Symplectic homogeneous space

A symplectic manifold together with a transitive Lie group of automorphisms of . The elements of the Lie algebra of can be regarded as symplectic vector fields on , i.e. fields that preserve the symplectic -form :

where the dot denotes the Lie derivative, is the operation of interior multiplication by and is the exterior differential. A symplectic homogeneous space is said to be strictly symplectic if all fields are Hamiltonian, i.e. , where is a function on (the Hamiltonian of ) that can be chosen in such a way that the mapping is a homomorphism from the Lie algebra to the Lie algebra of functions on with respect to the Poisson bracket. An example of a strictly-symplectic homogeneous space is the orbit of the Lie group relative to its co-adjoint representation in the space of linear forms on , passing through an arbitrary point . The invariant symplectic -form on is given by the formula

where , are the values of the vector fields at . The field has Hamiltonian .

For an arbitrary strictly-symplectic homogeneous space there is the -equivariant moment mapping

which maps onto the orbit of in and is a local isomorphism of symplectic manifolds. Thus, every strictly-symplectic homogeneous space of is a covering over an orbit of in the co-adjoint representation.

The simply-connected symplectic homogeneous spaces with a simply-connected, but not necessarily effectively-acting automorphism group are in one-to-one correspondence with the orbits of the natural action of on the space of closed -forms on its Lie algebra . The correspondence is defined in the following way. The kernel of any -form is a subalgebra of . The connected subgroup of the Lie group corresponding to is closed and defines a simply-connected homogeneous space . The form determines a non-degenerate -form on the tangent space at a point of the manifold , which extends to a -invariant symplectic form on . Thus, to the form one assigns the simply-connected symplectic homogeneous space . If contains no ideals of , then the action of on is locally effective. Two symplectic homogeneous spaces and are isomorphic if and only if the forms , belong to the same orbit of on . For an exact -form , the symplectic homogeneous space is identified with the universal covering of the symplectic homogeneous space , which is the orbit of a point in the co-adjoint representation. If , then the orbit of any point is canonically provided with the structure of a symplectic homogeneous space, and any symplectic homogeneous space of a simply-connected group is isomorphic to the covering over one of these orbits. In particular, is the universal covering of .

Let be a compact symplectic homogeneous space of a simply-connected connected group whose action is locally effective. Then is the direct product of a semi-simple compact group and a solvable group isomorphic to the semi-direct product of an Abelian subgroup and an Abelian normal subgroup, and the symplectic homogeneous space decomposes into the direct product of symplectic homogeneous spaces with automorphism groups and , 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 . 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 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.