The monoidal center and the character algebra

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) is an element of CF(C) for an object X is an element of 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 : Gr(k)(C) -> CF(C) given by taking the internal character is a well-defined injective homomorphism of k-algebras, where Gr(k)(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 Gr(k)(C) is semisimple if C is a non degenerate pivotal fusion category. If, moreover, Gr(k) (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. (C) 2016 Elsevier B.V. All rights reserved.