On compact homogeneous symplectic manifolds

Zwart, P. B., and Boothby, William M.. "On compact homogeneous symplectic manifolds." Annales de l'institut Fourier 30.1 (1980): 129-157. <http://eudml.org/doc/74438>.

@article{Zwart1980,
abstract = {In this paper the authors study compact homogeneous spaces $G/K$ (of a Lie group $G$) on which there if defined a $G$-invariant symplectic form $\Omega $. It is an important feature of the paper that very little is assumed concerning $G$ and $K$. The essential assumptions are: (1) $G$ is connected and (2) $K$ is uniform (i.e., $G/K$ is compact). Further, for convenience only and with no loss of generality, it is supposed that $G$ is simply connected and $K$ contains no connected normal subgroup of $G$, i.e., that $G$ acts almost effectively on $G/K$. It is then shown that $G=S\times R$, a direct product, where $S$ is compact semi-simple and $R$ is a semi-direct product $AN$ of a connected abelian subgroup $A$ and the maximal connected normal nilpotent group $N$, which is also abelian. Further $K=(K\cap S)\times (K\cap R)$ and $S/(K\cap S),R/K\cap R)$ each have a natural symplectic structure. Some further results on $R/(K\cap R)$ are given together with an example which shows that $R$ can actually possess this two step solvable structure, i.e., it need not be abelian, although $R/(K\cap R)$ is a torus topologically. Once it has been established that $S$ is compact and $S/(K\cap S)$ symplectic, then the structure of $S/(K\cap S)$ is well-known from the work of others.},
author = {Zwart, P. B., Boothby, William M.},
journal = {Annales de l'institut Fourier},
keywords = {semi-simple Lie group; homogeneous symplectic manifolds},
language = {eng},
number = {1},
pages = {129-157},
publisher = {Association des Annales de l'Institut Fourier},
title = {On compact homogeneous symplectic manifolds},
url = {http://eudml.org/doc/74438},
volume = {30},
year = {1980},
}

TY - JOUR
AU - Zwart, P. B.
AU - Boothby, William M.
TI - On compact homogeneous symplectic manifolds
JO - Annales de l'institut Fourier
PY - 1980
PB - Association des Annales de l'Institut Fourier
VL - 30
IS - 1
SP - 129
EP - 157
AB - In this paper the authors study compact homogeneous spaces $G/K$ (of a Lie group $G$) on which there if defined a $G$-invariant symplectic form $\Omega $. It is an important feature of the paper that very little is assumed concerning $G$ and $K$. The essential assumptions are: (1) $G$ is connected and (2) $K$ is uniform (i.e., $G/K$ is compact). Further, for convenience only and with no loss of generality, it is supposed that $G$ is simply connected and $K$ contains no connected normal subgroup of $G$, i.e., that $G$ acts almost effectively on $G/K$. It is then shown that $G=S\times R$, a direct product, where $S$ is compact semi-simple and $R$ is a semi-direct product $AN$ of a connected abelian subgroup $A$ and the maximal connected normal nilpotent group $N$, which is also abelian. Further $K=(K\cap S)\times (K\cap R)$ and $S/(K\cap S),R/K\cap R)$ each have a natural symplectic structure. Some further results on $R/(K\cap R)$ are given together with an example which shows that $R$ can actually possess this two step solvable structure, i.e., it need not be abelian, although $R/(K\cap R)$ is a torus topologically. Once it has been established that $S$ is compact and $S/(K\cap S)$ symplectic, then the structure of $S/(K\cap S)$ is well-known from the work of others.
LA - eng
KW - semi-simple Lie group; homogeneous symplectic manifolds
UR - http://eudml.org/doc/74438
ER -