Birational automorphisms of three-dimensional algebraic varieties

1980 ◽  
Vol 13 (6) ◽  
pp. 815-868 ◽  
Author(s):  
V. A. Iskovskikh
Keyword(s):  
Three Dimensional ◽  
Algebraic Varieties ◽  
Birational Automorphisms

scholarly journals ON THE SYMMETRIES OF INTEGRABILITY

1992 ◽  
Vol 06 (11n12) ◽  
pp. 1881-1903 ◽  
Author(s):  
M. BELLON ◽  
J-M. MAILLARD ◽  
C. VIALLET
Keyword(s):  
Quantum Groups ◽  
Discrete Group ◽  
Three Dimensional ◽  
Algebraic Combinatorics ◽  
Two Dimensional ◽  
Algebraic Varieties ◽  
Spin Models ◽  
Projective Transformations ◽  
Vertex Models ◽  
Higher Dimensional

We show that the Yang-Baxter equations for two-dimensional models admit as a group of symmetry the infinite discrete group [Formula: see text]. 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 qn=1 so often mentioned in the theory of quantum groups, when no q parameter is available.

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