3 research outputs found
Gaudin Models and Bending Flows: a Geometrical Point of View
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.Comment: 27 pages, LaTeX with amsmath and amssym
Limits of Gaudin algebras, quantization of bending flows, Jucys--Murphy elements and Gelfand--Tsetlin bases
Gaudin algebras form a family of maximal commutative subalgebras in the
tensor product of copies of the universal enveloping algebra U(\g) of a
semisimple Lie algebra \g. This family is parameterized by collections of
pairwise distinct complex numbers . We obtain some new commutative
subalgebras in U(\g)^{\otimes n} as limit cases of Gaudin subalgebras. These
commutative subalgebras turn to be related to the hamiltonians of bending flows
and to the Gelfand--Tsetlin bases. We use this to prove the simplicity of
spectrum in the Gaudin model for some new cases.Comment: 11 pages, references adde
An integrable discretization of the rational su(2) Gaudin model and related systems
The first part of the present paper is devoted to a systematic construction
of continuous-time finite-dimensional integrable systems arising from the
rational su(2) Gaudin model through certain contraction procedures. In the
second part, we derive an explicit integrable Poisson map discretizing a
particular Hamiltonian flow of the rational su(2) Gaudin model. Then, the
contraction procedures enable us to construct explicit integrable
discretizations of the continuous systems derived in the first part of the
paper.Comment: 26 pages, 5 figure