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

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.Comment: 8 pages. arXiv admin note: substantial text overlap with arXiv:2110.0599