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.

💡 Research Summary

This paper establishes a discrete variational identity that works for semi‑direct sums of Lie algebras equipped with a general non‑degenerate bilinear form. In the continuous setting, Hamiltonian structures of integrable systems are often obtained from the trace identity; in the discrete case, the trace is replaced by a bilinear form ⟨·,·⟩ that must satisfy two invariance properties: ⟨