Infinite discrete symmetry group for the Yang-Baxter equations. Vertex models

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Abstract We show that the Yang-Baxter equations for two-dimensional vertex 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 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 symmetries. Although generalizing very naturally the previous one, this is a much bigger hyperbolic Coxeter group. We indicate how this symmetry should be used to resolve the Yang-Baxter equations and their higher-dimensional generalizations and initiate the study of a family of three-dimensional vertex models.