Symplectic homogeneous space

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.