Invariant classification of second-order conformally flat superintegrable systems

Abstract

In this paper we continue the work of Kalnins et al in classifying all second-order conformally-superintegrable (Laplace-type) systems over conformally flat spaces, using tools from algebraic geometry and classical invariant theory. The results obtained show, through Staeckel equivalence, that the list of known nondegenerate superintegrable systems over three-dimensional conformally flat spaces is complete. In particular, a 7-dimensional manifold is determined such that each point corresponds to a conformal class of superintegrable systems. This manifold is foliated by the nonlinear action of the conformal group in three-dimensions. Two systems lie in the same conformal class if and only if they lie in the same leaf of the foliation. This foliation is explicitly described using algebraic varieties formed from representations of the conformal group. The proof of these results rely heavily on Groebner basis calculations using the computer algebra software packages Maple and Singular.

References in corpus (8)

• Completeness of superintegrability in two-dimensional constant curvature spaces
• Superintegrability in a two-dimensional space of non-constant curvature
• Nondegenerate 3D complex Euclidean superintegrable systems and algebraic varieties
• Classical ladder operators, polynomial Poisson algebras and classification of superintegrable systems
• Algebraic Integrability Conditions for Killing Tensors on Constant Sectional Curvature Manifolds
• The Variety of Integrable Killing Tensors on the 3-Sphere
• Quadratic algebra contractions and 2nd order superintegrable systems
• Superintegrability in a non-conformally-flat space

Cited by in corpus (22)

• Invariant Classification and Limits of Maximally Superintegrable Systems in 3D
• Racah Polynomials and Recoupling Schemes of $\mathfrak{su}(1,1)$
• Toward a classification of semidegenerate 3D superintegrable systems
• Superintegrable Systems on 3 Dimensional Conformally Flat Spaces
• Bôcher Contractions of Conformally Superintegrable Laplace Equations
• An Algebraic Geometric Foundation for a Classification of Superintegrable Systems in Arbitrary Dimension
• Algebraic Conditions for Conformal Superintegrability in Arbitrary Dimension
• Superintegrable quantum mechanical systems with position dependent masses invariant with respect to two parametric Lie groups
• (Super-)integrable systems associated to 2-dimensional projective connections with one projective symmetry
• Toward Classification of 2nd Order Superintegrable Systems in 3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent Symmetry Operators
• Stäckel Equivalence of Non-Degenerate Superintegrable Systems, and Invariant Quadrics
• Abundant Superintegrable Systems and Hessian Structures
• Projectively equivalent 2-dimensional superintegrable systems with projective symmetries
• Examples of Complete Solvability of 2D Classical Superintegrable Systems
• Separation equations for 2D superintegrable systems on constant curvature spaces
• Torsion-free connections of second-order maximally superintegrable systems
• Dynamical symmetry algebra of two superintegrable two-dimensional systems
• Integrable and superintegrable quantum mechanical systems with position dependent masses invariant with respect to one parametric Lie groups. 2. Systems with dilatation and shift symmetries
• Laplace equations, conformal superintegrability and Bôcher contractions
• Superintegrability of 3-dimensional Hamiltonian systems with conformally Euclidean metrics. Oscillator-related and Kepler-related systems
• Conformal Laplace superintegrable systems in 2D: polynomial invariant subspaces
• Superintegrability on the 3-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the Sphere $S^3$ and on the Hyperbolic space $H^3$