Complete commutative subalgebras in polynomial Poisson algebras: a proof of the Mischenko--Fomenko conjecture

Alexey V. Bolsinov

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.