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A [[Symplectic manifold|symplectic manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s0918301.png" /> together with a transitive Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s0918302.png" /> of automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s0918303.png" />. The elements of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s0918304.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s0918305.png" /> can be regarded as symplectic vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s0918306.png" />, i.e. fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s0918307.png" /> that preserve the symplectic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s0918309.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183010.png" />:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183011.png" /></td> </tr></table>
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{{TEX|done}}
  
where the dot denotes the Lie derivative, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183012.png" /> is the operation of interior multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183014.png" /> is the exterior differential. A symplectic homogeneous space is said to be strictly symplectic if all fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183015.png" /> are Hamiltonian, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183017.png" /> is a function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183018.png" /> (the Hamiltonian of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183019.png" />) that can be chosen in such a way that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183020.png" /> is a homomorphism from the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183021.png" /> to the Lie algebra of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183022.png" /> with respect to the Poisson bracket. An example of a strictly-symplectic homogeneous space is the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183023.png" /> of the Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183024.png" /> relative to its co-adjoint representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183025.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183026.png" /> of linear forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183027.png" />, passing through an arbitrary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183028.png" />. The invariant symplectic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183029.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183030.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183031.png" /> is given by the formula
+
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 $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183032.png" /></td> </tr></table>
+
$$
 +
X \cdot \omega  = di _ {X} \omega  = 0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183034.png" /> are the values of the vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183035.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183036.png" />. The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183037.png" /> has Hamiltonian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183038.png" />.
+
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
  
For an arbitrary strictly-symplectic homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183039.png" /> there is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183040.png" />-equivariant moment mapping
+
$$
 +
\omega ( X _  \beta  , Y _  \beta  )  = \
 +
d \beta ( X, Y)  \equiv  \beta ([ X, Y]),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183041.png" /></td> </tr></table>
+
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) $.
  
which maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183042.png" /> onto the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183043.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183044.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183045.png" /> and is a local isomorphism of symplectic manifolds. Thus, every strictly-symplectic homogeneous space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183046.png" /> is a covering over an orbit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183047.png" /> in the co-adjoint representation.
+
For an arbitrary strictly-symplectic homogeneous space $  ( M, \omega , G) $
 +
there is the $  G $-
 +
equivariant moment mapping
  
The simply-connected symplectic homogeneous spaces with a simply-connected, but not necessarily effectively-acting automorphism group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183048.png" /> are in one-to-one correspondence with the orbits of the natural action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183049.png" /> on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183050.png" /> of closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183051.png" />-forms on its Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183052.png" />. The correspondence is defined in the following way. The kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183053.png" /> of any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183054.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183055.png" /> is a subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183056.png" />. The connected subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183057.png" /> of the Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183058.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183059.png" /> is closed and defines a simply-connected homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183060.png" />. The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183061.png" /> determines a non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183062.png" />-form on the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183063.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183064.png" /> of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183065.png" />, which extends to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183066.png" />-invariant symplectic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183067.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183068.png" />. Thus, to the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183069.png" /> one assigns the simply-connected symplectic homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183070.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183071.png" /> contains no ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183072.png" />, then the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183073.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183074.png" /> is locally effective. Two symplectic homogeneous spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183076.png" /> are isomorphic if and only if the forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183078.png" /> belong to the same orbit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183079.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183080.png" />. For an exact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183081.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183082.png" />, the symplectic homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183083.png" /> is identified with the universal covering of the symplectic homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183084.png" />, which is the orbit of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183085.png" /> in the co-adjoint representation. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183086.png" />, then the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183087.png" /> of any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183088.png" /> is canonically provided with the structure of a symplectic homogeneous space, and any symplectic homogeneous space of a simply-connected group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183089.png" /> is isomorphic to the covering over one of these orbits. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183090.png" /> is the universal covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183091.png" />.
+
$$
 +
\mu : M  \rightarrow  \mathfrak g  ^ {*} ,\ \
 +
x  \mapsto  \mu _ {x} ,\ \
 +
\mu _ {x} ( X)  = H _ {X} ( x),
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183092.png" /> be a compact symplectic homogeneous space of a simply-connected connected group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183093.png" /> whose action is locally effective. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183094.png" /> is the direct product of a semi-simple compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183095.png" /> and a solvable group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183096.png" /> isomorphic to the semi-direct product of an Abelian subgroup and an Abelian normal subgroup, and the symplectic homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183097.png" /> decomposes into the direct product of symplectic homogeneous spaces with automorphism groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s09183099.png" />, respectively.
+
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.
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s091830100.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091830/s091830101.png" /> are solvable, but from dimension 6 onwards there are unsolvable symplectic group spaces [[#References|[3]]].
+
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====
 
====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>
 
<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====
 
====Comments====
 
See [[Lie differentiation|Lie differentiation]] for the definitions of Lie derivative and interior multiplication.
 
See [[Lie differentiation|Lie differentiation]] for the definitions of Lie derivative and interior multiplication.

Latest revision as of 08:24, 6 June 2020


A 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 [3].

References

[1] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)
[2] V. Guillemin, S. Sternberg, "Geometric asymptotics" , Amer. Math. Soc. (1977)
[3] B.-Y. Chu, "Symplectic homogeneous spaces" Trans. Amer. Math. Soc. , 197 (1974) pp. 145–159
[4] Ph.B. Zwart, W.M. Boothby, "On compact, homogeneous symplectic manifolds" Ann. Inst. Fourier , 30 : 1 (1980) pp. 129–157
[5] N.E. Hurt, "Geometric quantization in action" , Reidel (1983)
[6] 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)

Comments

See Lie differentiation for the definitions of Lie derivative and interior multiplication.

How to Cite This Entry:
Symplectic homogeneous space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symplectic_homogeneous_space&oldid=48933
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article