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Integrable Systems Associated to the Filtrations of Lie Algebras

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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 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.

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Notes

  1. The gradient is determined by an invariant metric: \(df(\xi)=\langle\nabla f(x),\xi\rangle\). Also, to simplify notation, the Lie brackets, the Lie – Poisson brackets and the gradients of the functions on \(\mathfrak{g}_{i}\) will be denoted by the same symbols as on \(\mathfrak{g}\), \(i=1,\dots,n\).

  2. In [6] the symplectic case is considered, but the proof can be easily modified to the Poisson case.

  3. \(\mathfrak{g}_{l}(x_{k})\) denotes the isotropy algebra of \(x_{k}\in\mathfrak{g}_{k}\) within \(\mathfrak{g}_{l}\):

    $$\displaystyle\mathfrak{g}_{l}(x_{k})=\{\xi\in\mathfrak{g}_{l}|[\xi,x_{k}]=0\},\qquad l\leqslant k.$$

    Generic means that the dimensions of the isotropy algebras \(\mathfrak{g}_{i}(x_{i})\) and \(\mathfrak{g}_{i-1}(x_{i})\) are minimal.

  4. By \(\langle\cdot,\cdot\rangle\) we also denote an invariant quadratic form on \(\mathfrak{g}^{\mathbb{C}}\), the extension of the invariant scalar product from \(\mathfrak{g}\) to \(\mathfrak{g}^{\mathbb{C}}\).

  5. Again, we use the same symbol for different objects. The restriction of \(\langle\cdot,\cdot\rangle\) to \(\mathfrak{g}\) is a positive definite invariant scalar product we are dealing with. Also, as above, we identify \(\mathfrak{h}\) and \(\mathfrak{h}^{*}\) by \(\langle\cdot,\cdot\rangle\).

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ACKNOWLEDGMENTS

We are grateful to the referees for a thorough review and constructive comments that have greatly improved quality of the paper.

Funding

This research is supported by Project 7744592 MEGIC, Integrability and Extremal Problems in Mechanics, Geometry and Combinatorics, of the Science Fund of Serbia.

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Authors and Affiliations

  1. Mathematical Institute, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11000, Belgrade, Serbia

    Božidar Jovanović

  2. Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000, Belgrade, Serbia

    Tijana Šukilović & Srdjan Vukmirović

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  1. Božidar Jovanović
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  2. Tijana Šukilović
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  3. Srdjan Vukmirović
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Correspondence to Božidar Jovanović, Tijana Šukilović or Srdjan Vukmirović.

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MSC2010

37J35, 17B63, 17B80, 53D20

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Jovanović, B., Šukilović, T. & Vukmirović, S. Integrable Systems Associated to the Filtrations of Lie Algebras. Regul. Chaot. Dyn. 28, 44–61 (2023). https://doi.org/10.1134/S1560354723010045

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