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JOURNALS // Uspekhi Matematicheskikh Nauk // Archive

Uspekhi Mat. Nauk, 2021 Volume 76, Issue 4(460), Pages 139–176 (Mi rm10009)

This article is cited in 13 papers

Surveys

Tetrahedron equation: algebra, topology, and integrability

D. V. Talalaevab

a Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Centre of Integrable Systems, P.G. Demidov Yaroslavl State University

Abstract: The Zamolodchikov tetrahedron equation inherits almost all the richness of structures and topics in which the Yang–Baxter equation is involved. At the same time, this transition symbolizes the growth of the order of the problem, the step from the Yang–Baxter equation to the local Yang–Baxter equation, from the Lie algebra to the 2-Lie algebra, from ordinary knots in $\mathbb{R}^3$ to 2-knots in $\mathbb{R}^4$. These transitions are followed in several examples, and there are also discussions of the manifestation of the tetrahedron equation in the long-standing question of integrability of the three-dimensional Ising model and a related model of neural network theory: the Hopfield model on a two-dimensional lattice.
Bibliography: 82 titles.

Keywords: tetrahedron equation, 2-knots, integrable models of statistical physics, Hopfield model.

UDC: 515.1+512+519.1

MSC: 16T25, 82B20

Received: 09.05.2021

DOI: 10.4213/rm10009


 English version:
Russian Mathematical Surveys, 2021, 76:4, 685–721

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© Steklov Math. Inst. of RAS, 2024