Tetrahedron maps and symmetries of three dimensional integrable discrete equations

Abstract

A relationship between the tetrahedron equation for maps and the consistency property of integrable discrete equations on ${\mathbb{Z}}^3$ is investigated. Our approach is a generalization of a method developed in the context of Yang-Baxter maps, based on the invariants of symmetry groups of the lattice equations. The method is demonstrated by a case-by-case analysis of the octahedron type lattice equations classified recently, leading to some new examples of tetrahedron maps and integrable coupled lattice equations.

arXiv comment22 pages

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Cited by in corpus (11)

• Non-commutative bi-rational maps satisfying Zamolodchikov equation, and Desargues lattices
• Yang--Baxter maps, Darboux transformations, and linear approximations of refactorisation problems
• Local Yang--Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Non-Abelian hierarchies of compatible maps, associated integrable difference systems and Yang-Baxter maps
• Matrix factorizations and pentagon maps
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Birational solutions to the set-theoretical 4-simplex equation
• Boundary from bulk integrability in three dimensions: 3D reflection maps from tetrahedron maps
• Set-theoretical solutions to the Zamolodchikov tetrahedron equation on associative rings and Liouville integrability
• Noncommutative solutions to Zamolodchikov's tetrahedron equation and matrix six-factorisation problems