Integrable Systems Associated to the Filtrations of Lie Algebras

In 1983 Bogoyavlenski conjectured that, if the Euler equations on a Lie algebra \(\mathfrak{g}_{0}\) are integrable, then their certain extensions to semisimple lie algebras \(\mathfrak{g}\) related to the filtrations of Lie algebras \(\mathfrak{g}_{0}\subset\mathfrak{g}_{1}\subset\mathfrak{g}_{2}\dots\subset\mathfrak{g}_{n-1}\subset\mathfrak{g}_{n}=\mathfrak{g}\) are integrable as well. In particular, by taking \(\mathfrak{g}_{0}=\{0\}\) and natural filtrations of \({\mathfrak{so}}(n)\) and \(\mathfrak{u}(n)\), we have Gel’fand – Cetlin integrable systems. We prove the conjecture for filtrations of compact Lie algebras \(\mathfrak{g}\): the system is integrable in a noncommutative sense by means of polynomial integrals. Various constructions of complete commutative polynomial integrals for the system are also given.