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Quantum groups and integrable systems

This project was devoted to the study of the links between the theory of Hopf (or, more generally, quasi-Hopf) algebras to various domains of mathematics, particularly to the theory of Riemann surfaces and Poisson geometry. The following results were obtained by the participants of the project
1) Construction of a deformed Yangian (i.e. two-parameter deformation of the enveloping algebra of sl2[u]), and of its quantum double and R-matrix.

2) Description of the fusion relations for the eigenvalues of the quantum transfer matrices; it is proved that they coincide with Hirota's classical difference equations.

3) Introduction of the discrete moduli spaces of algebraic curves; relation between the intersection theory for these moduli spaces and the Kontsevich matrix models.

4) Construction of quasi-Hopf algebras associated with curves of higher genus.

5) Construction of Poisson brackets within the category of Loday (also called Leibniz) algebras; applications to various types of brackets: Vinogradov's, big, Frohlicher-Nijenhuis, etc.

Reported by

Centre National de la Recherche Scientifique
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