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Quaternion-Kähler symmetric space

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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

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References

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  • 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, Bibcode:1982InMat..67..143S, doi:10.1007/BF01393378, MR 0664330, S2CID 118575943.