TY - JOUR
T1 - On unimodular finite tensor categories
AU - Shimizu, Kenichi
N1 - Funding Information:
This work was supported by Grant-in-Aid for JSPS Fellows (24-3606).
Publisher Copyright:
© The Author(s) 2016. Published by Oxford University Press. All rights reserved.
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.
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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 -