Noncommutative solutions to Zamolodchikov's tetrahedron equation and matrix six-factorisation problems

Abstract

It is known that the local Yang--Baxter equation is a generator of potential solutions to Zamolodchikov's tetrahedron equation. In this paper, we show under which additional conditions the solutions to the local Yang--Baxter equation are tetrahedron maps, namely solutions to the set-theoretical tetrahedron equation. This is exceptionally useful when one wants to prove that noncommutative maps satisfy the Zamolodchikov's tetrahedron equation. We construct new noncommutative maps and we prove that they possess the tetrahedron property. Moreover, by employing Darboux transformations with noncommutative variables, we derive noncommutative tetrahedron maps. In particular, we derive a noncommutative nonlinear Schrödinger type of tetrahedron map which can be restricted to a noncommutative version of Sergeev's map on invariant leaves. We prove that these maps are tetrahedron maps.

18 pages, 2 figures. Revised version, published in Physica D

References in corpus (6)

• On non-abelian quadrirational Yang-Baxter maps
• Burgers and KP hierarchies: A functional representation approach
• Poisson structures for lifts and periodic reductions of integrable lattice equations
• Relativistic Collisions as Yang-Baxter maps
• A discrete Darboux-Lax scheme for integrable difference equations
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations