Integrable Systems Associated to the Filtrations of Lie Algebras

Abstract

In 1983 Bogoyavlenski conjectured that, if the Euler equations on a Lie algebra g are integrable, then their certain extensions to semisimple lie algebras g related to the filtrations of Lie algebras (Formula presented.)are integrable as well.In particular, by taking g0={0} and natural filtrations of so(n) and u(n), we haveGel’fand – Cetlin integrable systems. We prove the conjecturefor filtrations of compact Lie algebras 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.