PIRSA:16090044

Monodromy of the Casimir connection and Coxeter categories

Appel, A. (2016). Monodromy of the Casimir connection and Coxeter categories. Perimeter Institute. https://pirsa.org/16090044

Appel, Andrea. Monodromy of the Casimir connection and Coxeter categories. Perimeter Institute, Sep. 08, 2016, https://pirsa.org/16090044 

          @misc{ pirsa_PIRSA:16090044,
            doi = {10.48660/16090044},
            url = {https://pirsa.org/16090044},
            author = {Appel, Andrea},
            keywords = {Mathematical physics},
            language = {en},
            title = {Monodromy of the Casimir connection and Coxeter categories},
            publisher = {Perimeter Institute},
            year = {2016},
            month = {sep},
            note = {PIRSA:16090044 see, \url{https://pirsa.org}}
          }

Andrea Appel

University of Southern California

Talk number
PIRSA:16090044
Collection
Talk Type
Subject
Abstract

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.