Table Of ContentGrundlehren der
mathematischen Wissenschaften 346
ASeriesofComprehensiveStudiesinMathematics
Serieseditors
M.Berger P.delaHarpe F.Hirzebruch
N.J. Hitchin L.Hörmander A.Kupiainen
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Editor-in-Chief
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Forfurthervolumes:
www.springer.com/series/138
Jean-Louis Loday (cid:2) Bruno Vallette
Algebraic Operads
Jean-LouisLoday(1946–2012) BrunoVallette
IRMA,CNRSetUniversitédeStrasbourg Lab.deMathématiquesJ.A.Dieudonné
Strasbourg,France UniversitédeNice-SophiaAntipolis
Nice,France
ISSN0072-7830 GrundlehrendermathematischenWissenschaften
ISBN978-3-642-30361-6 ISBN978-3-642-30362-3(eBook)
DOI10.1007/978-3-642-30362-3
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012944723
MathematicsSubjectClassification: 18D50,17AXX,18G50,55P48,57T30
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Àlamémoirede JLL
Preface
Anoperadisanalgebraicdevicewhichencodesatypeofalgebras.Insteadofstudy-
ingthepropertiesofaparticularalgebra,wefocusontheuniversaloperationsthat
canbeperformedontheelementsofanyalgebraofagiventype.Theinformation
containedinanoperadconsistsintheseoperationsandallthewaysofcomposing
them.Theclassicaltypesofalgebras,thatisassociativealgebras,commutativealge-
brasandLiealgebras,givethefirstexamplesofalgebraicoperads.Recently,there
hasbeenmuchinterestinothertypesofalgebras,tonameafew:Poissonalgebras,
Gerstenhaberalgebras,Jordanalgebras,pre-Liealgebras,Batalin–Vilkoviskyalge-
bras, Leibniz algebras, dendriform algebras and the various types of algebras up
to homotopy. The notion of operad permits us to study them conceptually and to
comparethem.
Theoperadicpointofviewhasseveraladvantages.First,manyresultsknownfor
classicaltypesofalgebras,whenwrittenintheoperadiclanguage,canbeappliedto
othertypesofalgebras.Second,theoperadiclanguagesimplifiesboththestatements
andtheproofs.So,itclarifiestheglobalunderstandingandallowsonetogofurther.
Third, even for classical algebras, the operad theory provides new results that had
not been unraveled before. Operadic theorems have been applied to prove results
in other fields, like the deformation-quantization of Poisson manifolds by Maxim
KontsevichandDmitryTamarkinforinstance.Nowadays,operadsappearinmany
differentthemes:algebraictopology,differentialgeometry,noncommutativegeom-
∗
etry, C -algebras, symplectic geometry, deformation theory, quantum field theory,
stringtopology,renormalizationtheory,combinatorialalgebra,categorytheory,uni-
versalalgebraandcomputerscience.
Historically,thetheoreticalstudyofcompositionsofoperationsappearedinthe
1950s in the work of Michel Lazard as “analyseurs”. Operad theory emerged as
an efficient tool in algebraic topology in the 1960s in the work of Frank Adams,
J.MichaelBoardmann,AndréJoyal,GregoryKelly,PeterMay,SaundersMcLane,
JimStasheff,RainerVogtandothertopologistsandcategorytheorists.Inthe1990s,
there was a “renaissance” of the theory in the development of deformation theory
andquantumfieldtheory,withashiftfromtopologytoalgebra,thatcanbefound
intheworkofEzraGetzler,VictorGinzburg,VladimirHinich,JohnJones,Mikhail
vii
viii Preface
Kapranov, Maxim Kontsevich, Yuri I. Manin, Martin Markl, Vadim Schechtman,
VladimirSmirnovandDmitryTamarkinforinstance.Tenyearslater,afirstmono-
graph[MSS02]onthissubjectwaswrittenbyMartinMarkl,SteveShniderandJim
Stasheffinwhichonecanfindmoredetailsonthehistoryofoperadtheory.
Now,20yearsaftertherenaissanceoftheoperadtheory,mostofthebasicaspects
ofithavebeensettledanditseemstobetherighttimetoprovideacomprehensive
accountofalgebraicoperadtheory.Thisisthepurposeofthisbook.
One of the main fruitful problems in the study of a given type of algebras is
itsrelationshipwithalgebraichomotopytheory.Forinstance,startingwithachain
complex equipped with some compatible algebraic structure, can this structure be
transferred to any homotopy equivalent chain complex? In general, the answer is
negative.However,onecanprovetheexistenceofhigheroperationsonthehomo-
topyequivalentchaincomplex,whichendowitwitharicheralgebraicstructure.In
theparticularcaseofassociativealgebras,thishigherstructureisencodedintothe
notionofassociativealgebrauptohomotopy,aliasA-infinityalgebra,unearthedby
Stasheffinthe1960s.IntheparticularcaseofLiealgebras,itgivesrisetothenotion
of L-infinityalgebras, whichwas successfullyusedin theproofof theKontsevich
formalitytheorem.Itisexactlytheproblemofgoverningthesehigherstructuresthat
promptedtheintroductionofthenotionofoperad.
Operad theory provides an explicit answer to this transfer problem for a large
familyoftypesofalgebras,forinstancethoseencodedbyKoszuloperads.Koszul
dualitywasfirstdevelopedatthelevelofassociativealgebrasbyStewartPriddyin
the 1970s. It was then extended to algebraic operads by Ginzburg and Kapranov,
andalsoGetzlerandJonesinthe1990s(partoftherenaissanceperiod).Theduality
between Lie algebras and commutative algebras in rational homotopy theory was
recognized to coincide with the Koszul duality theory between the operad encod-
ingLiealgebrasandtheoperadencodingcommutativealgebras.Theapplicationof
Koszuldualitytheoryforoperadstohomotopicalalgebraisafar-reachinggeneral-
izationoftheideasofDanQuillenandDennisSullivan.
Theaimofthisbookis,first,toprovideanintroductiontoalgebraicoperads,sec-
ond,togiveaconceptualtreatmentofKoszulduality,and,third,togiveapplications
tohomotopicalalgebra.
Webeginbydevelopingthegeneraltheoryoftwistingmorphisms,whosemain
application here is the Koszul duality theory for associative algebras. We do it in
such a way that this pattern can be adapted to the operad setting. After giving the
definitionandthemainpropertiesofthenotionofoperad,wedeveloptheoperadic
homologicalalgebra.Finally,Koszuldualitytheoryofoperadspermitsustostudy
thehomotopypropertiesofalgebrasoveranoperad.
Weareverygratefultothemanyfriendsandcolleagueswhohavehelpedusand
in particular to pioneers of the subject Jim Stasheff, Dennis Sullivan, and Yuri I.
Manin. We owe thanks to Olivia Bellier, Alexander Berglund, Emily Burgunder,
DamienCalaque,YongshanChen,Pierre-LouisCurien,VladimirDotsenko,Gabriel
Drummond-Cole,ClémentDupont,YaëlFrégier,BenoitFresse,HidekazuFurusho,
Ezra Getzler, Darij Grinberg, Moritz Groth, Li Guo, Kathryn Hess, Joseph Hirsh,
LaurentHofer,EricHoffbeck,RalfHolkamp,MagdalenaKe¸dziorek,MurielLiver-
net,JoanMillès,NikolayNikolov,TodorPopov,MariaRonco,HenrikStrohmayer,
Preface ix
Antoine Touzé, Christine Vespa, Yong Zhang, and to the referees for their helpful
andcriticalcomments.
We wish to express our appreciation to the Centre National de Recherche Sci-
entifique,theEidgenössischeTechnischeHochschule(Zürich),andtheMax-Planck
InstitutfürMathematik(Bonn)fortheirsupport.
Last but not least, nous sommes heureux de remercier tout particulièrement
Eliane et Catherine pour avoir su créer autour de nous l’environnement idéal à la
rédactiond’untelouvrage.
Strasbourg,France Jean-LouisLoday
18thJanuary2012 BrunoVallette
Contents
1 Algebras,Coalgebras,Homology . . . . . . . . . . . . . . . . . . . . 1
1.1 ClassicalAlgebras(Associative,Commutative,Lie) . . . . . . . 1
1.2 CoassociativeCoalgebras . . . . . . . . . . . . . . . . . . . . . 9
1.3 Bialgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Pre-LieAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5 DifferentialGradedAlgebra . . . . . . . . . . . . . . . . . . . . 22
1.6 Convolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.7 Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2 TwistingMorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1 TwistingMorphisms . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2 BarandCobarConstruction . . . . . . . . . . . . . . . . . . . . 41
2.3 KoszulMorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4 CobarConstructionandQuasi-isomorphisms . . . . . . . . . . . 49
2.5 ProofoftheComparisonLemma . . . . . . . . . . . . . . . . . 51
2.6 Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 KoszulDualityforAssociativeAlgebras . . . . . . . . . . . . . . . . 61
3.1 QuadraticData,QuadraticAlgebra,QuadraticCoalgebra . . . . . 62
3.2 KoszulDualofaQuadraticAlgebra . . . . . . . . . . . . . . . . 64
3.3 BarandCobarConstructiononaQuadraticData . . . . . . . . . 66
3.4 KoszulAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5 GeneratingSeries . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.6 KoszulDualityTheoryforInhomogeneousQuadraticAlgebras . 75
3.7 Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4 MethodstoProveKoszulityofanAlgebra . . . . . . . . . . . . . . . 89
4.1 RewritingMethod . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2 ReductionbyFiltration . . . . . . . . . . . . . . . . . . . . . . . 92
xi
