Monodromy of the Casimir connection and Coxeter categories

A Coxeter category is a braided tensor category which carries an action of a generalised braid group $B_W$ on its objects. The axiomatics 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 arerelated to the coherence of a family of fiber functors.

We will show how to construct two examples of such structure on the integrable category $\matcal{O}$ representations of a symmetrisable Kac–Moody algebra $\mathfrak{g}$, the first one arising from the quantum group $U_\hbar(\mathfrak{g})$, and the second one encoding the monodromy of the KZ and Casimir connections of $\mathfrak{g}$.

The rigidity of this structure, proved in the framework of $\mathsf{PROP}$ categories, implies in particular that the monodromy of the Casimir connection is given by the quantum Weyl group operators of $U_\hbar(\mathfrak{g})$.

This is a joint work with Valerio Toledano Laredo.