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What is a Lie group?

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Informally, a Lie group is a group of symmetries where the symmetries are continuous. A circle has a continuous group of symmetries: you can rotate the circle ... Whatisagroup? Mathematiciansinventedtheconceptofagrouptocapturethe essenceofsymmetry.Thecollectionofsymmetriesofanyobjectisa group,andeverygroupisthesymmetriesofsomeobject. E8isarathercomplicatedgroup:itisthesymmetriesofaparticular 57dimensionalobject,andE8itselfis248dimensional!  E8isevenmorespecial:itisaLiegroup, whichmeansthatitalsohas anicegeometricstructure. Thetheoryofgroupshasfoundwidespreadapplication.Itwas usedtodeterminethepossiblestructureofcrystals,andithas deepimplicationsforthetheoryofmolecularvibration. Theconservationlawsofphysics,suchas conservationofenergyandconservationofelectriccharge,allarise fromthesymmetriesintheequationsofphysics.Andoneofthesimplest groups,knownas"themultiplicativegroupmoduloN"isusedeverytime yousendsecureinformationovertheInternet. Foranintroductiontogroupsrequiringlittlemathematicsbackground,see GroupsandSymmetry byDavidFarmer. WhatisaLiegroup? Liegroupslieattheintersectionoftwofundamentalfieldsof mathematics:algebraandgeometry.ALiegroupisfirstofallagroup. Secondlyitisa smoothmanifoldwhichisaspecifickindofgeometric object.Thecircleandthesphereareexamplesofsmoothmanifolds. Finallythealgebraicstructureandthegeometric structuremustbecompatibleinapreciseway. Informally,aLiegroupisagroupofsymmetrieswherethe symmetriesarecontinuous.Acirclehasacontinuousgroupof symmetries:youcanrotatethecircleanarbitrarilysmallamount anditlooksthesame.Thisisincontrasttothehexagon,forexample. Ifyourotatethehexagonbyasmallamountthenitwilllook different.Onlyrotationsthataremultiplesofone-sixthofa fullturnaresymmetriesofahexagon. LiegroupswerestudiedbytheNorwegianmathematicianSophusLieat theendofthe19thcentury.Liewasinterestedinsolving equations.Atthattimetechniquesforsolvingequationswere basicallyabagoftricks.Atypicaltoolwastomakeacleverchange ofvariableswhichwouldmakeoneofthevariabledropoutofthe equations.Lie'sbasicinsightwasthatwhenthishappeneditwasdue toanunderlyingsymmetricoftheequations,andthatunderlyingthis symmetrywaswhatisnowcalledaLiegroup. Liegroupsareubiquitousinmathematicsandallareasof science.Associatedtoanysystemwhichhasacontinuousgroupof symmetriesisaLiegroup. ThebasicbuildingblocksofLiegroupsaresimpleLie groups.Theclassificationofthesegroupsstartswiththe classificationofthecomplex,simpleLiealgebras.Thesewere classifiedbyWilhelmKillingandElieCartaninthe1890s. CartanconstructedallthesimpleLiealgebras,whichcorrespondto thesimplerootsystems: An,Bn,Cn,andDn(n=1,2,3....) andtheexceptionalones: G2,F4,E6,E7,and E8. Theexceptionaloneshavedimensions14,52,78,133and248,respectively. ForaLiegroup,thesubscriptniscalledthe rankofthegroup,which isameasureofhowlargethegroupis. Inasense E8 isthemostcomplicatedLiegroup. AlthoughA1000,forexample, iscertainlylargerthan E8, mathematiciansknow howtodescribetherepresentationsof Anforeveryn,sothereisnothingspecial about A1000. Allof An, Bn, Cn,and Dnarewell-understood,sotheremainingchallenge istotackletheexceptionalgroups. All5oftheexceptional groupsneedtobetreatedseparately,and E8isthemostcomplicatedofthese. MainE8page

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