On the Geometric Origin of the Bi-Hamiltonian Structure of the Calogero–Moser System

Abstract

We show that the bi-Hamiltonian structure of the rational n-particle (attractive) Calogero-Moser system can be obtained by means of a double projection from a very simple Poisson pair on the cotangent bundle of gl(n, R). The relation with the Lax formalism is also discussed. Remark 3.

Reduction of a bi-Hamiltonian hierarchy on $$T^*\mathrm{U}(n)$$ to spin Ruijsenaars–Sutherland models

“…An interesting open problem that stems from our work is that the global structure of the full reduced phase space should be explored in the future, dropping the restriction to M reg ⊂ M. The issue of possible generalizations of the bi-Hamiltonian structure to the elliptic case and for other Lie groups should be also investigated. Finally, we wish to mention the question whether there is any relation between our results and the earlier studies [4,12] of a bi-Hamiltonian structure for the rational, spinless Calogero model.…”

Poisson–Nijenhuis structures on quiver path algebras

“…The path algebra of the quiver with two loops provides the natural noncommutative counterpart of Calogero-Moser phase space, as shown in [15]. In § 3.1 we define a noncommutative ωN structure on this path algebra and prove that it induces -first on the representation spaces, then on the quotient space -the ωN structures used in [3].…”

“…In a number of important cases -e.g. for the Calogero-Moser system [3] the manifold M is a cotangent bundle, M = T * X, π 0 is the inverse of the canonical symplectic form on M , and the recursion operator N = π 1 •π −1 0 turns out be the complete lift of a torsionless endomorphism L : T X → T X [34].…”

“…In section 3 we discuss two significant applications of our formalism, namely the noncommutative versions of the rational Calogero-Moser system and of the Gibbons-Hermsen system. The bihamiltonian structure of the Calogero-Moser system was first described in [25]; more recently, a geometric interpretation of that structure was given in [3] by means of a two-step reduction of the two Poisson bivectors. The path algebra of the quiver with two loops provides the natural noncommutative counterpart of Calogero-Moser phase space, as shown in [15].…”

“…The formalism is then applied to the study of the Calogero-Moser and Gibbons-Hermsen integrable systems. In the former case, we give a new interpretation of the bihamiltonian reduction performed in [3].…”