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
Here, Sp is the compact form of the SL-triple associated with the highest root of G, and K its centralizer in G. These are classified as follows.
GHquaternionic dimensiongeometric interpretation
pGrassmannian of complex 2-dimensional subspaces of
pGrassmannian of oriented real 4-dimensional subspaces of
pGrassmannian of quaternionic 1-dimensional subspaces of
10Space of symmetric subspaces of isometric to
16Rosenfeld projective plane over
28Space of symmetric subspaces of isomorphic to
7Space of the symmetric subspaces of which are isomorphic to
2Space of the subalgebras of the octonion algebra which are isomorphic to the quaternion algebra

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.