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In this talk, we provide a description of the quantum groups as analytic objects, natural receptacles for the monodromy of certain systems of differential equations arising in Lie theory.

We will first review the Drinfeld–Kohno theorem, describing the monodromy of the Knizhnik–Zamolodchikov equations associated to a simple Lie algebra in terms of the universal R–matrix of the corresponding quantum group. We will then present an extension of this result, providing a description of the monodromy of the Casimir equations associated to a simple Lie algebra (in fact, to any symmetrisable Kac–Moody algebra) in terms of the quantum Weyl group operators of the corresponding quantum group. The proof relies on the notion of generalised braided category (aka quasi–Coxeter category), which is to a generalised braid group what a braided monoidal category is to the standard braid group on n strands (this is a joint work with V. Toledano Laredo).