Hofer's metric in compact Lie groups

  • Gabriel Larotonda

    Universidad de Buenos Aires, Ciudad de Buenos Aires, Argentina
  • Martín Miglioli

    Consejo Nacional de Investigaciones Científicas y Técnicas, Ciudad de Buenos Aires, Argentina
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In this article, we study the Hofer geometry of a compact Lie group which acts by Hamiltonian diffeomorphisms on a symplectic manifold . Generalized Hofer norms on the Lie algebra of are introduced and analyzed with tools from group invariant convex geometry, functional and matrix analysis. Several global results on the existence of geodesics and their characterization in finite-dimensional Lie groups endowed with bi-invariant Finsler metrics are proved. We relate the conditions for being a geodesic in the group and in the group of Hamiltonian diffeomorphisms. These results are applied to obtain necessary and sufficient conditions on the moment polytope of the momentum map, for the commutativity of the Hamiltonians of geodesics. Particular cases are studied, where a generalized non-crossing of eigenvalues property of the Hamiltonians hold.

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Gabriel Larotonda, Martín Miglioli, Hofer's metric in compact Lie groups. Groups Geom. Dyn. 17 (2023), no. 3, pp. 839–898

DOI 10.4171/GGD/721