### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 022, 37 pages      arXiv:1708.07024      https://doi.org/10.3842/SIGMA.2018.022

### Poisson Algebras and 3D Superintegrable Hamiltonian Systems

Allan P. Fordy a and Qing Huang b
a) School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
b) School of Mathematics, Northwest University, Xi'an 710069, People's Republic of China

Received August 24, 2017, in final form March 06, 2018; Published online March 16, 2018

Abstract
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.

Key words: Hamiltonian system; super-integrability; Poisson algebra; conformal algebra; constant curvature.

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