Comomentum sections and Poisson maps in Hamiltonian Lie algebroids

TL;DR

This paper introduces the concept of comomentum sections in Hamiltonian Lie algebroids, showing they induce Lie algebroid morphisms and generalize momentum maps as Poisson and Dirac morphisms.

Contribution

It defines comomentum sections and demonstrates their role as Lie algebroid morphisms and their connection to Poisson and Dirac structures, extending classical momentum map theory.

Findings

Comomentum sections induce Lie algebroid morphisms.

Momentum sections are Poisson maps between Poisson manifolds.

Momentum sections can be viewed as Dirac morphisms.

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.