e subgroups of Aut the group of invertible linear transformations from to itself for an ndimensional vector space over a 64257eld Once a basis for has been chosen then elements of are invertible n by n matri ces with entries in and Gl nF Group mul ID: 10980
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1DierentialGeometry,aReviewAnn-dimensionalsmoothor(C1)manifoldisaspaceMcoveredbyopensetsUtogetherwith\coordinatemaps":U!RnsuchthatisahomeomorphismofUanditsrange,and1:(U\U)!(U\U)isaC1map.Inthiscourse,mapsshouldbeassumedtobesmoothunlessotherwisestated.Denition1(VectorField).AvectoreldonasmoothmanifoldMisaderivationX:C1(M)!C1(M)i.e.alinearmapsuchthatonaproductoffunctionsfandgX(fg)=f(Xg)+(Xf)gLocallyonecanchoosecoordinatessothatsuchaderivationisalinearcombinationofthederivativeswithrespecttothecoordinatesX=nXi=1ai(x1;;xn)@ @xiandwithrespecttothischoiceofcoordinates,avectoreldisgivenateachpointbythen-vector(a1;;an).Sometimeswe'llalsorefertothevalueofavectoreldatapointm2MasXm,thiscanbethoughtofintermsofderivationsactingongermsoffunctionsatm.ThespaceofsuchXmmakesupthetangentspacetoMatm,calledTm(M).Onthespaceofvectoreldsthereisananticommutativebilinearoperation:Denition2(LieBracket).TheLiebracketoftwovectoreldsXandYisdenedby[X;Y]=XYYXTheLiebracketofvectoreldssatisestheJacobiidentity[X;[Y;Z]]+[Z;[X;Y]]+[Y;[Z;X]]=0Vectoreldsbehave\covariantly"undersmoothmaps,i.e.forasmoothmapofsmoothmanifolds :M1!M2thereisa\push-forward"map ,thedierentialof ,denedby: X(g)=X(g )2 2LieGroups,LieAlgebrasandtheExponentialMapBasicallyaLiegroupisasmoothmanifoldwhosepointscanbe(smoothly)multipliedtogetherDenition4(LieGroup).ALiegroupisasmoothmanifoldGtogetherwithasmoothmultiplicationmap(g1;g2)2GG!g1g22Gandasmoothinversemapg2G!g12Gthatsatisfythegroupaxioms.Foreachelementg2G,therearetwomapsofGtoitself,givenbyrightandleftmultiplication.Lg(h)=gh;Rg(h)=hg1(theinverseistheresothatRgRh=Rgh).Asonanymanifold,there'saninnitedimensionalspaceofvectoreldsonG,butinthiscasewecanrestrictattentiontoinvariantones.Denition5(LieAlgebra).TheLiealgebragofGisthespaceofallleft-invariantvectoreldsonG,i.e.vectoreldssatisfyingXgh=(Lg)(Xh)TheLiebracketofoftwoleft-invariantvectoreldsisleftinvariant,soitdenesanantisymmetricbilinearproduct[X;Y]ongsatisfyingtheJacobiiden-tity.Vectoreldsarerst-orderdierentialoperators,theuniversalenvelopingalgebraU(g)=T(g)=(X YY X[X;Y])isthespaceofallleft-invariantdierentialoperatorsonG.Onecanidentifygmoreexplicitlywiththefollowingsecondwayofcharac-terizingtheLiealgebra:Theorem1.ThemapX2g!Xe2Te(G)givenbyrestrictionofavectoreldtoitsvalueattheidentityisabijection.Finally,thereisathirdwaytocharacterizetheLiealgebrausingtheexpo-nentialmap.4 forsomecomplexnumbersandsatisfyingjj2+jj2=1NotethatthisistheequationforS3C2.AbasisfortheLiealgebrasu(2)isgivenbytakingthePaulimatrices:1=0110;2=0ii0;3=1001andmultiplyingthembyi.References[1]Simon,B.,RepresentationsofFiniteandCompactGroups,AmericanMath-ematicalSociety,1996.[2]Warner,F.,FoundationsofDierentiableManifoldsandLieGroups,Springer,1983.6