Birational solutions to the set-theoretical 4-simplex equation

S. Konstantinou-Rizos

doi:10.1016/j.physd.2023.133696

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arXiv · 2211.16338

Birational solutions to the set-theoretical 4-simplex equation

S. Konstantinou-Rizos

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Abstract

The 4-simplex equation is a higher-dimensional analogue of Zamolodchikov's tetrahedron equation and the Yang--Baxter equation which are two of the most fundamental equations of mathematical physics. In this paper, we introduce a method for constructing 4-simplex maps, namely solutions to the set-theoretical 4-simplex equation, using Lax matrix refactorisation problems. Employing this method, we construct 4-simplex maps which at a certain limit give tetrahedron maps classified by Kashaev, Korepanov and Sergeev. Moreover, we construct a Kadomtsev--Petviashvili type of 4-simplex map. Finally, we introduce a method for constructing 4-simplex maps which can be restricted on level sets to parametric 4-simplex maps using Darboux transformations of integrable PDEs. We construct a nonlinear Schr\"odinger type parametric 4-simplex map which is the first parametric 4-simplex map in the literature.

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BibTeX RIS

S. Konstantinou-Rizos. 2022-11-29. Birational solutions to the set-theoretical 4-simplex equation. https://doi.org/10.1016/j.physd.2023.133696

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