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Almost multiplicity free subgroups of compact Lie groups and polynomial integrability of sub-Riemannian geodesic flows

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Abstract

We classify almost multiplicity free subgroups K of compact simple Lie groups G. The problem is related to the integrability of Riemannian and sub-Riemannian geodesic flows of left-invariant metrics defined by a specific extension of integrable systems from \(T^*K\) to \(T^*G\).

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We do not analyse or generate any datasets, because our work proceeds within a theoretical and mathematical approach.

Notes

  1. Alternatively, we can use a following general statement. Assume that compact group G acts freely in a Hamiltonian way on a symplectic manifold M. If a G–invariant Hamiltoniam system is integrable in a noncommutative sense on the reduced space M/G with \(\delta \)–dimensional invariant tori, then the original system on M is also integrable with \((\delta +{{\,\mathrm{\textrm{rank}}\,}}G)\)–dimensional invariant tori (see [19, 33])

  2. Here we take \(x_i\in {\mathfrak {p}}_i\), \(i=1,\dots ,n\), such that the dimensions of the isotropy algebras \({\mathfrak {g}}_{i}(x_0+\dots +x_i)\) and \({\mathfrak {g}}_{i-1}(x_0+\dots +x_i)\) are minimal.

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Acknowledgements

The research was supported by the Project no. 7744592 MEGIC “Integrability and Extremal Problems in Mechanics, Geometry and Combinatorics” of the Science Fund of Serbia. The authors would like to thank the reviewers for their thorough reviews and constructive comments, which greatly improved the quality of the paper.

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Authors and Affiliations

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

    Božidar Jovanović

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

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Jovanović, B., Šukilović, T. & Vukmirović, S. Almost multiplicity free subgroups of compact Lie groups and polynomial integrability of sub-Riemannian geodesic flows. Lett Math Phys 114, 14 (2024). https://doi.org/10.1007/s11005-023-01757-w

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