On the Symmetries of Integrability

TL;DR

This paper uncovers the symmetry groups underlying integrability in two- and three-dimensional models, revealing how these symmetries explain spectral parameters and aid in solving complex equations like Yang-Baxter and tetrahedron equations.

Contribution

It identifies infinite discrete symmetry groups for these models, including a new hyperbolic Coxeter group for three-dimensional cases, advancing understanding of integrability and model parametrization.

Findings

Identifies $A_2^{(1)}$ as symmetry group for 2D Yang-Baxter equations.

Discovers a larger hyperbolic Coxeter group symmetry for 3D models.

Provides a framework for solving and parametrizing integrable models.

Abstract

We show that the Yang-Baxter equations for two dimensional models admit as a group of symmetry the infinite discrete group $A_2^{(1)}$. The existence of this symmetry explains the presence of a spectral parameter in the solutions of the equations. We show that similarly, for three-dimensional vertex models and the associated tetrahedron equations, there also exists an infinite discrete group of symmetry. Although generalizing naturally the previous one, it is a much bigger hyperbolic Coxeter group. We indicate how this symmetry can help to resolve the Yang-Baxter equations and their higher-dimensional generalizations and initiate the study of three-dimensional vertex models. These symmetries are naturally represented as birational projective transformations. They may preserve non trivial algebraic varieties, and lead to proper parametrizations of the models, be they integrable or not. We mention the relation existing between spin models and the Bose-Messner algebras of algebraic combinatorics. Our results also yield the generalization of the condition $q^n=1$ so often mentioned in the theory of quantum groups, when no $q$ parameter is available.