# Hofer's metric in compact Lie groups

## TL;DR

This paper explores the geometry of compact Lie groups acting on symplectic manifolds using generalized Hofer norms, analyzing geodesics, their properties, and conditions for Hamiltonian commutativity.

## Contribution

It introduces generalized Hofer norms on Lie algebras, characterizes geodesics in compact Lie groups with bi-invariant metrics, and links these to Hamiltonian dynamics and moment polytopes.

## Key findings

- Established conditions for geodesic existence and characterization in Lie groups.
- Connected geodesic properties in Lie groups to Hamiltonian commutativity.
- Identified cases with non-crossing eigenvalues of Hamiltonians.

## Abstract

In this article we study the Hofer geometry of a compact Lie group $K$ which acts by Hamiltonian diffeomorphisms on a symplectic manifold $M$. Generalized Hofer norms on the Lie algebra of $K$ 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 $K$ endowed with bi-invariant Finsler metrics are proved. We relate the conditions for being a geodesic in the group $K$ 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.