### Abstract

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.

Original language | English |
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Journal | Journal of Pure and Applied Algebra |

DOIs | |

Publication status | Accepted/In press - 2016 Jan 19 |

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**The monoidal center and the character algebra.** / Shimizu, Kenichi.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - The monoidal center and the character algebra

AU - Shimizu, Kenichi

PY - 2016/1/19

Y1 - 2016/1/19

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.

UR - http://www.scopus.com/inward/record.url?scp=85009200747&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85009200747&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2016.12.037

DO - 10.1016/j.jpaa.2016.12.037

M3 - Article

AN - SCOPUS:85009200747

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

ER -