Quaternion-Kähler symmetric space

In differential geometry, a quaternion-Kähler symmetric space or Wolf space is a quaternion-Kähler manifold which, as a Riemannian manifold, is a Riemannian symmetric space. Any quaternion-Kähler symmetric space with positive Ricci curvature is compact and simply connected, and is a Riemannian product of quaternion-Kähler symmetric spaces associated to compact simple Lie groups.

For any compact simple Lie group G, there is a unique G/H obtained as a quotient of G by a subgroup

H=K\cdot {\mathrm {Sp}}(1).\,

Here, Sp(1) is the compact form of the SL(2)-triple associated with the highest root of G, and K its centralizer in G. These are classified as follows.

G	H	quaternionic dimension	geometric interpretation
${\mathrm {SU}}(p+2)\,$	${\mathrm {S}}({\mathrm {U}}(p)\times {\mathrm {U}}(2))$	p	Grassmannian of complex 2-dimensional subspaces of ${\mathbb {C}}^{{p+2}}$
${\mathrm {SO}}(p+4)\,$	${\mathrm {SO}}(p)\cdot {\mathrm {SO}}(4)$	p	Grassmannian of oriented real 4-dimensional subspaces of ${\mathbb {R}}^{{p+4}}$
${\mathrm {Sp}}(p+1)\,$	${\mathrm {Sp}}(p)\cdot {\mathrm {Sp}}(1)$	p	Grassmannian of quaternionic 1-dimensional subspaces of ${\mathbb {H}}^{{p+1}}$
$E_{6}\,$	${\mathrm {SU}}(6)\cdot {\mathrm {SU}}(2)$	10	Space of symmetric subspaces of $({\mathbb C}\otimes {\mathbb O})P^{2}$ isometric to $({\mathbb C}\otimes {\mathbb H})P^{2}$
$E_{7}\,$	${\mathrm {Spin}}(12)\cdot {\mathrm {Sp}}(1)$	16	Rosenfeld projective plane $({\mathbb H}\otimes {\mathbb O})P^{2}$ over ${\mathbb H}\otimes {\mathbb O}$
$E_{8}\,$	$E_{7}\cdot {\mathrm {Sp}}(1)$	28	Space of symmetric subspaces of $({\mathbb {O}}\otimes {\mathbb O})P^{2}$ isomorphic to $({\mathbb {H}}\otimes {\mathbb O})P^{2}$
$F_{4}\,$	${\mathrm {Sp}}(3)\cdot {\mathrm {Sp}}(1)$	7	Space of the symmetric subspaces of ${\mathbb {OP}}^{2}$ which are isomorphic to ${\mathbb {HP}}^{2}$
$G_{2}\,$	${\mathrm {SO}}(4)\,$	2	Space of the subalgebras of the octonion algebra $\mathbb {O}$ which are isomorphic to the quaternion algebra $\mathbb {H}$

The twistor spaces of quaternion-Kähler symmetric spaces are the homogeneous holomorphic contact manifolds, classified by Boothby: they are the adjoint varieties of the complex semisimple Lie groups.

These spaces can be obtained by taking a projectivization of a minimal nilpotent orbit of the respective complex Lie group. The holomorphic contact structure is apparent, because the nilpotent orbits of semisimple Lie groups are equipped with the Kirillov-Kostant holomorphic symplectic form. This argument also explains how one can associate a unique Wolf space to each of the simple complex Lie groups.

References

Besse, Arthur L. (2008), Einstein Manifolds, Classics in Mathematics, Berlin: Springer-Verlag, ISBN 978-3-540-74120-6, MR 2371700 . Reprint of the 1987 edition.
Salamon, Simon (1982), "Quaternionic Kähler manifolds", Inventiones Mathematicae, 67 (1): 143–171, MR 664330, doi:10.1007/BF01393378 .

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Quaternion-Kähler symmetric space

See also

References