Limits of Gaudin algebras, quantization of bending flows, Jucys--Murphy elements and Gelfand--Tsetlin bases

Abstract

Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of $n$ copies of the universal enveloping algebra $U(\g)$ of a semisimple Lie algebra $\g$. This family is parameterized by collections of pairwise distinct complex numbers $z_1,...,z_n$. We obtain some new commutative subalgebras in $U(\g)^{\otimes n}$ as limit cases of Gaudin subalgebras. These commutative subalgebras turn to be related to the hamiltonians of bending flows and to the Gelfand--Tsetlin bases. We use this to prove the simplicity of spectrum in the Gaudin model for some new cases.

11 pages, references added

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• Gaudin Models and Multipoint Conformal Blocks: General Theory
• From Gaudin Integrable Models to $d$-dimensional Multipoint Conformal Blocks
• Gaudin subalgebras and stable rational curves
• Limits of Gaudin Systems: Classical and Quantum Cases
• Bethe subalgebras of the group algebra of the symmetric group
• Self-dual Grassmannian, Wronski map, and representations of ${\mathfrak{gl}}_N$, ${\mathfrak{sp}}_{2r}$, ${\mathfrak{so}}_{2r+1}$