Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation

TL;DR

This paper introduces new algebraic and geometric methods to construct set-theoretical solutions to the Zamolodchikov tetrahedron equation, including polynomial, linear, and noncommutative maps, with applications to integrable systems.

Contribution

It presents novel constructions of tetrahedron maps, including polynomial, linear, and noncommutative versions, and establishes their integrability and connections to soliton equations.

Findings

New polynomial tetrahedron maps on matrix spaces.

Linear approximations of nonlinear tetrahedron maps.

Matrix generalizations in noncommutative variables.

Abstract

We present several algebraic and differential-geometric constructions of tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation. In particular, we obtain a family of new (nonlinear) polynomial tetrahedron maps on the space of square matrices of arbitrary size, using a matrix refactorisation equation, which does not coincide with the standard local Yang--Baxter equation. Liouville integrability is established for some of these maps. Also, we show how to derive linear tetrahedron maps as linear approximations of nonlinear ones, using Lax representations and the differentials of nonlinear tetrahedron maps on manifolds. We apply this construction to two nonlinear maps: a tetrahedron map obtained in [arXiv:1708.05694] in a study of soliton solutions of vector KP equations and a tetrahedron map obtained in [arXiv:2005.13574] in a study of a matrix trifactorisation problem related to a Darboux matrix associated with a Lax operator for the NLS equation. We derive parametric families of new linear tetrahedron maps (with nonlinear dependence on parameters), which are linear approximations for these nonlinear ones. Furthermore, we present (nonlinear) matrix generalisations of a tetrahedron map from Sergeev's classification [arXiv:solv-int/9709006]. These matrix generalisations can be regarded as tetrahedron maps in noncommutative variables. Besides, several tetrahedron maps on arbitrary groups are constructed.