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

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.

arXiv comment8 pages. arXiv admin note: substantial text overlap with arXiv:2110.05998

References in corpus (8)

• Zamolodchikov's Tetrahedron Equation and Hidden Structure of Quantum Groups
• Functional Tetrahedron Equation
• Darboux transformations, finite reduction groups and related Yang-Baxter maps
• Tetrahedron maps and symmetries of three dimensional integrable discrete equations
• Non-commutative bi-rational maps satisfying Zamolodchikov equation, and Desargues lattices
• Cohomologies of $n$-simplex relations
• Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Boundary from bulk integrability in three dimensions: 3D reflection maps from tetrahedron maps

Cited by in corpus (1)

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