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(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 |
---|---|---|---|
p | Grassmannian of complex 2-dimensional subspaces of | ||
p | Grassmannian of oriented real 4-dimensional subspaces of | ||
p | Grassmannian of quaternionic 1-dimensional subspaces of | ||
10 | Space of symmetric subspaces of isometric to | ||
16 | Rosenfeld projective plane over | ||
28 | Space of symmetric subspaces of isomorphic to | ||
7 | Space of the symmetric subspaces of which are isomorphic to | ||
2 | Space 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.
See also
References
- Besse, Arthur L. (2008), Einstein Manifolds, Classics in Mathematics, Berlin: Springer-Verlag, ISBN 978-3-540-74120-6, MR 2371700. Reprint of the 1987 edition.
- Salamon, Simon (1982), "Quaternionic Kähler manifolds", Inventiones Mathematicae, 67 (1): 143–171, doi:10.1007/BF01393378, MR 664330.