scholarly journals Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties

2007 ◽  
Vol 48 (11) ◽  
pp. 113518 ◽  
Author(s):  
E. G. Kalnins ◽  
J. M. Kress ◽  
W. Miller
Keyword(s):  
Three Dimensional ◽  
Algebraic Varieties ◽  
Superintegrable Systems

scholarly journals Carter's constant and superintegrability

2018 ◽  
Vol 27 (07) ◽  
pp. 1850066
Author(s):  
Payel Mukhopadhyay ◽  
K. Rajesh Nayak
Keyword(s):  
Dynamical System ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Integrals Of Motion ◽  
Superintegrable System ◽  
Simple Observation ◽  
Superintegrable Systems ◽  
Axisymmetric Spacetime ◽  
Three Degrees Of Freedom

Carter's constant is a nontrivial conserved quantity of motion of a particle moving in stationary axisymmetric spacetime. In the version of the theorem originally given by Carter, due to the presence of two Killing vectors, the system effectively has two degrees of freedom. We propose an extension to the first version of Carter's theorem to a system having three degrees of freedom to find two functionally independent Carter-like integrals of motion. We further generalize the theorem to a dynamical system with [Formula: see text] degrees of freedom. We further study the implications of Carter's constant to superintegrability and present a different approach to probe a superintegrable system. Our formalism gives another viewpoint to a superintegrable system using the simple observation of separable Hamiltonian according to Carter's criteria. We then give some examples by constructing some two-dimensional superintegrable systems based on this idea and also show that all three-dimensional simple classical superintegrable potentials are also Carter separable.

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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