Algebraic Operads [draft]

534 Pages·2012·4.625 MB·English

by  Jean-Louis Loday and Bruno Vallette

Algebraic Operads
Version 0.999(cid:13)c
Jean-Louis Loday and Bruno Vallette

Preface
An operad is an algebraic device which encodes a type of algebras. Instead of
studying the properties of a particular algebra, we focus on the universal opera-
tions that can be performed on the elements of any algebra of a given type. The
information contained in an operad consists in these operations and all the ways
of composing them. The classical types of algebras, that is associative algebras,
commutative algebras and Lie algebras, give the first examples of algebraic oper-
ads. Recently, there has been much interest in other types of algebras, to name
a few: Poisson algebras, Gerstenhaber algebras, Jordan algebras, pre-Lie algebras,
Batalin–Vilkovisky algebras, 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 compare them.
The operadic point of view has several advantages. First, many results known
for classical types of algebras, when written in the operadic language, can be ap-
plied to other types of algebras. Second, the operadic language simplifies both
the statements and the proofs. So, it clarifies the global understanding and allows
one to go further. Third, even for classical algebras, the operad theory provides
new results that had not been unraveled before. Operadic theorems have been ap-
plied to prove results in other fields, like the deformation-quantization of Poisson
manifolds by Maxim Kontsevich and Dmitry Tamarkin for instance. Nowadays,
operadsappearinmanydifferentthemes: algebraictopology,differentialgeometry,
noncommutative geometry, C∗-algebras, symplectic geometry, deformation theory,
quantum field theory, string topology, renormalization theory, combinatorial alge-
bra, category theory, universal algebra and computer science.
Historically, the theoretical study of compositions of operations appeared in
the 1950’s in the work of Michel Lazard as “analyseurs”. Operad theory emerged
as an efficient tool in algebraic topology in the 1960’s in the work of Frank Adams,
J.MichaelBoardmann, Andr´eJoyal, GregoryKelly, PeterMay, SaundersMcLane,
Jim Stasheff, Rainer Vogt and other topologists and category theorists. In the
1990’s, there was a “renaissance” of the theory in the development of deformation
theory and quantum field theory, with a shift from topology to algebra, that can
be found in the work of Ezra Getzler, Victor Ginzburg, Vladimir Hinich, John
Jones,MikhailKapranov,MaximKontsevich,YuriI.Manin,MartinMarkl,Vadim
Schechtman,VladimirSmirnovandDmitryTamarkinforinstance. Tenyearslater,
a first monograph [MSS02] on this subject was written by Martin Markl, Steve
ShniderandJimStasheffinwhichonecanfindmoredetailsonthehistoryofoperad
theory.
iii
iv PREFACE
Now, 20 years after the renaissance of the operad theory, most of the basic
aspects of it have been settled and it seems to be the right time to provide a com-
prehensive account of algebraic operad theory. This is the purpose of this book.
Oneofthemainfruitfulproblemsinthestudyofagiventypeofalgebrasisits
relationship with algebraic homotopy theory. For instance, starting with a chain
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, one can prove the existence of higher operations on the homo-
topyequivalentchaincomplex,whichendowitwitharicheralgebraicstructure. In
the particular case of associative algebras, this higher structure is encoded into the
notion of associative algebra up to homotopy, alias A-infinity algebra, unearthed
by Stasheff in the 1960’s. In the particular case of Lie algebras, it gives rise to
the notion of L-infinity algebras, which was successfully used in the proof of the
Kontsevich formality theorem. It is exactly the problem of governing these higher
structures that prompted the introduction of the notion of operad.
Operad theory provides an explicit answer to this transfer problem for a large
family of types of algebras, for instance those encoded by Koszul operads. Koszul
dualitywasfirstdevelopedatthelevelofassociativealgebrasbyStewartPriddyin
the 1970’s. It was then extended to algebraic operads by Ginzburg and Kapranov,
and also Getzler and Jones in the 1990’s (part of the renaissance period). The
duality between Lie algebras and commutative algebras in rational homotopy the-
ory was recognized to coincide with the Koszul duality theory between the operad
encoding Lie algebras and the operad encoding commutative algebras. The appli-
cationofKoszuldualitytheoryforoperadstohomotopicalalgebraisafar-reaching
generalization of the ideas of Dan Quillen and Dennis Sullivan.
The aim of this book is, first, to provide an introduction to algebraic oper-
ads, second, to give a conceptual treatment of Koszul duality, and, third, to give
applications to homotopical algebra.
We begin by developing the general theory of twisting morphisms, whose main
application here is the Koszul duality theory for associative algebras. We do it in
suchawaythatthispatterncanbeadaptedtotheoperadsetting. Aftergivingthe
definitionandthemainpropertiesofthenotionofoperad,wedeveloptheoperadic
homological algebra. Finally, Koszul duality theory of operads permits us to study
the homotopy properties of algebras over an operad.
We are very grateful to the many friends and colleagues who have helped us
andinparticulartopioneersofthesubjectJimStasheff, DennisSullivan, andYuri
I. Manin. We owe thanks to Olivia Bellier, Alexander Berglund, Emily Burgunder,
DamienCalaque,YongshanChen,Pierre-LouisCurien,VladimirDotsenko,Gabriel
Drummond-Cole, Cl´ement Dupont, Ya¨el Fr´egier, Benoit Fresse, Hidekazu Furusho,
Ezra Getzler, Darij Grinberg, Moritz Groth, Li Guo, Kathryn Hess, Joseph Hirsh,
Laurent Hofer, Eric Hoffbeck, Ralf Holkamp, Magdalena K¸edziorek, Muriel Liver-
net, JoanMill`es, NikolayNikolov, TodorPopov, MariaRonco, HenrikStrohmayer,
Antoine Touz´e, Christine Vespa, Yong Zhang, and to the referees for their helpful
and critical comments.
PREFACE v
We wish to express our appreciation to the Centre National de Recherche Sci-
entifique, the Eidgen¨ossische Technische Hochschule (Zu¨rich), and the Max-Planck
Institut fu¨r Mathematik (Bonn) for their support.
Last but not least, nous sommes heureux de remercier tout particuli`erement
Eliane et Catherine pour avoir su cr´eer autour de nous l’environnement id´eal `a la
r´edaction d’un tel ouvrage.
18th January 2012 Jean-Louis Loday and Bruno Vallette

Contents
Preface iii
Leitfaden xi
Introduction xiii
Chapter 1. Algebras, coalgebras, homology 1
1.1. Classical algebras (associative, commutative, Lie) 1
1.2. Coassociative coalgebras 7
1.3. Bialgebra 11
1.4. Pre-Lie algebras 15
1.5. Differential graded algebra 17
1.6. Convolution 24
1.7. R´esum´e 26
1.8. Exercises 26
Chapter 2. Twisting morphisms 29
2.1. Twisting morphisms 29
2.2. Bar and cobar construction 32
2.3. Koszul morphisms 38
2.4. Cobar construction and quasi-isomorphisms 39
2.5. Proof of the Comparison Lemma 40
2.6. R´esum´e 44
2.7. Exercises 46
Chapter 3. Koszul duality for associative algebras 47
3.1. Quadratic data, quadratic algebra, quadratic coalgebra 48
3.2. Koszul dual of a quadratic algebra 49
3.3. Bar and cobar construction on a quadratic data 51
3.4. Koszul algebras 53
3.5. Generating series 57
3.6. Koszul duality theory for inhomogeneous quadratic algebras 58
3.7. R´esum´e 64
3.8. Exercises 66
Chapter 4. Methods to prove Koszulity of an algebra 69
4.1. Rewriting method 70
4.2. Reduction by filtration 71
4.3. Poincar´e–Birkhoff–Witt bases and Gr¨obner bases 78
4.4. Koszul duality theory and lattices 85
4.5. Manin products for quadratic algebras 87
4.6. R´esum´e 90
vii
viii CONTENTS
4.7. Exercises 91
Chapter 5. Algebraic operad 95
5.1. S-module and Schur functor 97
5.2. Algebraic operad and algebra over an operad 104
5.3. Classical and partial definition of an operad 113
5.4. Various constructions associated to an operad 119
5.5. Free operad 122
5.6. Combinatorial definition of an operad 127
5.7. Type of algebras 129
5.8. Cooperad 132
5.9. Nonsymmetric operad 140
5.10. R´esum´e 148
5.11. Exercises 149
Chapter 6. Operadic homological algebra 155
6.1. Infinitesimal composite 156
6.2. Differential graded S-module 158
6.3. Differential graded operad 160
6.4. Operadic twisting morphism 167
6.5. Operadic Bar and Cobar construction 171
6.6. Operadic Koszul morphisms 176
6.7. Proof of the Operadic Comparison Lemmas 178
6.8. R´esum´e 181
6.9. Exercises 182
Chapter 7. Koszul duality of operads 185
7.1. Operadic quadratic data, quadratic operad and cooperad 186
7.2. Koszul dual (co)operad of a quadratic operad 188
7.3. Bar and cobar construction on an operadic quadratic data 190
7.4. Koszul operads 191
7.5. Generating series 194
7.6. Binary quadratic operads 195
7.7. Nonsymmetric binary quadratic operad 201
7.8. Koszul duality for inhomogeneous quadratic operads 203
7.9. R´esum´e 208
7.10. Exercises 209
Chapter 8. Methods to prove Koszulity of an operad 211
8.1. Rewriting method for binary quadratic ns operads 212
8.2. Shuffle operad 214
8.3. Rewriting method for operads 220
8.4. Reduction by filtration 223
8.5. PBW bases and Gr¨obner bases for operads 227
8.6. Distributive laws 232
8.7. Partition poset method 239
8.8. Manin products 249
8.9. R´esum´e 255
8.10. Exercises 257
CONTENTS ix
Chapter 9. The operads As and A 263
∞
9.1. Associative algebras and the operad Ass 264
9.2. Associative algebras up to homotopy 273
9.3. The bar–cobar construction on As 278
9.4. Homotopy Transfer Theorem for the operad As 280
9.5. An example of an A -algebra with nonvanishing m 286
∞ 3
9.6. R´esum´e 288
9.7. Exercises 288
Chapter 10. Homotopy operadic algebras 291
10.1. Homotopy algebras: definitions 292
10.2. Homotopy algebras: morphisms 299
10.3. Homotopy Transfer Theorem 305
10.4. Inverse of ∞-isomorphisms and ∞-quasi-isomorphisms 313
10.5. Homotopy operads 317
10.6. R´esum´e 324
10.7. Exercises 326
Chapter 11. Bar and cobar construction of an algebra over an operad 329
11.1. Twisting morphism for P-algebras 329
11.2. Bar and cobar construction for P-algebras 331
11.3. Bar–cobar adjunction for P-algebras 336
11.4. Homotopy theory of P-algebras 339
11.5. R´esum´e 344
11.6. Exercises 346
Chapter 12. (Co)homology of algebras over an operad 347
12.1. Homology of algebras over an operad 348
12.2. Deformation theory of algebra structures 352
12.3. Andr´e–Quillen (co)homology of algebras over an operad 363
12.4. Operadic cohomology of algebras with coefficients 377
12.5. R´esum´e 381
12.6. Exercises 383
Chapter 13. Examples of algebraic operads 387
13.1. Commutative algebras and the operad Com 388
13.2. Lie algebras and the operad Lie 395
13.3. Poisson algebras, Gerstenhaber algebras and their operad 405
13.4. Pre-Lie algebras and Perm-algebras 414
13.5. Leibniz algebras and Zinbiel algebras 418
13.6. Dendriform algebras and diassociative algebras 420
13.7. Batalin–Vilkovisky algebras and the operad BV 428
13.8. Magmatic algebras 434
13.9. Parametrized binary quadratic operads 435
13.10. Jordan algebras, interchange algebras 436
13.11. Multi-ary algebras 437
13.12. Examples of operads with 1-ary operations, 0-ary operation 442
13.13. Generalized bialgebras and triples of operads 444
13.14. Types of operads 449
x CONTENTS
Appendix A. The symmetric group 457
A.1. Action of groups 457
A.2. Representations of the symmetric group S 458
n
Appendix B. Categories 463
B.1. Categories and functors 463
B.2. Adjoint functors, free objects 466
B.3. Monoidal category 468
B.4. Monads 471
B.5. Categories over finite sets 472
B.6. Model categories 474
B.7. Derived functors and homology theories 478
Appendix C. Trees 481
C.1. Planar binary trees 481
C.2. Planar trees and Stasheff polytope 484
C.3. Trees and reduced trees 486
C.4. Graphs 488
Bibliography 491
Index 505
List of Notations 513