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

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

2006-08-23

arxiv.org/abs/math/0608586v3

UMN 60 (2005), No 2, pp.173--174 (Russian)

Mathematics

Quantum Algebra

4 pages, references added

Scientific paper

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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