## Poisson algebras and 3D superintegrable Hamiltonian systems.(English)Zbl 1416.17011

Summary: Using a Poisson bracket representation, in 3D, of the Lie algebra $$\mathfrak{sl}(2)$$, we first use highest weight representations to embed this into larger Lie algebras. These are then interpreted as symmetry and conformal symmetry algebras of the “kinetic energy”, related to the quadratic Casimir function. We then consider the potentials which can be added, whilst remaining integrable, leading to families of separable systems, depending upon arbitrary functions of a single variable. Adding further integrals, in the superintegrable case, restricts these functions to specific forms, depending upon a finite number of arbitrary parameters. The Poisson algebras of these superintegrable systems are studied. The automorphisms of the symmetry algebra of the kinetic energy are extended to the full Poisson algebra, enabling us to build the full set of Poisson relations.

### MSC:

 17B63 Poisson algebras 17B80 Applications of Lie algebras and superalgebras to integrable systems 37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics 70G65 Symmetries, Lie group and Lie algebra methods for problems in mechanics 70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
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