Uniform boundedness principle - Revision history

en>Gutworth: /* Example: pointwise convergence of Fourier series */ use boldface T everywhere

2014-12-12T03:37:34Z

Example: pointwise convergence of Fourier series: use boldface T everywhere

← Older revision		Revision as of 05:37, 12 December 2014
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	~~{{Group theory sidebar}}~~		52 yrs old Geophysicist Les from Rockland, has many hobbies that include snowshoeing, world at arms hack and dolls. Was exceptionally stimulated after visiting Bisotun.<br><br>My blog post :: [https://www.facebook.com/pages/World-at-Arms-Hack/215334922007147 world at arms ios Hack no survey]

	~~In [[algebraic geometry]]~~, an '''algebraic group''' (or '''group variety''') is a [[Group (mathematics)\|group]] that is an [[algebraic variety]], such that the multiplication and inversion operations are given by [[regular function]]s on the variety.

	~~In terms of [[category theory]], an algebraic group is a [[group object]] in the [[Category (mathematics)\|category]] of algebraic varieties.~~

	~~== Classes ==~~
	~~Several important classes of groups are algebraic groups, including:~~
	* [[Finite group]]s
	* GL(''n'', '''C'''), the [[general linear group]] of [[invertible matrices]] over '''C'''
	* [[Elliptic curve]]s.

	Two important classes of algebraic groups arise, that for the most part are studied separately: ''[[abelian variety\|abelian varieties]]'' (the 'projective' theory) and ''[[linear algebraic group]]s'' (the 'affine' theory). There are certainly examples that are neither one nor the other — these occur for example in the modern theory of [[differential of the first kind\|integrals of the second and third kinds]] such as the [[Weierstrass zeta function]], or the theory of [[generalized Jacobian]]s. But according to [[Chevalley's structure theorem]] any algebraic group is an extension of an [[abelian variety]] by a linear algebraic group. This is a result of [[Claude Chevalley]]: if ''K'' is a [[perfect field]], and ''G'' an algebraic group over ''K'', there exists a unique normal closed subgroup ''H'' in ''G'', such that ~~''H'' is a linear group and ''G''/''H'' an abelian variety.~~

	~~According to another basic theorem~~, any group in the category of [[affine variety\|affine varieties]] has a [[Faithful representation\|faithful]] finite dimensional [[linear representation]]: we can consider it to be a [[matrix group]] over ''K'', defined by polynomials over ''K'' and ~~with matrix multiplication as the group operation. For that reason a concept of ''affine algebraic group'' is redundant over a field — we may as well use a very concrete definition~~. Note that this means that algebraic group is narrower than [[Lie group]], when working over the field of real numbers: there are examples such as the [[universal cover]] of the 2×2 special linear group that are Lie groups, but have no faithful linear representation. ~~A more obvious difference between the two concepts arises because the [[identity component]] of an affine algebraic group ''G'' is necessarily of finite [[Index of a subgroup\|index]] in ''G''.~~

	When one wants to work over a base ring ''R'' (commutative), there is the [[group scheme]] concept: that is, a [[group object]] in the category of [[scheme (mathematics)\|scheme]]s over ''R''. ''Affine group scheme'' is the concept dual to a type of [[Hopf algebra]]. There is quite a refined theory of group schemes, that enters for example in the contemporary theory of abelian varieties.

	~~==Algebraic subgroup==~~
	~~An '''algebraic subgroup''' of an algebraic group is a [[Zariski topology\|Zariski closed]] [[subgroup]].~~
	~~Generally these are taken to be connected (or irreducible as a variety) as well.~~

	~~Another way of expressing the condition is as a [[subgroup]] which is also a [[algebraic variety\|subvariety]].~~

	This may also be generalized by allowing [[scheme (mathematics)\|schemes]] in place of varieties. The main effect of this in practice, apart from allowing subgroups in which the [[connected space\|connected component]] is of finite index > 1, is to admit non-[[reduced scheme]]s, in characteristic ''p''.

	~~==Coxeter groups==~~
	~~{{main\|Coxeter group}}~~
	~~{{further\|Field with one element}}~~
	~~There are a number of analogous results between algebraic groups and [[Coxeter group]]s – for instance, the number of elements of the symmetric group is~~ <~~math~~>n!<~~/math~~>, and the number of elements of the general linear group over a finite field is the [[q-factorial\|''q''-factorial]] <math>[n]_q!</math>; thus the symmetric group behaves as though it were a linear group over "the field with one element". This is formalized by the [~~[field with one element]], which considers Coxeter groups to be simple algebraic groups over the field with one element.~~

	~~==See also==~~
	*[[Algebraic topology (object)]]
	*[[Borel subgroup]]
	*[[Tame group]]
	*[[Morley rank]]
	*[[Cherlin–Zilber conjecture]]
	*[[Adelic algebraic group]]
	*[[Glossary of algebraic groups]]

	~~==Notes==~~
	~~<references/>~~

	~~== References ==~~
	*{{Citation \| editor1-last=Chevalley \| editor1-first=Claude \| title=Séminaire C. Chevalley, 1956--1958. Classification des groupes de Lie algébriques \| url=http://www.~~numdam~~.~~org~~/~~numdam-bin~~/~~browse?id=SCC_1956~~-~~1958__1_ \| publisher=Secrétariat Mathématique \| location=Paris \| series=2 vols \| mr=0106966 \|id= Reprinted as volume 3 of Chevalley's collected works. \| year=1958}}~~
	* {{Citation \| last1=Humphreys \| first1=James E. \| title=Linear Algebraic Groups \| publisher=[[Springer-~~Verlag]] \| location=Berlin, New York \| series=Graduate Texts in Mathematics \| isbn=978-0-387-90108-4 \| mr=0396773 \| year=1972 \| volume=21}}~~
	* {{Citation \| last1=Lang \| first1=Serge \| author1-link=Serge Lang \| title=Abelian varieties \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| isbn=978-0-387-90875-~~5 \| year=1983}}~~
	* Milne, J. S., ''[http:/~~/www.jmilne.org/math/CourseNotes/ala.html Affine Group Schemes; Lie Algebras; Lie Groups; Reductive Groups; Arithmetic Subgroups]''~~
	* {{Citation \| last1=Mumford \| first1=David \| author1-link=David Mumford \| title=Abelian varieties \| publisher=[[Oxford University Press]] \| isbn=978-0-19-560528-0 \| oclc=138290 \| year=1970}}
	* {{Citation \| last1=Springer \| first1=Tonny A. \| title=Linear algebraic groups \| publisher=Birkhäuser Boston \| location=Boston, MA \| edition=2nd \| series=Progress in Mathematics \| isbn=978-0-8176-4021-7 \| mr=1642713 \| year=1998 \| volume=9}}
	* {{Citation \| last1=Waterhouse \| first1=William C. \| authorlink = William C. Waterhouse \| title=Introduction to affine group schemes \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Graduate Texts in Mathematics \| isbn=978-0-387-90421-4 \| year=1979 \| volume=66}}
	* {{Citation \| last1=Weil \| first1=André \| author1-link=André Weil \| title=Courbes algébriques et variétés abéliennes \| publisher=Hermann \| location=Paris \| oclc=322901 \| year=1971}}

	~~[[Category:Algebraic groups\| ]]~~
	~~[[Category:Properties of groups]]~~
	~~[[Category:Algebraic geometry]]~~
	~~[[Category:Algebraic varieties]~~]

en>Bdmy: /* Uniform boundedness principle */ more cleaning

2013-07-11T11:59:51Z

Uniform boundedness principle: more cleaning

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+In [[algebraic geometry]], an '''algebraic group''' (or '''group variety''') is a [[Group (mathematics)|group]] that is an [[algebraic variety]], such that the multiplication and inversion operations are given by [[regular function]]s on the variety.
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+In terms of [[category theory]], an algebraic group is a [[group object]] in the [[Category (mathematics)|category]] of algebraic varieties.

en>Tassedethe: WPCleaner v1.13 - Repaired link to disambiguation page - (You can help) - Hans Hahn

2012-04-29T16:49:07Z

WPCleaner v1.13 - Repaired link to disambiguation page - (You can help) - Hans Hahn

New page

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