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Ergebnisse der Mathematik Volume 58 und ihrer Grenzgebiete 3.Folge A Series of Modern Surveys in Mathematics EditorialBoard L.Ambrosio,Pisa G.-M.Greuel,Kaiserslautern M.Gromov,Bures-sur-Yvette J.Jost,Leipzig J.Kollár,Princeton S.S.Kudla,Toronto G.Laumon,Orsay H.W.Lenstra,Jr.,Leiden J.Tits,Paris D.B.Zagier,Bonn ManagingEditor R.Remmert,Münster Forfurthervolumes: www.springer.com/series/728 Eckhard Meinrenken Clifford Algebras and Lie Theory EckhardMeinrenken DepartmentofMathematics UniversityofToronto Toronto,ON,Canada ISSN0071-1136 ErgebnissederMathematikundihrerGrenzgebiete.3.Folge/ASeries ofModernSurveysinMathematics ISBN978-3-642-36215-6 ISBN978-3-642-36216-3(eBook) DOI10.1007/978-3-642-36216-3 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013933367 MathematicsSubjectClassification: 15A66,17B20,17B35,17B55 ©Springer-VerlagBerlinHeidelberg2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) DedicatedtomyfamilyNozomiandEmma Preface Given a symmetric bilinear form B on a vector space V, one defines the Clifford algebraCl(V;B)tobetheassociativealgebrageneratedbytheelementsofV,with relations v v +v v =2B(v ,v ), v ,v ∈V. 1 2 2 1 1 2 1 2 IfB=0thisisjusttheexterioralgebra∧(V),andforgeneralB thereisanisomor- phism(thequantizationmap) q:∧(V)→Cl(V;B). Hence,theCliffordalgebramayberegardedas∧(V)withanew,“deformed”prod- uct. Clifford algebras enter the world of Lie groups and Lie algebras from multi- pledirections.Forexample,theyareusedtogiveconstructionsofthespingroups Spin(n),thesimplyconnecteddoublecoveringsofSO(n)forn≥3.Goingalittle further,onethenobtainsthespinrepresentationsofSpin(n),whichisanirreducible representationifnisoddandbreaksupintotwoinequivalentirreduciblerepresen- tations if n is even. The “accidental” isomorphisms of Lie groups in low dimen- ∼ sions, such as Spin(6)=SU(4), all find natural explanations using the spin repre- sentation. Furthermore, there are explicit constructions of the exceptional groups E ,E ,E ,F ,G ,usingspecialfeaturesofthespingroupsandthespinrepresen- 6 7 8 4 2 tation.Therearemanyremarkableaspectsoftheseconstructions,seee.g.,Adams’ book[1]orthesurveyarticlebyBaez[20]. Further relationships between Lie groups and Clifford algebras come from the theory of Dirac operators on homogeneous spaces. Recall that by the Borel–Weil Theorem,anyirreduciblerepresentationofacompactLiegroupGisrealizedasa spaceofholomorphicsectionsofalinebundleoveranappropriatecoadjointorbit. These sections may be regarded as solutions of the Dolbeault–Dirac operator for thestandardcomplexstructureonthecoadjointorbit.Fornon-compactLiegroups, therearemanyconstructionsof representationsusingmoregeneraltypesof Dirac operators,beginningwiththeworkofAtiyah–Schmid[16]andParthasarathy[106]. SeethebookbyHuang–Pandzic[66]forfurtherreferencesandmorerecentdevel- opments. vii viii Preface Given a Lie algebra g with a non-degenerate invariant symmetric bilinear form B (e.g., g semisimple, with B the Killing form), it is also of interest to consider the Clifford algebra of g itself. Let φ∈∧3g be the structure constant tensor of g, definedusingthemetric.ByabeautifulobservationofKostantandSternberg[90], thequantizedelementq(φ)∈Cl(g)squarestoascalar.Ifgiscomplexsemisimple, thecubicelementisoneofthebasiselements φ ∈∧2mi+1(g), i=1,...,rank(g), i of the primitive subspace P(g)⊆∧(g)g of the invariant subalgebra of the exterioralgebra.AccordingtotheHopf–Koszul–SamelsonTheorem,∧(g)g isitselfan exterioralgebra∧P(g)withgeneratorsφ ,...,φ .In[86],Kostantprovedthemar- 1 l velousresultthat,similarly,theinvariantsubalgebraoftheCliffordalgebraCl(g)g isitselfaCliffordalgebraCl(P(g))overthequantizedgeneratorsq(φ ).Inparticu- i lar,theelementsq(φ )allsquaretoscalars—asurprisingfactgiventhattheφ can i i haveveryhighdegree. My own involvement with this subject goes back to work with Anton Alek- seev on group-valued moment maps [11]. We had developed a non-standard non- commutative equivariant de Rham theory tailored towards these applications, and werelookingforanaturalframeworkinordertounderstanditsrelationtothestan- dard equivariant de Rham theory. Inspired by [86, 90], we were led to consider a quantumWeilalgebra W(g)=U(g)⊗Cl(g), whereU(g)istheenvelopingalgebraofg,andequippedwithadifferentialsimilar totheusualWeildifferentialonthestandardWeilalgebraW(g)=S(g∗)⊗∧(g∗). Thedifferentialwasexplicitlygivenasagradedcommutationwithacubicelement (cid:2) D = ei⊗e +1⊗q(φ)∈W(g), g i i wheree isabasisofg,ei thedualbasis,andφ∈∧3gasabove.Thefactthat[D ,·] i g squarestozeroisduetothefactthatD2liesinthecenterofW(g)—infactonefinds g that 1 D2= Cas ⊗1+const., g 2 g where Cas ∈U(g) is thequadraticCasimir element.In [4] andits sequel [7], we g usedthistheorytogiveanewproofforthecaseofquadraticLiealgebrasofDuflo’s Theorem,givinganalgebraisomorphism (cid:3) sym◦J1/2:S(g)g→U(g)g (cf. Section 5.6 for notation) between invariants in the symmetric and enveloping algebrasofg. Independently,Kostant[87]hadintroducedamoregeneralcubicelement D ∈U(g)⊗Cl(p), g,k Preface ix associatedtoaquadraticLiesubalgebrak⊆gwithorthogonalcomplementp=k⊥, which he called the cubic Dirac operator. Taking k=0, one recovers D =D . g,0 g For the case where g and k are the Lie algebras of compact Lie groups G and K, withK amaximalranksubgroupofG,heusedthecubicDiracoperatortoobtain analgebraicversionoftheBorel–Weilconstruction[87,88],notrequiringinvariant complexstructuresonG/K.(Inthecomplexcase,analgebraicversionoftheBorel– WeilmethodhadbeenaccomplishedseveraldecadesearlierinKostant’swork[84].) OtherbeautifulapplicationsincludeageneralizationoftheWeylcharacterformula, andthediscoveryofmultipletsofrepresentationsduetoGross–Kostant–Ramond– Sternberg[57]. It is possible to go in the opposite direction and interpret D as a geometric g,k Dirac operator on a homogeneous space. As such, it corresponds to a particular affineconnectionwithanon-zerotorsion.SuchgeometricDiracoperatorshadbeen studied by Slebarski [114, 115] in the 1980s; the precise relation with Kostant’s cubic Dirac operator was clarified in [2]. There are also precursors in conformal fieldtheoryandKac–Moodytheory;seeinparticularKac–Todorov[73],Kazami– Suzuki[77]andWassermann[119]. This book had its beginnings as lecture notes for a graduate course at the Uni- versityofToronto,taughtinthefallof2005.Theplanforthelectureswastogive anintroductiontothecubicDiracoperator,coveringsomeoftheapplicationstoLie theorydescribedabove.Tosetthestage,itwasnecessarytodevelopsomefounda- tionalmaterialonCliffordalgebras.Inasimilargraduatecourseinthefallof2009, we included other aspects such as the theory of pure spinors and Petracci’s new proof [107] of thePoincaré–Birkhoff–WittTheorem. Thebookitself containsfur- thertopicsnotcoveredinthelectures;inparticularitdescribesKostant’sstructure theory of Cl(g) for a complex reductive Lie algebra and surveys the recent devel- opments concerning the properties of the Harish-Chandra projection for Clifford algebras,hcCl:Cl(g)→Cl(t). Anumberofinterestingtopicsrelatedtothethemeofthisbookhadtobeomit- ted. For example, we did not include applications of the cubic Dirac operator to Kac–Moodyalgebras[53,73,79,94,103,109,119].Wealsodecidedtoomitadis- cussionofDiracgeometry[13,43],group-valuedmomentmaps[11]orgeneralized complexgeometry[59,60,63],eventhoughsomeofthematerialpresentedhereis motivatedbythosetopics.Furthermore,thebookdoesnotcovertheapplicationsto theclassicaldynamicalYang–Baxterequation[6, 51] or totheKashiwara–Vergne conjecture[5,8,75]. Manyofthetopicsinthisbookplayaroleintheoreticalphysics.Wehavealready indicatedsomerelationshipswithconformalfieldtheory.TheprototypeofaDirac operator(withrespecttotheMinkowskimetric)appearedinDirac’s1928articleon his quantum theory of the electron [46]. More general Dirac operators on pseudo- Riemannianmanifoldswerestudiedlater,andbecameastandardtoolindifferential geometrywiththeadventofAtiyah–Singer’sindextheory[18,97].Theroleofrep- resentationtheoryinquantummechanicswasexploredinWeyl’s1929book[120]. The“multi-valued”representationsoftherotationgroup,i.e.,representationsofthe spingroup,areofbasicimportanceinquantumelectrodynamics,correspondingto x Preface particleswithpossiblyhalf-integerspin.Asystematicstudy,introducingthegroup Spin(n),wasmadein1935intheworkofBrauer–Weyl[29].Detailedinformation on the numerous applications of Clifford algebras, spinors and Dirac operators in physicsmaybefoundinthebooks[23,64]. Acknowledgments Largeportionsofthisbookarebasedonmaterialfromjoint publicationswithAntonAlekseev,anditisapleasuretothankhimforhiscollab- oration.IamindebtedtoBertramKostantforhiswritingsaswellasinspiringcon- versations;hisexplanationsof[86]greatlyinfluencedthefinalchapterofthisbook. Iamgratefultotheanonymousrefereesfornumerouscommentsandsuggestions. I would also like to thank Henrique Bursztyn, Marco Gualtieri, Victor Guillemin, BorisKhesin,GregLandweber,DavidLi-Bland,ReyerSjamaar,andShlomoStern- bergfordiscussions. I thank the University of Toronto, University of Geneva, Keio University, and theUniversityofAdelaidefortheirsupportduringthetimeofwritingofthisbook. Finally,IwouldliketothankAnnKostantfromSpringerforadvice,patience,and generoushelpincopyeditingandproducingthefinalversionofthisbook. Toronto,ON,Canada EckhardMeinrenken Conventions ThroughoutthisbookKwilldenoteagroundfield.Thepreciseconventionwillusu- allybestatedatthebeginningofachapter—initiallyK isanyfieldofcharacteris- tic(cid:12)=2;towardstheendofthebookwetakeKtobetherealorcomplexnumbers. Wewillmakefrequentuseofsuper-conventions,asoutlinedinAppendixA.Forin- stance,givenaZ -gradedalgebra,wewillreferto“commutators”wheresomeau- 2 thorsprefertheterm“supercommutator”,andwewillcall“derivations”whatsome authorswouldcall“superderivations”oreven“anti-derivations”.Unlessspecified otherwise or clear from the context, vector spaces or Lie algebras are taken to be finitedimensional,andalgebrasaretakentobeassociativealgebraswithunit.Some backgroundmaterialandconventionsaboutreductiveLiealgebrasmaybefoundin AppendixB. xi