Dynamical quantum groups, deformation quantization of Lie bialgebroids and integrable systems in mathematical physics
Start date: Oct 1, 2007,
End date: Sep 30, 2008
The main goal of this proposal is two-fold: Study the quantization problem for Lie bialgebroids on one hand and pay attention to many examples coming from physics which are globally identified as dynamical quantum groups on the other hand. We take this problem as a pretext to develop a general mathematical theory of dynamical equations like the Yang-Baxter; one that first appeared in the work of Felder on integrable systems and conformal field theory. This will be the occasion to explicit the precise links with some other mathematical objects from category theory (especially tensor and module categories), geometry (especially Lie algebroids and groupoids) and mathematical physics (especially integrable systems).The two leading purposes we will keep in mind are:- the study of Knizhnik-Zamolodchikov-Bernard equations (coming from physics) and their relations with some group theoretical problems on one hand and number theory on the other hand. Here the classical dynamical Yang-Baxter equation is equivalent to the compatibility condition for these equations.- Deformation quantization of integrable systems and Lie bialgebroids coming from classical dynamical r-matrices and relations with the quantization of dynamical r-matrices by dynamical twists.
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