en>Omnipaedista: expand 'see also' section

2014-12-21T17:40:42Z

expand 'see also' section

en>Len loker: /* Orbifolds in string theory */

2013-12-24T04:33:34Z

Orbifolds in string theory

← Older revision		Revision as of 06:33, 24 December 2013
Line 1:		Line 1:
	~~Roberto~~ is ~~what~~'~~s written concerning his birth certificate rather he never really preferred that name~~. ~~South Carolina~~ is ~~michael~~'~~s birth place~~. The ~~favorite hobby~~ for ~~him~~ and ~~his kids~~ is so that ~~you can fish~~ and he's ~~resulted in~~ being ~~doing it~~ for a ~~long time~~. [~~http~~://En.~~Search~~.~~wordpress~~.~~com~~/?q=~~Auditing Auditing~~] is ~~how he supports~~ the ~~puppy~~'~~s family~~. Go for ~~his website~~ to ~~find out more~~: ~~http~~://~~prometeu~~.~~net~~<br><br>~~Here~~ is ~~my web site~~ - [~~http:~~/~~/prometeu~~.~~net clash~~ of ~~clans cheat no survey~~]		In [[mathematics]], the '''indefinite orthogonal group''', O(''p'',''q'') is the [[Lie group]] of all [[linear transformation]]s of a ''n''-[[dimension of a vector space\|dimensional]] real [[vector space]] which leave invariant a [[nondegenerate form\|nondegenerate]], [[symmetric bilinear form]] of [[signature of a quadratic form\|signature]] (''p'',''q''), where {{nowrap\|1=''n'' = ''p'' + ''q''}}. The dimension of the group is {{nowrap\|''n''(''n'' − 1)/2}}.

			The '''indefinite special orthogonal group''', SO(''p'',''q'') is the [[subgroup]] of O(''p'',''q'') consisting of all elements with [[determinant]] 1. Unlike in the definite case, SO(''p'',''q'') is not connected – it has 2 components – and there are two additional finite index subgroups, namely the connected SO<sup>+</sup>(''p'',''q'') and O<sup>+</sup>(''p'',''q''), which has 2 components – see the [[#Topology\|topology section]] for definition and discussion.

			The signature of the form determines the group up to [[isomorphism]]; interchanging ''p'' with ''q'' amounts to replacing the metric by its negative, and so gives the same group. If either ''p'' or ''q'' equals zero, then the group is isomorphic to the ordinary [[orthogonal group]] O(''n''). We assume in what follows that both ''p'' and ''q'' are positive.

			The group O(''p'',''q'') is defined for vector spaces over the [[real number\|real]]s. For [[Complex number\|complex]] spaces, all groups {{nowrap\|O(''p'',''q''; '''C''')}} are isomorphic to the usual [[orthogonal group]] {{nowrap\|O(''p'' + ''q''; '''C''')}}, since the transform <math>z_j \mapsto iz_j</math> changes the signature of a form.

			In even dimension, the middle group O(''n'',''n'') is known as the [[#Split orthogonal group\|split orthogonal group]], and is of particular interest. In odd dimension, split form is the almost-middle group {{nowrap\|O(''n'',''n'' + 1)}}.

			== Examples ==
			[[File:Squeeze r=1.5.svg\|thumb\|[[Squeeze mapping]]s, here ''r'' = 3/2, are the basic hyperbolic symmetries.]]
			The basic example is the [[squeeze mapping]]s, which is the group SO<sup>+</sup>(1,1) of (the identity component of) linear transforms preserving the [[unit hyperbola]]. Concretely, these are the matrices <math>\left[\begin{smallmatrix} \lambda & 0 \\ 0 & 1/\lambda\end{smallmatrix}\right],</math> and can be interpreted as ''hyperbolic rotations,'' just as the group SO(2) can be interpreted as ''circular rotations.''

			In physics, the [[Lorentz group]] O(1,3) is of central importance, being the setting for [[electromagnetism]] and [[special relativity]].

			==Matrix definition==
			One can define O(''p'',''q'') as a group of [[matrix (mathematics)\|matrices]], just as for the classical orthogonal group O(''n''). The standard inner product on '''R'''<sup>''p'',''q''</sup> is given in coordinates by the [[diagonal matrix]]:
			:<math>\eta = \mathrm{diag}(\underbrace{1,\cdots,1}_{p},\underbrace{-1,\cdots,-1}_{q}).\,</math>
			As a quadratic form,
			<math>Q(x_1,\dots,x_n) = x_1^2 + \cdots + x_p^2 - x_{p+1}^2 - \cdots - x_{p+q}^2.</math>

			The group O(''p'',''q'') is then the group of a ''n''×''n'' matrices ''M'' (where ''n'' = ''p''+''q'') such that <math>Q(Mv)=Q(v)</math>; as a bilinear form,
			:<math>M^T\eta M = \eta.\,</math>
			Here ''M''<sup>''T''</sup> denotes the [[transpose]] of the matrix ''M''. One can easily verify that the set of all such matrices forms a group. The inverse of ''M'' is given by
			:<math>M^{-1} = \eta^{-1}M^T\eta.\,</math>

			One obtains an isomorphic group (indeed, a conjugate subgroup of GL(V)) by replacing η with any [[symmetric matrix]] with ''p'' positive eigenvalues and ''q'' negative ones (such a matrix is necessarily [[nonsingular matrix\|nonsingular]]); equivalently, any quadratic form with signature (''p'',''q''). Diagonalizing this matrix gives a conjugation of this group with the standard group O(''p'',''q'').

			==Topology==
			Assuming both ''p'' and ''q'' are nonzero, neither of the groups O(''p'',''q'') or SO(''p'',''q'') are [[connected space\|connected]], having four and two components respectively.
			{{nowrap\|1=''π''<sub>0</sub>(O(''p'',''q'')) &cong; ''C''<sub>2</sub> × ''C''<sub>2</sub>}} is the [[Klein four-group]], with each factor being whether an element preserves or reverses the respective orientations on the ''p'' and ''q'' dimensional subspaces on which the form is definite; note that reserving orientation on only one of these subspaces reverses orientation on the whole space. The special orthogonal group has components {{nowrap\|1=''π''<sub>0</sub>(SO(''p'',''q'')) = {(1,1),(−1,−1)}}} which either preserves both orientations or reverses both orientations, in either case preserving the overall orientation.

			The [[identity component]] of O(''p'',''q'') is often denoted SO<sup>+</sup>(''p'',''q'') and can be identified with the set of elements in SO(''p'',''q'') which preserves both orientations. This notation is related to the notation O<sup>+</sup>(1,3) for the [[orthochronous Lorentz group]], where the + refers to preserving the orientation on the first (temporal) dimension.

			The group O(''p'',''q'') is also not [[compact space\|compact]], but contains the compact subgroups O(''p'') and O(''q'') acting on the subspaces on which the form is definite. In fact, {{nowrap\|1=O(''p'') × O(''q'')}} is a [[maximal compact subgroup]] of O(''p'',''q''), while {{nowrap\|1=S(O(''p'') × O(''q''))}} is a maximal compact subgroup of SO(''p'',''q'').
			Likewise, {{nowrap\|1=SO(''p'') × SO(''q'')}} is a maximal compact subgroup of SO<sup>+</sup>(''p'', ''q'').
			Thus up to homotopy, the spaces are products of (special) orthogonal groups, from which algebro-topological invariants can be computed.

			In particular, the [[fundamental group]] of SO<sup>+</sup>(''p'',''q'') is the product of the fundamental groups of the components, {{nowrap\|1=''π''<sub>1</sub>(SO<sup>+</sup>(''p'',''q'')) = ''π''<sub>1</sub>(SO(''p'')) × ''π''<sub>1</sub>(SO(''q''))}}, and is given by:
			:{\| border="1" cellpadding="11" style="border-collapse: collapse; border: 1px #aaa solid;"
			!style="background:#efefef;"\| ''π''<sub>1</sub>(SO<sup>+</sup>(''p'',''q''))
			!style="background:#efefef;"\| ''p'' = 1
			!style="background:#efefef;"\| ''p'' = 2
			!style="background:#efefef;"\| ''p'' ≥ 3
			\|-
			!style="background:#efefef;"\| ''q'' = 1
			\| {1} \|\| '''Z''' \|\| '''Z'''<sub>2</sub>
			\|-
			!style="background:#efefef;"\| ''q'' = 2
			\| '''Z''' \|\| '''Z''' × '''Z'''
			\|\| '''Z''' × '''Z'''<sub>2</sub>
			\|-
			!style="background:#efefef;"\| ''q'' ≥ 3
			\| '''Z'''<sub>2</sub> \|\| '''Z'''<sub>2</sub> × '''Z'''
			\|\| '''Z'''<sub>2</sub> × '''Z'''<sub>2</sub>
			\|}

			==Split orthogonal group==
			In even dimension, the middle group O(''n'',''n'') is known as the '''split orthogonal group''', and is of particular interest. It is the [[split Lie group]] corresponding to the complex [[Lie algebra]] so<sub>2''n''</sub> (the Lie group of the [[split real form]] of the Lie algebra); more precisely, the identity component is the split Lie group, as non-identity components cannot be reconstructed from the Lie algebra. In this sense it is opposite to the definite orthogonal group O(''n'') := O(''n'',0) = O(0,''n''), which is the [[compact real form\|''compact'' real form]] of the complex Lie algebra.

			The case (1,1) corresponds to the [[split-complex number]]s.

			In terms of being a [[group of Lie type]] – i.e., construction of an algebraic group from a Lie algebra – split orthogonal groups are [[Chevalley group]]s, while the non-split orthogonal groups require a slightly more complicated construction, and are [[Steinberg group (Lie theory)\|Steinberg groups]].

			Split orthogonal groups are used to construct the [[generalized flag variety]] over non-algebraically closed fields.

			{{Expand section\|date=March 2011}}
			In odd dimension, the split form is the almost-middle group O(''n'',''n''+1).

			==See also==
			*[[Pin group]]

			==References==
			{{refbegin}}
			*{{springer\|id=O/o070300\|title=Orthogonal group\|author=[[Vladimir L. Popov\|V. L. Popov]]}}
			*[[Anthony Knapp]], ''Lie Groups Beyond an Introduction'', Second Edition, Progress in Mathematics, vol. 140, Birkhäuser, Boston, 2002. ISBN 0-8176-4259-5 – see page 372 for a description of the indefinite orthogonal group
			*[[Joseph A. Wolf]], ''Spaces of constant curvature'', (1967) page. 335.
			{{refend}}

			[[Category:Lie groups]]

en>Gaius Cornelius: Format dates and tidy using AWB

2012-05-05T06:33:06Z

Format dates and tidy using AWB

New page

Roberto is what's written concerning his birth certificate rather he never really preferred that name. South Carolina is michael's birth place. The favorite hobby for him and his kids is so that you can fish and he's resulted in being doing it for a long time. [http://En.Search.wordpress.com/?q=Auditing Auditing] is how he supports the puppy's family. Go for his website to find out more: http://prometeu.net<br><br>Here is my web site - [http://prometeu.net clash of clans cheat no survey]

Orbifold - Revision history

en>Omnipaedista: expand 'see also' section

en>Len loker: /* Orbifolds in string theory */

en>Gaius Cornelius: Format dates and tidy using AWB