Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation

Author:

Igonin SergeiORCID,Konstantinou-Rizos SotirisORCID

Abstract

Abstract We present several algebraic and differential-geometric constructions of tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation. In particular, we obtain a family of new (nonlinear) polynomial tetrahedron maps on the space of square matrices of arbitrary size, using a matrix refactorisation equation, which does not coincide with the standard local Yang–Baxter equation. Liouville integrability is established for some of these maps. Also, we show how to derive linear tetrahedron maps as linear approximations of nonlinear ones, using Lax representations and the differentials of nonlinear tetrahedron maps on manifolds. We apply this construction to two nonlinear maps: a tetrahedron map obtained in Dimakis and Müller-Hoissen (2019 Lett. Math. Phys. 109 799–827) in a study of soliton solutions of vector Kadomtsev–Petviashvili equations and a tetrahedron map obtained in Konstantinou-Rizos (2020 Nucl. Phys. B 960 115207) in a study of a matrix trifactorisation problem related to a Darboux matrix associated with a Lax operator for the nonlinear Schrödinger equation. We derive parametric families of new linear tetrahedron maps (with nonlinear dependence on parameters), which are linear approximations for these nonlinear ones. Furthermore, we present (nonlinear) matrix generalisations of a tetrahedron map from Sergeev’s classification Sergeev (1998 Lett. Math. Phys. 45 113–9). These matrix generalisations can be regarded as tetrahedron maps in noncommutative variables. Besides, several tetrahedron maps on arbitrary groups are constructed.

Funder

Ministry of Science and Higher Education of The Russian Federation

Russian Science Foundation

Publisher

IOP Publishing

Subject

General Physics and Astronomy,Mathematical Physics,Modeling and Simulation,Statistics and Probability,Statistical and Nonlinear Physics

Reference42 articles.

Cited by 3 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Matrix factorizations and pentagon maps;Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences;2023-12

2. Local Yang–Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation;Journal of Physics A: Mathematical and Theoretical;2023-06-14

3. Birational solutions to the set-theoretical 4-simplex equation;Physica D: Nonlinear Phenomena;2023-06

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