# Limits of Gaudin systems: classical and quantum cases

@article{Chervov2009LimitsOG, title={Limits of Gaudin systems: classical and quantum cases}, author={Alexander Chervov and Gregorio Falqui and Leonid Rybnikov}, journal={Symmetry Integrability and Geometry-methods and Applications}, year={2009}, volume={5}, pages={029} }

We consider the XXX homogeneous Gaudin system with N sites, both in classical and the quantum case. In particular we show that a suitable limiting procedure for letting the poles of its Lax matrix collide can be used to define new families of Liouville integrals (in the classical case) and new "Gaudin" algebras (in the quantum case). We will especially treat the case of total collisions, that gives rise to (a generalization of) the so called Bending flows of Kapovich and Millson. Some aspects… Expand

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#### References

SHOWING 1-10 OF 64 REFERENCES

Gaudin models with irregular singularities

- Mathematics, Physics
- 2010

Abstract We introduce a class of quantum integrable systems generalizing the Gaudin model. The corresponding algebras of quantum Hamiltonians are obtained as quotients of the center of the enveloping… Expand

Gaudin Models and Bending Flows: a Geometrical Point of View

- Mathematics, Physics
- 2003

In this paper, we discuss the bi-Hamiltonian formulation of the (rational XXX) Gaudin models of spin–spin interaction, generalized to the case of sl(r)-valued 'spins'. We only consider the classical… Expand

Algebraic Extensions of Gaudin Models

- Mathematics, Physics
- 2004

Abstract We perform a Inönü–Wigner contraction on Gaudin models, showing how the integrability property is preserved by this algebraic procedure. Starting from Gaudin models we obtain new integrable… Expand

Hitchin systems, higher Gaudin operators and $r$-matrices

- Mathematics, Physics
- 1995

We adapt Hitchin's integrable systems to the case of a punctured curve. In the case of $\CC P^{1}$ and $SL_{n}$-bundles, they are equivalent to systems studied by Garnier. The corresponding quantum… Expand

Quantization of the Gaudin System

- Physics, Mathematics
- 2004

In this article we exploit the known commutative family in Y(gl(n)) - the Bethe subalgebra - and its special limit to construct quantization of the Gaudin integrable system. We give explicit… Expand

Universal G-oper and Gaudin eigenproblem

- Physics, Mathematics
- 2004

This paper is devoted to the eigenvalue problem for the quantum Gaudin system. We prove the universal correspondence between eigenvalues of Gaudin Hamiltonians and the so-called G-opers without… Expand

Gaudin model, Bethe Ansatz and critical level

- Mathematics, Physics
- 1994

We propose a new method of diagonalization oif hamiltonians of the Gaudin model associated to an arbitrary simple Lie algebra, which is based on the Wakimoto modules over affine algebras at the… Expand

Uniqueness of higher Gaudin Hamiltonians

- Mathematics
- 2006

For any semisimple Lie algebra g , the universal enveloping algebra of the infinite-dimensional pro-nilpotent Lie algebra g _:= g ⊗ t − 1 ℂ [ t − 1 ] contains a large commutative subalgebra A ⊂ U ( g… Expand

Rational Lax operators and their quantization

- Physics, Mathematics
- 2004

We investigate the construction of the quantum commuting hamiltonians for the Gaudin integrable model. We prove that [Tr L^k(z), Tr L^m(u) ]=0, for k,m < 4 . However this naive receipt of… Expand

A systematic construction of completely integrable Hamiltonians from coalgebras

- Mathematics, Physics
- 1998

A universal algorithm to construct N-particle (classical and quantum) completely integrable Hamiltonian systems from representations of coalgebras with Casimir elements is presented. In particular,… Expand