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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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