(Super)integrability from coalgebra symmetry: formalism and applications

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.

33 pages. Review-contribution to the "Workshop on higher symmetries in Physics", 6-8 November 2008, Madrid, Spain

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• From constants of motion to superposition rules for Lie-Hamilton systems
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• Lie--Hamilton systems: theory and applications
• Superintegrable systems with spin induced by coalgebra symmetry
• Integrable deformations of Rössler and Lorenz systems from Poisson-Lie groups
• Racah Algebra $R(n)$ from Coalgebraic Structures and Chains of $R(3)$ Substructures
• Poisson-Hopf algebra deformations of Lie-Hamilton systems
• Higher Order Quantum Superintegrability: a new "Painlevé conjecture"
• Exact solutions and superposition rules for Hamiltonian systems generalizing time-dependent SIS epidemic models with stochastic fluctuations
• Multisymplectic structures and invariant tensors for Lie systems
• Coalgebra symmetry for discrete systems
• The ${\mathfrak{sl}}_2(\mathbb{R})$ coalgebra symmetry and the superintegrable discrete-time systems
• Poisson-Hopf deformations of Lie-Hamilton systems revisited: deformed superposition rules and applications to the oscillator algebra
• Poisson-Lie groups, bi-Hamiltonian systems and integrable deformations
• Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system
• Massless geodesics in $Ad{S}_5\times Y(p,q)$ as a superintegrable system
• Classification of real three-dimensional Poisson-Lie groups
• Integrable perturbations of the N-dimensional isotropic oscillator
• Discrete-time systems in quasi-standard form and the ${\mathfrak{h}}_6$ coalgebra symmetry
• Quantum group deformation of the Kittel--Shore model
• Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum
• Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras
• Curvature as an integrable deformation
• Solutions by quadratures of complex Bernoulli differential equations and their quantum deformation
• Non-coboundary Poisson-Lie structures on the book group