TY - JOUR
T1 - The pivotal cover and Frobenius-Schur indicators
AU - Shimizu, Kenichi
N1 - Funding Information:
The study of the “adjoint object” was greatly motivated by a talk given by Michihisa Wakui at a workshop held at Kyoto University in September 2012. The author thanks him for valuable comments. The author also thanks the referee for careful reading of the manuscript and pointing out a number of errors in the previous version. The author is supported by Grant-in-Aid for JSPS Fellows ( 24⋅3606 ).
Publisher Copyright:
© 2015 Elsevier Inc.
PY - 2015/4/5
Y1 - 2015/4/5
N2 - In this paper, we introduce the notion of the pivotal cover Cpiv of a left rigid monoidal category C to develop a theoretical foundation for the theory of Frobenius-Schur (FS) indicators in "non-pivotal" settings. For an object V∈Cpiv, the (n, r)-th FS indicator νn,r(V) is defined by generalizing that of an object of a pivotal monoidal category. This notion gives a categorical viewpoint to some recent results on generalizations of FS indicators.Based on our framework, we also study the FS indicators of the "adjoint object" in a finite tensor category, which can be considered as a generalization of the adjoint representation of a Hopf algebra. The indicators of this object closely relate to the space of endomorphisms of the iterated tensor product functor.
AB - In this paper, we introduce the notion of the pivotal cover Cpiv of a left rigid monoidal category C to develop a theoretical foundation for the theory of Frobenius-Schur (FS) indicators in "non-pivotal" settings. For an object V∈Cpiv, the (n, r)-th FS indicator νn,r(V) is defined by generalizing that of an object of a pivotal monoidal category. This notion gives a categorical viewpoint to some recent results on generalizations of FS indicators.Based on our framework, we also study the FS indicators of the "adjoint object" in a finite tensor category, which can be considered as a generalization of the adjoint representation of a Hopf algebra. The indicators of this object closely relate to the space of endomorphisms of the iterated tensor product functor.
KW - Frobenius-Schur indicators
KW - Hopf algebras
KW - Pivotal monoidal category
KW - Rigid monoidal category
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U2 - 10.1016/j.jalgebra.2015.01.014
DO - 10.1016/j.jalgebra.2015.01.014
M3 - Article
AN - SCOPUS:84922470488
SN - 0021-8693
VL - 428
SP - 357
EP - 402
JO - Journal of Algebra
JF - Journal of Algebra
ER -