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

TL;DR

This paper explores conditions under which solutions to the local Yang--Baxter equation become tetrahedron maps, constructs new noncommutative tetrahedron maps, and demonstrates their properties, including a noncommutative nonlinear Schr"odinger type map.

Contribution

It introduces new noncommutative tetrahedron maps derived via Darboux transformations and establishes their tetrahedron property, advancing understanding of noncommutative solutions to Zamolodchikov's equation.

Findings

Constructed new noncommutative tetrahedron maps.

Proved these maps satisfy the tetrahedron property.

Derived a noncommutative nonlinear Schr"odinger type tetrahedron map.

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 Schr\"odinger 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.