Hamiltonian systems defined by Euler equations on Lie algebras arise in various problems in mathematical physics. In 1978, A. S. Mischenko and A. T. Fomenko presented a so-called argument shift method. This method can be used to construct a family of polynomial functions in involution with respect to a Lie-Poisson bracket on a Lie algebra; these functions are exactly the integrals of such Hamiltonian system. It turns out that these functions commute with respect to another Poisson bracket on a Lie algebra. It is natural to ask whether there exists a complete family of polynomials in involution with respect to both Poisson brackets.
Jordan--Kronecker invariants of a Lie algebra were first introduced by A. V. Bolsinov and P. Zhang in 2011. By definition, these invariants describe the canonical block-diagonal decomposition of a pair of skew-symmetric forms defined by the generic pair of elements of dual Lie algebra with blocks of Jordan and Kronecker types. A pair of skew-symmetric forms corresponds to a pair of Poisson brackets mentioned earlier. It was proved by Bolsinov that the completeness of commutative family of shifts for a Lie algebra implies that this Lie algebra is of Kronecker type, i.e. the canonical decomposition of two forms contains only Kronecker blocks.
The talk will cover the recent developments in this area of research.