; TeX output 2005.07.01:1057 s html: html:o html: html:4;**src:152Loop_groups_and_string_topology.texDt G G cmr17Losop7tgroupsandstringtopology (Lectures7tforthesummerscqhosolalgebraicgroupscG9ottingen,7tJuly2005 jXQ cmr12ThomasScrhick2K cmsy8 jĺUniG ottingen Germanry3AILastcompiledJuly1,2005;lasteditedJune30,2005orlater*html: html:b)N ff cmbx121LIntros3ductionQ*src:157Loop_groups_and_string_topology.texK`y
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cmmi10GbGeacompactLiegroup..ThenthespaceLGofallmapsfromthecircle*src:158Loop_groups_and_string_topology.texS ^ٓR cmr71 to#uGbGecomesagroupbypointwise#umultiplication.a'Actually*,-otherearedierentvqariants`Zof*src:159Loop_groups_and_string_topology.texLG,YdepGendingontheclassesofmapsoneconsiders,andthetopGologytobGeputonthemappingspace.+Intheselectures,֩wewillalwaysloGokatthespace˶ofsmoGoth(i.e.*src:162Loop_groups_and_string_topology.texC ^O! cmsy71 0)maps,Nwiththetopologyofuniformconvergence˶ofallUUderivqatives. *src:165Loop_groups_and_string_topology.texThese[groupscertainlyarenotalgebraicgroupsintheusualsenseoftheword.LxNevertheless,theygsharemanypropGertiesofalgebraicgroups(concerninge.g."theirrepresentationtheory).wThereareactuallyanalogousob 8jectswhichareRveryalgebraic(comparee.g.[html:1 html: ]),SanditturnsoutthatthosehavepropGertiesremarkqablyUUclosetothoseofthesmoGothloopgroups. *src:172Loop_groups_and_string_topology.texTheUUlecturesareorganizedasfollows.}html: html:8(1) *src:174Loop_groups_and_string_topology.texLecture˯1:,ReviewofcompactLiegroupsandtheirrepresentations,7basics ofUUloGopgroupsofcompactLiegroups.2html: html:8(2) *src:176Loop_groups_and_string_topology.texLectureUU2:qFinerpropGertiesofloopgroupskhtml: html:
k8(3) *src:177Loop_groups_and_string_topology.texLectureUU3:qtherepresentationsofloGopgroups(ofpositiveenergy)! *src:180Loop_groups_and_string_topology.texTheRlecturesandthesenotesaremainlybasedontheexcellentmonograph\LoGopUUgroups"byPressleyandSegal[html:3 html: ].Dhtml: html:2LBasicsffabs3outcompactLiegroupsghtml: html: ;1"V
cmbx102.1DDenition.*src:187Loop_groups_and_string_topology.texAׅLieפgroupGisasmoGothmanifoldGwithagroupstructure,suchUUthatthemap*src:188Loop_groups_and_string_topology.texG8
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cmsy10G!G;(g[;h)7!gh^ 1ɲisUUsmoGoth. = ff v @
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cmsl10ThomasUUSchicko ^ *src:190Loop_groups_and_string_topology.texTheWhgroupactsonitselfbyleftmultiplication:еl 0er cmmi7g(h)=g[h.#AW(vectorWheld*src:191Loop_groups_and_string_topology.texX2 O (TcG)iscalled2 ':
cmti10leftXinvariant,ƴif*src:192Loop_groups_and_string_topology.tex(lg)X'#=^AXxforeachg2G.ThespaceofallO left܍invqariantvectoreldsiscalledtheLie$algebr}'a*src:193Loop_groups_and_string_topology.texLie(G).IIfweconsidervectorO eldscasderivqations,fthenthecommutatoroftwoleftinvqariantvectoreldsO againZisaleftinvqariantZvectoreld.[ThisdenestheLiebracket*src:197Loop_groups_and_string_topology.tex[;]p:8ݵLie(G)O Lie(G)!Lie(G);[X:;Y8]=X Yqĸ 8YX.^ *src:200Loop_groups_and_string_topology.texByNleftinvqariance,OeachNleftinvariantNvectoreldisdetermineduniquelybyO itsqvqalueat*src:201Loop_groups_and_string_topology.tex1皸2G,-thereforeqwegettheidentication*src:202Loop_groups_and_string_topology.texT1|sGT͍皸+3皲=RLie(G);XwewillO frequentlyUUusebGothvqariants.^ *src:205Loop_groups_and_string_topology.texT*oeachleftinvqariantvectoreldXcmweassoGciateits
ow*src:206Loop_groups_and_string_topology.tex X:Gg-
msbm10R:r!GO (alpriori,itmightonlybGedenedonanopensubsetof*src:207Loop_groups_and_string_topology.texGf0g).U$W*eldenetheO exp}'onentialmap exp :
Lie(G)!G;X7! X$(1;1):)bO *src:212Loop_groups_and_string_topology.texThisisdenedonanopGensubsetof02G.QThedierential*src:213Loop_groups_and_string_topology.texd0'exp:ѵLie(G)!O T1|s(G)=Lie(G)istheidentity*,ʹthereforeonasuitablysmallopGenneighborhoodO ofUU*src:214Loop_groups_and_string_topology.tex0,expisadieomorphismontoitsimage.^ *src:217Loop_groups_and_string_topology.texAmaximaltorusTNJofthecompactLiegroupGisaLiesubgroup*src:218Loop_groups_and_string_topology.texT#GO whichisisomorphictoatorusTc^nײ(i.e.aproGductofcircles)andwhichhasO maximal1rankamongallsuch.eIt'satheoremthatforaconnectedcompactLieO groupD)*src:221Loop_groups_and_string_topology.texGandagivenmaximaltorusT*G,GanarbitraryconnectedabGelianLieO subgroupUU*src:222Loop_groups_and_string_topology.texAGisconjugatetoasubgroupof*src:223Loop_groups_and_string_topology.texTc.O ӟhtml: html:
92.2Example.*src:227Loop_groups_and_string_topology.texThe`groupU(n)\:=fA2M(n;C)jAA^s@=1g`isaLiegroup,caLieUUsubmanifoldofthegroup*src:228Loop_groups_and_string_topology.texGl2`(n;C)ofallinvertibleUUmatrices. *src:231Loop_groups_and_string_topology.texIn\thiscase,ݵT1|sU(n)w=fA2M(n;C)jA9+A^?[=w0g.The\commutatorof*src:232Loop_groups_and_string_topology.texT1|sU(n)=L(U(n))istheusualcommutatorofmatrices:Rc*src:233Loop_groups_and_string_topology.tex[A;B q]=AB;ø RBAforA;BG2T1|sU(n). *src:235Loop_groups_and_string_topology.texTheexpGonentialmapfortheLiegroupU(n)istheusualexponentialmapofUUmatrices,givenbythepGowerseries:DexpT:[T1|sU(n)!U(n);A7!exp7(A)=o1 u
cmex10Xk+B=0SAk됵=kP!:!n*src:241Loop_groups_and_string_topology.texTheUUfunctionalequationshowsthattheimageindeedbGelongstoU(n). *src:243Loop_groups_and_string_topology.texSimilarly*,UULie(S U(n))=fA2M(n;C)jA8+A^_=0;tr x(A)=0g. ahtml: html: 2.3Theorem.*src:248Loop_groups_and_string_topology.texIfGisac}'onnectedcompactLiegroup,?thenexp#:2ǵLie(G),!Gissurje}'ctive.html: html:
92.4Denition.*src:252Loop_groups_and_string_topology.texEvery>'compactconnectedLiegroupGcanbGerealizedasaLie-subgroup[Jof*src:253Loop_groups_and_string_topology.texS U(n)forbigenoughn.ItfollowsthatitsLiealgebra*src:254Loop_groups_and_string_topology.texLie(G)isxasub-LiealgebraofLie(U(n)) =fA2M(n;C)jA^Ȳ= Ag.Therefore,thecomplexicatione*src:256Loop_groups_and_string_topology.texLie(G)
.qy msbm7R7CisasubLiealgebraof*src:257Loop_groups_and_string_topology.texLie(U(n))
R7C=fA+iBGjA;BƯ2F>M(n;C);A^"= A;(iB q)^"= (iB)g=M(n;C).VNotethatthebracket(givenUUbythecommutator)iscomplexlinearonthesecomplexvectorspaces. *src:262Loop_groups_and_string_topology.texThe^correspGondingsubLiegroupGCɲofGl2`(n;C)(thesimplyconnectedLiegroup^withLiealgebra*src:263Loop_groups_and_string_topology.texM(n;C))withLiealgebra*src:264Loop_groups_and_string_topology.texLie(G)
Rx+CiscalledthecomplexicationUUof*src:265Loop_groups_and_string_topology.texG. *src:267Loop_groups_and_string_topology.texGC]is*(ac}'omplexl.Liegroup,2i.e.*(themanifoldGC]hasanaturalstructureofacomplexmanifold(chartswithholomorphictransitionmaps),1andthecompGo-sitionUU*src:269Loop_groups_and_string_topology.texGC8GCUM!GC㊲isholomorphic. s html: html:LoGopUUgroupsandstringtopology ģv3o html: html:
2.5Exer}'cise.;/*src:274Loop_groups_and_string_topology.texChecktheassertionsmadeinDenitionhtml:2.4 html:,ТinparticularabGout theUUcomplexstructure._~html: html:
{2.68Example. *src:279Loop_groups_and_string_topology.texThe8complexiedLiealgebraLie(S U(n))
R5C=fA2M(n;C)jtr:(A)=0g,UUandthecomplexicationof*src:281Loop_groups_and_string_topology.texS U(n)isSl2`(n;C). [nhtml: html: 2.7Denition.i8*src:285Loop_groups_and_string_topology.texLet̵GbGeacompactLiegroup.9Itactsonitselfbyconjugation:*src:286Loop_groups_and_string_topology.texG8G!G;(g[;h)7!ghg^ 1M. *src:288Loop_groups_and_string_topology.texF*orcNxedg:;2bG,fwecantakethedierentialofthecorrespGondingmap*src:289Loop_groups_and_string_topology.texhb7!g[hg^ 1matUUh=1.qThisdenestheadjointrepresentation*src:290Loop_groups_and_string_topology.texadp:8ݵG!Gl2`(Lie(G)). *src:292Loop_groups_and_string_topology.texW*enowdecompGoseLie(G)intoirreduciblesub-representationsforthisac-tion,*src:293Loop_groups_and_string_topology.texLie(G)=4%n
eufm10g1""gk됲.EachйoftheseareLiesubalgebras,andwehave*src:295Loop_groups_and_string_topology.tex[giTL;gj6]=0UUifi6=j . *src:297Loop_groups_and_string_topology.texGiscalledsemi-simpleifnonofthesummandsisone-dimensional,AandsimpleNifthereisonlyonesummand,whichadditionallyisrequirednottobGeone-dimensional.nhtml: html:2.8LR}'emark.
M*src:303Loop_groups_and_string_topology.texTheksimplyconnectedsimplecompactLiegroupshavekbGeenclas-sied,Athey|consistof*src:304Loop_groups_and_string_topology.texS U(n),SOG(n),the|symplecticgroupsS pnrandveexcep-tionalUUgroups(called*src:305Loop_groups_and_string_topology.texG2|s,F4,E6,E7,E8).html: html:2.9Denition.w*src:309Loop_groups_and_string_topology.texLetGbGeacompactLiegroupwithamaximaltorusTc.ֵGacts(inducedfromconjugation)on*src:310Loop_groups_and_string_topology.texLie(G)viatheadjointrepresentation,Nwhichinducesiarepresentationon*src:312Loop_groups_and_string_topology.texLie(G)a`
RC=:gC5.#2ThisiLiealgebracontainstheLiealgebraUU*src:313Loop_groups_and_string_topology.textC㊲ofthemaximaltorus,onwhich*src:314Loop_groups_and_string_topology.texTactstrivially(sinceTisabGelian). *src:316Loop_groups_and_string_topology.texSinceӒT7!isamaximaltorus,!itactsnon-trivialoneverynon-zerovectoroftheUUcomplement. *src:319Loop_groups_and_string_topology.texAs]'everynitedimensionalrepresentationofatorus,thecomplementdecom-pGosesintoadirectsum*src:320Loop_groups_and_string_topology.texLg,whereoneach*src:321Loop_groups_and_string_topology.texg,t5`2Tactsviamultiplicationwith_C*src:322Loop_groups_and_string_topology.tex z(t)פ2S ^1 ,awhere%:ET;3!S ^1oCisahomomorphism,acalledtheweightRofthesummandUU*src:323Loop_groups_and_string_topology.texg. *src:325Loop_groups_and_string_topology.texW*e3cantranslatethehomomorphisms%:AT:!8S ^1 Cintotheirderivqativeattheidentity*,+thusgettingalinearmap*src:327Loop_groups_and_string_topology.tex z^0#:
It?!R,+i.e.anelementofthedualspaceUU*src:328Loop_groups_and_string_topology.text^,theyarerelatedby*src:329Loop_groups_and_string_topology.tex z(expG(x))=e^ir0n cmsy50s(x). *src:331Loop_groups_and_string_topology.texThisway*,"wethinkofthegroupofcharactersx䍑^T;Բ=UHom*(T V;S ^1 )asalatticeinXu*src:332Loop_groups_and_string_topology.text^,Y=calledthelatticeofweights.{'ItXucontainsthesetofr}'oots,Y=i.e.thenon-zeroweightsUUoGccurringintheadjointrepresentationof*src:334Loop_groups_and_string_topology.texG.5html: html:
92.10>dR}'emark.o0*src:338Loop_groups_and_string_topology.texAsgC ۲isthecomplexicatoinofarealrepresentation,