Set-theoretical solutions to the Zamolodchikov tetrahedron equation on associative rings and Liouville integrability

TL;DR

This paper constructs a family of set-theoretical solutions to the Zamolodchikov tetrahedron equation on associative rings, explaining known matrix solutions and establishing Liouville integrability for some maps.

Contribution

It introduces a new family of tetrahedron maps on associative rings, generalizing matrix solutions and providing algebraic insights into their properties.

Findings

Matrix tetrahedron maps are special cases of the new construction

Liouville integrability is proven for some of the maps

Provides an algebraic explanation for existing matrix solutions

Abstract

This paper is devoted to tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation. We construct a family of tetrahedron maps on associative rings. We show that matrix tetrahedron maps presented in [arXiv:2110.05998] are a particular case of our construction. This provides an algebraic explanation of the fact that the matrix maps from [arXiv:2110.05998] satisfy the tetrahedron equation. Also, Liouville integrability is established for some of the constructed maps.