# Integrable Systems Associated to the Filtrations of Lie Algebras

*2023, Volume 28, Number 1, pp. 44-61*

Author(s):

**Jovanović B., Šukilović T., Vukmirović S.**

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.

Access to the full text on the Springer website |