Non symmetric operads Symmetric operads
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79
pages
English
Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
- algebras
- symmetric operads
- chapoton algebraic
- permutations ?? associative
- trees ??
- integer partitions
- trees
- ?? pre-lie
Publié par
Langue
English
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
