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Abstract:
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.
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