Geodesic
Complete commutative subalgebras in polynomial Poisson algebras: a proof of the Mischenko--Fomenko conjecture
Theoretical and applied mechanics, Tome 43 (2016) no. 2, p. 145

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The Mishchenko--Fomenko conjecture says that for each real or complex finite-dimensional Lie algebra $\mathfrak g$ there exists a complete set of commuting polynomials on its dual space $\mathfrak g^*$. In terms of the theory of integrable Hamiltonian systems this means that the dual space $\mathfrak g^*$ endowed with the standard Lie--Poisson bracket admits polynomial integrable Hamiltonian systems. This conjecture was proved by S. T.~Sadetov in 2003. Following his idea, we give an explicit geometric construction for commuting polynomials on $\mathfrak g^*$ and consider some examples.
DOI : 10.2298/TAM161111012B
Classification : 37J35, 17B80, 70H06, 53D17, 17B63
Keywords: Poisson-Lie bracket, complete integrability, field extension, Mischenko-Fomenko conjecture, chains of subalgebras, shifting of argument 
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Alexey V. Bolsinov. Complete commutative subalgebras in polynomial Poisson algebras: a proof of the Mischenko--Fomenko conjecture. Theoretical and applied mechanics, Tome 43 (2016) no. 2, p. 145 . doi: 10.2298/TAM161111012B

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