A quasi-Coxeter algebra is a bialgebra which carries actions of a given generalised braid group and of Artin’s braid groups on the tensor products of its representations. A basic example is the quantum group U_h(g) endowed with Lusztig’s quantum Weyl group elements and its universal R-matrix.

I will explain how to construct such a structure on the enveloping algebra Ug of a semisimple Lie algebra g, which accounts for the monodromy of the Casimir and KZ connections of g. An interesting ingredient of this construction is the use of irregular singularities to produce a (transcendental) twist which kills Drinfeld’s KZ associator.

Given that quasi-Coxeter structures on Ug are rigid, this implies in particular that the monodromy of the rational Casimir connection of g is described by the quantum Weyl group operators of U_h(g).

Joint work with Andrea Appel extends these results to the case of an arbitrary Kac-Moody algebra.