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 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, doi:10.1007/BF01393378, MR 0664330.