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1、Physics195SupplementaryNotesGroups,Liealgebras,andLiegroups020922F.PorterThisnotedefinessomemathematicalstructureswhichareusefulinthediscussionofangularmomentuminquantummechanics(amongotherthings).Def:Apair(G,◦),whereGisanon-emptyset,and◦isabinaryoperationdefinedonG,
2、iscalledagroupif:1.Closure:Ifa,b∈G,thena◦b∈G.2.Associativity:Ifa,b,c∈G,thena◦(b◦c)=(a◦b)◦c.3.Existenceofrightidentity:Thereexistsanelemente∈Gsuchthata◦e=aforalla∈G.4.Existenceofrightinverse:Forsomerightidentitye,andforanya∈G,thereexistsanelementa−1∈Gsuchthata◦a−1=e
3、.The◦operationistypicallyreferredtoas“multiplication”.Theabovemaybetermeda“minimal”definitionofagroup.Itisamusing(anduseful)toprovethat:1.Therightidentityelementisunique.2.Therightinverseelementofanyelementisunique.3.Therightidentityisalsoaleftidentity.4.Therightinv
4、erseisalsoaleftinverse.5.Thesolutionforx∈Gtotheequationa◦x=bexistsandisunique,foranya,b∈G.Wewillusuallydroptheexplicit◦symbol,andmerelyusejuxtapositiontodenotegroupmultiplication.NotethatbothG(theset)and◦(the“mul-tiplicationtable”)mustbespecifiedinordertospecifyagro
5、up.Wheretheoperationisclear,wewillusuallyjustreferto“G”asagroup.Def:Anabelian(orcommutative)groupisoneforwhichthemultiplica-tioniscommutative:ab=ba∀a,b∈G.(1)1Def:TheorderofagroupisthenumberofelementsinthesetG.Ifthisnumberisinfinite,wesayitisan“infinitegroup”.Inthedis
6、cussionofinfinitegroupsofrelevancetophysics(inparticular,Liegroups),itisusefultoworkinthecontextofaricherstructurecalledanalgebra.Forbackground,westartbygivingsomemathematicaldefinitionsoftheunderlyingstructures:Def:AringisatripletR,+,◦consistingofanon-emptysetofel
7、ements(R)withtwobinaryoperations(+and◦)suchthat:1.R,+isanabeliangroup.2.(◦)isassociative.3.Distributivityholds:foranya,b,c∈Ra◦(b+c)=a◦b+a◦c(2)and(b+c)◦a=b◦a+c◦a(3)Conventions:Weuse0(“zero”)todenotetheidentityofR,+.Wespeakof(+)asad-ditionandof(◦)asmultiplication
8、,typicallyomittingthe(◦)symbolentirely(i.e.,ab≡a◦b).Def:Aringiscalledafieldifthenon-zeroelementsofRformanabeliangroupunder(◦).Def:AnabeliangroupV
