Local Yang--Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation

TL;DR

This paper explores new solutions to the Zamolodchikov tetrahedron equation via local Yang--Baxter correspondences, expanding classification, and establishing integrability of certain tetrahedron maps with applications to nonlinear Schrödinger equations.

Contribution

It introduces non-Sergeev solutions to the local Yang--Baxter equation, broadening the classification of tetrahedron maps and providing new integrable maps with matrix Lax representations.

Findings

Discovered non-Sergeev tetrahedron maps from simple 2x2 matrix functions.

Proved Liouville integrability for some derived tetrahedron maps.

Constructed new birational tetrahedron maps, including those related to the NLS equation.

Abstract

We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation, and their matrix Lax representations defined by the local Yang--Baxter equation. Sergeev [S.M. Sergeev 1998 Lett. Math. Phys. 45, 113--119] presented classification results on three-dimensional tetrahedron maps obtained from the local Yang--Baxter equation for a certain class of matrix-functions in the situation when the equation possesses a unique solution which determines a tetrahedron map. In this paper, using correspondences arising from the local Yang--Baxter equation for some simple $2\times 2$ matrix-functions, we show that there are (non-unique) solutions to the local Yang--Baxter equation which define tetrahedron maps that do not belong to the Sergeev list; this paves the way for a new, wider classification of tetrahedron maps. We present invariants for the derived tetrahedron maps and prove Liouville integrability for some of them. Furthermore, using the approach of solving correspondences arising from the local Yang--Baxter equation, we obtain several new birational tetrahedron maps, including maps with matrix Lax representations on arbitrary groups, a $9$-dimensional map associated with a Darboux transformation for the derivative nonlinear Schr\"odinger (NLS) equation, and a $9$-dimensional generalisation of the $3$-dimensional Hirota map.