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.
    Keywords: noncommutative integrability, invariant polynomials, Gel’fand – Cetlin systems
    Citation: Jovanović B., Šukilović  T., Vukmirović S., Integrable Systems Associated to the Filtrations of Lie Algebras, Regular and Chaotic Dynamics, 2023, Volume 28, Number 1, pp. 44-61

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