General linear group - Wikipedia

In mathematics, the general linear group of degree n is the set of n×n ; To be more precise, it is necessary to specify what kind of objects may appear in the ... Generallineargroup FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Algebraicstructure→GrouptheoryGrouptheory Basicnotions Subgroup Normalsubgroup Quotientgroup (Semi-)directproduct Grouphomomorphisms kernel image directsum wreathproduct simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable action Glossaryofgrouptheory Listofgrouptheorytopics Finitegroups Classificationoffinitesimplegroups cyclic alternating Lietype sporadic Cauchy'stheorem Lagrange'stheorem Sylowtheorems Hall'stheorem p-group Elementaryabeliangroup Frobeniusgroup Schurmultiplier SymmetricgroupSn Kleinfour-groupV DihedralgroupDn QuaterniongroupQ DicyclicgroupDicn DiscretegroupsLattices Integers( Z {\displaystyle\mathbb{Z}} ) Freegroup ModulargroupsPSL(2, Z {\displaystyle\mathbb{Z}} )SL(2, Z {\displaystyle\mathbb{Z}} ) Arithmeticgroup Lattice Hyperbolicgroup TopologicalandLiegroups Solenoid Circle GenerallinearGL(n) SpeciallinearSL(n) OrthogonalO(n) EuclideanE(n) SpecialorthogonalSO(n) UnitaryU(n) SpecialunitarySU(n) SymplecticSp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop InfinitedimensionalLiegroupO(∞)SU(∞)Sp(∞) Algebraicgroups Linearalgebraicgroup Reductivegroup Abelianvariety Ellipticcurve vte Setofnxninvertiblematrices Liegroups Classicalgroups GenerallinearGL(n) SpeciallinearSL(n) OrthogonalO(n) SpecialorthogonalSO(n) UnitaryU(n) SpecialunitarySU(n) SymplecticSp(n) SimpleLiegroups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8 OtherLiegroups Circle Lorentz Poincaré Conformalgroup Diffeomorphism Loop Euclidean Liealgebras Liegroup–Liealgebracorrespondence Exponentialmap Adjointrepresentation KillingformIndex SimpleLiealgebra SemisimpleLiealgebra Dynkindiagrams Cartansubalgebra RootsystemWeylgroup RealformComplexification SplitLiealgebra CompactLiealgebra Representationtheory Liegrouprepresentation Liealgebrarepresentation RepresentationtheoryofsemisimpleLiealgebras RepresentationsofclassicalLiegroups Theoremofthehighestweight Borel–Weil–Botttheorem Liegroupsinphysics Particlephysicsandrepresentationtheory Lorentzgrouprepresentations Poincarégrouprepresentations Galileangrouprepresentations Scientists SophusLie HenriPoincaré WilhelmKilling ÉlieCartan HermannWeyl ClaudeChevalley Harish-Chandra ArmandBorel Glossary TableofLiegroupsvte Inmathematics,thegenerallineargroupofdegreenisthesetofn×ninvertiblematrices,togetherwiththeoperationofordinarymatrixmultiplication.Thisformsagroup,becausetheproductoftwoinvertiblematricesisagaininvertible,andtheinverseofaninvertiblematrixisinvertible,withidentitymatrixastheidentityelementofthegroup.Thegroupissonamedbecausethecolumns(andalsotherows)ofaninvertiblematrixarelinearlyindependent,hencethevectors/pointstheydefineareingenerallinearposition,andmatricesinthegenerallineargrouptakepointsingenerallinearpositiontopointsingenerallinearposition. Tobemoreprecise,itisnecessarytospecifywhatkindofobjectsmayappearintheentriesofthematrix.Forexample,thegenerallineargroupoverR(thesetofrealnumbers)isthegroupofn×ninvertiblematricesofrealnumbers,andisdenotedbyGLn(R)orGL(n,R). Moregenerally,thegenerallineargroupofdegreenoveranyfieldF(suchasthecomplexnumbers),oraringR(suchastheringofintegers),isthesetofn×ninvertiblematriceswithentriesfromF(orR),againwithmatrixmultiplicationasthegroupoperation.[1]TypicalnotationisGLn(F)orGL(n,F),orsimplyGL(n)ifthefieldisunderstood. Moregenerallystill,thegenerallineargroupofavectorspaceGL(V)istheabstractautomorphismgroup,notnecessarilywrittenasmatrices. Thespeciallineargroup,writtenSL(n,F)orSLn(F),isthesubgroupofGL(n,F)consistingofmatriceswithadeterminantof1. ThegroupGL(n,F)anditssubgroupsareoftencalledlineargroupsormatrixgroups(theabstractgroupGL(V)isalineargroupbutnotamatrixgroup).Thesegroupsareimportantinthetheoryofgrouprepresentations,andalsoariseinthestudyofspatialsymmetriesandsymmetriesofvectorspacesingeneral,aswellasthestudyofpolynomials.ThemodulargroupmayberealisedasaquotientofthespeciallineargroupSL(2,Z). Ifn≥2,thenthegroupGL(n,F)isnotabelian. Contents 1Generallineargroupofavectorspace 2Intermsofdeterminants 3AsaLiegroup 3.1Realcase 3.2Complexcase 4Overfinitefields 4.1History 5Speciallineargroup 6Othersubgroups 6.1Diagonalsubgroups 6.2Classicalgroups 7Relatedgroupsandmonoids 7.1Projectivelineargroup 7.2Affinegroup 7.3Generalsemilineargroup 7.4Fulllinearmonoid 8Infinitegenerallineargroup 9Seealso 10Notes 11References 12Externallinks Generallineargroupofavectorspace[edit] IfVisavectorspaceoverthefieldF,thegenerallineargroupofV,writtenGL(V)orAut(V),isthegroupofallautomorphismsofV,i.e.thesetofallbijectivelineartransformationsV→V,togetherwithfunctionalcompositionasgroupoperation.IfVhasfinitedimensionn,thenGL(V)andGL(n,F)areisomorphic.Theisomorphismisnotcanonical;itdependsonachoiceofbasisinV.Givenabasis(e1,...,en)ofVandanautomorphismTinGL(V),wehavethenforeverybasisvectoreithat T ( e i ) = ∑ j = 1 n a i j e j {\displaystyleT(e_{i})=\sum_{j=1}^{n}a_{ij}e_{j}} forsomeconstantsaijinF;thematrixcorrespondingtoTisthenjustthematrixwithentriesgivenbytheaij. Inasimilarway,foracommutativeringRthegroupGL(n,R)maybeinterpretedasthegroupofautomorphismsofafreeR-moduleMofrankn.OnecanalsodefineGL(M)foranyR-module,butingeneralthisisnotisomorphictoGL(n,R)(foranyn). Intermsofdeterminants[edit] OverafieldF,amatrixisinvertibleifandonlyifitsdeterminantisnonzero.Therefore,analternativedefinitionofGL(n,F)isasthegroupofmatriceswithnonzerodeterminant. OveracommutativeringR,morecareisneeded:amatrixoverRisinvertibleifandonlyifitsdeterminantisaunitinR,thatis,ifitsdeterminantisinvertibleinR.Therefore,GL(n,R)maybedefinedasthegroupofmatriceswhosedeterminantsareunits. Overanon-commutativeringR,determinantsarenotatallwellbehaved.Inthiscase,GL(n,R)maybedefinedastheunitgroupofthematrixringM(n,R). AsaLiegroup[edit] Realcase[edit] ThegenerallineargroupGL(n,R)overthefieldofrealnumbersisarealLiegroupofdimensionn2.Toseethis,notethatthesetofalln×nrealmatrices,Mn(R),formsarealvectorspaceofdimensionn2.ThesubsetGL(n,R)consistsofthosematriceswhosedeterminantisnon-zero.Thedeterminantisapolynomialmap,andhenceGL(n,R)isanopenaffinesubvarietyofMn(R)(anon-emptyopensubsetofMn(R)intheZariskitopology),andtherefore[2] asmoothmanifoldofthesamedimension. TheLiealgebraofGL(n,R),denoted g l n , {\displaystyle{\mathfrak{gl}}_{n},} consistsofalln×nrealmatriceswiththecommutatorservingastheLiebracket. Asamanifold,GL(n,R)isnotconnectedbutratherhastwoconnectedcomponents:thematriceswithpositivedeterminantandtheoneswithnegativedeterminant.Theidentitycomponent,denotedbyGL+(n,R),consistsoftherealn×nmatriceswithpositivedeterminant.ThisisalsoaLiegroupofdimensionn2;ithasthesameLiealgebraasGL(n,R). ThegroupGL(n,R)isalsononcompact.“The”[3]maximalcompactsubgroupofGL(n,R)istheorthogonalgroupO(n),while"the"maximalcompactsubgroupofGL+(n,R)isthespecialorthogonalgroupSO(n).AsforSO(n),thegroupGL+(n,R)isnotsimplyconnected(exceptwhenn=1),butratherhasafundamentalgroupisomorphictoZforn=2orZ2forn>2. Complexcase[edit] Thegenerallineargroupoverthefieldofcomplexnumbers,GL(n,C),isacomplexLiegroupofcomplexdimensionn2.AsarealLiegroup(throughrealification)ithasdimension2n2.ThesetofallrealmatricesformsarealLiesubgroup.Thesecorrespondtotheinclusions GL(n,R)2. Othersubgroups[edit] Diagonalsubgroups[edit] ThesetofallinvertiblediagonalmatricesformsasubgroupofGL(n,F)isomorphicto(F×)n.InfieldslikeRandC,thesecorrespondtorescalingthespace;theso-calleddilationsandcontractions. Ascalarmatrixisadiagonalmatrixwhichisaconstanttimestheidentitymatrix.ThesetofallnonzeroscalarmatricesformsasubgroupofGL(n,F)isomorphictoF×.ThisgroupisthecenterofGL(n,F).Inparticular,itisanormal,abeliansubgroup. ThecenterofSL(n,F)issimplythesetofallscalarmatriceswithunitdeterminant,andisisomorphictothegroupofnthrootsofunityinthefieldF. Classicalgroups[edit] Theso-calledclassicalgroupsaresubgroupsofGL(V)whichpreservesomesortofbilinearformonavectorspaceV.Theseincludethe orthogonalgroup,O(V),whichpreservesanon-degeneratequadraticformonV, symplecticgroup,Sp(V),whichpreservesasymplecticformonV(anon-degeneratealternatingform), unitarygroup,U(V),which,whenF=C,preservesanon-degeneratehermitianformonV. ThesegroupsprovideimportantexamplesofLiegroups. Relatedgroupsandmonoids[edit] Projectivelineargroup[edit] Mainarticle:Projectivelineargroup TheprojectivelineargroupPGL(n,F)andtheprojectivespeciallineargroupPSL(n,F)arethequotientsofGL(n,F)andSL(n,F)bytheircenters(whichconsistofthemultiplesoftheidentitymatrixtherein);theyaretheinducedactionontheassociatedprojectivespace. Affinegroup[edit] Mainarticle:Affinegroup TheaffinegroupAff(n,F)isanextensionofGL(n,F)bythegroupoftranslationsinFn.Itcanbewrittenasasemidirectproduct: Aff(n,F)=GL(n,F)⋉Fn whereGL(n,F)actsonFninthenaturalmanner.TheaffinegroupcanbeviewedasthegroupofallaffinetransformationsoftheaffinespaceunderlyingthevectorspaceFn. Onehasanalogousconstructionsforothersubgroupsofthegenerallineargroup:forinstance,thespecialaffinegroupisthesubgroupdefinedbythesemidirectproduct,SL(n,F)⋉Fn,andthePoincarégroupistheaffinegroupassociatedtotheLorentzgroup,O(1,3,F)⋉Fn. Generalsemilineargroup[edit] Mainarticle:Generalsemilineargroup ThegeneralsemilineargroupΓL(n,F)isthegroupofallinvertiblesemilineartransformations,andcontainsGL.Asemilineartransformationisatransformationwhichislinear“uptoatwist”,meaning“uptoafieldautomorphismunderscalarmultiplication”.Itcanbewrittenasasemidirectproduct: ΓL(n,F)=Gal(F)⋉GL(n,F) whereGal(F)istheGaloisgroupofF(overitsprimefield),whichactsonGL(n,F)bytheGaloisactionontheentries. ThemaininterestofΓL(n,F)isthattheassociatedprojectivesemilineargroupPΓL(n,F)(whichcontainsPGL(n,F))isthecollineationgroupofprojectivespace,forn>2,andthussemilinearmapsareofinterestinprojectivegeometry. Fulllinearmonoid[edit] Thissectionneedsexpansionwith:basicproperties.Youcanhelpbyaddingtoit.(April2015) Ifoneremovestherestrictionofthedeterminantbeingnon-zero,theresultingalgebraicstructureisamonoid,usuallycalledthefulllinearmonoid,[6][7][8]butoccasionallyalsofulllinearsemigroup,[9]generallinearmonoid[10][11]etc.Itisactuallyaregularsemigroup.[7] Infinitegenerallineargroup[edit] TheinfinitegenerallineargrouporstablegenerallineargroupisthedirectlimitoftheinclusionsGL(n,F)→GL(n+1,F)astheupperleftblockmatrix.ItisdenotedbyeitherGL(F)orGL(∞,F),andcanalsobeinterpretedasinvertibleinfinitematriceswhichdifferfromtheidentitymatrixinonlyfinitelymanyplaces.[12] ItisusedinalgebraicK-theorytodefineK1,andovertherealshasawell-understoodtopology,thankstoBottperiodicity. Itshouldnotbeconfusedwiththespaceof(bounded)invertibleoperatorsonaHilbertspace,whichisalargergroup,andtopologicallymuchsimpler,namelycontractible–seeKuiper'stheorem. Seealso[edit] Listoffinitesimplegroups SL2(R) RepresentationtheoryofSL2(R) RepresentationsofclassicalLiegroups Notes[edit] ^Hereringsareassumedtobeassociativeandunital. ^ SincetheZariskitopologyiscoarserthanthemetrictopology;equivalently,polynomialmapsarecontinuous. ^Amaximalcompactsubgroupisnotunique,butisessentiallyunique,henceoneoftenrefersto“the”maximalcompactsubgroup. ^Galois,Évariste(1846)."LettredeGaloisàM.AugusteChevalier".JournaldeMathématiquesPuresetAppliquées.XI:408–415.Retrieved2009-02-04,GL(ν,p)discussedonp.410.{{citejournal}}:CS1maint:postscript(link) ^Suprunenko,D.A.(1976),Matrixgroups,TranslationsofMathematicalMonographs,AmericanMathematicalSociety,TheoremII.9.4 ^JanOkniński(1998).SemigroupsofMatrices.WorldScientific.Chapter2:Fulllinearmonoid.ISBN 978-981-02-3445-4. ^abMeakin(2007)."GroupsandSemigroups:Connectionsandcontrast".InC.M.Campbell(ed.).GroupsStAndrews2005.CambridgeUniversityPress.p. 471.ISBN 978-0-521-69470-4. ^JohnRhodes;BenjaminSteinberg(2009).Theq-theoryofFiniteSemigroups.SpringerScience&BusinessMedia.p. 306.ISBN 978-0-387-09781-7. ^EricJespers;JanOkniski(2007).NoetherianSemigroupAlgebras.SpringerScience&BusinessMedia.2.3:Fulllinearsemigroup.ISBN 978-1-4020-5810-3. ^MeinolfGeck(2013).AnIntroductiontoAlgebraicGeometryandAlgebraicGroups.OxfordUniversityPress.p. 132.ISBN 978-0-19-967616-3. ^MahirBilenCan;ZhenhengLi;BenjaminSteinberg;QiangWang(2014).AlgebraicMonoids,GroupEmbeddings,andAlgebraicCombinatorics.Springer.p. 142.ISBN 978-1-4939-0938-4. ^Milnor,JohnWillard(1971).IntroductiontoalgebraicK-theory.AnnalsofMathematicsStudies.Vol. 72.Princeton,NJ:PrincetonUniversityPress.p. 25.MR 0349811.Zbl 0237.18005. References[edit] Springer,TonnyAlbert(1998).LinearAlgebraicGroups(2nd ed.).Birkhäuser.ISBN 978-0-8176-4839-8. Externallinks[edit] "Generallineargroup",EncyclopediaofMathematics,EMSPress,2001[1994] "GL(2,p)andGL(3,3)ActingonPoints"byEdPegg,Jr.,WolframDemonstrationsProject,2007. 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