The quantum Gaudin system

In this paper, we explicitly construct the quantum $mathfrak{g}mathfrak{l}_n $ Gaudin model for general n and for a general number N of particles. To this end, we construct a commutative family in $U(mathfrak{g}mathfrak{l}_n )^{ otimes N} $ . When passing to the classical limit (which is the projection onto the associated graded algebra), our family gives the entire family of classical Gaudin Hamiltonians. The construction is based on the special limit of the Bethe subalgebra in the Yangian $Y(mathfrak{g}mathfrak{l}_n )$ .