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

Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of $n$ copies of the universal enveloping algebra $U(\text{g})$ of a semisimple Lie algebra $\text{g}$. This family is parameterized by collections of pairwise distinct complex numbers $z_1,\mathrm{.}\mathrm{.}\mathrm{.},z_n$. We obtain some new commutative subalgebras in $U(\text{g}{)}^{\otimes n}$ 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.