Seminar of November 8th

Representation theory and category theory are useful tools for the study of groups and Hopf algebras. In this talk, we shall introduce the background to understand the notion of characters in the context of tensor categories. This concept was introduced by K. Shimizu, who also extended many related notions and properties about classical characters to this categorical context, such as the orthogonality relations, the relation with conjugacy classes, etc.

Notice that a character of a finite group \( G\) can be viewed as a morphism of G-modules \(\chi: kG \to k\), where\(kG\) is equipped with the adjoint action. So the first step in the definition of characters for tensor categories is the generalization of the notion of adjoint algebra for arbitrary (finite) tensor categories.  This generalization plays a fundamental roll in the theory of  Lyubashenko on the action of the modular group on no-semisimple tensor categories. In order to introduce the adjoint algebra, we will first introduce the concepts of (co)monads and of (co)ends in categories.