Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 029, 17 pages      arXiv:0903.1604
Contribution to the Proceedings of the XVIIth International Colloquium on Integrable Systems and Quantum Symmetries

Limits of Gaudin Systems: Classical and Quantum Cases

Alexander Chervov a, Gregorio Falqui b and Leonid Rybnikov a
a) Institute for Theoretical and Experimental Physics, 25 Bolshaya Cheremushkinskaya Str., 117218 Moscow, Russia
b) Dipartimento di Matematica e Applicazioni, Università di Milano - Bicocca, via R. Cozzi, 53, 20125 Milano, Italy

Received November 01, 2008, in final form February 25, 2009; Published online March 09, 2009

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 of multi-Poisson geometry will be addressed (in the classical case). We will make use of properties of ''Manin matrices'' to provide explicit generators of the Gaudin Algebras in the quantum case.

Key words: Gaudin models; Hamiltonian structures; Gaudin algebras.

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