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 ⋅ 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.
| width=10% | G | H | width=10% | quaternionic dimension | geometric interpretation |
|---|---|---|---|---|---|
SU(p+2) | S(U(p) x U(2)) | p | Grassmannian of complex 2-dimensional subspaces of Cp+2 | ||
SO(p+4) | SO(p) ⋅ SO(4) | p | Grassmannian of oriented real 4-dimensional subspaces of Rp+4 | ||
Sp(p+1) | Sp(p) ⋅ Sp(1) | p | Grassmannian of quaternionic 1-dimensional subspaces of Hp+1 | ||
E6 | SU(6) ⋅ SU(2) | 10 | Space of symmetric subspaces of (C ⊗ O)P2 (C ⊗ H)P2 | ||
E7 | Spin(12) ⋅ Sp(1) | 16 | Rosenfeld projective plane (H ⊗ O)P2 H ⊗ O | ||
E8 | E7 ⋅ Sp(1) | 28 | Space of symmetric subspaces of (O ⊗ O)P2 (H ⊗ O)P2 | ||
F4 | Sp(3) ⋅ Sp(1) | 7 | Space of the symmetric subspaces of OP2 HP2 | ||
G2 | SO(4) | 2 | Space of the subalgebras of the octonion algebra O 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 ofa minimal nilpotent orbit of the respective complex Lie group.The holomorphic contact structure is apparent, becausethe nilpotent orbits of semisimple Lie groups are equipped with the Kirillov-Kostant holomorphic symplectic form. This argument also explains how onecan associate a unique Wolf space to each of the simplecomplex Lie groups.