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Noncommutative Integrable Systems

Le : 11/01/2016 16h15
Par : Semeon Arthamonov (Rutgers et ITEP, Moscou)
Lieu : I 001
Lien web :
Résumé : One of the main advantages of the Hamilton formalism is that it reduces the problem of finding the first integrals of ODE (conserved quantities) to the purely algebraic problem of finding certain maximal Poisson-commuting subalgebra of algebra of smooth functions on a manifold. It is natural to ask what happens if we use the idea of noncommutative geometry and replace the commutative algebra of smooth functions on a manifold with some abstract (in general noncommutative) algebra. Recently Kontsevich proposed an example of noncommutative ODE on the group algebra of free group with two generators and conjectured that it should be integrable in certain sense. Based on my research I will provide an explicit example of the modified double Poisson bracket, Casimir elements, Lax elements and infinite family of generalized Hamiltonians which generate commuting flows on this associative algebra. If time permits I will describe the relation to Liouville integrability and quantum integrable systems. In particular, I will show that any modified double Poisson bracket defines a usual Poisson bracket on the moduli space of representations of the underlying associative algebra. Generalized Hamiltonians provide a family of Hamiltonians in involution.