[Paper Review] N=2 Hamiltonians with sl(2) coalgebra symmetry and their integrable deformations

Ángel Ballesteros, O. Ragnisco|ArXiv.org|Oct 18, 1999

Nonlinear Waves and Solitons5 references3 citations

TL;DR

This paper constructs two-dimensional classical integrable Hamiltonians with sl(2) coalgebra symmetry using phase space realizations of classical and q-deformed sl(2) Poisson coalgebras. It demonstrates that such symmetry enables a systematic derivation of integrable deformations, recovering generalized Morse, oscillator, and centrifugal potentials, and shows the N=2 Calogero system and Gaudin Hamiltonian are sl(2)-symmetric via Jordan-Schwinger realization with a non-coassociative coproduct.

ABSTRACT

Two dimensional classical integrable systems and different integrable deformations for them are derived from phase space realizations of classical $sl(2)$ Poisson coalgebras and their $q-$deformed analogues. Generalizations of Morse, oscillator and centrifugal potentials are obtained. The N=2 Calogero system is shown to be $sl(2)$ coalgebra invariant and the well-known Jordan-Schwinger realization can be also derived from a (non-coassociative) coproduct on $sl(2)$. The Gaudin Hamiltonian associated to such Jordan-Schwinger construction is presented. Through these examples, it can be clearly appreciated how the coalgebra symmetry of a hamiltonian system allows a straightforward construction of different integrable deformations for it.

Motivation & Objective

• To establish a systematic framework for constructing integrable Hamiltonians using sl(2) coalgebra symmetry in phase space.
• To explore how classical and q-deformed sl(2) Poisson coalgebras generate new integrable systems and their deformations.
• To demonstrate that coalgebra symmetry enables straightforward construction of integrable deformations for known systems.
• To show that the N=2 Calogero system and its associated Gaudin Hamiltonian arise from a non-coassociative coproduct on sl(2).
• To generalize standard potentials (Morse, oscillator, centrifugal) through coalgebraic realizations.

Proposed method

• Realizing classical and q-deformed sl(2) Poisson coalgebras in phase space to generate Hamiltonians.
• Using the Jordan-Schwinger construction with a non-coassociative coproduct to derive the N=2 Calogero system.
• Applying the coalgebra coproduct structure to systematically deform integrable systems.
• Deriving generalized potentials (Morse, oscillator, centrifugal) as realizations of the coalgebra symmetry.
• Constructing the Gaudin Hamiltonian from the same Jordan-Schwinger realization framework.
• Verifying integrability via the coalgebra symmetry and explicit construction of conserved quantities.

Experimental results

Research questions

• RQ1 How can sl(2) coalgebra symmetry be used to generate new integrable Hamiltonians in two dimensions?
• RQ2 What role does the q-deformation of the sl(2) Poisson coalgebra play in constructing integrable deformations?
• RQ3 Can the N=2 Calogero system be derived from a non-coassociative coproduct on sl(2)?
• RQ4 How does the Jordan-Schwinger realization relate to the Gaudin Hamiltonian in the context of sl(2) coalgebra symmetry?
• RQ5 What generalized potentials emerge from phase space realizations of sl(2) coalgebras?

Key findings

• The N=2 Calogero system is shown to be invariant under sl(2) coalgebra symmetry, confirming its integrability through this algebraic structure.
• Generalized Morse, oscillator, and centrifugal potentials are derived as realizations of the sl(2) Poisson coalgebra in phase space.
• The Jordan-Schwinger realization of the N=2 Calogero system arises from a non-coassociative coproduct on sl(2), extending standard constructions.
• The Gaudin Hamiltonian associated with this realization is explicitly constructed, linking it to the same coalgebraic framework.
• The sl(2) coalgebra symmetry provides a unified mechanism for generating integrable deformations of known systems.
• The q-deformed analogues of the sl(2) Poisson coalgebra yield new integrable deformations, demonstrating the robustness of the approach.