The infinitesimal Hopf algebra and the operads of planar forests

0.35 MB·English

by  Loïc Foissy

The in(cid:28)nitesimal Hopf algebra and the operads of planar forests
Loï
Foissy
Laboratoire de Mathématiques, FRE3111, Université de Reims
Moulin de la Housse - BP 1039 - 51687 REIMS Cedex 2, Fran
e
e-mail : loi
.foissyuniv-reims.fr
ABSTRACT. We introdu
e two operads whi
h own the set of planar forests as a basis. With
9 its usual produ
t and two other produ
ts de(cid:28)ned by di(cid:27)erent types of graftings, the algebra
0 of planar rooted trees H be
omes an algebra over these operads. The
ompatibility with the
0
in(cid:28)nitesimal
oprodu
t of H and these stru
tures is studied. As an appli
ation, an indu
tive
2
way of
omputing the dual basis of H for its in(cid:28)nitesimal pairing is given. Moreover, three
n
a Cartier-Quillen-Milnor-Moore theorems are given for the operads of planar forests and a rigidity
J theorem for one of them.

5
1
KEYWORDS. In(cid:28)nitesimal Hopf algebra, Planar rooted trees, Operads.

]
A
AMS CLASSIFICATION. 16W30, 05C05, 18D50.
R
.
h
t Contents
a
m
1 Planar rooted forests and their in(cid:28)nitesimal Hopf algebra 3
[

1.1 Planar trees and forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1
v 1.2 In(cid:28)nitesimal Hopf algebra of planar forests . . . . . . . . . . . . . . . . . . . . . . 4
2 1.3 Pairing on H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
0
2
2 The operads of forests and graftings 5
2 Σ
. 2.1 A few re
alls on non- -operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1
0 2.2 Presentations of the operads of forests . . . . . . . . . . . . . . . . . . . . . . . . 7
9 2.3 Grafting on the root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
0
2.4 Grafting on the left leave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
: P P
v ց ր
2.5 Dimensions of and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
i
X
2.6 A
ombinatorial des
ription of the
omposition . . . . . . . . . . . . . . . . . . . 11
r
a
3 Appli
ations to the in(cid:28)nitesimal Hopf algebra H 13
3.1 Antipode of H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
γ
3.2 Inverse of the appli
ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Elements of the dual basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Primitive suboperads 18
4.1 Compatibilities between produ
ts and
oprodu
ts . . . . . . . . . . . . . . . . . . 18
PRIM(1)
4.2 Suboperad ր . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Prim( )
4.3 Another basis of H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.4 From the basis (ft)t∈T to the basis (pt)t∈Tb . . . . . . . . . . . . . . . . . . . . . 23
PRIM(2)
4.5 Suboperad ր . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
PRIM
ց
4.6 Suboperad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1
P
ր
5 A rigidity tPheorem for -algebras 27
ր
5.1 Double -in(cid:28)nitePsimal bialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 27
ր
5.2 Conne
ted double -in(cid:28)nitesimal bialgebras . . . . . . . . . . . . . . . . . . . . 28
Introdu
tion
The Connes-Kreimer Hopf algebra of rooted trees, introdu
ed in [1, 6, 7, 8℄, is a
ommutative,
non
o
ommutative Hopf algebra, its
oprodu
t being given by admissible
uts of trees. A non

ommutative version, the Hopf algebra of planar rooted trees, is introdu
ed in [3, 5℄. We fur-
themore introdu
e in [4℄ an in(cid:28)nitesimal version of this obje
t, repla
ing admissible
uts by left
admissible
uts: this last obje
t is here denoted by H. Similarly with the Hopf
ase, H is a
,
self-dual obje
t and it owns a non-degenerate, symmetri
Hopf pairing, denoted by h− −i. This
pairing is related to a partial order on the set of planar rooted forests, making it isomorphi
to
(fF)F∈F
theFTamari poset. As a
onsequen
e, H isgiven a dua(lfbt)at∈siTsdenoted by , indexed by the
set Tof planar forest. In parti
ular, the sub-family indexed by the set of planar rooted
trees is a basis of the spa
e of primitive elements of H.
The aim of this text is to introdu
e twoPstru
turePs of operad on the spa
e of planar forests.
ց ր
We introdu
e two (non-symmetri
) operads and de(cid:28)ned in the following way:
P m P (2)
ց ց
1. is generated by and ց∈ , with relations:
m ( ,I) = (I,m),
◦ ց ց ◦
m (m,I) = m (I,m),
 ◦ ◦
(m,I) = (I, ).

ց ◦ ց ◦ ց
P m P (2)
ր ր
2. is generated by and ր∈ , with relations:
m ( ,I) = (I,m),
◦ ր ր ◦
m (m,I) = m (I,m),
 ◦ ◦
( ,I) = (I, ).

ր ◦ ր ր ◦ ր

We then introdu
e two produ
ts on H or on its augmentation ideal M, denoted by ր and ց.
F G F G F G
Theprodu
Ft ր
onsists ofgGrafting ontheleft leave of and tmheprodu
t ց
onsPists
ց
of graftPing on the left root of . Together with itPs usual prPodu
t , M be
omes both a -
ր ց ր
and a q -algebra. More pre
Pisely, MPis the free - and -algebra generated by a single
ց ր
element . As a
onsequen
e, and inherits a
ombinatorial representation using planar
forests, with
omposition iteratively des
ribed using the produ
ts ց and ր.
We then give several appli
ations oPf these operadi
stru
tures. For example,fthe antiptodeTof
ց t
H is des
ribed in tePrm of the operad .fWe sFhowFhow to
ompute elements 's, with ∈ ,
ց F
usiPnրg the a
tion of , and the elements 's, ∈ from the pre
eding ones using(ftFhe)Fa∈
Ftion
of . Combining all these results, it is possible tom
ompute by indu
tion t∆h˜e basis .
∆˜ We(cid:28)nallystudythe
ompatibilitiesofprodu
ts ,ր,ցP,the
oprodu
t andthe
oprodPu
t
ր ր ց
dual of ր. This leads to the de(cid:28)nition of twoPtypesPof -bialgebras, and one type of -
ր ց
bialgebras. Ea
h type then de(cid:28)ne a suboperad of or
orresponding to primitive elements
of M, whi
h are expli
itively des
ribed:
qq q q P (2)
ր
1. The (cid:28)rst one is a free operad, generated by the element − ∈ . As a
onsequen
e,
the spa
e of primitive elements of H admits a basis (pt)t∈Tb indexed by the set of planar
(ft)t∈T
binary trees. The link with the basis is given with the help of the Tamari order.
2. The se
ond one admits a
combiPnat(onri)alnrepr2esentation in terms of planar rooted trkee,sl. It2is
n ր
generated by the
orollas ∈ , ≥ , with the following relations: for all ≥ ,
c (c , I,...,I ) = c (I,...,I,c ).
k l l k
◦ ◦
k−1 l−1
times times
| {z } | {z }
2
3. The third one admits aqq
oPmbi(n2a)torial representation in terms of planar rooted trees, and
ց
is freely generated by ∈ .
P P
ր ր
We also give the de(cid:28)nition of a double -bialgebra,
ombining the two tPypes of -
ր
bialgebras already introdu
ed. We then prove a rigidity theorem: any double -bialgebra

onne
ted as a
oalgebra is isomorphi
to a de
orated version of M.
This text is organised as follows: the (cid:28)rst se
tion gives several re
alls on the in(cid:28)nitesimal
Hopf algebra of planar rooted trees and its pairing. The two produ
ts ց and ր are introdu
ed
in se
tion 2, as well as the
ombinatorial representation of the two asso
iated operads. The ap-
(fF)F∈F
pli
ations to the
omputation of is given in se
tion 3. Se
tion 4 is devoted to the study
of the sPuboperads of primitive elements and the last se
tion deals with the rigidity theorem for
ր
double -bialgebras.
Notations.
K
1. We shall denote by a
ommutative (cid:28)eld, of any
hara
teristi
. Every ve
tor spa
e,
K
algebra,
oalgebra, et
, will be taken over .
(A,∆,ε) 1 A ∆(1) = 1 1
2. Let be a
ounitary
oalgebra. Let ∈ , non zero, su
h that ⊗ . We
then de(cid:28)ne the non
ounitary
oprodu
t:
Ker(ε) Ker(ε) Ker(ε)
∆˜ : −→ ⊗
a ∆˜(a) =∆(a) a 1 1 a.
(cid:26) −→ − ⊗ − ⊗
∆(a) = a(1) a(2) ∆˜(a) = a′ a′′
We shall use the Sweedler notations ⊗ and ⊗ .
1 Planar rooted forests and their in(cid:28)nitesimal Hopf algebra
We here re
all some results and notations of [4℄.
1.1 Planar trees and forests
T F
1. The set of planar trees is denoted by , and the set of planar fonrestsNis denoted by .FT(hne)
weight of a planar forest is the number of its verti
es. For all ∈ , we denote by
n
the set of planar forests of weight .
5
Examples. Planar rooted trees of weight ≤ :
q q q q q q q
q, qq, qqq, qqq, qqqq, qqqq, qqqq, ∨qqqq, qqqq ,Hq qq(cid:8)q q, qqqqq, qqqqq, qqqqq, qqqqq, ∨qqqqq, q∨qqqq, qqqq, qqqq, ∨qqqqq, ∨qqqq, ∨qqqq,∨qqq, qqqq.
∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨
4
Planar rooted forests of weight ≤ :
q q q q q q q qq
q q q q q q q q q q q q q q q q q qqq q q q q q q
1, q, qq, q, qq q, q q, q q, q , q, qq qq, q q q, q q q, q q q, q q, q q , qq, qq, q q, q , q , q , ∨q , q .
∨ ∨ ∨ ∨ ∨ ∨
T
2. The algebra H is the free asso
Fiative, unitary algebra generated by . As a
onsequen
e,
a linear basis of H is given by , and its produ
t is given by the
on
atenation of planar
forests.
Vert(F)
3. FWe sFhall also need two partial orFde=rstan.d..at total ordesr,so′n the set oFf verti
es of
1 n
∈ , de(cid:28)ned in [3, 4℄. We put , and let be two verti
es of .
s s′ s′ s F F
high
(a) We shall say that ≥ if there exists a path from to in , the edges of
high
being oriented from the roots to the leaves. Note that ≥ is a partial order, whose
F
Hasse graph is the forest .
3
s s′ s s′
high left
(b) If and are not
omparable for ≥ , we shall say that ≥ if one of these
assertions is satis(cid:28)ed:
s t s′ t i < j
i j
i. is a vertex of and is a vertex of , with .
s s′ t s s′ t
i left i
ii. and are verti
es of the same , and ≥ in the forest obtained from
by deleting its root.
F
left
This de(cid:28)nes the partial order ≥ for all forests , by indu
tion on the the weight.
s s′ s s′ s s′
h,l high left
(
) We shall say that ≥ if ≥ or ≥ . This de(cid:28)nes a total order on the
F
verti
es of .
1.2 In(cid:28)nitesimal Hopf algebra of planar forests
F F F
1. Let ∈ . An admissible
ut is a non empty
ut of
ertain edges and trees of , su
h
F
that ea
h path in a non-
ut tree of meets at most one
ut edge. The set of admissible
F Adm(F) c F

uts of will be denoted by . If is an admissible
ut of , the forest of the
c Pc(t) c
verti
es whi
h are over the
uts of will be denoted by (bran
h of the
ut ), and
Rc(t) F
the remaining forest will be denoted by (trunk of the
ut). An admissible
ut of
x y F x Pc(F) x y
left
will be said to be left-admissible if, for all verti
es and of , ∈ and ≤
y Pc(F) F Adml(F)
imply that ∈ . The set of left-admissible
uts of will be denoted by .
F F
2. H is given a
oprodu
t by the following formula: for all ∈ ,
∆(F)= Pc(F) Rc(F)+F 1+1 F.
⊗ ⊗ ⊗
c∈AXdml(F)
( ,∆) x,y
Then H is an in(cid:28)nitesimal bialgebra, that is to say: for all ∈ H,
∆(xy)= (x 1)∆(y)+∆(x)(1 y) x y.
⊗ ⊗ − ⊗
Examples.
∆(q) = q 1+1 q,
⊗ ⊗
∆(q q) = q q 1+1 qq + q q,
⊗ ⊗ ⊗
q q q
∆(q) = q 1+1 q + q q,
⊗ ⊗ ⊗
q q q q
∆(q q) = q q 1+1 q q + q qq + q q,
q q q q⊗ ⊗ q q ⊗ ⊗ q
∆( q ) = q 1+1 q + qq q + q q,
∨q q∨ ⊗ ⊗q ∨ ⊗ ⊗
q q q q q
∆(q) = q 1+1 q + q q + q q,
⊗ ⊗ ⊗ ⊗
∆(q qq q) = q qq q 1+1 q q qq + qq q q + q qq q,
⊗ ⊗ ⊗ ⊗ ⊗
q q q q q
∆(q q q) = q q q 1+1 q qq + q qq q + q q q + q q q,
⊗ ⊗ ⊗ ⊗ ⊗
q q q q q
∆(q q q) = q q q 1+1 q q q + q q q + q q q q + q q q,
⊗ ⊗ ⊗ ⊗ ⊗
q q q q q
∆(q q q) = q q q 1+1 qq q + q q q + q q q + qq q q,
q q q q⊗ ⊗ q q ⊗ q q ⊗ q ⊗
∆(q q ) = q q 1+1 q q + q q + q q q + q qq q,
∨q q∨ ⊗ ⊗q ∨ q ⊗ ∨ ⊗ ⊗
q q q q q q
∆(q q) = q q 1+1 qq + q q + qq q + q q q,
q q q q⊗ ⊗ q q ⊗ q ⊗ ⊗ q q
∆( q q) = q q 1+1 q + q q q + qq qq + q q,
∨q q∨ ⊗ ⊗q ∨ ⊗ ⊗ q ∨ ⊗
q q q q q q
∆(q q) = q q 1+1 q + q q q + q qq + q q,
⊗ ⊗ ⊗ ⊗ ⊗
q q q q q q q q q q
∆(q q) = q q 1+1 q q + q q q + q q + q q q,
⊗ ⊗ ⊗ ⊗ ⊗
4
qqq qqq qqq q q q
∆( q ) = q 1+1 q + q q + q q q + q q q q,
∨q ∨q ⊗ ⊗ ∨q ⊗ ∨ ⊗ ⊗
q q q q q q q q q q q
∆( q ) = q 1+1 q + q q + q q + q q q,
∨q ∨q ⊗ ⊗ ∨q ⊗ ∨ ⊗ ⊗
q
q q q q q q q q q
∆( q ) = q 1+1 q + q q + q q q + q q q,
∨ ∨ ⊗ ⊗ ∨ ⊗ ⊗ ⊗
qqq qqq qqq qq q q q
∆( ∨q ) = ∨q 1+1 ∨q q q + q q q + q q,
q q ⊗ ⊗q ⊗ ⊗ ∨ ⊗
q q q q q
q q q q q q q
∆(q ) = q 1+1 q + q q + q q + q q.
⊗ ⊗ ⊗ ⊗ ⊗
S We proved in [4℄ that HS(i1s)a=n i1n(cid:28)nSi(tte)simaPl rHiomp(f a)lgebra, thtat iTs to saySh(aFs)a=n a0ntipode
F. TFhis(aTntipo1de)satis(cid:28)es , ∈ H for all ∈ , and for all
∈ − ∪{ } .
1.3 Pairing on H
B+ : F F
1. Wede(cid:28)netheoperator H −→ H,whi
hasso
iates,toaforest ∈ ,thetreeobtainqeqd
q q q q
F B+( q q) = ∨q
by grafting the rooqtqs of the trees of on a
ommon root. For example, ∨ ∨,
q q q q
B+(q q ) = ∨q
and ∨ ∨ .
γ
2. The appli
ation is de(cid:28)ned by:
γ : H −→ H
(cid:26) t1...tn ∈ F −→ δt1,qt2...tn.
, : K
3. There exists a unique pairing h− −i H×H −→ , satisfying:
1,x = ε(x) x
i. h i for all ∈ H.
xy,z = y x,∆(z) x,y,z
ii. h i h ⊗ i for all ∈H.
B+(x),y = x,γ(y) x,y
iii. h i h i for all ∈ H.
Moreover:
,
iv. h− −i is symmetri
and non-degenerate.
x y x,y = 0
v. If and are homogeneous of di(cid:27)erent weights, h i .
S(x),y = x,S(y) x,y
vi. h i h i for all ∈ H.
left high
This pairing admits a
ombinatorial interpretation using the partial orders ≥ and ≥
and is related to the Tamari order on planar binary trees, see [4℄.
(fF)F∈F ,
4. We denote by F F fthe dual basis of tfhe,bGasis=ofδforests for the pairGing h−F −i. In other
F F F,G
terms, for all ∈ , is de(cid:28)ned by h i , for all forest ∈ . The family
(ft)t∈T Prim( )
is a basis of the spa
e H of primitive elements of H.
2 The operads of forests and graftings
Σ
2.1 A few re
alls on non- -operads
Σ P =
1. (WPe(ns)h)anl∈lNwork here with non- -operads [11℄. Re
all that su
hna,nk1o,b.j.e.
,tknis aNfamily
of ve
tor spa
es, together with a
omposition for all ∈ :
P(n) P(k ) ... P(k ) P(k +...+k )
1 n 1 n
⊗ ⊗ ⊗ −→
p p ... p p (p ,...,p ).
1 n 1 n
(cid:26) ⊗ ⊗ ⊗ −→ ◦
5
p P(n) p P(k ) ... p
1 1 n
TP(hke )folplow,in..g.,apsso
iativPity
ondition is satis(cid:28)ed: for all ∈ , ∈ , , ∈
n , 1,1 n,kn ∈ ,
(p (p ,...,p )) (p ,...,p ,...,p ,...,p )
◦ 1 n ◦ 1,1 1,k1 n,1 n,kn
= p (p (p ,...,p ),...,p (p ,...,p )).
◦ 1◦ 1,1 1,k1 n◦ n,1 n,kn
I P(1) p P(n)
Moreover, there exists a unit element ∈ , satisfying: for all ∈ ,
p (I,...,I) = p,
◦
I p = p.
(cid:26) ◦
Σ P S P(n)
n
An operad is a non- -operad with a right a
tion of the symmetri
group on for
n
all , satisfying a
ertain
ompatibility with the
omposition.
P Σ P A P
2. Let be a non- -operad. A -algebra is a ve
tor spa
e , together with an a
tion of :
P(n) A⊗n A
⊗ −→
p a ... a p.(a ,...,a ),
1 n 1 n
(cid:26) ⊗ ⊗ ⊗ −→
p P(n) p P(k ) ... p P(k )
1 1 n n
satisfying the following
ompatibility: for all ∈ , ∈ , , ∈ , for all
a ,...,a A
1,1 n,kn ∈ ,
(p (p ,...,p )).(a ,...,a ,...,a ...,a )
◦ 1 n 1,1 1,k1 n,1 n,kn
= p.(p .(a ,...,a ),...,p .(a ,...,a )).
1 1,1 1,k1 n n,1 n,kn
I.a = a a A
Moreover, for all ∈ .
V P V
In parti
ular, if is a ve
tor spa
e, the free -algebra generated by is:
FP(V) = P(n) V⊗n,
⊗
n∈N
M
P
with the a
tion of given by:
p.((p a ... a ),...,(p a ... a ))
1⊗ 1,1⊗ ⊗ 1,k1 n⊗ n,1⊗ ⊗ n,kn
= (p (p ,...,p )) a ... a ... a ... a .
◦ 1 n ⊗ 1,1⊗ ⊗ 1,k1 ⊗ ⊗ n,1⊗ ⊗ n,kn
T
b
3. Let be the set of planar binary trees:
∨ ∨ ∨ ∨H(cid:8)
T = , ,∨ , ∨ ,∨ ,∨ , ∨, ∨, ... .
b  ∨ ∨ ∨ ∨ ∨ ∨ ∨ 
 
 
n N T (n)   T n
b b
For all ∈ , is the ve
tor spa
e generated by the elements of with leaves:
T (0) = (0),
b
T (1) = Vect( ),
b
T (2) = Vect ,
b ∨
(cid:18) (cid:19)
T (3) = Vect ∨ , ∨ ,
b ∨ ∨
!
∨ ∨ ∨ ∨H(cid:8)
T (4) = Vect ∨ ,∨ , ∨, ∨, .
b  ∨ ∨ ∨ ∨ 
 
 
6
T Σ
b
The family of ve
tor spat
,ets,...i,stgiveTn atstru
tunre of non- -otpe(rtad,.b.y.,gtra)ftings on the
1 n b 1 n
leaves. More pre
isely, if ∈ , with leaves, then ◦ is the binary
t t t t
1 2
tree obtained by grafting on the (cid:28)rst leave of , on the se
ond leave of , and so on
t
(note that the leaves of are ordered from left to right). The unit is .
T Σ T (2)
b ∨ b
It is knowmn ,t.h.a.t,m is tPh(e2)free non- -operad generated by ∈ Σ . SimPilarly, given
1 k
elements in , it is possible to des
ribe the free non- -operad generated
by these elements in terms of planar binary trees whose internal verti
es are de
orated by
m ,...,m
1 k
.
2.2 Presentations of the operads of forests
De(cid:28)nition 1
P Σ m P (2)
ց ց
1. is the non- -operad generated by and ց∈ , with relations:
m ( ,I) = (I,m),
◦ ց ց ◦
m (m,I) = m (I,m),
 ◦ ◦
(m,I) = (I, ).

ց ◦ ց ◦ ց
P Σ  m P (2)
ր ր
2. is the non- -operad generated by and ր∈ , with relations:
m ( ,I) = (I,m),
◦ ր ր ◦
m (m,I) = m (I,m),
 ◦ ◦
( ,I) = (I, ).

ր ◦ ր ր ◦ ր

Remark. We shall prove in [2℄ that these quadrati
operads are Koszul.
2.3 Grafting on the root
F,G F 1 G = t ...t t = B+(G )
1 n 1 1
Let ∈ −{ }. We put and . We de(cid:28)ne:
F G= B+(FG )t ...t .
1 2 n
ց
F G
In other terms, is grafted on the root of the (cid:28)rst tree of , on the left. In parti
ular,
F q = B+(F)
ց .
Examples.
q
q q q qq = Hq qq(cid:8)q q qq qq q = qq qq q q q qq = qqq q q q qq qq = qqqq q
ց ∨q ց ց ∨ ց ∨q
q q qqq q q qq q q q q q q q q
q q q = q q q q = q q q q q q = q q q q qq = q q
ց ∨q ց q ց ∨ ց ∨q
q q qqq q q q q q qqq q q q
q q q = q q q q = q q q q q q = q q q q qq = q q
ց q∨q ց ∨q ց ∨ ց ∨q q
qqq qq = ∨qqq qq qqq = qqqq q q qqq = Hq qq(cid:8)q q qqq qq = ∨qq q
∨ ց q∨ ց ∨ ∨ ց ∨ ∨ ∨ ց
q q q q q
qq q q q q qq q q qq qqq qq qq
q q = q q q = q q q q = q q qq = q q.
ց ∨ ց ∨ ց ∨ ց
F,G,H F 1
Obviously, ց
an be indu
tively de(cid:28)ned in the following way: for ∈ −{ },
F q = B+(F),
ց
F (GH) = (F G)H
 ց ց
F B+(G) = B+(FG).

ց

We denote bFy M1the augmentation:ideal of H, that is to say the ve
tor spa
e generated by
the elements of −{ }. We extend ց M⊗M −→ M by linearity.
7
x,y,z
Proposition 2 For all ∈ M:
x (yz) = (x y)z,
ց ց (1)
x (y z) = (xy) z.
ց ց ց (2)
x,y,z F 1
Proof. We
azn=resBtr+i
(zt o)uzrsezlv,ezs to F ∈ −{ }. Then (1) is immediate. In order to
1 2 1 2
prove (2), we put , ∈ . Then:
x (y z)= x (B+(yz )z )= B+(xyz )z = (xy) (B+(z )z )= (xy) z,
1 2 1 2 1 2
ց ց ց ց ց
(cid:3)
whi
h proves (2).
P m
ց
Corollary 3 M is given a graded -algebra stru
ture by its produ
ts and by ց.
(cid:3)
Proof. Immediate, by proposition 2.
2.4 Grafting on the left leave
F,G F G = 1 F G F
Let ∈ . Suppose that 6 . Then ր is the planar forest obtained by grafting
G G = 1 F 1 = F
on the leave of whi
h is at most on the left. For , we put ր . In parti
ular,
F q = B+(F)
ր .
Examples.
qqq q
q q q q q q qqq
qq q q = ∨q q qq q = q qq qq qq q = q q q qq q q q = q q
ր ր ր ∨ ր ∨
q
q q q q
q q q q q q q q q q q q q q
q q q = ∨q q q q = q q qq q q = q q q q q q = q q
ր ր ր ∨ ր ∨
q
q q qqq q q qqq q qqq q qq q
q q q = ∨q q q q = q q qq q q = ∨q q q q q q = q q
ր q q ր q ր ր ∨
q q q q q q
q q q ∨q q q q q q q q q q q q q
q q = q q q = q qq q = ∨q q q q = ∨q q
∨ ր q ր ∨ ∨q ր ∨ q q∨ ∨ ր
q q q q
qq q qq q qq qq qq ∨q qq qq
q q = q q q = q qq q = q q q q = q q.
ր ր ր ր
F,G,H F
In an obvious way, ր
an be indu
tively de(cid:28)ned in the following way: for ∈ ,
F 1 = F,
ր
F (GH) = (F G)H G = 1,
 ր ր if 6
F B+(G) = B+(F G).

ր ր
: 
We extend ր H⊗H −→ H by linearity.
x,z y
Proposition 4 1. For all ∈ H, ∈ M:
x (yz) = (x y)z.
ր ր (3)
x,y,z
2. For all ∈ H:
x (y z)= (x y) z.
ր ր ր ր
( , ) 1
So H ր is an asso
iative algebra, with unitary element .
8
x,y,z F y = 1
Proof. Note that (3) is immediate for ∈ , with 6 . This implies the (cid:28)rst point.
In order to prove the se
ond point, we
onsider:
Z = z / x,y , x (y z) =(x y) z .
{ ∈ H ∀ ∈H ր ր ր ր }
1 Z x,y
Let us (cid:28)rst prove that ∈ : for all ∈ H,
x (y 1) = x y = (x y) 1.
ր ր ր ր ր
z ,z Z z z Z
1 2 1 2
Let ∈ . Let us show that ∈ . By linearity, we
an separate the proof into two

ases:
z = 1
1
1. . Then it is obvious.
ε(z ) = 0 x,y
1
2. . Let ∈ H. By the (cid:28)rst point:
x (y (z z )) = x ((y z )z ))
1 2 1 2
ր ր ր ր
= (x (y z ))z
1 2
ր ր
= ((x y) z )z
1 2
ր ր
= (x y) (z z ).
1 2
ր ր
Z B+ z Z x,y
So is a subalgebra of H. Let us show that it is stable by . Let ∈ , ∈ H. Then:
x (y B+(z)) = x B+(y z)
ր ր ր ր
= B+(x (y z))
ր ր
= B+((x y) z)
ր ր
= (x y) B+(z).
ր ր
Z B+ Z = (cid:3)
So is a subalgebra of H, stable by . Hen
e, H.
Remarks.
x,y,z
1. (3) is equivalent to: for any ∈ H,
x (yz) ε(y)x z =(x y)z ε(y)xz.
ր − ր ր −
F F 1 (qF ,..., qF ) F
1 n
2. Let ∈ −{ }. There exists a unique family of elements of su
h that:
F = (qF ) ... (qF ).
1 n
ր ր
q q
q q q q
∨q q q = (qq) (qq) (q q q) ( , )
ForqeFxample, ∨ ր ր . As a
onsequen
e, H ր is freely generated
by as an asso
iative algebra.
P m
ր
Corollary 5 M is given a graded -algebra stru
ture by its produ
t and by ր.
(cid:3)
Proof. Immediate, by proposition 4.
P P
ց ր
2.5 Dimensions of and
P (n) P (n) n
ց ր
WP e now
Pompute the dimensions ofq and for all and dedu
e that M is the free
ց ր
- and -algebra generated by .
r R(X) =
n
Notation. We denote by the number of planar rooted forests and we put
+∞
1 2X √1 4X
r Xn R(X) = − − −
n
. It is well-known (see [3, 13℄) that 2X .
n=1
X
9
, n N∗ P
?
Proposition 6 For → ∈ {ց ր} and all ∈ , in the →-algebra M:
P (n).(q,..., q) =Vect( n).
?
→ planar forests of weight
P q
?
As a
onsequen
e, M is generated as a →-algebra by .
P
?
Proof. ⊆. Immediate, as M is a graded →-algebra.
n n = 1 I.(q)= q n 2
⊇. Indu
tion on . For , . For ≥ , two
ases are possible.
F = F F weight(F ) = n < n p ,p
1 2 i i 1 2
1. P , F = p .(q,..., q) . BFy =thep i.n(dq,u.
.t.io,nq)hypothe(smis, t(hper,epe)x)i.s(tqs,..., q) =∈
? 1 1 2 2 1 2
→, su
h that and . Then ◦
m.(F ,F )= F F
1 2 1 2
.
F T F = B+(G) p P p.(q,..., q) = G
?
2. ∈ . Let us put . Then there exists ∈ →, su
h that .
Then: ( (p,I)).(q,..., q) = G q = F,
ց ◦ ց
( (p,I)).(q,..., q) = G q = F.
(cid:26) ր ◦ ր
F P (n).(q,..., q) (cid:3)
?
Hen
e, in both
ases, ∈ → .
? , n N∗ dim(P (n)) r
? n
Corollary 7 For all → ∈ {ց ր}, ∈ , → ≥ .
Proof. Be
ause we proved the surje
tivity of the following appli
ation:
P (n) Vect( n)
?
ev : → −→ planar forests of weight
?
→ p p.(q,..., q).
(cid:26) −→
(cid:3)
? , n N∗ dim(P (n)) r
? n
Lemma 8 For all → ∈ {ց ր}, ∈ , → ≤ .
? = n N∗ P (n)
ր
Proof. We prove it for → ր. Let us (cid:28)x ∈ . Then is linearly generated by
m
planar binary trees whose internal verti
es are de
orated by and ր. The following relations
hold:
@(cid:0) @(cid:0) @(cid:0) @(cid:0) @(cid:0) @(cid:0)
ր@(cid:0) @(cid:0)ր ր@(cid:0) @(cid:0)m m@(cid:0) @(cid:0)m
ր ր m ր m m
= , = , = .
In the sequel of the proof, we shall say that su
h a tree is admissible if it satis(cid:28)es the following

onditions:
s m s
1. For ea
h internal vertex de
orated by , the left
hild of is a leave.
s s
2. For ea
h internal vertex de
orated by ր, the left
hild of is a leave or is de
orated by
m
.
1 2 3
For example, here are the admissible trees with , or leaves:
@(cid:0) @(cid:0) @(cid:0) @(cid:0) @(cid:0)
@(cid:0) @(cid:0) m@(cid:0) @(cid:0)m @(cid:0)m @(cid:0)ր @(cid:0)ր
m ր ր m ր m ր
, , , , , , , .
P (n) n
ր
The pred
iemdi(nPg r(enla)t)ions imply that a is linearly generated by admissiblentrees with
ր n
leaves. So is smaller than , the number of admissible trees with leaves. For
n 2 b n
n
≥ , we denote by the number of admissible trees with leaves whose root is de
orated by
10