Bogoyavlenski conjecture and classification of multiplicity free and almost multiplicity free subgroups

Abstract

In [Bogoyavlenski, O.I.: Integrable Euler equations associated with filtrations of Lie algebras, Mat. Sb. 121(163) (1983) 233–242] is conjectured that if the Euler equations on a Lie algebra g0 are integrable, then their certain extensions to semisimple lie algebras g related to the filtrations of Lie algebras g0 ⊂ g1 ⊂ g2 · · · ⊂ gn−1 ⊂ gn = g are integrable as well. In particular, by taking g0 = {0} and natural filtrations of so(n) and u(n), we have Gel’fand-Cetlin integrable systems. We proved the conjecture for filtrations of compact Lie algebras g: the system are integrable in a noncommutative sense by means of polynomial integrals. Various constructions of complete commutative polynomial integrals for the system are also given. In addition, related to commutative polynomial integrability, we classify almost multiplicity free subgroups of compact simple Lie groups, see [Guillemin, V and Sternberg, S.: Multiplicity-free spaces, J. Diff. Geometry 19 (1984) 31–56], [Krämer, M.: Multiplicity free subgroups of compact connected Lie groups. Arch. Math. 27 (1976) 28–36].