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A [[Symplectic manifold|symplectic manifold]]  $  ( M, \omega ) $
together with a transitive Lie group  $  G $
of automorphisms of  $  M $. 
The elements of the Lie algebra  $  \mathfrak g $
of  $  G $
can be regarded as symplectic vector fields on  $  M $, 
i.e. fields  $  X $
that preserve the symplectic  $  2 $-
form  $  \omega $:

$$ 
X \cdot \omega  =  di _ {X} \omega  =  0,
$$

where the dot denotes the Lie derivative,  $  i _ {X} $
is the operation of interior multiplication by  $  X $
and  $  d $
is the exterior differential. A symplectic homogeneous space is said to be strictly symplectic if all fields  $  X \in \mathfrak g $
are Hamiltonian, i.e.  $  i _ {X} \omega = dH _ {X} $, 
where  $  H _ {X} $
is a function on  $  M $(
the Hamiltonian of  $  X $) 
that can be chosen in such a way that the mapping  $  X \mapsto H _ {X} $
is a homomorphism from the Lie algebra  $  \mathfrak g $
to the Lie algebra of functions on  $  M $
with respect to the Poisson bracket. An example of a strictly-symplectic homogeneous space is the orbit  $  M _  \alpha  = (  \mathop{\rm Ad}  ^ {*} G) \alpha $
of the Lie group  $  G $
relative to its co-adjoint representation  $   \mathop{\rm Ad}  ^ {*} G $
in the space  $  \mathfrak g  ^ {*} $
of linear forms on  $  \mathfrak g $, 
passing through an arbitrary point  $  \alpha \in \mathfrak g  ^ {*} $. 
The invariant symplectic  $  2 $-
form  $  \omega $
on  $  M _  \alpha  $
is given by the formula

$$ 
\omega ( X _  \beta  , Y _  \beta  )  = \ 
d \beta ( X, Y)  \equiv  \beta ([ X, Y]),
$$

where  $  X _  \beta  $, 
$  Y _  \beta  $
are the values of the vector fields  $  X, Y \in \mathfrak g $
at  $  \beta \in M _  \alpha  $. 
The field  $  X \in \mathfrak g $
has Hamiltonian  $  H _ {X} ( \beta ) = \beta ( X) $.

For an arbitrary strictly-symplectic homogeneous space  $  ( M, \omega , G) $
there is the  $  G $-
equivariant moment mapping

$$ 
\mu :  M  \rightarrow  \mathfrak g  ^ {*} ,\ \ 
x  \mapsto  \mu _ {x} ,\ \ 
\mu _ {x} ( X)  =  H _ {X} ( x),
$$

which maps  $  M $
onto the orbit  $  \mu ( M) $
of  $  G $
in  $  \mathfrak g  ^ {*} $
and is a local isomorphism of symplectic manifolds. Thus, every strictly-symplectic homogeneous space of  $  G $
is a covering over an orbit of  $  G $
in the co-adjoint representation.

The simply-connected symplectic homogeneous spaces with a simply-connected, but not necessarily effectively-acting automorphism group  $  G $
are in one-to-one correspondence with the orbits of the natural action of  $  G $
on the space  $  Z  ^ {2} ( \mathfrak g ) $
of closed  $  2 $-
forms on its Lie algebra  $  \mathfrak g $. 
The correspondence is defined in the following way. The kernel  $  \mathfrak K  ^  \sigma  $
of any  $  2 $-
form  $  \sigma \in Z  ^ {2} ( \mathfrak g ) $
is a subalgebra of  $  \mathfrak g $. 
The connected subgroup  $  K  ^  \sigma  $
of the Lie group  $  G $
corresponding to  $  \mathfrak K  ^  \sigma  $
is closed and defines a simply-connected homogeneous space  $  M  ^  \sigma  = G/K  ^  \sigma  $. 
The form  $  \sigma $
determines a non-degenerate  $  2 $-
form on the tangent space  $  T _ {O} M  ^  \sigma  \simeq \mathfrak g / \mathfrak K  ^  \sigma  $
at a point  $  O = eK  ^  \sigma  $
of the manifold  $  M  ^  \sigma  $, 
which extends to a  $  G $-
invariant symplectic form  $  \omega  ^  \sigma  $
on  $  M  ^  \sigma  $. 
Thus, to the form  $  \sigma $
one assigns the simply-connected symplectic homogeneous space  $  ( M  ^  \sigma  , \omega  ^  \sigma  ) $. 
If  $  \mathfrak K  ^  \sigma  $
contains no ideals of  $  \mathfrak g $, 
then the action of  $  G $
on  $  M  ^  \sigma  $
is locally effective. Two symplectic homogeneous spaces  $  M  ^  \sigma  $
and  $  M ^ {\sigma  ^  \prime  } $
are isomorphic if and only if the forms  $  \sigma $, 
$  \sigma  ^  \prime  $
belong to the same orbit of  $  G $
on  $  Z  ^ {2} ( \mathfrak g ) $. 
For an exact  $  2 $-
form  $  \sigma = d \alpha $, 
the symplectic homogeneous space  $  M  ^  \sigma  $
is identified with the universal covering of the symplectic homogeneous space  $  M _  \alpha  $, 
which is the orbit of a point  $  \alpha $
in the co-adjoint representation. If  $  [ \mathfrak g , \mathfrak g ] = \mathfrak g $, 
then the orbit  $  G \sigma $
of any point  $  \sigma \in Z  ^ {2} ( \mathfrak g ) $
is canonically provided with the structure of a symplectic homogeneous space, and any symplectic homogeneous space of a simply-connected group  $  G $
is isomorphic to the covering over one of these orbits. In particular,  $  M  ^  \sigma  $
is the universal covering of  $  G \sigma $.

Let  $  ( M, \omega ) $
be a compact symplectic homogeneous space of a simply-connected connected group  $  G $
whose action is locally effective. Then  $  G $
is the direct product of a semi-simple compact group  $  S $
and a solvable group  $  R $
isomorphic to the semi-direct product of an Abelian subgroup and an Abelian normal subgroup, and the symplectic homogeneous space  $  ( M, \omega ) $
decomposes into the direct product of symplectic homogeneous spaces with automorphism groups  $  S $
and  $  R $, 
respectively.

A symplectic group space is a special type of symplectic homogeneous space. It consists of a Lie group together with a left-invariant symplectic form  $  \omega $. 
It is known that for a Lie group admitting a left-invariant symplectic form, reductivity implies commutativity, and unimodularity implies solvability. All such groups of dimension  $  \leq  4 $
are solvable, but from dimension 6 onwards there are unsolvable symplectic group spaces [[#References|[3]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Kirillov,   "Elements of the theory of representations" , Springer  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V. Guillemin,   S. Sternberg,   "Geometric asymptotics" , Amer. Math. Soc.  (1977)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.-Y. Chu,   "Symplectic homogeneous spaces"  ''Trans. Amer. Math. Soc.'' , '''197'''  (1974)  pp. 145–159</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Ph.B. Zwart,   W.M. Boothby,   "On compact, homogeneous symplectic manifolds"  ''Ann. Inst. Fourier'' , '''30''' :  1  (1980)  pp. 129–157</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.E. Hurt,   "Geometric quantization in action" , Reidel  (1983)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  D.V. Alekseevskii,   A.M. Vinogradov,   V.V. Lychagin,   "The principal ideas and methods of differential geometry" , ''Encycl. Math. Sci.'' , '''28''' , Springer  (Forthcoming)  pp. Chapt. 4, Sect. 5  (Translated from Russian)</TD></TR></table>

====Comments====
See [[Lie differentiation|Lie differentiation]] for the definitions of Lie derivative and interior multiplication.