Tohoku Mathematical Journal

Structure of symplectic Lie groups and momentum map

Alberto Medina

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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.

Article information

Tohoku Math. J. (2), Volume 67, Number 3 (2015), 419-431.

First available in Project Euclid: 6 November 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D20: Momentum maps; symplectic reduction
Secondary: 70G65: Symmetries, Lie-group and Lie-algebra methods

Symplectic Lie groups Hamiltonian Lie groups symplectic reduction symplectic double extension


Medina, Alberto. Structure of symplectic Lie groups and momentum map. Tohoku Math. J. (2) 67 (2015), no. 3, 419--431. doi:10.2748/tmj/1446818559.

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