Hopf algebras, we recover the result of Fischman, Montgomery and Schneider on the Frobenius type property of A/B. We also apply our results to obtain a “braided” version and a “bosonization” version of the result of Fischman et al.

Original language | English |
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Pages (from-to) | 75-112 |

Number of pages | 38 |

Journal | Journal of Algebra |

Volume | 471 |

DOIs | |

Publication status | Published - 2017 Feb 1 |

### Keywords

- Frobenius extensions
- Frobenius functors
- Hopf algebras
- Tensor categories

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**The relative modular object and Frobenius extensions of finite Hopf algebras.** / Shimizu, Kenichi.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - The relative modular object and Frobenius extensions of finite Hopf algebras

AU - Shimizu, Kenichi

PY - 2017/2/1

Y1 - 2017/2/1

N2 - For a certain kind of tensor functor F:C→D, we define the relative modular object χF∈D as the “difference” between a left adjoint and a right adjoint of F. Our main result claims that, if C and D are finite tensor categories, then χF can be written in terms of a categorical analogue of the modular function on a Hopf algebra. Applying this result to the restriction functor associated to an extension A/B of finite-dimensional Hopf algebras, we recover the result of Fischman, Montgomery and Schneider on the Frobenius type property of A/B. We also apply our results to obtain a “braided” version and a “bosonization” version of the result of Fischman et al.

AB - For a certain kind of tensor functor F:C→D, we define the relative modular object χF∈D as the “difference” between a left adjoint and a right adjoint of F. Our main result claims that, if C and D are finite tensor categories, then χF can be written in terms of a categorical analogue of the modular function on a Hopf algebra. Applying this result to the restriction functor associated to an extension A/B of finite-dimensional Hopf algebras, we recover the result of Fischman, Montgomery and Schneider on the Frobenius type property of A/B. We also apply our results to obtain a “braided” version and a “bosonization” version of the result of Fischman et al.

KW - Frobenius extensions

KW - Frobenius functors

KW - Hopf algebras

KW - Tensor categories

UR - http://www.scopus.com/inward/record.url?scp=84988908384&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84988908384&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2016.09.017

DO - 10.1016/j.jalgebra.2016.09.017

M3 - Article

AN - SCOPUS:84988908384

VL - 471

SP - 75

EP - 112

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -