Structure of Symplectic Lie groups and momentum map

TL;DR

This paper characterizes symplectic Lie groups using semi-direct products, symplectic reduction, and affine bundles, especially focusing on Hamiltonian cases and specific classes like nilpotent and Frobenius groups.

Contribution

It provides a comprehensive structural description of symplectic Lie groups and characterizes Hamiltonian symplectic Lie groups, including special cases and classical examples.

Findings

Structural description of symplectic Lie groups via semi-direct products

Characterization of Hamiltonian symplectic Lie groups

Analysis of classical affine Lie group as a symplectic Lie group

Abstract

We describe the structure of the Lie groups endowed with a left-invariant symplectic form, called symplectic Lie groups, in terms of semi-direct products of Lie groups, symplectic reduction and principal bundles with affine fiber. This description is particularly nice if the group is Hamiltonian, that is, if the left canonical action of the group on itself is Hamiltonian. The principal tool used for our description is a canonical affine structure associated with the symplectic form. We also characterize the Hamiltonian symplectic Lie groups among the connected symplectic Lie groups. We specialize our principal results to the cases of simply connected Hamiltonian symplectic nilpotent Lie groups or Frobenius symplectic Lie groups. Finally we pursue the study of the classical affine Lie group as a symplectic Lie group. MSC Classes 53D20,70G65 Key words:Symplectic Lie groups,Hamiltonian Lie groups, Symplectic reduction,Symplectic double extension.