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 $\omega$:

$$X\cdot \omega =d{i}_X\omega =0,$$

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. $i_X\omega =d{H}_X$, 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 $M_{\alpha }=({\mathrm{Ad}}^{\ast }G)\alpha$ of the Lie group $G$ relative to its co-adjoint representation ${\mathrm{Ad}}^{\ast }G$ in the space ${\mathfrak{g}}^{\ast }$ of linear forms on $\mathfrak{g}$, passing through an arbitrary point $\alpha \in {\mathfrak{g}}^{\ast }$. 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

$$\mu :M\to {\mathfrak{g}}^{\ast },x\mapsto {\mu }_x,{\mu }_x(X)=H_X(x),$$

which maps $M$ onto the orbit $\mu (M)$ of $G$ in ${\mathfrak{g}}^{\ast }$ 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 $O=e{K}^{\sigma }$ 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 $\le 4$ are solvable, but from dimension 6 onwards there are unsolvable symplectic group spaces [3].

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