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

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.