TY - JOUR

T1 - The monoidal center and the character algebra

AU - Shimizu, Kenichi

N1 - Funding Information:
The author thanks Victor Ostrik and Makoto Yamashita for letting him know about [26] and [25], respectively, and giving him some valuable comments. The author also thanks Takahiro Hayashi for valuable comments and suggestions. The author appreciates the referee for careful reading of the manuscript. The author is currently supported by JSPS KAKENHI Grant Number JP16K17568.
Publisher Copyright:
© 2016 Elsevier B.V.

PY - 2017/9/1

Y1 - 2017/9/1

N2 - For a pivotal finite tensor category C over an algebraically closed field k, we define the algebra CF(C) of class functions and the internal character ch(X)∈CF(C) for an object X∈C by using an adjunction between C and its monoidal center Z(C). We also develop the theory of integrals and the Fourier transform in a unimodular finite tensor category by using the same adjunction. Our main result is that the map ch:Grk(C)→CF(C) given by taking the internal character is a well-defined injective homomorphism of k-algebras, where Grk(C) is the scalar extension of the Grothendieck ring of C to k. Moreover, under the assumption that C is unimodular, the map ch is an isomorphism if and only if C is semisimple. As an application, we show that the algebra Grk(C) is semisimple if C is a non-degenerate pivotal fusion category. If, moreover, Grk(C) is commutative, then we define the character table of C based on the integral theory. It turns out that the character table is obtained from the S-matrix if C is a modular tensor category. Generalizing corresponding results in the finite group theory, we prove the orthogonality relations and the integrality of the character table.

AB - For a pivotal finite tensor category C over an algebraically closed field k, we define the algebra CF(C) of class functions and the internal character ch(X)∈CF(C) for an object X∈C by using an adjunction between C and its monoidal center Z(C). We also develop the theory of integrals and the Fourier transform in a unimodular finite tensor category by using the same adjunction. Our main result is that the map ch:Grk(C)→CF(C) given by taking the internal character is a well-defined injective homomorphism of k-algebras, where Grk(C) is the scalar extension of the Grothendieck ring of C to k. Moreover, under the assumption that C is unimodular, the map ch is an isomorphism if and only if C is semisimple. As an application, we show that the algebra Grk(C) is semisimple if C is a non-degenerate pivotal fusion category. If, moreover, Grk(C) is commutative, then we define the character table of C based on the integral theory. It turns out that the character table is obtained from the S-matrix if C is a modular tensor category. Generalizing corresponding results in the finite group theory, we prove the orthogonality relations and the integrality of the character table.

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U2 - 10.1016/j.jpaa.2016.12.037

DO - 10.1016/j.jpaa.2016.12.037

M3 - Article

AN - SCOPUS:85009200747

SN - 0022-4049

VL - 221

SP - 2338

EP - 2371

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

IS - 9

ER -