Abstract
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 µ, z 1, ..., z n , where µ ∈ \(\mathfrak{g}\) * and z i ∈ 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.
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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 40, No. 3, pp. 30–43, 2006
Original Russian Text Copyright © by L. G. Rybnikov
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Rybnikov, L.G. The argument shift method and the Gaudin model. Funct Anal Its Appl 40, 188–199 (2006). https://doi.org/10.1007/s10688-006-0030-3
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DOI: https://doi.org/10.1007/s10688-006-0030-3