Minimal volume

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 ${ℂ}^{p+2}$
$\mathrm{S}\mathrm{O}(p+4)$	$\mathrm{S}\mathrm{O}(p)\cdot \mathrm{S}\mathrm{O}(4)$	p	Grassmannian of oriented real 4-dimensional subspaces of ${ℝ}^{p+4}$
$\mathrm{S}\mathrm{p}(p+1)$	$\mathrm{S}\mathrm{p}(p)\cdot \mathrm{S}\mathrm{p}(1)$	p	Grassmannian of quaternionic 1-dimensional subspaces of ${ℍ}^{p+1}$
$E_6$	$\mathrm{S}\mathrm{U}(6)\cdot \mathrm{S}\mathrm{U}(2)$	10	Space of symmetric subspaces of $(ℂ\otimes 𝕆)P^2$ isometric to $(ℂ\otimes ℍ)P^2$
$E_7$	$\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(12)\cdot \mathrm{S}\mathrm{p}(1)$	16	Rosenfeld projective plane $(ℍ\otimes 𝕆)P^2$ over $ℍ\otimes 𝕆$
$E_8$	$E_7\cdot \mathrm{S}\mathrm{p}(1)$	28	Space of symmetric subspaces of $(𝕆\otimes 𝕆)P^2$ isomorphic to $(ℍ\otimes 𝕆)P^2$
$F_4$	$\mathrm{S}\mathrm{p}(3)\cdot \mathrm{S}\mathrm{p}(1)$	7	Space of the symmetric subspaces of ${𝕆\mathbb{P}}^2$ which are isomorphic to ${ℍ\mathbb{P}}^2$
$G_2$	$\mathrm{S}\mathrm{O}(4)$	2	Space of the subalgebras of the octonion algebra $𝕆$ which are isomorphic to the quaternion algebra $ℍ$

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 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 Lancelot, Einstein Manifolds, Springer-Verlag, New York (1987).
• Salamon, Simon, Quaternionic Kähler manifolds, Invent. Math. 67 (1982), 143–171.