Nonabelian differential-difference integrable systems: Hamiltonian operators and double Poisson algebras

Dr Matteo Casati, University of Kent

In a joint work with JP Wang [1] we have recently presented a list of integrable nonabelian differential-difference equations, together with their Lax representation and recursion operator. As already observed in the differential case by Olver and Sokolov [2], the nonabelian counterparts of local Abelian operators are in general not local. For nonabelian systems of ODEs, the algebraic framework representing Hamiltonian equations is given by double Poisson algebras: within that context, one can obtain "Hamiltonian systems" which are defined by non Poisson (albeit "quasi-Poisson") brackets, like Kontsevich system. I will present the notion of double Poisson vertex algebra corresponding to differential-difference Hamiltonian operators and discuss its possible "quasi"-generalisation.