Symplectic homogeneous spaces

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].