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