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

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

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

The twistor spaces of quaternion-Kähler symmetric spaces are the homogeneous holomorphic contact manifolds, classified by Boothby: they are the 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 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, doi:10.1007/BF01393378, MR 0664330.