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

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Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of n copies of the universal enveloping algebra U() of a semisimple Lie algebra. This family is parameterized by collections of pairwise distinct complex numbers z₁,...,zₙ. We obtain some new commutative subalgebras in U()⊗ 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.