Argument Shift Method and Gaudin Model

Abstract

We construct 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 μ; z_1,...,z_n, where μ\in g^*, and z_1,...,z_n are pairwise distinct complex numbers. The construction presented here generalizes the famous construction of the higher Gaudin hamiltonians due to Feigin, Frenkel, and Reshetikhin. For n=1, our construction gives a quantization of the family of maximal Poisson-commutative subalgebras in S(g) obtained by the argument shift method. Next, we describe natural representations of commutative algebras of our family in tensor products of finite-dimensional g-modules as certain degenerations of the Gaudin model. In the case of g=sl_r we prove that our commutative subalgebras have simple spectrum in tensor products of finite-dimensional g-modules for generic μand z_i. This implies simplicity of spectrum in the "generic" sl_r Gaudin model.