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Centralizers of certain quadratic elements in Poisson--Lie algebras and Argument Shift method

Leonid Rybnikov
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Abstract

We study maximal Poisson-commutative subalgebras in the Poisson algebra $S(\mathfrak{g})$ of a semisimple Lie algebra $\mathfrak{g}$ constructed by Mischenko and Fomenko with the help of the argument shift method. We prove that these subalgebras are Poisson centralizers of certain quadratic elements of $S(\mathfrak{g})$. We deduce from this that there is a unique quantization of Mischenko--Fomenko subalgebras, i.e. there is a unique way to lift Mischenko--Fomenko subalgebras to commutative subalgebras of the universal enveloping algebra $U(\mathfrak{g})$.

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Leonid Rybnikov. 2006-11-08. Centralizers of certain quadratic elements in Poisson--Lie algebras and Argument Shift method. https://arxiv.org/abs/math/0608586

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