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The argument shift method and the Gaudin model. (English. Russian original) Zbl 1112.17018
Funct. Anal. Appl. 40, No. 3, 188-199 (2006); translation from Funkts. Anal. Prilozh. 40, No. 3, 30-43 (2005).
Summary: 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 $$\mu$$ , $$z_1,\ldots,z_n$$, where $$\mu \in \mathfrak{g}^*$$ and $$z_i \in \mathbb C$$. 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.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 17B63 Poisson algebras 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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