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Limits of Gaudin systems: classical and quantum cases. (English) Zbl 1160.82316
Summary: We consider the XXX homogeneous Gaudin system with \(N\) sites, both in the 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.

82B23 Exactly solvable models; Bethe ansatz
81R12 Groups and algebras in quantum theory and relations with integrable systems
17B80 Applications of Lie algebras and superalgebras to integrable systems
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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