The argument shift method and the Gaudin model

We construct a family of maximal commutative subalgebras in the tensor product of n copies of the universal enveloping algebra U ( $\mathfrak{g}$ ) of a semisimple Lie algebra $\mathfrak{g}$ . This family is parameterized by finite sequences µ, z ₁, ..., z _n , where µ ∈ $\mathfrak{g}$ ^* and z _i ∈ ?. The construction presented here generalizes the famous construction of the higher Gaudin Hamiltonians due to Feigin, Frenkel, and Reshetikhin. For n = 1, the corresponding commutative subalgebras in the Poisson algebra S( $\mathfrak{g}$ ) were obtained by Mishchenko and Fomenko with the help of the argument shift method. For commutative algebras of our family, we establish a connection between their representations in the tensor products of finite-dimensional $\mathfrak{g}$ -modules and the Gaudin model.