Clifford Algebras and Lie Theory (December 2012 version)

343 Pages·2012·1.333 MB·English

by  Eckhard Meinrenken

Eckhard Meinrenken
Clifford Algebras and Lie Theory
Springer

Thisbookisdedicatedtomyfamily:
NozomiandEmma.

Preface
Given a symmetric bilinear form B on a vector spaceV, 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),andforgeneralBthereisanisomor-
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,suchasSpin(6)∼=SU(4),allfindnaturalexplanationsusingthespinrepre-
sentation. Furthermore, there are explicit constructions of the exceptional groups
E ,E ,E ,F ,G , using special features of the spin groups and the spin represen-
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,
therearemany constructions ofrepresentationsusingmoregeneral typesofDirac
operators,beginningwiththeworkofAtiyah–Schmid[16]andParthasarathy[106].
vii
viii
SeethebookbyHuang–Pandzic[66]forfurtherreferencesandmorerecentdevel-
opments.
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
theCliffordalgebraofgitself.Letφ ∈∧3gbethestructureconstantstensorofg,
definedusingthemetric.ByabeautifulobservationofKostantandSternberg[90],
thequantizedelementq(φ)∈Cl(g)squarestoascalar.Ifgiscomplexsemisimple,
thecubicelementisoneofthebasiselements
φ ∈∧2mi+1(g), i=1,...,rank(g),
i
of the primitive subspaceP(g)⊆∧(g)g of the invariantsubalgebra of the exterior
algebra.AccordingtotheHopf–Koszul–SamelsonTheorem,∧(g)gisitselfanexte-
rioralgebra∧P(g)withgeneratorsφ ,...,φ.In[86],Kostantprovedthemarvelous
1 l
resultthat,similarly,theinvariantsubalgebraoftheCliffordalgebraCl(g)gisitself
a Clifford algebra Cl(P(g)) over the quantized generators q(φ). In particular, the
i
elements q(φ) all square to scalars — a surprising fact given that the φ can have
i i
veryhighdegree.
My own involvement with this subject goes back to work with Anton Alek-
seev on group-valued moment maps [5]. 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
to the usual Weil differential on the standard Weil algebraW(g)=S(g∗)⊗∧(g∗).
Thedifferentialwasexplicitlygivenasagradedcommutationwithacubicelement
D =∑ei⊗e +1⊗q(φ)∈W(g),
g i
i
wheree isabasisofg,eithedualbasis,andφ ∈∧3gasabove.Thefactthat[D ,·]
i g
squarestozeroisduetothefactthatD2 liesinthecenterofW(g)—infactone
g
findsthat
1
D2= Cas ⊗1+const.,
g 2 g
where Cas ∈U(g)is thequadratic Casimirelement. In[6] andits sequel[9], we
g
usedthistheorytogiveanewproofforthecaseofquadraticLiealgebrasofDuflo’s
Theorem,givinganalgebraisomorphism
sym◦J(cid:100)1/2: S(g)g→U(g)g
(cf. Section 5.6 for notation) between invariants in the symmetric and enveloping
algebrasofg.
ix
Independently,Kostant[87]hadintroducedamoregeneralcubicelement
D ∈U(g)⊗Cl(p),
g,k
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,
with K a maximal rank subgroup of G, he used the cubic Dirac operator to ob-
tainanalgebraicversionoftheBorel–Weilconstruction[87,88],notrequiringin-
variant complex structures on G/K. (In the complex case, an algebraic version of
theBorel–WeilmethodhadbeenaccomplishedseveraldecadesearlierinKostant’s
work[84].)OtherbeautifulapplicationsincludeageneralizationoftheWeylcharac-
terformula,andthediscoveryofmultipletsofrepresentationsdueGross–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[74],Kazami–
Suzuki[77]andWassermann[119].
ThisbookhaditsbeginningsaslecturenotesforagraduatecoursesattheUni-
versityofToronto,taughtinthefallof2005.Theplanforthelectureswastogive
anintroductiontothecubicDiracoperator,coveringsomeoftheapplicationstoLie
theorydescribedabove.Tosetthestage,itwasnecessarytodevelopsomefounda-
tionalmaterialonCliffordalgebras.Inasecondeditionofthecourse,inthefallof
2009, we included other aspects such as the theory of pure spinors and Petracci’s
newproof[107]ofthePoincare´–Birkhoff–WittTheorem.Thebookitselfcontains
furthertopicsnotcoveredinthelectures;inparticularitdescribesKostant’sstruc-
turetheoryofCl(g)foracomplexreductiveLiealgebraandsurveystherecentde-
velopmentsconcerningthepropertiesoftheHarish-ChandraprojectionforClifford
algebras,hc : Cl(g)→Cl(t).
Cl
Anumberofinterestingtopicsrelatedtothethemeofthisbookhadtobeomit-
ted. For example, we did not include applications of the cubic Dirac operator to
Kac–Moodyalgebras[74,94,119,53,79,109,103].Wealsodecidedtoomitadis-
cussion of Dirac geometry [43, 4], group-valued moment maps [5] or generalized
complexgeometry[59,60,63],eventhoughsomeofthematerialpresentedhereis
motivatedbythosetopics.Furthermore,thebookdoesnotcovertheapplicationsto
theclassicaldynamicalYang–Baxterequation[51,8]ortotheKashiwara–Vergne
conjecture[75,7,10].
Manyofthetopicsinthisbookplayaroleintheoreticalphysics.Wehavealready
indicatedsomerelationshipswithconformalfieldtheory.TheprototypeofaDirac
operator(withrespecttotheMinkowskimetric)appearedinDirac’s1928articleon
his quantum theory of the electron [46]. More general Dirac operators on pseudo-
Riemannianmanifoldswerestudiedlater,andbecameastandardtoolindifferential
x
geometrywiththeadventofAtiyah–Singer’sindextheory[19,97].Theroleofrep-
resentationtheoryinquantummechanicswasexploredinWeyl’s1929book[120].
The“multi-valued”representationsoftherotationgroup,i.e.,representationsofthe
spingroup,areofbasicimportanceinquantumelectrodynamics,correspondingto
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.
Large portions of this book are based on material from joint publications with
Anton Alekseev, and it is a pleasure to thank him for his collaboration. I am in-
debted to Bertram Kostant for his writings as well as inspiring conversations; his
explanationsof[86]greatlyinfluencedthefinalchapterofthisbook.Iamgrateful
to the anonymous referees for numerous comments and suggestions. I would also
liketothankHenriqueBursztyn,MarcoGualtieri,VictorGuillemin,BorisKhesin,
Greg Landweber, David Li-Bland, Reyer Sjamaar, and Shlomo Sternberg for dis-
cussions.
I thank the University of Toronto, University of Geneva, Keio University, and
theUniversityofAdelaidefortheirsupportduringthetimeofwritingofthisbook.
Finally,IwouldliketothankAnnKostantfromSpringerforadvice,patience,and
editorialhelpincopyeditingandhelpingproducethefinalversionofthisbook.
Toronto,December2012 EckhardMeinrenken
Conventions
Throughout this book K will denote a ground field. The precise convention will
usually be stated at the beginning of a chapter – initially K is any field of char-
acteristic (cid:54)=2; towards the end of the book we take K to be the real or complex
numbers.Wewillmakefrequentuseofsuper-conventions,asoutlinedinAppendix
A. For instance, given a Z -graded algebra, we will refer to commutators where
2
some authors prefer the term super-commutator, and we will call derivations what
some authors would call super derivations or even anti-derivations. Unless speci-
fiedotherwiseorclearfromthecontext,vectorspacesorLiealgebrasaretakento
be finite-dimensional, and algebras are taken to be associative algebras with unit.
Some background material and conventions about reductive Lie algebras may be
foundinAppendixB.
xi