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 (1998 Lett. Math. Phys. 45 113–9) 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 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 with Lax representations and invariants, including maps on arbitrary groups, a nine-dimensional map associated with a Darboux transformation for the derivative nonlinear Schrödinger equation, and a nine-dimensional generalisation of the three-dimensional Hirota map.