Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 003, 21 pages      arXiv:2405.03533      https://doi.org/10.3842/SIGMA.2025.003

Comomentum Sections and Poisson Maps in Hamiltonian Lie Algebroids

Yuji Hirota a and Noriaki Ikeda b
a) Division of Integrated Science, Azabu University, Sagamihara, Kanagawa 252-5201, Japan
b) Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan

Received June 16, 2024, in final form December 29, 2024; Published online January 05, 2025

Abstract
In a Hamiltonian Lie algebroid over a pre-symplectic manifold and over a Poisson manifold, we introduce a map corresponding to a comomentum map, called a comomentum section. We show that the comomentum section gives a Lie algebroid morphism among Lie algebroids. Moreover, we prove that a momentum section on a Hamiltonian Lie algebroid is a Poisson map between proper Poisson manifolds, which is a generalization that a momentum map is a Poisson map between the symplectic manifold to dual of the Lie algebra. Finally, a momentum section is reinterpreted as a Dirac morphism on Dirac structures.

Key words: Poisson geometry; momentum maps; Poisson maps; Dirac structures.

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