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

TL;DR

This paper establishes a discrete variational identity for semi-direct sums of Lie algebras, linking it to Hamiltonian structures in integrable systems like the Volterra lattice.

Contribution

It introduces a new variational identity applicable to semi-direct sums of Lie algebras and connects it to integrable couplings and Hamiltonian structures.

Findings

Derived the discrete variational identity using bilinear forms.

Connected the constant γ to solutions of the stationary discrete zero curvature equation.

Applied the identity to obtain Hamiltonian structures for Volterra lattice couplings.

Abstract

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.