The monoidal center and the character algebra

For a pivotal finite tensor category $\mathcal{C}$ over an algebraically closed field $k$, we define the algebra $\mathsf{CF}(\mathcal{C})$ of class functions and the internal character $\mathsf{ch}(X) \in \mathsf{CF}(\mathcal{C})$ for an object $X \in \mathcal{C}$ by using an adjunction between $\mathcal{C}$ and its monoidal center $\mathcal{Z}(\mathcal{C})$. We also develop the integral theory in a unimodular finite tensor category by using the same adjunction. By utilizing these tools, we extend some results in the character theory of finite-dimensional Hopf algebras to this category-theoretical setting. Our main result is that the map $\mathsf{ch}: \mathsf{Gr}_k(\mathcal{C}) \to \mathsf{CF}(\mathcal{C})$ given by taking the internal character is a well-defined injective algebra map, where $\mathsf{Gr}_k(\mathcal{C})$ is the scalar extension of the Grothendieck ring of $\mathcal{C}$ to $k$. Moreover, under the assumption that $\mathcal{C}$ is unimodular, the map $\mathsf{ch}$ is an isomorphism if and only if $\mathcal{C}$ is semisimple. As an application, we show that the algebra $\mathsf{Gr}_{k}(\mathcal{C})$ is semisimple if $\mathcal{C}$ is a non-degenerate pivotal fusion category. If, moreover, $\mathsf{Gr}_k(\mathcal{C})$ is commutative, then the character table of $\mathcal{C}$ is defined based on the integral theory. It turns out that the character table is obtained from the $S$-matrix if $\mathcal{C}$ is a modular tensor category. Generalizing corresponding results in the finite group theory, we prove the orthogonality relations and the integrality.