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

Abstract

We present several algebraic and differential-geometric constructions of tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation. In particular, we obtain a family of new (nonlinear) polynomial tetrahedron maps on the space of square matrices of arbitrary size, using a matrix refactorisation equation, which does not coincide with the standard local Yang--Baxter equation. Liouville integrability is established for some of these maps. Also, we show how to derive linear tetrahedron maps as linear approximations of nonlinear ones, using Lax representations and the differentials of nonlinear tetrahedron maps on manifolds. We apply this construction to two nonlinear maps: a tetrahedron map obtained in [arXiv:1708.05694] in a study of soliton solutions of vector KP equations and a tetrahedron map obtained in [arXiv:2005.13574] in a study of a matrix trifactorisation problem related to a Darboux matrix associated with a Lax operator for the NLS equation. We derive parametric families of new linear tetrahedron maps (with nonlinear dependence on parameters), which are linear approximations for these nonlinear ones. Furthermore, we present (nonlinear) matrix generalisations of a tetrahedron map from Sergeev's classification [arXiv:solv-int/9709006]. These matrix generalisations can be regarded as tetrahedron maps in noncommutative variables. Besides, several tetrahedron maps on arbitrary groups are constructed.

21 pages; v3: minor corrections in Example 5.12, references updated

References in corpus (8)

• Yang-Baxter maps: dynamical point of view
• Simplex and Polygon Equations
• Quantum geometry of 3-dimensional lattices
• Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Set-theoretical solutions of simplex equations
• 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

Cited by in corpus (4)

• Local Yang--Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Matrix factorizations and pentagon maps
• Birational solutions to the set-theoretical 4-simplex equation
• Set-theoretical solutions to the Zamolodchikov tetrahedron equation on associative rings and Liouville integrability