# Symplectic Lie Groups

###### Oliver Baues and Vicente Cortés
Publisher:
Société Mathématique de France
Publication Date:
2016
Number of Pages:
90
Format:
Paperback
Series:
Astérisque 379
Price:
52.00
ISBN:
9782856298343
Category:
Monograph
[Reviewed by
Felipe Zaldivar
, on
08/22/2016
]

A symplectic Lie group is a symplectic manifold that is also a Lie group whose symplectic form is left-invariant. Symplectic Lie groups appear naturally when one considers symplectic manifolds $M$ that admit a transitive Lie group $G$ of symplectic automorphisms, and in this case the manifold is a homogeneous symplectic manifold $M=G/H$. Thus, a symplectic Lie group $G$ is a symplectic homogeneous manifold $G/H$ with trivial stabilizer $H$.

The book under review, actually a long article as is usually the case with the Astérisque series, has as its main goals to further develop the structure theory of symplectic Lie groups by means of their isotropic normal subgroups. Moreover, since the study of simply connected groups can be reduced to the study of their symplectic Lie algebras, most of the results of the article are formulated for these Lie algebras. The main themes treated could be summarized as: Reduction of symplectic Lie algebras, Lagrangian extensions of Lie algebras, and the existence of Lagrangian ideals of symplectic Lie algebras.

The book will of interest to researchers working on symplectic Lie groups or geometric quantization.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx