Mathematics Colloquia and Seminars

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Description

In this talk, I will describe how quantum groups serve as a useful means of expressing the monodromy of certain integrable, first order PDE's. A fundamental, and paradigmatic result in this context is the Kohno-Drinfeld theorem. Roughly speaking, it asserts that the representations of Artin's braid groups on n strings given by the universal R-matrix of a quantum group describe the monodromy of the Knizhnik-Zamolodchikov (KZ) equations, a flat connection on the configuration space on n points in the complex plane.

I will describe an analogue of the Kohno-Drinfeld theorem which I recently proved, and had been independently conjectured by De Concini and myself. In this analogue, the R-matrix representations are replaced by the quantum Weyl group representations constructed by Lusztig, Kirillov-Reshetikhin and Soibelman. Accordingly, the KZ equations are replaced by the flat connection on generalised configuration spaces associated with root systems which I constructed in collaboration with J. Millson.