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

$H=K\cdot \mathrm{S}\mathrm{p}(1).$

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
$\mathrm{S}\mathrm{U}(p+2)$	$\mathrm{S}(\mathrm{U}(p)\times \mathrm{U}(2))$	p	Grassmannian of complex 2-dimensional subspaces of ${\mathbb{ℂ}}^{p+2}$
$\mathrm{S}\mathrm{O}(p+4)$	$\mathrm{S}\mathrm{O}(p)\cdot \mathrm{S}\mathrm{O}(4)$	p	Grassmannian of oriented real 4-dimensional subspaces of ${\mathbb{ℝ}}^{p+4}$
$\mathrm{S}\mathrm{p}(p+1)$	$\mathrm{S}\mathrm{p}(p)\cdot \mathrm{S}\mathrm{p}(1)$	p	Grassmannian of quaternionic 1-dimensional subspaces of ${\mathbb{ℍ}}^{p+1}$
$E_6$	$\mathrm{S}\mathrm{U}(6)\cdot \mathrm{S}\mathrm{U}(2)$	10	Space of symmetric subspaces of $(\mathbb{ℂ}\otimes \mathbb{𝕆})P^2$ isometric to $(\mathbb{ℂ}\otimes \mathbb{ℍ})P^2$
$E_7$	$\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(12)\cdot \mathrm{S}\mathrm{p}(1)$	16	Rosenfeld projective plane $(\mathbb{ℍ}\otimes \mathbb{𝕆})P^2$ over $\mathbb{ℍ}\otimes \mathbb{𝕆}$
$E_8$	$E_7\cdot \mathrm{S}\mathrm{p}(1)$	28	Space of symmetric subspaces of $(\mathbb{𝕆}\otimes \mathbb{𝕆})P^2$ isomorphic to $(\mathbb{ℍ}\otimes \mathbb{𝕆})P^2$
$F_4$	$\mathrm{S}\mathrm{p}(3)\cdot \mathrm{S}\mathrm{p}(1)$	7	Space of the symmetric subspaces of ${\mathbb{𝕆}\mathbb{\mathbb{P}}}^2$ which are isomorphic to ${\mathbb{ℍ}\mathbb{\mathbb{P}}}^2$
$G_2$	$\mathrm{S}\mathrm{O}(4)$	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 . Reprint of the 1987 edition.
• Salamon, Simon (1982), "Quaternionic Kähler manifolds", Inventiones Mathematicae 67 (1): 143–171, doi:10.1007/BF01393378, Bibcode: 1982InMat..67..143S .