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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Symplectic homogeneous spaces


Author: Bon Yao Chu
Journal: Trans. Amer. Math. Soc. 197 (1974), 145-159
MSC: Primary 22E15; Secondary 53C30, 57F99
MathSciNet review: 0342642
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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].


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DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0342642-7
PII: S 0002-9947(1974)0342642-7
Keywords: Symplectic structure, homogeneous space, differential form, regular involutive distribution, affine transformation group, cohomology, semisimple, solvable and unimodular Lie groups
Article copyright: © Copyright 1974 American Mathematical Society