Integrable systems associated to the filtrations of Lie algebras

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\m...

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Bibliographic Details
Published in:arXiv.org 2024-03
Main Authors: Jovanovic, Bozidar, Sukilovic, Tijana, Vukmirovic, Srdjan
Format: Article
Language:English
Subjects:
Algebra
Euler-Lagrange equation
Group theory
Integrals
Lie groups
Polynomials
Set theory
Subgroups
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1912.03199