HOME > Table of Contents and Abstracts > Vol. 67, No. 3
Tohoku Mathematical Journal
SECOND SERIES VOL. 67, NO. 3
|Tohoku Math. J.|
67 (2015), 419-431
STRUCTURE OF SYMPLECTIC LIE GROUPS AND MOMENTUM MAP
(Received November 5, 2013, revised June 11, 2014)
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.
Mathematics Subject Classification.
Primary 53D20; Secondary 70G65.
Key words and phrases.
Symplectic Lie groups, Hamiltonian Lie groups, symplectic reduction, symplectic double extension.
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