The monoidal center and the character algebra

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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 languageEnglish
JournalJournal of Pure and Applied Algebra
DOIs
Publication statusAccepted/In press - 2016 Jan 19

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ASJC Scopus subject areas

  • Algebra and Number Theory

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