Complete commutative subalgebras in polynomial poisson algebras: A proof of the Mischenko-Fomenko conjecture

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.The Mishchenko–Fomenko conjecture says that for each real or\udcomplex finite-dimensional Lie algebra g there exists a complete set of commuting\udpolynomials on its dual space g*. In terms of the theory of integrable\udHamiltonian systems this means that the dual space g* endowed with the standard\udLie–Poisson bracket admits polynomial integrable Hamiltonian systems.\udThis conjecture was proved by S. T. Sadetov in 2003. Following his idea, we\udgive an explicit geometric construction for commuting polynomials on g* and\udconsider some examples