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 \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 Poissoncommutative 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 finitedimensional gmodules 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 finitedimensional gmodules for generic \mu and z_i. This implies simplicity of spectrum in the "generic" sl_r Gaudin model.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 DOI:
 10.48550/arXiv.math/0606380
 arXiv:
 arXiv:math/0606380
 Bibcode:
 2006math......6380R
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Quantum Algebra
 EPrint:
 15 pages, references added