Spin Calogero models obtained from dynamical r-matrices and geodesic motion

Abstract

We study classical integrable systems based on the Alekseev-Meinrenken dynamical r-matrices corresponding to automorphisms of self-dual Lie algebras, $\mathcal{G}$. We prove that these r-matrices are uniquely characterized by a non-degeneracy property and apply a construction due to Li and Xu to associate spin Calogero type models with them. The equation of motion of any model of this type is found to be a projection of the natural geodesic equation on a Lie group $G$ with Lie algebra $\mathcal{G}$, and its phase space is interpreted as a Hamiltonian reduction of an open submanifold of the cotangent bundle $T^{\ast }G$, using the symmetry arising from the adjoint action of $G$ twisted by the underlying automorphism. This shows the integrability of the resulting systems and gives an algorithm to solve them. As illustrative examples we present new models built on the involutive diagram automorphisms of the real split and compact simple Lie algebras, and also explain that many further examples fit in the dynamical r-matrix framework.

25 pages, with minor stylistic changes and updated references in v2

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Cited by in corpus (20)

• Physics and Mathematics of Calogero particles
• Poisson-Lie analogues of spin Sutherland models
• Spin Calogero models associated with Riemannian symmetric spaces of negative curvature
• Notes on Collective Field Theory of Matrix and Spin Calogero Models
• Hamiltonian reductions of free particles under polar actions of compact Lie groups
• Singular cotangent bundle reduction and spin Calogero-Moser systems
• Geometry of non-holonomic diffusion
• Twisted spin Sutherland models from quantum Hamiltonian reduction
• Bi-Hamiltonian structure of a dynamical system introduced by Braden and Hone
• Superintegrability of Calogero-Moser systems associated with the cyclic quiver
• Calogero-Sutherland system with two types interacting spins
• Liouville integrability of a class of integrable spin Calogero-Moser systems and exponents of simple Lie algebras
• Reduction of a bi-Hamiltonian hierarchy on $T^{\ast }\mathrm{U}(n)$ to spin Ruijsenaars--Sutherland models
• Spinning particles in a Yang-Mills field
• Spin Calogero models and dynamical r-matrices
• Poisson reductions of master integrable systems on doubles of compact Lie groups
• Generalized spin Sutherland systems revisited
• Geometry of Higgs bundles over elliptic curves related to automorphisms of simple Lie algebras, Calogero-Moser systems and KZB equations
• On the bi-Hamiltonian structure of the trigonometric spin Ruijsenaars--Sutherland hierarchy
• Symmetry reduction of Brownian motion and Quantum Calogero-Moser systems