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It is known that the local Yang--Baxter equation is a generator of potentialsolutions to Zamolodchikov's tetrahedron equation. In this paper, we show underwhich additional conditions the solutions to the local Yang--Baxter equationare tetrahedron maps, namely solutions to the set-theoretical tetrahedronequation. This is exceptionally useful when one wants to prove thatnoncommutative maps satisfy the Zamolodchikov's tetrahedron equation. Weconstruct new noncommutative maps and we prove that they possess thetetrahedron property. Moreover, by employing Darboux transformations withnoncommutative variables, we derive noncommutative tetrahedron maps. Inparticular, we derive a noncommutative nonlinear Schr\"odinger type oftetrahedron map which can be restricted to a noncommutative version ofSergeev's map on invariant leaves. We prove that these maps are tetrahedronmaps.

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