On elliptic Lax systems on the lattice and a compound theorem for hyperdeterminants

Abstract

A general elliptic N × N matrix Lax scheme is presented, leading to two classes of elliptic lattice systems, one which we interpret as the higher-rank analogue of the Landau–Lifschitz equations, while the other class we characterize as the higher-rank analogue of the lattice Krichever–Novikov equation (or Adlerʼs lattice). We present the general scheme, but focus mainly on the latter type of models. In the case N = 2 we obtain a novel Lax representation of Adlerʼs elliptic lattice equation in its so-called 3-leg form. The case of rank N = 3 is analyzed using Cayleyʼs hyperdeterminant of format , yielding a multi-component system of coupled 3-leg quad-equations.