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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1Di erentialGeometry,aReviewAnn-dimensionalsmoothor(C1)manifoldisaspaceMcoveredbyopensetsU togetherwith\coordinatemaps" :U !Rnsuchthat isahomeomorphismofU anditsrange,and �1: (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]=XY�YXTheLiebracketofvectoreldssatisestheJacobiidentity[X;[Y;Z]]+[Z;[X;Y]]+[Y;[Z;X]]=0Vectoreldsbehave\covariantly"undersmoothmaps,i.e.forasmoothmapofsmoothmanifolds :M1!M2thereisa\push-forward"map ,thedi erentialof ,denedby: X(g)=X(g )2 2LieGroups,LieAlgebrasandtheExponentialMapBasicallyaLiegroupisasmoothmanifoldwhosepointscanbe(smoothly)multipliedtogetherDenition4(LieGroup).ALiegroupisasmoothmanifoldGtogetherwithasmoothmultiplicationmap(g1;g2)2GG!g1g22Gandasmoothinversemapg2G!g�12Gthatsatisfythegroupaxioms.Foreachelementg2G,therearetwomapsofGtoitself,givenbyrightandleftmultiplication.Lg(h)=gh;Rg(h)=hg�1(theinverseistheresothatRgRh=Rgh).Asonanymanifold,there'saninnitedimensionalspaceofvectoreldsonG,butinthiscasewecanrestrictattentiontoinvariantones.Denition5(LieAlgebra).TheLiealgebragofGisthespaceofallleft-invariantvectoreldsonG,i.e.vectoreldssatisfyingXgh=(Lg)(Xh)TheLiebracketofoftwoleft-invariantvectoreldsisleftinvariant,soitdenesanantisymmetricbilinearproduct[X;Y]ongsatisfyingtheJacobiiden-tity.Vectoreldsarerst-orderdi erentialoperators,theuniversalenvelopingalgebraU(g)=T(g)=(X Y�Y X�[X;Y])isthespaceofallleft-invariantdi erentialoperatorsonG.Onecanidentifygmoreexplicitlywiththefollowingsecondwayofcharac-terizingtheLiealgebra:Theorem1.ThemapX2g!Xe2Te(G)givenbyrestrictionofavectoreldtoitsvalueattheidentityisabijection.Finally,thereisathirdwaytocharacterizetheLiealgebrausingtheexpo-nentialmap.4 � forsomecomplexnumbers andsatisfyingj j2+jj2=1NotethatthisistheequationforS3C2.AbasisfortheLiealgebrasu(2)isgivenbytakingthePaulimatrices:1=0110;2=0�ii0;3=100�1andmultiplyingthembyi.References[1]Simon,B.,RepresentationsofFiniteandCompactGroups,AmericanMath-ematicalSociety,1996.[2]Warner,F.,FoundationsofDi erentiableManifoldsandLieGroups,Springer,1983.6