Comomentum sections and Poisson maps in Hamiltonian Lie algebroids
Yuji Hirota, Noriaki Ikeda
TL;DR
The paper extends momentum maps to momentum sections for Hamiltonian Lie algebroids on both pre-symplectic and Poisson bases, introducing bracket-compatible comomentum sections and showing they induce Lie algebroid morphisms. It proves that, under a curvature-compatibility condition ${}\langle {}^A S,\mu\rangle=0$, momentum sections yield Poisson maps between appropriate Poisson or graded Poisson spaces, including a Poisson map from $T^*M\oplus\mathbb{R}$ to $A^*$ and a dual statement from $TM\oplus\mathbb{R}$ to $A^*$. The work further reinterprets momentum sections as Dirac morphisms, connecting Lie algebroid morphisms, Poisson maps, and Dirac structures within a unified framework and enabling potential reductions and quantization studies for Hamiltonian Lie algebroids.
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.