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.

See also

• Quaternionic discrete series representation

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, Bibcode:1982InMat..67..143S, doi:10.1007/BF01393378, MR 0664330, S2CID 118575943.