Hamiltonian Structures for Integrable Nonabelian Difference Equations

In this paper we extensively study the notion of Hamiltonian structure for nonabelian differential-difference systems, exploring link between different algebraic (in terms double Poisson algebras and vertex algebras) geometric bivectors) definitions. We introduce multiplicative (PVAs) as suitable noncommutative counterpart to PVAs, used describe equations in commutative setting, prove that these are one-to-one correspondence with structures defined by difference operators, providing a sufficient condition fulfilment Jacobi identity. Moreover, define polyvector fields their Schouten brackets, both finitely generated infinitely ones: allows us provide unified characterisation bivectors quasi-Poisson algebra structures. Finally, an application obtain some results towards classification local scalar construct Kaup, Ablowitz-Ladik Chen-Lee-Liu integrable lattices.