Non symmetric operads Symmetric operads

F. Chapoton

Description

Non-symmetric operads Symmetric operads From operads to groups Algebraic combinatorics and trees F. Chapoton March 22, 2007 F. Chapoton Algebraic combinatorics and trees

• algebras


• symmetric operads


• chapoton algebraic


• permutations ?? associative


• trees ??


• integer partitions


• trees


• ?? pre-lie

Sujets

• Algebra
• Partition (number theory)

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 12
• Langue: English

Extrait

seertdMarch22,2007

nF.Chapoton

aseert

sdna

ccombinatorics

iAlgebraic

rotanibmocciarbeglAnotopahC.FspuorgotsdarepomorFsdarepocirtemmySsdarepocirtemmys-noN
Letusrecalltheclassicalrelationbetween
words
and
associativity
.
Analphabet
A
isasetofletters
{
a
,
b
,
c
,...
}
.Aword
w
inthe
alphabet
A
isasequenceofletters
w
=(
w
1
,
w
2
,...,
w
k
).Thereis
abasicoperationonwordsgivenbyconcatenation,whichis
associative.Infact,thesetofwordsisexactlythefreeassociative
monoidontheset
A
.Sothestudyofwordsnaturallytakesplace
inthesettingofassociativealgebras.
Considernowthealphabet
{
a
1
,...,
a
n
}
.Thenthesetofwords
whereeachletter
a
i
appearsexactlyoncecanbeseenasthesetof
permutations
of
{
1
,...,
n
}
.

Wordsandpermutations

seertdnascirotanibmocciarbeglAnotopahC.FspuorgotsdarepomorFsdarepocirtemmySsdarepocirtemmys-noN
ocirtemmys-noNThenaturalsettingofthisgeneralisationisthetheoryof
operads
.

planarbinarytrees
←→
dendriformalgebras.

rootedtrees
←→

wordsorpermutations
←→
associativealgebras,

Correspondence

Onecanfindasimilarrelationbetweensomekindsoftreesand
somenewkindsofalgebraicstructures.

seertdnascirotanibmocciarbeglAnotopahC.F,sarbeglaeiL-erpspuorgotsdarepomorFsdarepocirtemmySsdarep
,spuorgcirtemmysehtrevoseludom1
→
2
→
3
→∙∙∙→
n
.

(1)

Thereisanaturalbijectionbetweenplanarbinarytreesandtilting
modulesoverthefollowingquivers:

partitions
←→

planarbinarytrees
←→
tiltingmodulesonquiversoftypeA.

Integerpartitionsareclassicalincombinatoricsandareimportant
tooinrepresentationtheory.
TheHopfalgebraofsymmetricfunctionscanbeseenasa
descriptionofrepresentationsofsymmetricgroups.
Thesetofplanarbinarytreesshouldhavealsosuchadualnature.

Partitions

spuorgotsdarepomorFsdarepocirtemmySsdarepocirtemmys-noNseertdnascirotanibmocciarbeglAnotopahC.F
seertdnascirotanibmocciarbeglAnotopahC.FOperadssometimesprovideawaytounderstandalltheseobjects.

Algebraicstructuresontreesdidalreadyappearalongtimeago,
forinstanceintheworkofButcherinnumericalanalysis.
Manynewalgebraicstructuresontreeshavebeenintroducedmore
recently,notablybyConnesandKreimer.Amongthem,onecan
dnfi

operads.

groups,

Liealgebras,

Hopfalgebras,

Grafting,cutting,pruning,gluing,etc

spuorgotsdarepomorFsdarepocirtemmySsdarepocirtemmys-noN
spuorgotsdarepomorFsdarepocirtemmySsdarepocirtemmys-noNgeneratingseriesandtheLambertWfunction(randomgraph)

proofofLagrangeinversionusingtrees

randomtrees

bijectionsormorphismsbetweentreesandpermutations

statisticsontrees(likepermutations)

posetsonsomesetsoftrees

Obviously,treesareusedeverywhereincombinatorics.For
example,

seertdnascirotanibmocciarbeglAnotopahC.F
seertdnascirotanibmocciarbeglAnotopahC.Frootedtreesandtworelatedoperads

planarbinarytreesandtworelatedoperads

Theaimoftheselecturesistointroducethenotionofoperad,ina
combinatorialcontext.
Wegivedefinitionsofseveralvariantsofthenotionofoperadand
illustrateeachofthembysomespecificexample.
Wealsoexplainhowonecanbuildfromanoperadotheralgebraic
structures,suchasagroupof”invertibleformalpowerseries”.
Wewillconcentrateontwoparticularlynicekindsoftrees.

spuorgotsdarepomorFsdarepocirtemmySsdarepocirtemmys-noN
spuorgotsdarepomorFsdarepocirtemmySsdarepocirtemmys-noNobjects.
Thisseconddichotomycorrespondsalso(insomesense)to
non-planarorplanartrees.

non-symmetricorunlabeled(withoutactionsofsymmetric
groups)

symmetricorlabeled(withactionsofsymmetricgroups)or

andalsoeitherwith

inthecategoryofvectorspaces,

inthecategoryofsetsor

Onehastodistinguishfourkindsofoperads:eitherwework

Fourflavoursofoperads

seertdnascirotanibmocciarbeglAnotopahC.F
spuorgotsdarepomorFsdarepocirtemmySsdarepocirtemmys-noNseertInVect,symmetric:pre-Lie

dInVect,non-symmetric:Dendriform,Mould

nInSet,symmetric:Commutative,NAP

aInSet,non-symmetric:Associative,OverUnder

sWewillconsiderexamplesofoperadsofallfourkinds.

cirotanibmocciarbeglAnotopahC.F
secapsrotcevfoyrogetacehtnisdarepOstesfoyrogetacehtnisdarepOspuorgotsdarepomorFsdarepocirtemmySsdarepocirtemmys-noNInthissection,thenotionofoperadisintroduced,firstinthe
categoryofsets,theninthecategoryofvectorspaces.Wegivetwo
differentdefinitionsandexplainhowtheyarerelatedtoeachother.
Operadswerefirstintroducedinalgebraictopologyinthe1960’s.
Morerecently,thetheoryofoperadshasknownfurther
developmentsinmanydirections.Operadsareusefultodescribe
andworkwithcomplicatednewkindsofalgebrasandalgebrasup
tohomotopy.

seertdnascirotanibmocciarbeglAnotopahC.F