The matrix realization of affine Jacobi varieties and the extended Lotka-Volterra lattice

TL;DR

This paper establishes a connection between integrable Hamiltonian systems and algebraic geometry by representing affine Jacobi varieties through matrix realizations, and applies this to prove integrability of the extended Lotka-Volterra lattice.

Contribution

It constructs an explicit isomorphism between gauge equivalence classes of polynomial matrices and affine Jacobi varieties, linking algebraic geometry with integrable systems.

Findings

Constructed an isomorphism to affine Jacobi varieties.

Proved algebraic complete integrability of the extended Lotka-Volterra lattice.

Linked monodromy matrices to algebraic curves and their Jacobians.

Abstract

We study completely integrable Hamiltonian systems whose monodromy matrices are related to the representatives for the set of gauge equivalence classes $\boldsymbol{\mathcal{M}}_F$ of polynomial matrices. Let $X$ be the algebraic curve given by the common characteristic equation for $\boldsymbol{\mathcal{M}}_F$. We construct the isomorphism from the set of representatives to an affine part of the Jacobi variety of $X$. This variety corresponds to the invariant manifold of the system, where the Hamiltonian flow is linearized. As the application, we discuss the algebraic completely integrability of the extended Lotka-Volterra lattice with a periodic boundary condition.