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Lie Group -- from Wolfram MathWorld

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A Lie group is a smooth manifold obeying the group properties and that satisfies the additional condition that the group operations are differentiable. Algebra AppliedMathematics CalculusandAnalysis DiscreteMathematics FoundationsofMathematics Geometry HistoryandTerminology NumberTheory ProbabilityandStatistics RecreationalMathematics Topology AlphabeticalIndex NewinMathWorld Algebra GroupTheory LieTheory LieGroups MathWorldContributors Rowland,Todd LieGroup ALiegroupisasmoothmanifoldobeyingthegrouppropertiesandthatsatisfiestheadditionalconditionthatthegroupoperations aredifferentiable. ThisdefinitionisrelatedtothefifthofHilbert'sproblems,whichasksiftheassumptionofdifferentiabilityforfunctionsdefining acontinuoustransformationgroupcanbeavoided. ThesimplestexamplesofLiegroupsareone-dimensional.Underaddition,thereallineisaLiegroup.Afterpickingaspecificpointtobetheidentity element,thecircleisalsoaLiegroup.Anotherpoint onthecircleatanglefromtheidentitythenactsbyrotating thecirclebytheangleIngeneral,aLiegroupmayhave amorecomplicatedgroupstructure,suchastheorthogonal group(i.e.,theorthogonal matrices),orthegenerallineargroup(i.e.,theinvertible matrices).TheLorentzgroupisalsoaLiegroup. ThetangentspaceattheidentityofaLiegroupalwayshasthestructureofaLiealgebra,andthis Liealgebradeterminesthelocalstructureofthe Liegroupviatheexponentialmap.Forexample, thefunctiongivestheexponential mapfromthecircle'stangentspace(i.e.,thereals),tothecircle,thought ofasaunitcirclein.Amoredifficult exampleistheexponentialmapfromantisymmetricmatricestothespecial orthogonalgroup,thesubsetofwithdeterminant 1. ThetopologyofaLiegroupisfairlyrestricted.Forexample,therealwaysexistsanonvanishingvectorfield.Thisstructurehasallowed completeclassificationofthefinitedimensionalsemisimple Liegroupsandtheirrepresentations. SeealsoCompactLieGroup,ContinuousGroup,Group,LieAlgebra, LieGroupoid,Lie-Type Group,LorentzGroup,Nil Geometry,OrthogonalGroup,Semisimple LieGroup,SmoothManifold,Sol Geometry,TangentSpace,Vector FieldExplorethis topicintheMathWorldclassroom ThisentrycontributedbyTodd Rowland ExplorewithWolfram|Alpha Morethingstotry: birthdayproblem35people div[x^2siny,y^2sinxz,xysin(cosz)] interval[-sqrt(5),1+sqrt(5)] Citethisas: Rowland,Todd."LieGroup."FromMathWorld--AWolframWebResource,createdbyEric W.Weisstein.https://mathworld.wolfram.com/LieGroup.html Subjectclassifications Algebra GroupTheory LieTheory LieGroups MathWorldContributors Rowland,Todd Created,developedandnurturedbyEricWeissteinatWolframResearch

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