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
- [math]\displaystyle{ H = K \cdot \mathrm{Sp}(1).\, }[/math]
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 |
|---|---|---|---|
| [math]\displaystyle{ \mathrm{SU}(p+2)\, }[/math] | [math]\displaystyle{ \mathrm{S}(\mathrm{U}(p) \times \mathrm{U}(2)) }[/math] | p | Grassmannian of complex 2-dimensional subspaces of [math]\displaystyle{ \mathbb{C}^{p+2} }[/math] |
| [math]\displaystyle{ \mathrm{SO}(p+4)\, }[/math] | [math]\displaystyle{ \mathrm{SO}(p) \cdot \mathrm{SO}(4) }[/math] | p | Grassmannian of oriented real 4-dimensional subspaces of [math]\displaystyle{ \mathbb{R}^{p+4} }[/math] |
| [math]\displaystyle{ \mathrm{Sp}(p+1)\, }[/math] | [math]\displaystyle{ \mathrm{Sp}(p) \cdot \mathrm{Sp}(1) }[/math] | p | Grassmannian of quaternionic 1-dimensional subspaces of [math]\displaystyle{ \mathbb{H}^{p+1} }[/math] |
| [math]\displaystyle{ E_6\, }[/math] | [math]\displaystyle{ \mathrm{SU}(6)\cdot\mathrm{SU}(2) }[/math] | 10 | Space of symmetric subspaces of [math]\displaystyle{ (\mathbb C\otimes\mathbb O)P^2 }[/math] isometric to [math]\displaystyle{ (\mathbb C\otimes \mathbb H)P^2 }[/math] |
| [math]\displaystyle{ E_7\, }[/math] | [math]\displaystyle{ \mathrm{Spin}(12)\cdot\mathrm{Sp}(1) }[/math] | 16 | Rosenfeld projective plane [math]\displaystyle{ (\mathbb H\otimes\mathbb O)P^2 }[/math] over [math]\displaystyle{ \mathbb H\otimes\mathbb O }[/math] |
| [math]\displaystyle{ E_8\, }[/math] | [math]\displaystyle{ E_7\cdot\mathrm{Sp}(1) }[/math] | 28 | Space of symmetric subspaces of [math]\displaystyle{ (\mathbb{O}\otimes\mathbb O)P^2 }[/math] isomorphic to [math]\displaystyle{ (\mathbb{H}\otimes\mathbb O)P^2 }[/math] |
| [math]\displaystyle{ F_4\, }[/math] | [math]\displaystyle{ \mathrm{Sp}(3)\cdot\mathrm{Sp}(1) }[/math] | 7 | Space of the symmetric subspaces of [math]\displaystyle{ \mathbb{OP}^2 }[/math] which are isomorphic to [math]\displaystyle{ \mathbb{HP}^2 }[/math] |
| [math]\displaystyle{ G_2\, }[/math] | [math]\displaystyle{ \mathrm{SO}(4)\, }[/math] | 2 | Space of the subalgebras of the octonion algebra [math]\displaystyle{ \mathbb{O} }[/math] which are isomorphic to the quaternion algebra [math]\displaystyle{ \mathbb{H} }[/math] |
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. Reprint of the 1987 edition.
- Salamon, Simon (1982), "Quaternionic Kähler manifolds", Inventiones Mathematicae 67 (1): 143–171, doi:10.1007/BF01393378, Bibcode: 1982InMat..67..143S.
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