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1、CliffordAlgebras,CliffordGroups,andaGeneralizationoftheQuaternions:ThePinandSpinGroupsJeanGallierDepartmentofComputerandInformationScienceUniversityofPennsylvaniaPhiladelphia,PA19104,USAe-mail:jean@saul.cis.upenn.eduFebruary18,20042Contents1CliffordAlgebras,CliffordGroups,Pinand
2、Spin51.1Introduction:RotationsAsGroupActions...................51.2CliffordAlgebras.................................71.3CliffordGroups..................................161.4TheGroupsPin(n)andSpin(n)........................221.5TheGroupsPin(p,q)andSpin(p,q).....................
3、.281.6PeriodicityoftheCliffordAlgebrasClp,q....................301.7TheComplexCliffordAlgebrasCl(n,C)....................341.8TheGroupsPin(p,q)andSpin(p,q)asdoublecovers.............341.9MoreontheTopologyofO(p,q)andSO(p,q).................3934CONTENTSChapter1CliffordAlgebras,Cl
4、iffordGroups,andtheGroupsPin(n)andSpin(n)1.1Introduction:RotationsAsGroupActionsOneofthemaingoalsofthesenotesistoexplainhowrotationsinRnareinducedbytheactionofacertaingroup,Spin(n),onRn,inawaythatgeneralizestheactionoftheunitcomplexnumbers,U(1),onR2,andtheactionoftheunitquate
5、rnions,SU(2),onR3(i.e.,theactionisdefinedintermsofmultiplicationinalargeralgebracontainingboththegroupSpin(n)andRn).ThegroupSpin(n),calledaspinorgroup,isdefinedasacertainsubgroupofunitsofanalgebra,Cl,theCliffordalgebraassociatedwithRn.Furthermore,forn≥3,nwearelucky,becausethegr
6、oupSpin(n)istopologicallysimplerthanthegroupSO(n).Indeed,forn≥3,thegroupSpin(n)issimplyconnected(afactthatitnotsoeasytoprovewithoutsomemachinery),whereasSO(n)isnotsimplyconnected.Intuitivelyspeaking,SO(n)ismoretwistedthanSpin(n).Infact,wewillseethatSpin(n)isadoublecoverofSO(
7、n).SincethespinorgroupsarecertainwellchosensubroupsofunitsofCliffordalgebras,itisnecessarytoinvestigateCliffordalgebrastogetafirmunderstandingofspinorgroups.ThesenotesprovideatutorialonCliffordalgebraandthegroupsSpinandPin,includingastudyofthestructureoftheCliffordalgebraClp,qass
8、ociatedwithanondegeneratesymmetricbilinearformofsignature(p,q)andculminatin
