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 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.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- June 2006
- DOI:
- 10.48550/arXiv.math/0606380
- arXiv:
- arXiv:math/0606380
- Bibcode:
- 2006math......6380R
- Keywords:
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- Mathematics - Representation Theory;
- Mathematics - Quantum Algebra
- E-Print:
- 15 pages, references added