Argument Shift Method and Gaudin Model

Details Argument Shift Method and Gaudin Model Argument Shift Method and Gaudin Model

2006-06-15

arxiv.org/abs/math/0606380v3

Func. Anal. Appl. 40 (2006), No 3, pp. 30--43

Mathematics

Representation Theory

15 pages, references added

Scientific paper

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 \mu; z_1,...,z_n, where \mu \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 \mu and z_i. This implies simplicity of spectrum in the "generic" sl_r Gaudin model.

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