Table Of ContentExponential series without denominators
Jean-LouisLoday
2
1
0
2
n
a
J
4
2 AbstractForacommutativealgebrawhichcomesfromaZinbielalgebratheexpo-
nentialseriescanbewrittenwithoutdenominators.Whenliftedtodendriformalge-
] brasthisnewseriessatisfiesafunctionalequationanalogoustotheBaker-Campbell-
A
Hausdorffformula.Wemakeitexplicitbyshowingthattheobstructionseriesisthe
R
sumofthebraceproducts.InthemultilinearcaseweshowthattheroletheEulerian
.
h idempotentisplayedbytheiteratedpre-Lieproduct.
t
a
m
[ Introduction
1
v
Theclassicalexponentialseries
3
4
x2 xn
0 exp(x)=1+x+ +···+ +···
5 2! n!
.
1 canbewrittenwithoutdenominatorsprovidedoneassumessomepropertiesonthe
0 commutativealgebra.ForinstanceinaZinbielalgebratheterm xn canbereplaced
2 n!
bytheiteratedproductx≺n :=x≺(x≺n−1), withx≺1=x. Recallthata Zinbielal-
1
: gebra is a commutativealgebrain which the productsplits as xy=x≺y+y≺x.
v
Matriceswithcoefficientsinacommutativealgebra(resp.Zinbielalgebra)areen-
i
X dowedwithastructureofassociativealgebra(resp.dendriformalgebra).Recallthat
r adendriformalgebraisanassociativealgebrawhoseproductisthesumoftwobi-
a
naryoperations:xy=x≺y+x≻y,supposedtosatisfysomerelations.Thereisa
uniquewaytoextendtheexponentialseriesfromcommutativealgebrastoassocia-
tivealgebras.Howeverthereareseveralwaystoextendtheexponentialseriesfrom
J.-L.Loday
InstitutdeRechercheMathe´matiqueAvance´e,CNRSetUniversite´deStrasbourg
ZinbielInstituteofMathematics
e-mail:[email protected]
1
2 J.-L.Loday
Zinbielalgebrastodendriformalgebras:eithertake xn orx≺norx≻n.Thesechoices
n!
are equalin a Zinbielalgebra,butdifferentin a dendriformalgebra.We adoptthe
notation
e(x):=1+x+x≺2+···+x≺n+···, e′(x):=1+x+x≻2+···+x≻n+···
fortheexponentialserieswithoutdenominators.Theclassicalfunctionalequation
exp(x)exp(y)=exp(x+y)
holdsonlywhenweareinthecommutativeorintheZinbielsetting.Inthislastcase
itcanbewritten:
e(x)e(y)=e(x+y).
In the associative setting there is an error term for exp which is called the Baker-
Campbell-Hausdorffformula. We prove that, in the dendriformcontext, the func-
tionalequationfortheexponentialwithoutdenominatorsis:
e(x)e(y)=e x+e(x)≻y≺e′(−x) ,
(cid:0) (cid:1)
whichcanbewritten
(cid:229)
e(x)e(y)=e(x+y+ {x,...,x;y}),
n≥1
n
| {z }
where{−,···,−;−}isthebraceproduct.
IntheassociativeframeworktheBCH-formulaisbestunderstoodbylookingat
themultilinearizedcase.Then,themultilinearpolynomialinvolvedinthefunctional
equationoftheexponentialturnsouttobetheEulerianidempotent.Inthedendri-
formcase,weshowthattheanalogueoftheEulerianidempotentfortheexponential
withoutdenominatorsisplayedbytheiteratedpre-Lieproduct
h(x ,...,x )={x ,{x ,{···{x ,x }·}}},
1 n 1 2 n−1 n
wherethepre-Lieproductis{x,y}:=x≻y−y≺x.
Here is the content of this paper. In the first section we recall the notions of
Zinbiel algebras, dendriformalgebras and some of their properties. In the second
sectionweintroduceandstudytheexponentialwithoutdenominators.Inparticular
wecomputeitsinverse,thelogarithmwithoutdenominators,whichiscloselyrelated
to the 11-logarithm of Kontsevich. In section 3 we prove the BCH-type formula
2
for the exponentialseries without denominators. As a corollary we show that the
multilinear obstruction to the additivity of the exponential is the iterated pre-Lie
product.
InthispaperKdenotesaunitalcommutativering(forinstanceZ)whichisthe
groundring.WesometimesneedtosupposethatKcontainsQ.Thetensorproduct
overKissimplydenotedby⊗.
Exponentialserieswithoutdenominators 3
1 Zinbiel anddendriform algebras
WeintroducethenotionofZinbielalgebra(calleddualLeibnizalgebrain[7])and
its relationship with commutative algebras. We also introduce the notion of den-
driform algebras because matrices over a Zinbiel algebra bear the structure of a
dendriformalgebra.
1.1 Definition
AZinbielalgebraisamoduleAoverKequippedwithabinaryoperationx≺ysuch
that
(x≺y)≺z=x≺(y≺z+z≺y).
Symmetrizingthe Zinbieloperation,thatisdefiningxy:=x≺y+y≺x, we geta
productonAwhichiscommutativeandassociative.Indeed,onegets
(xy)z=(x≺y+y≺x)≺z+z≺(xy)=x≺(yz)+y≺(xz)+z≺(xy)=
x≺(yz)+y≺(zx)+z≺(yx)=x≺(yz)+(y≺z)≺x+(z≺y)≺x=x(yz).
Hencethereisaforgetfulfunctorbetweencategoriesofalgebras:
Zinb-alg→Com-alg.
1.2 Free Zinbielalgebra
Itisshownin[8]thatthefreeZinbielalgebraoverthevectorspaceV isthereduced
tensormodule
T¯(V):=V⊕···⊕V⊗n⊕···
where the generic element v ···v ∈V⊗n corresponds to the product v ≺(v ≺
1 n 1 2
(···≺v )). As a consequencethe space of n-aryoperationsof the operadZinb is
n
theS -moduleZinb(n)=K[S ].Morepreciselywehave:
n n
Theorem1.3 LetV beavectorspaceandletT¯(V)bethereducedtensormodule
overV.Thehalf-shuffle:
(cid:229)
v1···vp≺vp+1···vp+q:= v1vs −1(2)···vs −1(p+q)
s
where s is a (p−1,q)-shuffle acting on {2,...,p+q}, makes T¯(V) into the free
ZinbielalgebraonV.Theassociatedcommutativealgebraisthe(nonunital)shuffle
algebra,denotedbyT¯sh(V).TheZinbielalgebra(T¯(V),≺)isinfactthefreeZinbiel
algebraoverV,denotedbyZinb(V).
4 J.-L.Loday
Proof.See[8]. (cid:3)
1.4 Example
IfV isone-dimensionalspannedbyx,thenZinb(Kx)isspannedbyx≺n forn≥1.
Theoperationsaregivenby
p+q−1 p+q
x≺p≺x≺q= x≺p+q and x≺px≺q= x≺p+q.
(cid:18) q−1 (cid:19) (cid:18) q (cid:19)
1.5 Dendriformalgebra
AdendriformalgebraisaK-moduleAequippedwithtwolinearmaps(binaryop-
erations)
≺:A⊗A→A and ≻:A⊗A→A
calledtheleftoperationandtherightoperationrespectively,satisfyingthefollowing
threerelations
(x≺y)≺z=x≺(y≺z)+x≺(y≻z),
(x≻y)≺z=x≻(y≺z),
(x≺y)≻z+(x≻y)≻z=x≻(y≻z).
Fromtheseaxiomsitfollowsreadilythatthebinaryoperation
xy:=x≺y+x≻y.
isassociative.Underthisnotationthefirstrelationbecomes
(x≺y)≺z=x≺(yz).
In the proof of the main Theorem we allow ourselves to write x≺yz in place of
x≺(yz).
LetusmentionthatnumerouscombinatorialHopfalgebrascomewithadendri-
formstructure,cf.forinstance[1,2,9,10].
1.6 Commutativedendriformalgebra
Bydefinitionacommutativedendriformalgebraisadendriformalgebrawhoseleft
andrightoperationsarerelatedbythefollowingsymmetryrelation
Exponentialserieswithoutdenominators 5
x≺y=y≻x, foranyxandy.
Bydirectinspectionwesee thataZinbielalgebraisa commutativedendriform
algebraandvice-versa(cf.[8]).
Proposition1.7 Themoduleofn×n-matricesM (A)withcoefficientsintheZin-
n
bielalgebraAisadendriformalgebra.
Proof.Itisstraightforwardtocheckthatthematricesoveradendriformalgebrais
stilladendriformalgebra.Theformulasarelikeintheclassicalcase.SinceaZinbiel
algebraisaparticularcaseofadendriformalgebra,wearedone. (cid:3)
1.8 Dendriformcalculus
Werecallsomeresultsfrom[8]aboutcomputationinadendriformalgebraA.
It is often helpful to suppose that A is equipped with a unit 1. So A=K1⊕I
whereIisadendriformalgebra,andtherelationshipbetween1andtheelementsof
I aregivenby:
x≺1=x, 1≺x=0,
x≻1=0, 1≻x=x.
Observe that 1≺1 and 1≻1 are not defined, so, in order for an expression like
(1+u)(1+v)tomakesense,wehavetowriteitas
(1+u)(1+v)=1+u+v+uv=1+u+v+u≺v+u≻v.
The free unital dendriform algebra on one generator is spanned by the planar
rootedbinarytrees.WedenotebyPBT thesetofplanarrootedbinarytreest with
n
nleaves.Inlowdimensionwehave:
PBT1={|}, PBT2= ❄❄⑧⑧ , PBT3= ❄❄❄❄⑧❄⑧❄⑧⑧ , ❄❄⑧❄⑧❄⑧⑧⑧⑧ ,
(cid:8) (cid:9) n o
PBT4=(cid:26) ❄❄⑧⑧❄❄⑧❄⑧❄⑧⑧⑧⑧⑧⑧⑧⑧ , ❄❄❄❄⑧❄⑧❄❄❄⑧⑧⑧⑧⑧⑧⑧⑧ , ❄❄⑧⑧❄❄❄❄⑧⑧⑧⑧❄❄⑧⑧ , ❄❄❄❄❄❄❄❄⑧⑧⑧❄⑧❄⑧⑧⑧⑧ , ❄❄❄❄❄❄❄❄⑧❄⑧❄⑧⑧❄❄⑧⑧ (cid:27).
Whent hasn+1leaves(i.e.t∈PBT ),itdeterminesann-aryoperation.Ap-
n+1
pliedtothegenericn-tuplex ···x wedenotetheresultbyt(x ···x ).Forinstance:
1 n 1 n
|()=1, ❄❄⑧⑧ (x)=x, ❄❄⑧❄⑧❄⑧⑧⑧⑧ (xy)=x≻y, ❄❄❄❄⑧❄⑧❄⑧⑧ (xy)=x≺y.
Moregenerally,ifthetreet isthegraftingoftl andtr,denotedbyt=tl∨tr,then
6 J.-L.Loday
t(x ···x )=tl(x ···x )≻x ≺tr(x ···x ).
1 n 1 i−1 i i+1 n
ThefreedendriformalgebraoverthevectorspaceV isDend(V)= Dend ⊗
n n
V⊗n where Dendn =K[PBTn+1]. In order to check an equality of muLltilinear el-
ements in the free dendriform algebra over the set {x ,...,x } it is equivalent to
1 n
checktheequalityforeachpermutationofthevariablesindividually(inotherterms
theoperadDendisanonsymmetricoperad).
Itissometimeshelpfultoadoptthenotationxt:=t(x···x)orsimplytwhenthere
isonlyonevariableintoplay.Forinstancewehave:xt =x≺n=x≺(x≺(···≺x))
fort therightcombwithn+1leaves.Itcanbeshownthat
xn= (cid:229) xt
t∈PBTn+1
inanydendriformalgebra.
Wealreadyintroducedthenotationx≺n.Similarlywedefinex≻n:=(x≻n−1)≻x,
withx≻1:=x.
IfwenowworkinaZinbielalgebra,thenitcanbeshownthatxt =#j −1(t)x≺n
where j :S →PBT is the surjective map constructed in loc.cit. As a conse-
n n+1
quencewehave
xn= (cid:229) xt = (cid:229) #j −1(t)x≺n= (cid:229) x≺n=n!x≺n,
t∈PBTn+1 t∈PBTn+1 s ∈Sn
aswealreadyknow.
1.9 Pre-Lieproductand braceproducts
Recallthatabinaryoperationissaidtobeleftpre-Lieifitsassociatorissymmetric
inthefirsttwovariables.Inadendriformalgebrathebinaryoperation
{x,y}:=x≻y−y≺x
isaleftpre-Lieproduct.Itisadirectconsequenceofthedendriformaxioms.More
generally,onecanformthefollowing(n+1)-aryoperation:
n
{x ,...,x ;y}:=(cid:229) (−1)n−i(x ≺(x ≺(···≺x)))≻y≺(((x ≻···)≻x )≻x ).
1 n 1 2 i i+1 n−1 n
i=0
In[11]M.RoncoshowedthattheseoperationsareprimitivefortheHopfstructureof
thefreedendriformalgebraandthattheysatisfytheaxiomsofabracealgebra.They
arecalledbraceproducts.Observethat{x;y}={x,y}istheleftpre-Lieproduct.
Exponentialserieswithoutdenominators 7
1.10 Acommutativediagramofalgebras
Therelationshipbetweenallthesetypesofalgebrascanbesummarizedbytheex-
istenceofacommutativediagramofcategoriesofalgebras[8,11]:
Zinb-alg // Dend-alg //Brace-alg
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
Com-alg // Ass-alg //Lie-alg
Considering free algebras on one generator x we get the polynomials K[x] for
Zinb,ComandAss,andweget K[PBT ]forDend:
n n+1
L
K[x] oooo K[PBT ] n!x≺n oo ✤(cid:229) xt
OO n OO n+1 OO tOO
L
K[OOx] oo ∼= K[OOx] x❴n oo ✤x❴n
Observethattherearevariouspossibilitiestolifttheelement xn tothefreeden-
n!
drifromalgebra,forinstance xt foranyt∈PBT .
#j −1(t) n+1
2 Exponential series
We introduce the exponential series without denominators and we compute their
inversefortheassociativeproductandforcomposition(logarithmwithoutdenomi-
nators).
2.1 Exponential serieswithout denominators
LetV beamoduleoverKandletDend(V)∧ betheinfiniteproduct
Dend(V)∧:=(cid:213) Dend ⊗V⊗n.
n
n
Foranyx∈V aseriesinxisanelementofDend(V)∧ madeoutofproductsofthe
onlyvariablex∈V.
Thereareseveraldifferentwaystoextendtheexponentialseriesexp(x)=e(x)=
e′(x)onZinbielalgebrastodendriformalgebras.Forinstancewehave
1. exp(x)=1+x+x2+···+xn+···,theclassicalexponentialserieswhenQ⊂K,
2! n!
8 J.-L.Loday
2. e(x):=1+(cid:229) x≺n, e′(x):=1+(cid:229) x≻n, the exponentialserieswithoutde-
n≥1 n≥1
nominators,
3. ee(x):=1+(cid:229) 1(x≺n+x≻n),when2isinvertibleinK.
n≥12
Observethatwehavee(x)=1+x≺e(x).
Lemma2.2 Inthedendriformcontextwehave:
e′(−x)e(x)=1=e(x)e′(−x).
Proof.Letusfirstprovethatx≻i≻x≺j=x≻i+1≺x≺j−1.Wehave
x≻i≻x≺j=x≻i≻(x≺x≺j−1)
=(x≻i≻x)≺x≺j−1
=x≻i+1≺x≺j−1.
Expandingtheproducte′(−x)e(x)weget
x≺n+···+(−1)j(x≻i≻x≺j+x≻i≺x≺j)+···+(−1)nx≺n
indegreen>0.Thiselementis0asaconsequenceoftheprecedingformula.The
secondformulaisanimmediateconsequenceofthefirstone. (cid:3)
Proposition2.3 In the dendriform context, the inverse of the exponential series
E(x):=e(x)−1forcompositionistheseries
L(x):=x≺(1−x+x2−x3+···+(−1)nxn+···) ,
calledthelogarithmwithoutdenominators.
Proof.LetuswriteL(x)=x+j (x)+···+j (x)+··· fortheinverseofE(x).Since
2 n
E(x)=x≺(1+E(x)),byreplacingxbyL(x)inthisequalityweget
x=L(x)≺(1+x).
Hence we have j (x)=−x≺x and j (x)+j (x)≺x =0. By induction we
2 n+1 n
supposethatj (x)=(−1)n−1x≺xn−1.Wecompute:
n
j (x)=−j (x)≺x,
n+1 n
=−(−1)n−1(x≺xn−1)≺x,
=(−1)nx≺(xn−1≺x+x≻xn−1),
=(−1)nx≺(xn).
(cid:3)
Exponentialserieswithoutdenominators 9
2.4 Remark
Thispropositiongivesaquickproofofthefactthatexpandlogareinversetoeach
other. This proof is similar to the proof which uses the integral definition of the
logarithm:log(1+x)= dx .
1+x
R
2.5 Zinbielalgebrasincharacteristic p
LetussupposethatKisacharacteristicpfieldandletusworkintheZinbielframe-
work.Sincexn=n!x≺nthelogarithmbecomesapolynomialwhichcanbewritten
log(1+x)=L(x):=p(cid:229)−1(−1)i−1xi +(−1)p−1(p−1)!x≺p.
i
i=1
First,theelementx≺pisadividedpower,thatis(A,g (x):=x≺p)isadividedpower
algebra,cf. forinstance[3]. Second,the firstpartof thelogarithmisthe so-called
“oneandahalflogarithm”introducedbyMaximKontsevichin[4].Inotherwords
wehavethefollowingresult.
Proposition2.6 Inacharacteristicpdividedpoweralgebra(resp.Zinbielalgebra)
theoneandahalflogarithmplus(−1)p−1(p−1)!g (x)isaninvertibleserieswhose
inverseistheexponentialseries. (cid:3)
3 BCH-type formulaforthe exponential series without
denominators
In a Zinbiel algebra we have the functional equation e(x)e(y)=e(x+y). But, in
a dendriform algebra, x≻y6=y≺x (even for x=y), hence there are nontrivial
dendriformpolynomialsH (x,y)inxandysuchthat
n
e(x)e(y)=e(x+y+···+H (x,y)+···).
n
ForinstanceweobviouslyhaveH (x,y)=x≻y−y≺x,whichisthe(left)pre-Lie
2
product {x,y}. Our aim is to compute H (x,y), in fact to show that it is a brace
n
product.
Theorem3.1 Inadendriformalgebrathefollowingequalitieshold:
e(x)e(y)=e x+e(x)≻y≺e′(−x) ,
(cid:0) (cid:1)
e′(x)e′(y)=e′ e′(−y)≻x≺e′(y)+y ,
(cid:0) (cid:1)
10 J.-L.Loday
fore(x):=1+(cid:229) x≺nande′(x):=1+(cid:229) x≻n.
n≥1 n≥1
Corollary3.2 ThedendriformpolynomialH isthebraceproduct:
n
H (x,y)= (cid:229) (−1)j(x≺i)≻y≺(x≻j)={x,...,x;y}.
n
i+j=n−1
n−1
| {z }
Proof.[ofTheorem3.1]Weprovethefirstrelation.Letusdefine
R(x,y):=e(x)e(y)−e x+e(x)≻y≺e′(−x) .
(cid:0) (cid:1)
TheaimistoprovethatR(x,y)=0.
We use the following two relations: e(z) = 1+z ≺ e(z) and e′(−x)e(x) = 1
(Lemma2.2).Ononehandwehave
e x+e(x)≻y≺e′(−x) =1+ x+e(x)≻y≺e′(−x) ≺(e(x)e(y)−R(x,y))
(cid:0) (cid:1)=1+(cid:0)x≺(e(x)e(y))+(e(x)≻(cid:1) y)≺(e′(−x)e(x)e(y))
−F (x,y),
whereF (x,y):=(x+e(x)≻y≺e′(−x))≺R(x,y).Thenwehave
e x+e(x)≻y≺e′(−x) =1+x≺(e(x)e(y))+e(x)≻y≺e(y)−F (x,y)
(cid:0) (cid:1)=1+x≺(e(x)e(y))+e(x)≻E(y)−F (x,y),
byLemma2.2.Ontheotherhandwehave:
e(x)e(y) =e(x)(1+E(y))
=e(x)+e(x)≺E(y)+e(x)≻E(y).
Hencewecompute
R(x,y)=e(x)+e(x)≺E(y))−1−x≺(e(x)e(y))+F (x,y)
=e(x)+e(x)≺E(y)−1−x≺(e(x)(1+E(y))+F (x,y)
=e(x)+e(x)≺E(y)−1−x≺e(x)−x≺(e(x)E(y))+F (x,y)
=e(x)≺E(y)−x≺(e(x)E(y))+F (x,y)
=e(x)≺E(y)−(x≺e(x))≺E(y)+F (x,y)
=1≺E(y)+F (x,y)
=F (x,y).
SowehaveprovedthatR(x,y)satisfiesthefunctionalequation:
R(x,y)=(x+e(x)≻y≺e′(−x))≺R(x,y).
