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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 space|compact]] and [[simply connected]], and is a Riemannian product of quaternion-Kähler symmetric spaces associated to compact [[simple Lie group]]s.
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| For any compact simple Lie group ''G'', there is a unique ''G''/''H'' obtained as a quotient of ''G'' by a subgroup
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| :<math> H = K \cdot \mathrm{Sp}(1).\, </math>
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| 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.
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| {| class="wikitable"
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| |-
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| ! width=10% | ''G''
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| ! ''H''
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| ! width=10% | quaternionic dimension
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| ! geometric interpretation
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| |-
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| | <math>\mathrm{SU}(p+2)\,</math>
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| | <math>\mathrm{S}(\mathrm{U}(p) \times \mathrm{U}(2))</math>
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| | ''p''
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| | [[Grassmannian]] of complex ''2''-dimensional subspaces of <math>\mathbb{C}^{p+2}</math>
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| |-
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| | <math>\mathrm{SO}(p+4)\,</math>
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| | <math>\mathrm{SO}(p) \cdot \mathrm{SO}(4)</math>
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| | ''p''
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| | [[Grassmannian]] of oriented real ''4''-dimensional subspaces of <math>\mathbb{R}^{p+4}</math>
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| |-
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| | <math>\mathrm{Sp}(p+1)\,</math>
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| | <math>\mathrm{Sp}(p) \cdot \mathrm{Sp}(1)</math>
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| | ''p''
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| | [[Grassmannian]] of quaternionic ''1''-dimensional subspaces of <math>\mathbb{H}^{p+1}</math>
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| |-
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| | <math>E_6\,</math>
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| | <math>\mathrm{SU}(6)\cdot\mathrm{SU}(2)</math>
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| | 10
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| | Space of symmetric subspaces of <math>(\mathbb C\otimes\mathbb O)P^2</math> isometric to <math>(\mathbb C\otimes \mathbb H)P^2</math>
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| |-
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| | <math>E_7\,</math>
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| | <math>\mathrm{Spin}(12)\cdot\mathrm{Sp}(1)</math>
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| | 16
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| | [[Rosenfeld projective plane]] <math>(\mathbb H\otimes\mathbb O)P^2</math> over <math>\mathbb H\otimes\mathbb O</math>
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| |-
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| | <math>E_8\,</math>
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| | <math>E_7\cdot\mathrm{Sp}(1)</math>
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| | 28
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| | Space of symmetric subspaces of <math>(\mathbb{O}\otimes\mathbb O)P^2</math> isomorphic to <math>(\mathbb{H}\otimes\mathbb O)P^2</math>
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| |-
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| | <math>F_4\,</math>
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| | <math>\mathrm{Sp}(3)\cdot\mathrm{Sp}(1)</math>
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| | Space of the symmetric subspaces of <math>\mathbb{OP}^2</math> which are isomorphic to <math>\mathbb{HP}^2</math>
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| |-
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| | <math>G_2\,</math>
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| | <math>\mathrm{SO}(4)\,</math>
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| | 2
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| | Space of the subalgebras of the [[octonion|octonion algebra]] <math>\mathbb{O}</math> which are isomorphic to the [[quaternion|quaternion algebra]] <math>\mathbb{H}</math>
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| |}
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| The [[quaternion-Kähler manifold#Twistor spaces|twistor spaces]] of quaternion-Kähler symmetric spaces are the homogeneous holomorphic [[contact manifold]]s, classified by Boothby: they are the [[adjoint variety|adjoint varieties]] of the complex [[semisimple Lie group]]s.
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| These spaces can be obtained taking a [[projectivization]] of
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| a minimal [[nilpotent orbit]] of the respective complex Lie group.
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| The holomorphic contact structure is apparent, because
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| the nilpotent orbits of semisimple Lie groups
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| are equipped with the [[Kirillov-Kostant form|Kirillov-Kostant]] holomorphic symplectic form. This argument also explains how one
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| can associate a unique Wolf space to each of the simple
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| complex Lie groups.
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| ==See also==
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| *[[Quaternionic discrete series representation]]
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| ==References==
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| * Besse, Arthur Lancelot, ''Einstein Manifolds'', Springer-Verlag, New York (1987).
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| * Salamon, Simon, ''Quaternionic Kähler manifolds'', Invent. Math. '''67''' (1982), 143–171.
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| {{DEFAULTSORT:Quaternion-Kahler symmetric space}}
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| [[Category:Differential geometry]]
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| [[Category:Structures on manifolds]]
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| [[Category:Riemannian geometry]]
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| [[Category:Homogeneous spaces]]
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| [[Category:Lie groups]]
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