Integrable systems associated to the filtrations of Lie algebras
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2019
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| _version_ | 1866917603151183872 |
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| author | Jovanovic, Bozidar Sukilovic, Tijana Vukmirovic, Srdjan |
| author_facet | Jovanovic, Bozidar Sukilovic, Tijana Vukmirovic, Srdjan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1912_03199 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Integrable systems associated to the filtrations of Lie algebras Jovanovic, Bozidar Sukilovic, Tijana Vukmirovic, Srdjan Exactly Solvable and Integrable Systems Group Theory Symplectic Geometry 37J35, 17B63, 17B80, 53D20 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. |
| title | Integrable systems associated to the filtrations of Lie algebras |
| topic | Exactly Solvable and Integrable Systems Group Theory Symplectic Geometry 37J35, 17B63, 17B80, 53D20 |
| url | https://arxiv.org/abs/1912.03199 |