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en>Colonies Chris: /* Related topological invariants */sp, date & link fixes; unlinking common words, replaced: Chern-Weil → Chern–Weil using AWB

2014-03-09T22:29:00Z

Related topological invariants: sp, date & link fixes; unlinking common words, replaced: Chern-Weil → Chern–Weil using AWB

← Older revision		Revision as of 00:29, 10 March 2014
Line 1:		Line 1:
	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~~.		[http://www.edmposts.com/build-a-beautiful-organic-garden-using-these-ideas/ good psychic] Andrew Berryhill is what his wife enjoys to contact him and he totally digs that title. My working day job is an invoicing officer but I've currently utilized for an additional one. My spouse and I reside in Mississippi and I adore every day residing right here. I am truly fond of to go to karaoke but I've been using on new issues recently.<br><br>my blog post :: love [http://www.prayerarmor.com/uncategorized/dont-know-which-kind-of-hobby-to-take-up-read-the-following-tips/ real psychic readings] readings ([https://www-ocl.gist.ac.kr/work/xe/?document_srl=605236 click the next website page])

	~~For any compact simple Lie group ''G'', there is a unique ''G''/''H'' obtained as a quotient of ''G'' by a subgroup~~

	:<~~math~~> ~~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~~.

	~~{\| class="wikitable"~~
	\|-
	~~! width=10% \| ''G''~~
	~~! ''H''~~
	~~! width=10% \| quaternionic dimension~~
	~~! geometric interpretation~~
	\|-
	~~\| <math>\mathrm{SU}(p+2)\,<~~/~~math>~~
	~~\| <math>\mathrm{S}(\mathrm{U}(p) \times \mathrm{U}(2))<~~/~~math>~~
	~~\| ''p''~~
	~~\| [[Grassmannian]] of complex ''2''~~-~~dimensional subspaces of <math>\mathbb{C}^{p+2}</math>~~
	\|-
	~~\| <math>\mathrm{SO}(p+4)\,</math>~~
	~~\| <math>\mathrm{SO}(p) \cdot \mathrm{SO}(4)</math>~~
	~~\| ''p''~~
	~~\| [[Grassmannian]] of oriented real ''4''~~-~~dimensional subspaces of <math>\mathbb{R}^{p+4}</math>~~
	\|-
	~~\| <math>\mathrm{Sp}(p+1)\,</math>~~
	~~\| <math>\mathrm{Sp}(p) \cdot \mathrm{Sp}(1)</math>~~
	~~\| ''p''~~
	~~\| [[Grassmannian]]~~ of ~~quaternionic ''1''~~-~~dimensional subspaces of <math>\mathbb{H}^{p+1}</math>~~
	\|-
	~~\| <math>E_6\,</math>~~
	~~\| <math>\mathrm{SU}(6)\cdot\mathrm{SU}(2)</math>~~
	~~\| 10~~
	~~\| 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>~~
	\|-
	~~\| <math>E_7\,</math>~~
	~~\| <math>\mathrm{Spin}(12)\cdot\mathrm{Sp}(1)</math>~~
	~~\| 16~~
	~~\| [[Rosenfeld projective plane]] <math>(\mathbb H\otimes\mathbb O)P^2</math> over <math>\mathbb H\otimes\mathbb O</math>~~
	\|-
	~~\| <math>E_8\,</math>~~
	~~\| <math>E_7\cdot\mathrm{Sp}(1)</math>~~
	~~\| 28~~
	~~\| 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>~~
	\|-
	~~\| <math>F_4\,</math>~~
	~~\| <math>\mathrm{Sp}(3)\cdot\mathrm{Sp}(1)</math>~~
	~~\| 7~~
	~~\| Space of~~ the ~~symmetric subspaces of <math>\mathbb{OP}^2</math> which are isomorphic to <math>\mathbb{HP}^2</math>~~
	\|-
	~~\| <math>G_2\,<~~/~~math>~~
	~~\| <math>\mathrm{SO}~~(~~4)\,</math>~~
	~~\| 2~~
	~~\| 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>~~
	\|}

	~~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~~.

	~~These spaces can be obtained 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 form\|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==~~

	*[[Quaternionic discrete series representation]]

	~~==References==~~
	* Besse, Arthur Lancelot, ''Einstein Manifolds'', Springer-Verlag, New York (1987).
	* Salamon, Simon, ''Quaternionic Kähler manifolds'', Invent. Math. '''67''' (1982)~~, 143–171.~~

	~~{{DEFAULTSORT:Quaternion-Kahler symmetric space}}~~
	~~[[Category:Differential geometry]]~~
	~~[[Category:Structures on manifolds]]~~
	~~[[Category:Riemannian geometry]]~~
	~~[[Category:Homogeneous spaces]]~~
	~~[[Category:Lie groups]]~~

en>Mark viking: invariant -> topological invariant, for clarity

2013-10-14T20:24:35Z

invariant -> topological invariant, for clarity

← Older revision		Revision as of 22:24, 14 October 2013
Line 1:		Line 1:
	~~The writer~~ is ~~known~~ as ~~Wilber Pegues~~. ~~Since I was 18 I~~'~~ve been operating~~ as a ~~bookkeeper but quickly my wife and I will start our own business~~. ~~My spouse~~ and ~~I reside~~ in ~~Mississippi and I love each day living right here~~. ~~The preferred pastime for him and his children is to play lacross and he~~'~~ll be starting some thing else along with it.~~<br><br>~~Feel~~ [~~http:~~//m-~~card.co.kr~~/xe/~~mcard_2013_promote01~~/~~29877 free psychic~~] to ~~surf to my page :: telephone [http:~~//~~www.taehyuna.net~~/~~xe/?document_srl=78721 psychic readings]~~ [[~~http:~~//~~www.herandkingscounty.com~~/~~content~~/~~information~~-~~and~~-~~facts~~-~~you~~-~~must~~-~~know~~-~~about~~-~~hobbies http://www~~.~~herandkingscounty~~.~~com/~~]]		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.

			For any compact simple Lie group ''G'', there is a unique ''G''/''H'' obtained as a quotient of ''G'' by a subgroup

			:<math> 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.

			{\| class="wikitable"
			\|-
			! width=10% \| ''G''
			! ''H''
			! width=10% \| quaternionic dimension
			! geometric interpretation
			\|-
			\| <math>\mathrm{SU}(p+2)\,</math>
			\| <math>\mathrm{S}(\mathrm{U}(p) \times \mathrm{U}(2))</math>
			\| ''p''
			\| [[Grassmannian]] of complex ''2''-dimensional subspaces of <math>\mathbb{C}^{p+2}</math>
			\|-
			\| <math>\mathrm{SO}(p+4)\,</math>
			\| <math>\mathrm{SO}(p) \cdot \mathrm{SO}(4)</math>
			\| ''p''
			\| [[Grassmannian]] of oriented real ''4''-dimensional subspaces of <math>\mathbb{R}^{p+4}</math>
			\|-
			\| <math>\mathrm{Sp}(p+1)\,</math>
			\| <math>\mathrm{Sp}(p) \cdot \mathrm{Sp}(1)</math>
			\| ''p''
			\| [[Grassmannian]] of quaternionic ''1''-dimensional subspaces of <math>\mathbb{H}^{p+1}</math>
			\|-
			\| <math>E_6\,</math>
			\| <math>\mathrm{SU}(6)\cdot\mathrm{SU}(2)</math>
			\| 10
			\| 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>
			\|-
			\| <math>E_7\,</math>
			\| <math>\mathrm{Spin}(12)\cdot\mathrm{Sp}(1)</math>
			\| 16
			\| [[Rosenfeld projective plane]] <math>(\mathbb H\otimes\mathbb O)P^2</math> over <math>\mathbb H\otimes\mathbb O</math>
			\|-
			\| <math>E_8\,</math>
			\| <math>E_7\cdot\mathrm{Sp}(1)</math>
			\| 28
			\| 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>
			\|-
			\| <math>F_4\,</math>
			\| <math>\mathrm{Sp}(3)\cdot\mathrm{Sp}(1)</math>
			\| 7
			\| Space of the symmetric subspaces of <math>\mathbb{OP}^2</math> which are isomorphic to <math>\mathbb{HP}^2</math>
			\|-
			\| <math>G_2\,</math>
			\| <math>\mathrm{SO}(4)\,</math>
			\| 2
			\| 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>
			\|}

			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.

			These spaces can be obtained 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 form\|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==

			*[[Quaternionic discrete series representation]]

			==References==
			* Besse, Arthur Lancelot, ''Einstein Manifolds'', Springer-Verlag, New York (1987).
			* Salamon, Simon, ''Quaternionic Kähler manifolds'', Invent. Math. '''67''' (1982), 143–171.

			{{DEFAULTSORT:Quaternion-Kahler symmetric space}}
			[[Category:Differential geometry]]
			[[Category:Structures on manifolds]]
			[[Category:Riemannian geometry]]
			[[Category:Homogeneous spaces]]
			[[Category:Lie groups]]

en>LarRan at 16:14, 7 January 2010

2010-01-07T16:14:14Z

New page

The writer is known as Wilber Pegues. Since I was 18 I've been operating as a bookkeeper but quickly my wife and I will start our own business. My spouse and I reside in Mississippi and I love each day living right here. The preferred pastime for him and his children is to play lacross and he'll be starting some thing else along with it.<br><br>Feel [http://m-card.co.kr/xe/mcard_2013_promote01/29877 free psychic] to surf to my page :: telephone [http://www.taehyuna.net/xe/?document_srl=78721 psychic readings] [[http://www.herandkingscounty.com/content/information-and-facts-you-must-know-about-hobbies http://www.herandkingscounty.com/]]

← Older revision		Revision as of 00:29, 10 March 2014
Line 1:		Line 1:
	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~~.		[http://www.edmposts.com/build-a-beautiful-organic-garden-using-these-ideas/ good psychic] Andrew Berryhill is what his wife enjoys to contact him and he totally digs that title. My working day job is an invoicing officer but I've currently utilized for an additional one. My spouse and I reside in Mississippi and I adore every day residing right here. I am truly fond of to go to karaoke but I've been using on new issues recently.<br><br>my blog post :: love [http://www.prayerarmor.com/uncategorized/dont-know-which-kind-of-hobby-to-take-up-read-the-following-tips/ real psychic readings] readings ([https://www-ocl.gist.ac.kr/work/xe/?document_srl=605236 click the next website page])

	~~For any compact simple Lie group ''G'', there is a unique ''G''/''H'' obtained as a quotient of ''G'' by a subgroup~~

	:<~~math~~> ~~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~~.

	~~{\| class="wikitable"~~
	\|-
	~~! width=10% \| ''G''~~
	~~! ''H''~~
	~~! width=10% \| quaternionic dimension~~
	~~! geometric interpretation~~
	\|-
	~~\| <math>\mathrm{SU}(p+2)\,<~~/~~math>~~
	~~\| <math>\mathrm{S}(\mathrm{U}(p) \times \mathrm{U}(2))<~~/~~math>~~
	~~\| ''p''~~
	~~\| [[Grassmannian]] of complex ''2''~~-~~dimensional subspaces of <math>\mathbb{C}^{p+2}</math>~~
	\|-
	~~\| <math>\mathrm{SO}(p+4)\,</math>~~
	~~\| <math>\mathrm{SO}(p) \cdot \mathrm{SO}(4)</math>~~
	~~\| ''p''~~
	~~\| [[Grassmannian]] of oriented real ''4''~~-~~dimensional subspaces of <math>\mathbb{R}^{p+4}</math>~~
	\|-
	~~\| <math>\mathrm{Sp}(p+1)\,</math>~~
	~~\| <math>\mathrm{Sp}(p) \cdot \mathrm{Sp}(1)</math>~~
	~~\| ''p''~~
	~~\| [[Grassmannian]]~~ of ~~quaternionic ''1''~~-~~dimensional subspaces of <math>\mathbb{H}^{p+1}</math>~~
	\|-
	~~\| <math>E_6\,</math>~~
	~~\| <math>\mathrm{SU}(6)\cdot\mathrm{SU}(2)</math>~~
	~~\| 10~~
	~~\| 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>~~
	\|-
	~~\| <math>E_7\,</math>~~
	~~\| <math>\mathrm{Spin}(12)\cdot\mathrm{Sp}(1)</math>~~
	~~\| 16~~
	~~\| [[Rosenfeld projective plane]] <math>(\mathbb H\otimes\mathbb O)P^2</math> over <math>\mathbb H\otimes\mathbb O</math>~~
	\|-
	~~\| <math>E_8\,</math>~~
	~~\| <math>E_7\cdot\mathrm{Sp}(1)</math>~~
	~~\| 28~~
	~~\| 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>~~
	\|-
	~~\| <math>F_4\,</math>~~
	~~\| <math>\mathrm{Sp}(3)\cdot\mathrm{Sp}(1)</math>~~
	~~\| 7~~
	~~\| Space of~~ the ~~symmetric subspaces of <math>\mathbb{OP}^2</math> which are isomorphic to <math>\mathbb{HP}^2</math>~~
	\|-
	~~\| <math>G_2\,<~~/~~math>~~
	~~\| <math>\mathrm{SO}~~(~~4)\,</math>~~
	~~\| 2~~
	~~\| 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>~~
	\|}

	~~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~~.

	~~These spaces can be obtained 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 form\|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==~~

	*[[Quaternionic discrete series representation]]

	~~==References==~~
	* Besse, Arthur Lancelot, ''Einstein Manifolds'', Springer-Verlag, New York (1987).
	* Salamon, Simon, ''Quaternionic Kähler manifolds'', Invent. Math. '''67''' (1982)~~, 143–171.~~

	~~{{DEFAULTSORT:Quaternion-Kahler symmetric space}}~~
	~~[[Category:Differential geometry]]~~
	~~[[Category:Structures on manifolds]]~~
	~~[[Category:Riemannian geometry]]~~
	~~[[Category:Homogeneous spaces]]~~
	~~[[Category:Lie groups]]~~