Regulated integral - Revision history

en>Yobot: WP:CHECKWIKI error fixes using AWB (10093)

2014-05-05T12:25:36Z

WP:CHECKWIKI error fixes using AWB (10093)

← Older revision		Revision as of 14:25, 5 May 2014
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	In mathematics, the '''root datum''' ('''donnée radicielle''' in French) of a connected split [[reductive group\|reductive]] [[algebraic group]] over a field is a generalization of a [[root system]] that determines the group up to isomorphism. They were introduced by [[Michel Demazure]] in [[Grothendieck's Séminaire de géométrie algébrique\|SGA III]], published in 1970.		The writer is called Araceli Gulledge. Delaware is the only location I've been residing in. To play badminton is something he really enjoys doing. Managing people is what I do in my day job.<br><br>Feel free to visit my website ... [http://newdayz.de/index.php?mod=users&action=view&id=16038 newdayz.de]

	~~==Definition==~~
	~~A '''root datum''' consists of a quadruple~~
	~~:<math>(X^\ast, \Phi, X_\ast, \Phi^\vee)</math>,~~
	~~where~~
	* <math>X^\ast</math> and <math>X_\ast</math> are free abelian groups of finite [[Rank (linear algebra)\|rank]] together with a [[perfect pairing]] between them with values in <math>\mathbb{Z}</math> which we denote by ( , ) (in other words, each is identified with the [[dual lattice]] of the other).
	* <math>\Phi</math> is a finite subset of <math>X^\ast</math> and <math>\Phi^\vee</math> is a finite subset of <math>X_\ast</math> and there is a bijection from <math>\Phi</math> onto <math>\Phi^\vee</math>, denoted by <math>\alpha\mapsto\alpha^\vee</math>.
	* For each <math>\alpha</math>, <math>(\alpha, \alpha^\vee)=2</math>.
	* For each <math>\alpha</math>, the map <math>x\mapsto x-(x,\alpha^\vee)\alpha</math> induces an automorphism of the root datum (in other words it maps <math>\Phi</math> to <math>\Phi</math> and the induced action on <math>X_\ast</math> maps <math>\Phi^\vee</math> to <math>\Phi^\vee</math>)

	~~The elements of <math>\Phi</math> are called the '''roots''' of the root datum, and the elements of <math>\Phi^\vee</math> are called the '''coroots'''.~~ The ~~elements of <math>X^\ast</math> are sometimes called '''[[Weight_(representation_theory)\|weights]]''' and those of <math>X_\ast</math> accordingly '''coweights'''.~~

	~~If <math>\Phi</math> does not contain <math>2\alpha</math> for any <math>\alpha\in\Phi</math>, then the root datum~~ is called ~~'''reduced'''~~.

	~~==The root datum of an algebraic group==~~
	~~If ''G'' is a reductive algebraic group over an [[algebraically closed field]] ''K'' with a split maximal torus ''T'' then its '''root datum''' is a quadruple~~
	~~:(''X''<sup></sup>, Φ, ''X''<sub></sub>, Φ<sup>v</sup>),~~
	~~where~~
	''X''<sup></sup> is the ~~lattice of characters of the maximal torus,~~
	'~~'X''<sub></sub> is the dual lattice (given by the 1-parameter subgroups),~~
	*Φ is a set of roots,
	*Φ<sup>v</sup> is the corresponding set of coroots.

	~~A connected split reductive algebraic group over ''K''~~ is ~~uniquely determined (up to isomorphism) by its root datum, which is always reduced~~. ~~Conversely for any root datum there~~ is ~~a reductive algebraic group. A root datum contains slightly more information than the [[Dynkin diagram]], because it also determines the center of the group~~.

	~~For any root datum (''X''~~<~~sup~~><~~/sup~~>, Φ,''X''<sub></sub>, Φ<sup>v</sup>), we can define a '''dual root datum''' (''X''<sub></sub>, Φ<sup>v</sup>,''X''<sup></sup>, Φ) by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.

	If ''G'' is a connected reductive algebraic group over the algebraically closed field ''K'', then its [[Langlands dual group]] <sup>''L''</sup>''G'' is the complex connected reductive group whose root datum is dual to that of ''G''.

	~~==References==~~
	*[[Michel Demazure]], Exp. ~~XXI in~~ [http://~~modular~~.~~fas.harvard.edu/sga/sga/3-3~~/index.~~html SGA 3 vol 3]~~
	*[[T. A. Springer]], [http://www.ams.org/online_bks/pspum331/pspum331-ptI-1.pdf ''Reductive groups''], in [http://www.ams.org/online_bks/pspum331/ ''Automorphic forms, representations, and L-functions'' vol 1] ISBN 0-8218-3347-2
	~~[[Category:Representation theory]]~~
	~~[[Category:Algebraic groups]~~]

en>Mgkrupa: /* References */ Added {{Functional Analysis}} footer

2014-01-10T03:37:17Z

References: Added {{Functional Analysis}} footer

← Older revision		Revision as of 05:37, 10 January 2014
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	~~Golda~~ is ~~what~~'s ~~written on my birth certificate even though it~~ is ~~not~~ the ~~title on my birth certificate~~. As a ~~woman what she truly likes~~ is ~~fashion~~ and ~~she~~'~~s been doing it~~ for ~~quite~~ a ~~while~~. ~~Ohio~~ is ~~exactly where her house~~ is. ~~Distributing production~~ is ~~how he makes~~ a ~~living~~.<br><br>~~Here~~ is ~~my homepage phone psychic -~~ [http://~~Fashionlinked~~.~~com~~/index.~~php?do=~~/~~profile-13453~~/~~info~~/ http://~~Fashionlinked~~.~~com~~/~~index.php?do=~~/~~profile~~-~~13453/info~~] -		In mathematics, the '''root datum''' ('''donnée radicielle''' in French) of a connected split [[reductive group\|reductive]] [[algebraic group]] over a field is a generalization of a [[root system]] that determines the group up to isomorphism. They were introduced by [[Michel Demazure]] in [[Grothendieck's Séminaire de géométrie algébrique\|SGA III]], published in 1970.

			==Definition==
			A '''root datum''' consists of a quadruple
			:<math>(X^\ast, \Phi, X_\ast, \Phi^\vee)</math>,
			where
			* <math>X^\ast</math> and <math>X_\ast</math> are free abelian groups of finite [[Rank (linear algebra)\|rank]] together with a [[perfect pairing]] between them with values in <math>\mathbb{Z}</math> which we denote by ( , ) (in other words, each is identified with the [[dual lattice]] of the other).
			* <math>\Phi</math> is a finite subset of <math>X^\ast</math> and <math>\Phi^\vee</math> is a finite subset of <math>X_\ast</math> and there is a bijection from <math>\Phi</math> onto <math>\Phi^\vee</math>, denoted by <math>\alpha\mapsto\alpha^\vee</math>.
			* For each <math>\alpha</math>, <math>(\alpha, \alpha^\vee)=2</math>.
			* For each <math>\alpha</math>, the map <math>x\mapsto x-(x,\alpha^\vee)\alpha</math> induces an automorphism of the root datum (in other words it maps <math>\Phi</math> to <math>\Phi</math> and the induced action on <math>X_\ast</math> maps <math>\Phi^\vee</math> to <math>\Phi^\vee</math>)

			The elements of <math>\Phi</math> are called the '''roots''' of the root datum, and the elements of <math>\Phi^\vee</math> are called the '''coroots'''. The elements of <math>X^\ast</math> are sometimes called '''[[Weight_(representation_theory)\|weights]]''' and those of <math>X_\ast</math> accordingly '''coweights'''.

			If <math>\Phi</math> does not contain <math>2\alpha</math> for any <math>\alpha\in\Phi</math>, then the root datum is called '''reduced'''.

			==The root datum of an algebraic group==
			If ''G'' is a reductive algebraic group over an [[algebraically closed field]] ''K'' with a split maximal torus ''T'' then its '''root datum''' is a quadruple
			:(''X''<sup></sup>, Φ, ''X''<sub></sub>, Φ<sup>v</sup>),
			where
			''X''<sup></sup> is the lattice of characters of the maximal torus,
			''X''<sub></sub> is the dual lattice (given by the 1-parameter subgroups),
			*Φ is a set of roots,
			*Φ<sup>v</sup> is the corresponding set of coroots.

			A connected split reductive algebraic group over ''K'' is uniquely determined (up to isomorphism) by its root datum, which is always reduced. Conversely for any root datum there is a reductive algebraic group. A root datum contains slightly more information than the [[Dynkin diagram]], because it also determines the center of the group.

			For any root datum (''X''<sup></sup>, Φ,''X''<sub></sub>, Φ<sup>v</sup>), we can define a '''dual root datum''' (''X''<sub></sub>, Φ<sup>v</sup>,''X''<sup></sup>, Φ) by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.

			If ''G'' is a connected reductive algebraic group over the algebraically closed field ''K'', then its [[Langlands dual group]] <sup>''L''</sup>''G'' is the complex connected reductive group whose root datum is dual to that of ''G''.

			==References==
			*[[Michel Demazure]], Exp. XXI in [http://modular.fas.harvard.edu/sga/sga/3-3/index.html SGA 3 vol 3]
			*[[T. A. Springer]], [http://www.ams.org/online_bks/pspum331/pspum331-ptI-1.pdf ''Reductive groups''], in [http://www.ams.org/online_bks/pspum331/ ''Automorphic forms, representations, and L-functions'' vol 1] ISBN 0-8218-3347-2
			[[Category:Representation theory]]
			[[Category:Algebraic groups]]

en>Jason Quinn: MOS:APPENDIX fix

2012-07-16T04:41:07Z

MOS:APPENDIX fix

New page

Golda is what's written on my birth certificate even though it is not the title on my birth certificate. As a woman what she truly likes is fashion and she's been doing it for quite a while. Ohio is exactly where her house is. Distributing production is how he makes a living.<br><br>Here is my homepage phone psychic - [http://Fashionlinked.com/index.php?do=/profile-13453/info/ http://Fashionlinked.com/index.php?do=/profile-13453/info] -