(Super)integrability from coalgebra symmetry: formalism and applications

TL;DR

This paper reviews the coalgebra method for constructing integrable and superintegrable classical systems, highlighting symplectic realizations, examples on curved spaces, and extensions to quantum systems.

Contribution

It introduces a comprehensive coalgebra symmetry framework for classical and quantum integrable systems, including new generalizations using comodule and loop algebras.

Findings

Many Hamiltonians exhibit Liouville superintegrability.

Superintegrable systems on curved spaces are analyzed in detail.

The coalgebra approach is extended to quantum mechanical systems.

Abstract

The coalgebra approach to the construction of classical integrable systems from Poisson coalgebras is reviewed, and the essential role played by symplectic realizations in this framework is emphasized. Many examples of Hamiltonians with either undeformed or q-deformed coalgebra symmetry are given, and their Liouville superintegrability is discussed. Among them, (quasi-maximally) superintegrable systems on N-dimensional curved spaces of nonconstant curvature are analysed in detail. Further generalizations of the coalgebra approach that make use of comodule and loop algebras are presented. The generalization of such a coalgebra symmetry framework to quantum mechanical systems is straightforward.