# On compact homogeneous symplectic manifolds

P. B. Zwart; William M. Boothby

Annales de l'institut Fourier (1980)

- Volume: 30, Issue: 1, page 129-157
- ISSN: 0373-0956

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topZwart, 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 -

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