Integrable systems associated to the filtrations of Lie algebras

TL;DR

This paper proves Bogoyavlenski's conjecture that integrability of Euler equations on a Lie algebra extends to certain larger Lie algebras related to filtrations, establishing noncommutative integrability and constructing polynomial integrals.

Contribution

It confirms the conjecture for filtrations of compact Lie algebras and develops methods to construct complete polynomial integrals for these integrable systems.

Findings

Proved noncommutative integrability for systems associated with filtrations of compact Lie algebras.

Constructed explicit polynomial integrals for the integrable systems.

Extended the understanding of integrable systems related to Lie algebra filtrations.

Abstract

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 proved the conjecture for filtrations of compact Lie algebras $\mathfrak g$: 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.