Superintegrability on sl(2)-coalgebra spaces

TL;DR

This paper reviews superintegrable Hamiltonian systems on sl(2)-coalgebra spaces, highlighting their algebraic structure, construction methods, and new examples, including spaces with non-constant curvature.

Contribution

It introduces a unified algebraic framework for superintegrable systems on curved spaces using sl(2) coalgebra symmetry, with new explicit N=2 examples and potential functions.

Findings

Construction of N-dimensional superintegrable systems using sl(2) coalgebra

Introduction of potentials preserving superintegrability

Explicit examples of spaces with non-constant curvature

Abstract

We review a recently introduced set of N-dimensional quasi-maximally superintegrable Hamiltonian systems describing geodesic motions, that can be used to generate "dynamically" a large family of curved spaces. From an algebraic viewpoint, such spaces are obtained through kinetic energy Hamiltonians defined on either the sl(2) Poisson coalgebra or a quantum deformation of it. Certain potentials on these spaces and endowed with the same underlying coalgebra symmetry have been also introduced in such a way that the superintegrability properties of the full system are preserved. Several new N=2 examples of this construction are explicitly given, and specific Hamiltonians leading to spaces of non-constant curvature are emphasized.