Andrea Appel (University of Southern California), Geometric Methods in Representation Theory

Title: Monodromy of the Casimir connection and Coxeter categories

Abstract: A Coxeter category is a braided tensor category which carries an action of a generalised braid group B_w on its objects. The axioms of a Coxeter category and the data defining the action of B_W are similar in flavor to the associativity and commutativity constraints in a monoidal category, but are related to the coherence of a family of fiber functors. We will show how to construct two examples of such structure on the integrable category O representations of a symmetrisable Kac–Moody algebra g, the first one arising from the associated quantum group, and the second one encoding the monodromy of the KZ and Casimir connections of g. The rigidity of this structure implies in particular that the monodromy of the Casimir connection is given by the quantum Weyl group operators. This is a joint work with Valerio Toledano Laredo.