A discrete variational identity on semi-direct sums of Lie algebras

The discrete variational identity under general bilinear forms on semi-direct sums of Lie algebras is established. The constant $\gamma$ involved in the variational identity is determined through the corresponding solution to the stationary discrete zero curvature equation. An application of the resulting variational identity to a class of semi-direct sums of Lie algebras in the Volterra lattice case furnishes Hamiltonian structures for the associated integrable couplings of the Volterra lattice hierarchy.