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SIGMA 5 (2009), 029, 17 pages arXiv:0903.1604
https://doi.org/10.3842/SIGMA.2009.029
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
Abstract
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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