## Symplectic homogeneous spaces

HTML articles powered by AMS MathViewer

- by Bon Yao Chu
- Trans. Amer. Math. Soc.
**197**(1974), 145-159 - DOI: https://doi.org/10.1090/S0002-9947-1974-0342642-7
- PDF | Request permission

## Abstract:

It is proved in this paper that for a given simply connected Lie group*G*with Lie algebra $\mathfrak {g}$, every left-invariant closed 2-form induces a symplectic homogeneous space. This fact generalizes the results in [7] and [12] that if ${H^1}(\mathfrak {g}) = {H^2}(\mathfrak {g}) = 0$, then the most general symplectic homogeneous space covers an orbit in the dual of the Lie algebra $\mathfrak {g}$. A one-to-one correspondence can be established between the orbit space of equivalent classes of 2-cocycles of a given Lie algebra and the set of equivalent classes of simply connected symplectic homogeneous spaces of the Lie group. Lie groups with left-invariant symplectic structure cannot be semisimple; hence such groups of dimension four have to be solvable, and connected unimodular groups with left-invariant symplectic structure are solvable [4].

## References

- Claude Chevalley,
*Theory of Lie groups. I*, Princeton University Press, Princeton, N. J., 1946 1957. MR**0082628**, DOI 10.1515/9781400883851
—, - Claude Chevalley and Samuel Eilenberg,
*Cohomology theory of Lie groups and Lie algebras*, Trans. Amer. Math. Soc.**63**(1948), 85–124. MR**24908**, DOI 10.1090/S0002-9947-1948-0024908-8 - Jun-ichi Hano,
*On Kaehlerian homogeneous spaces of unimodular Lie groups*, Amer. J. Math.**79**(1957), 885–900. MR**95979**, DOI 10.2307/2372440 - G. Hochschild,
*The structure of Lie groups*, Holden-Day, Inc., San Francisco-London-Amsterdam, 1965. MR**0207883** - Shoshichi Kobayashi and Katsumi Nomizu,
*Foundations of differential geometry. Vol I*, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1963. MR**0152974** - Bertram Kostant,
*Quantization and unitary representations. I. Prequantization*, Lectures in Modern Analysis and Applications, III, Lecture Notes in Mathematics, Vol. 170, Springer, Berlin, 1970, pp. 87–208. MR**0294568**
Yozo Matsushima, - Richard S. Palais,
*A global formulation of the Lie theory of transformation groups*, Mem. Amer. Math. Soc.**22**(1957), iii+123. MR**121424** - L. S. Pontryagin,
*Topological groups*, Gordon and Breach Science Publishers, Inc., New York-London-Paris, 1966. Translated from the second Russian edition by Arlen Brown. MR**0201557** - Shlomo Sternberg,
*Lectures on differential geometry*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR**0193578** - J.-M. Souriau,
*Structure des systèmes dynamiques*, Dunod, Paris, 1970 (French). Maîtrises de mathématiques. MR**0260238** - È. B. Vinberg,
*The theory of homogeneous convex cones*, Trudy Moskov. Mat. Obšč.**12**(1963), 303–358 (Russian). MR**0158414**

*Théorie des groupes de Lie*. II.

*Groupes algèbres*. Actualités Sci. Indust., no. 1152, Hermann, Paris, 1951. MR

**14**, 448.

*Differential geometry*, Dekker, New York, 1972.

## Bibliographic Information

- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**197**(1974), 145-159 - MSC: Primary 22E15; Secondary 53C30, 57F99
- DOI: https://doi.org/10.1090/S0002-9947-1974-0342642-7
- MathSciNet review: 0342642