On compact homogeneous symplectic manifolds

Zwart, P. B.; Boothby, William M.

doi:10.5802/aif.778

Zwart, P. B. ; Boothby, William M.

Annales de l'Institut Fourier, Tome 30 (1980) no. 1, pp. 129-157.

Résumé
Abstract

Dans cet article, les auteurs considèrent les espaces homogènes compacts d’un groupe de Lie $G$ sur lesquels est donnée une structure symplectique $G$ -invariante $Ω$ . Un point important de l’article est le fait que les hypothèses sur $G$ et $K$ sont minimales : (1) $G$ est connexe et (2) $K$ est uniforme (i.e. $G / K$ est compact). Pour faciliter l’exposition mais sans diminuer la généralité des conclusions, on a aussi supposé que $G$ est simplement connexe et que $K$ ne contient aucun sous-groupe invariant connexe de $G$ , i.e. l’action de $G$ sur $G / K$ est presque effective. Alors, il est démontré que $G = S \times R$ , produit direct de $S$ semi-simple et compact et $R$ produit semi-direct $A N$ d’un sous-groupe abélien $A$ avec un sous-groupe invariant abélien $N$ . De plus, $K = (K \cap S) \times (K \cap R)$ et les espaces homogènes $S (K \cap S)$ et $R / (K \cap R)$ héritent chacun d’une structure homogène symplectique. Quelques résultats supplémentaires sur $R / (K \cap R)$ sont donnés avec un exemple qui montre que $R$ peut, en fait, être muni d’une structure de groupe résoluble non-abélien mais que $R / (K \cap R)$ doit être homéomorphe au tore $T^{n}$ . Quant à $S / (K \cap R)$ , une fois établi que $S$ est compact, semi-simple, la structure de cet espace est bien connue.

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 $Ω$ . 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 $A N$ 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.

MR Zbl

DOI : 10.5802/aif.778

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     author = {Zwart, P. B. and Boothby, William M.},
     title = {On compact homogeneous symplectic manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {129--157},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {30},
     number = {1},
     year = {1980},
     doi = {10.5802/aif.778},
     mrnumber = {81g:53040},
     zbl = {0417.53028},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.778/}
}

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Zwart, P. B.; Boothby, William M. On compact homogeneous symplectic manifolds. Annales de l'Institut Fourier, Tome 30 (1980) no. 1, pp. 129-157. doi : 10.5802/aif.778. https://www.numdam.org/articles/10.5802/aif.778/

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