Wolf space

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

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

$H = K \times SU(2)$ .

Here, $S U (2)$ is the $S L (2)$ -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 $\mathbb{C}^{p+2}$
$\mathrm{SO}(p+4)/\mathrm{SO}(p) \times \mathrm{SO}(4)$	$p$	Grassmannian of oriented real 4-dimensional subspaces of $\mathbb{R}^{p+q}$
$\mathrm{Sp}(p+1)/\mathrm{Sp}(p) \times \mathrm{Sp}(1)$	$p$	Grassmannian of quaternionic 1-dimensional subspaces of $\mathbb{H}^{p+1}$
$E_6/(\mathrm{SU}(6)\times\mathrm{SU}(2))$	10
$E_7/\mathrm{SO}(12)\times\mathrm{SU}(2)$	16
$E_8/E_7\times\mathrm{SU}(2)$	28
$F_4/\mathrm{Sp}(3)\times\mathrm{SU}(2)$	7	Space of the symmetric subspaces of $\mathbb{OP}^2$ which are isomorphic to $\mathbb{HP}^2$
$G_2/\mathrm{SU}(2)\times\mathrm{SU}(2)$	2	Space of the subalgebras of the octonion algebra $\mathbb{O}$ which are isomorphic to the quaternion algebra $\mathbb{H}$