Local Yang--Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation

Abstract

We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation, and their matrix Lax representations defined by the local Yang--Baxter equation. Sergeev [S.M. Sergeev 1998 Lett. Math. Phys. 45, 113--119] presented classification results on three-dimensional tetrahedron maps obtained from the local Yang--Baxter equation for a certain class of matrix-functions in the situation when the equation possesses a unique solution which determines a tetrahedron map. In this paper, using correspondences arising from the local Yang--Baxter equation for some simple $2\times 2$ matrix-functions, we show that there are (non-unique) solutions to the local Yang--Baxter equation which define tetrahedron maps that do not belong to the Sergeev list; this paves the way for a new, wider classification of tetrahedron maps. We present invariants for the derived tetrahedron maps and prove Liouville integrability for some of them. Furthermore, using the approach of solving correspondences arising from the local Yang--Baxter equation, we obtain several new birational tetrahedron maps, including maps with matrix Lax representations on arbitrary groups, a $9$-dimensional map associated with a Darboux transformation for the derivative nonlinear Schrödinger (NLS) equation, and a $9$-dimensional generalisation of the $3$-dimensional Hirota map.

arXiv comment18 pages. New results added (section 4), and also the references list was updated

References in corpus (19)

• Zamolodchikov's Tetrahedron Equation and Hidden Structure of Quantum Groups
• Functional Tetrahedron Equation
• Lax matrices for Yang-Baxter maps
• Permutation-type solutions to the Yang-Baxter and other n-simplex equations
• Darboux transformations, finite reduction groups and related Yang-Baxter maps
• Simplex and Polygon Equations
• On non-multiaffine consistent-around-the-cube lattice equations
• Quantum geometry of 3-dimensional lattices
• Reduction groups and related integrable difference systems of NLS type
• On non-abelian quadrirational Yang-Baxter maps
• A noncommutative discrete potential KdV lift
• Tetrahedron maps and symmetries of three dimensional integrable discrete equations
• Non-commutative bi-rational maps satisfying Zamolodchikov equation, and Desargues lattices
• Poisson Yang-Baxter maps with binomial Lax matrices
• On the $3D$ consistency of a Grassmann extended lattice Boussinesq system
• Cohomologies of $n$-simplex relations
• Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Set-theoretical solutions of simplex equations
• Boundary from bulk integrability in three dimensions: 3D reflection maps from tetrahedron maps