Limits of Gaudin Systems: Classical and Quantum Cases

Details Limits of Gaudin Systems: Classical and Quantum Cases Limits of Gaudin Systems: Classical and Quantum Cases

2009-03-09

arxiv.org/abs/0903.1604v1

SIGMA 5 (2009), 029, 17 pages

Mathematics

Quantum Algebra

Scientific paper

10.3842/SIGMA.2009.029

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 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.

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