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 |
|---|---|
| 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.
In: International Mathematics Research Notices, Vol. 2017, No. 1, 2017, p. 277-322.Research output: Contribution to journal › Article
}
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
AN - SCOPUS:85014471090
VL - 2017
SP - 277
EP - 322
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
SN - 1073-7928
IS - 1
ER -