Misplaced Pages

Quaternion-Kähler symmetric space

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.

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 ⋅ S p ( 1 ) . {\displaystyle H=K\cdot \mathrm {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.

G H quaternionic dimension geometric interpretation
S U ( p + 2 ) {\displaystyle \mathrm {SU} (p+2)\,} S ( U ( p ) × U ( 2 ) ) {\displaystyle \mathrm {S} (\mathrm {U} (p)\times \mathrm {U} (2))} p Grassmannian of complex 2-dimensional subspaces of C p + 2 {\displaystyle \mathbb {C} ^{p+2}}
S O ( p + 4 ) {\displaystyle \mathrm {SO} (p+4)\,} S O ( p ) ⋅ S O ( 4 ) {\displaystyle \mathrm {SO} (p)\cdot \mathrm {SO} (4)} p Grassmannian of oriented real 4-dimensional subspaces of R p + 4 {\displaystyle \mathbb {R} ^{p+4}}
S p ( p + 1 ) {\displaystyle \mathrm {Sp} (p+1)\,} S p ( p ) ⋅ S p ( 1 ) {\displaystyle \mathrm {Sp} (p)\cdot \mathrm {Sp} (1)} p Grassmannian of quaternionic 1-dimensional subspaces of H p + 1 {\displaystyle \mathbb {H} ^{p+1}}
E 6 {\displaystyle E_{6}\,} S U ( 6 ) ⋅ S U ( 2 ) {\displaystyle \mathrm {SU} (6)\cdot \mathrm {SU} (2)} 10 Space of symmetric subspaces of ( C ⊗ O ) P 2 {\displaystyle (\mathbb {C} \otimes \mathbb {O} )P^{2}} isometric to ( C ⊗ H ) P 2 {\displaystyle (\mathbb {C} \otimes \mathbb {H} )P^{2}}
E 7 {\displaystyle E_{7}\,} S p i n ( 12 ) ⋅ S p ( 1 ) {\displaystyle \mathrm {Spin} (12)\cdot \mathrm {Sp} (1)} 16 Rosenfeld projective plane ( H ⊗ O ) P 2 {\displaystyle (\mathbb {H} \otimes \mathbb {O} )P^{2}} over H ⊗ O {\displaystyle \mathbb {H} \otimes \mathbb {O} }
E 8 {\displaystyle E_{8}\,} E 7 ⋅ S p ( 1 ) {\displaystyle E_{7}\cdot \mathrm {Sp} (1)} 28 Space of symmetric subspaces of ( O ⊗ O ) P 2 {\displaystyle (\mathbb {O} \otimes \mathbb {O} )P^{2}} isomorphic to ( H ⊗ O ) P 2 {\displaystyle (\mathbb {H} \otimes \mathbb {O} )P^{2}}
F 4 {\displaystyle F_{4}\,} S p ( 3 ) ⋅ S p ( 1 ) {\displaystyle \mathrm {Sp} (3)\cdot \mathrm {Sp} (1)} 7 Space of the symmetric subspaces of O P 2 {\displaystyle \mathbb {OP} ^{2}} which are isomorphic to H P 2 {\displaystyle \mathbb {HP} ^{2}}
G 2 {\displaystyle G_{2}\,} S O ( 4 ) {\displaystyle \mathrm {SO} (4)\,} 2 Space of the subalgebras of the octonion algebra O {\displaystyle \mathbb {O} } which are isomorphic to the quaternion algebra H {\displaystyle \mathbb {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 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

Categories: