# Symplectic homogeneous space

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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)