Gaudin Models and Bending Flows: a Geometrical Point of View

Abstract

In this paper we discuss the bihamiltonian formulation of the (rational XXX) Gaudin models of spin-spin interaction, generalized to the case of sl(r)-valued spins. In particular, we focus on the homogeneous models. We find a pencil of Poisson brackets that recursively define a complete set of integrals of the motion, alternative to the set of integrals associated with the 'standard' Lax representation of the Gaudin model. These integrals, in the case of su(2), coincide wih the Hamiltonians of the 'bending flows' in the moduli space of polygons in Euclidean space introduced by Kapovich and Millson. We finally address the problem of separability of these flows and explicitly find separation coordinates and separation relations for the r=2 case.

arXiv comment27 pages, LaTeX with amsmath and amssymb

References in corpus (12)

• Separation of Variables. New Trends.
• Darboux Coordinates and Liouville-Arnold Integration in Loop Algebras
• Quasi-BiHamiltonian Systems and Separability
• Separation of Variables in the Classical Integrable SL(3) Magnetic Chain
• Bihamiltonian geometry and separation of variables for Toda lattices
• Classical Functional Bethe Ansatz for $SL(N)$: separation of variables for the magnetic chain
• On a Poisson reduction for Gel'fand--Zakharevich manifolds
• Exact Solution of the Quantum Calogero-Gaudin System and of its q-Deformation
• Superintegrability, Lax matrices and separation of variables
• The spin 1/2 Calogero-Gaudin System and its q-Deformation
• Projections of Jordan bi-Poisson structures that are Kronecker, diagonal actions, and the classical Gaudin systems
• Darboux Coordinates on Coadjoint Orbits of Lie Algebras

Cited by in corpus (11)

• From Gaudin Integrable Models to $d$-dimensional Multipoint Conformal Blocks
• Limits of Gaudin algebras, quantization of bending flows, Jucys--Murphy elements and Gelfand--Tsetlin bases
• Gaudin subalgebras and stable rational curves
• Rational Lax operators and their quantization
• Bäcklund Transformations as exact integrable time-discretizations for the trigonometric Gaudin model
• Limits of Gaudin Systems: Classical and Quantum Cases
• Poisson-Hopf limit of quantum algebras
• Gaudin models with ${\CU}_q(\mathfrak{osp}(1 | 2))$ symmetry
• Universal G-oper and Gaudin eigenproblem
• An integrable discretization of the rational su(2) Gaudin model and related systems
• Quantisation of bending flows