Hamiltonian Forms for a Hierarchy of Discrete Integrable Coupling Systems

A semi-direct sum of two Lie algebras of four-by-fourmatrices is presented, and a discrete four-by-four matrix spectral problemis introduced. A hierarchy of discrete integrable coupling systemsis derived. The obtained integrable coupling systems are all written intheir Hamiltonian forms by the discrete variational identity. Finally, weprove that the lattice equations in the obtained integrable coupling systemsare all Liouville integrable discrete Hamiltonian systems.