Local Yang-Baxter relations associated with Hirota's discrete equation

Summary

Abstract

The discrete Hirota equation is associated with infinite dimensional solutions ofthe local Yang-Baxter equation both in classical and quantum cases. The corresponding solution to the tetrahedron equation reproduces the infinite dimensional solution found recently by Sergeev et al.

Introduction

The local Yang-Baxter equation (LYBE) has been introduced by Maillet and Nijhoff as a generalization of the discrete zero curvature (Lax) relation to three dimensions. Recently Korepanov has constructed a family of (free-fermion) solutions for LYBE and has demonstrated by algebro-geometric methods integrability of the corresponding dynamical systems. The simplest solution in this family is related with the star-triangle relation (STR) in the Ising model. In an infinite dimensional solution to LYBE has been constructed. It is related to the STR in electrical networks, and the corresponding dynamical system appeared to be nothing else than Miwa's discrete equation of the BKP hierachy.

In this article we construct an infinite dimensional solution for LYBE, which directlyleads to the Hirota's discrete equation of the generalized Toda system. The quantized Hirota equation can be obtained from the straightforward q-deformation of the obtained solution. From the latter one can derive also a solution to the tetrahedron equation found by Sergeev et al.

The author is grateful to L.D. Faddeev, J.M. Maillet, A.Yu. Volkov for useful discussions.