Constructing nonlinear discrete integrable Hamiltonian couplings

Beginning with Lax pairs from special non-semisimple matrix Lie algebras, we establish a scheme for constructing nonlinear discrete integrable couplings. Discrete variational identities over the associated loop algebras are used to build Hamiltonian structures for the resulting integrable couplings. We illustrate the application of the scheme by means of an enlarged Volterra spectral problem and present an example of nonlinear discrete integrable Hamiltonian couplings for the Volterra lattice equations.