6 Dec 2002, U1, 14:30-16:00.

 Vladimir Roubtsov (Université d'Angers, France).

"Classical and Quantum Commuting Families in Skew Fields and their Applications".

A simple and straightforward construction of commuting element families in skew fields of tensor degrees of associative (Poisson) algebras is proposed. This construction can be considered as an algebraic version of separation of variables in a broad class of quantum (classical) integrable systems.

Among them are the so-called Beaville systems -- the Lagrangian foliations of (blowed up) symmetric product of projective surfaces, whose fibers are abelian varieties. The systems include the (deformation of) Hitchin systems, Toda-like systems, geodesic flows (Neumann systems) and generalizations of the Calogero-Moser-Ruijsenaars systems.

For all these systems our main theorem implies some classical integrability result (involutivity in the sense of Arnold-Liouville theorem). We give an explicit quantization for a special Beaville system associated to the cone of canonical curve in terms of (pseudo-) differential operators.