  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.
