N=2 Hamiltonians with sl(2) coalgebra symmetry and their integrable deformations

TL;DR

This paper explores how $sl(2)$ coalgebra symmetry facilitates the construction of integrable deformations of classical systems, including potentials and Calogero models, highlighting the role of coalgebra invariance in integrability.

Contribution

It introduces a method to derive integrable deformations of classical systems from $sl(2)$ Poisson coalgebras and their $q$-deformations, including new potentials and models.

Findings

Calogero system is $sl(2)$ coalgebra invariant

Jordan-Schwinger realization derived from non-coassociative coproduct

Coalgebra symmetry enables straightforward construction of integrable deformations

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.