Alexey Bolsinov (University of Loughborough)
A.Bolsinov@lboro.ac.uk
It is well known that the dual space g* of any finite dimensional Lie algebra g possesses a natural Poisson structure (the Lie-Poisson structure or, which is essentially the same, the Kirillov-Kostant-Berezin-... symplectic form on the coadjoint orbits). The Mischenko-Fomenko conjecture states that every finite-dimensional Lie algebra g admits an integrable system (on g*) with polynomial integrals. In algebraic terms this means that the Poisson-Lie algebra P(g) associated with g always admits a complete commutative subalgebra. This conjecture was proved in 2003 by S. Sadetov. My talk will be around this conjecture: history, generalizations, related results and a new geometric version of Sadetov's construction.