BOLSINOV, A., 2016. Complete commutative subalgebras in polynomial poisson algebras: a proof of the Mischenko-Fomenko conjecture. Theoretical and Applied Mechanics [= Teorijska i primenjena mehanika], 43 (2), pp.145-168.
The Mishchenko–Fomenko conjecture says that for each real or
complex finite-dimensional Lie algebra g there exists a complete set of commuting
polynomials on its dual space g*. In terms of the theory of integrable
Hamiltonian systems this means that the dual space 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 g* and
consider some examples.
This paper is a revised version of: BOLSINOV, A.V., 2006. Complete commutative families of polynomials in Poisson-Lie algebras: a proof of the Mischenko-Fomenko conjecture. Trudy seminara po vektornomu i tenzornomu analizu (ISSN: 0373-4870), 26, pp.87-109.