Table Of ContentINFINITE MAGMATIC BIALGEBRAS
6
0 EMILYBURGUNDER
0
2
Abstract. Aninfinitemagmaticbialgebraisavectorspaceendowedwithan
n n-ary operation, and an n-ary cooperation, for each n, verifying some com-
a patibility relations. We prove arigiditytheorem, analogue to the Hopf-Borel
J theorem for commutative bialgebras: any connected infinite magmatic bial-
4 gebra is of the form Mag∞(Prim H), where Mag∞(V) is the free infinite
magmaticalgebraoverthevector spaceV.
]
A
R
. 1. Introduction
h
t The Hopf-Borel theorem is a rigidity theorem for connected bialgebras which
a
m are both commutative and cocommutative. It takes the following form in the non-
graded case:
[
Theorem. (Hopf-Borel) Let H be a commutative and cocommutative bialgebra,
1
over a field K of characteristic zero. The following are equivalent:
v
8 (1) H is connected,
6 (2) H is isomorphic to S(Prim H).
0
Here S(V) is the symmetric algebra over the vector space V, which can also be
1
0 seen as the polynomial algebra.
6
This theorem has already been generalised to other types of bialgebras, see for
0
example [5], [10] . A particular type of bialgebras, verifying a theorem analogue
/
h to the Hopf-Borel one, are magmatic bialgebras, see [1]. They are vector spaces
t
a endowed with an unitary binary operation and a counitary binary co-operation
m relatedbyamagmaticcompatibilityrelation. Wegeneralisethemtobialgebrasen-
: dowedwithunitaryn-aryoperationsforeachn≥2,co-unitaryn-aryco-operations
v
, ∆ for each n≥2, related by some infinite magmatic compatibility relation. We
i n
X denote Mag∞(V) the free infinite magmatic algebra over a vector space V.
r We define the primitive part of a bialgebra H to be :
a
Prim H:=∩ x∈H | ∆¯ (x)=0 ,
n≥2 n
where, (cid:8) (cid:9)
n−1
∆ (x):=∆ (x)− 1⊗i⊗x⊗1⊗j − σ◦(∆ (x),1⊗m−n),
n n m
i+jX=n−1 mX=2σ∈ShX(m,m−n)
and, Sh are the (m,m−n)-shuffles.
(m,m−n)
The rigidity theorem for infinite magmatic bialgebras is as follows:
Theorem 17 (p.9) Let H be an infinite magmatic bialgebra over a field K of any
chararacteristic. The following are equivalent:
Key words and phrases. Bialgebra, Hopf algebra, Cartier-Milnor-Moore, Poincar´e-Birkhoff-
Witt,operad.
1
2 E.BURGUNDER
(1) H is connected,
(2) H is isomorphic to Mag∞(Prim H).
The proof is based on the construction of an idempotent projector from the
bialgebra to its primitive part, as in [5], [10], [1].
Acknowledgement 1. I am debtful to L. Gerritzen who raised out this ques-
tion in Bochum’s seminar. I would like to thank J.-L. Loday for his advisory, D.
Guin,R.HoltkampforacarefulreadingofafirstversionandA.Bruguieresforsome
remarks.
2. Infinite magmatic algebra
Definition 1. An infinite magmatic algebra A is a vectorspace endowedwith one
n-ary unitary operation µ for all n≥2 (one for each n) such that:
n
every µ admits the same unit, denoted by 1, and that,
n
µ (x ,··· ,x )=µ (x ,··· ,x ,x ,··· ,x ) where x =1 and x ∈A, ∀j.
n 1 n n−1 1 i−1 i+1 n i j
Diagrammatically this condition is the commutativity of:
Id⊗···⊗u⊗···⊗Id
µAnA⊗(cid:15)(cid:15) nrreeoo eeeeeeeeeeeAe⊗eeµien⊗e−e1Kee⊗eeAee⊗ene−eie−e1eeeA⊗n−1
.
where u:K−→A is the unit map.
2.1. Free infinite magmatic algebra.
Definition 2. An infinite magmatic algebra A is said to be free over the vector
0
space V, if it satisfies the following universalproperty. Any linear map f :V →A,
whereAisanyinfinitemagmaticalgebra,extendsinauniquemorphismofalgebras
f˜:A →A:
0
i
V //A
AAA 0
AAA f˜
f AA
(cid:15)(cid:15)
A .
2.1.1. Planar trees and n-ary products. A planar tree T is a planar graph which
is assumed to be simple (no loops nor multiple edges), connected and rooted. We
denote by Y the set of planar trees with n leaves. In low dimensions one gets:
n
Y0 ={∅}, Y1 ={|}, Y2 = ??(cid:127)(cid:127) ,Y3 = ??(cid:127)?(cid:127)?(cid:127)(cid:127)(cid:127)(cid:127) ????(cid:127)?(cid:127)?(cid:127)(cid:127) ////(cid:15)(cid:15)(cid:15)(cid:15) ,
(cid:26) (cid:27) ( )
Y4 = ??(cid:127)(cid:127)??(cid:127)?(cid:127)?(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ????(cid:127)?(cid:127)???(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ??(cid:127)(cid:127)????(cid:127)(cid:127)(cid:127)(cid:127)??(cid:127)(cid:127) ????????(cid:127)(cid:127)(cid:127)?(cid:127)?(cid:127)(cid:127)(cid:127)(cid:127) ????????(cid:127)?(cid:127)?(cid:127)(cid:127)??(cid:127)(cid:127)
??(cid:127)?(cid:127)?(cid:127)(cid:127)(cid:127)(cid:127) ????(cid:127)?(cid:127)?(cid:127)(cid:127) ??????(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ??(cid:127)?(cid:127)?(cid:127)(cid:127)(cid:127)(cid:127) ????(cid:127)?(cid:127)?(cid:127)(cid:127) ???/?//?/(cid:127)(cid:15)(cid:15)(cid:127)(cid:15)(cid:15)(cid:127)(cid:127)(cid:127) ,···
INFINITE MAGMATIC BIALGEBRAS 3
The n-grafting of n trees is the gluing of the root of each tree on a new root. For
example the 2-grafting of the two trees t and s is:
t s
∨2(t,s):= FFFwww ,
the 3-grafting of three trees t, s and u is:
t s u
∨3(t,s,u):= KKKqqq .
Remark 3. From our definition of a planar tree, any t∈Y is of the form
n
t=∨ (t ,··· ,t )
k 1 k
for uniquely determined trees t ,··· ,t .
1 k
Let V be a vector space. A labelled tree of degree n, n ≥ 1, denoted by
(t,v ···v ), is a tree t endowed with the labelling of the leaves by the elements
1 n
v ,...,v , represented as:
1 n
v v ... v
1 2 n
...
QQQQQQQrrrrr
.
Moreover one can define the n-grafting of labelled trees by the n-grafting of the
trees, where one keeps the labellings on the leaves.
2.1.2. Construction of the free infinite magmatic algebra. Wedenote byMag∞(V)
the vector space spanned by the labelled planar trees:
Mag∞(V):=⊕∞ Mag∞⊗V⊗n ,
n=0 n
where Mag∞ =K[Y ].
n n
The following result is well-known:
Proposition 4. Let V be a vector space. The space Mag∞(V) endowed with the
n-grafting of labelled trees, for all n≥2, is a infinite magmatic algebra. Moreover
it is the free infinite magmatic algebra over V.
(cid:3)
3. Infinite magmatic coalgebra
Definition 5. An infinite magmatic coalgebra C is a vector space endowed with
one n-ary co-unitary co-operation ∆ :C →C⊗n for all n≥2 such that:
n
every ∆ admits the same co-unit c : C −→ K and that the following diagram
n
is commutative:
Id⊗···⊗c⊗···⊗Id
CC⊗OO∆neneeeeeeeeeee//eCe⊗eeie∆⊗ene−Ke1e⊗eeCee⊗ene−eie−e1eee22C⊗n−1
.
4 E.BURGUNDER
3.1. Construction of the connected cofree infinite magmatic coalgebra.
We denote Sh(p,q) the set of (p,q)-shuffles. It is a permutation of (1,··· ,p;p+
1,···q)suchthattheimageoftheelements1topandoftheelementsp+1top+q
are in order.
We define
∆ (x):=∆ (x)− 1⊗i⊗x⊗1⊗j − σ◦(∆ (x),1⊗m−n).
n n m
i+jX=n−1 Xm σ∈ShX(m,m−n)
Let T denote the n-corolla. Then ∆ (T )=0 for all m6=n and ∆ (T )=|⊗n.
n m n n n
Definition 6. An infinite magmatic co-augmented coalgebra is connected if it
verifies the following property:
H= F H where F H:=K1
r≥0 r 0
and, by induction F H:=∩ x∈H | ∆¯ (x)∈F H⊗n ,
r n≥2 n r−1
S
Remark that connectedness only depends(cid:8)on the unit and co-operation(cid:9)s.
We define the primitive part of H as Prim H:=∩ x∈H | ∆¯ (x)=0 .
n≥2 n
Definition 7. An infinite magmatic coalgebra C is cofr(cid:8)ee on the vector spa(cid:9)ce V
0
if there exists a linear map p:C →V satisfying the following universal property:
0
anylinearmapφ:C →V,whereC isanyconnectedinfinitemagmaticcoalgebra
such that φ(1)=0, extends in a unique coalgebra morphism φ˜:C →C :
0
C
C
C
CCφ
φ˜ CC
CC
(cid:15)(cid:15) p !!
C // V .
0
3.1.1. Planartreesandn-arycoproducts. Weendowthevectorspaceofplanartrees
with the following n-ary co-operations, for n≥2: for any planar tree t we define:
∆ (t):= t ⊗···⊗t
n 1 n
where the sum is extended on all theXways to write t as ∨ (t ,··· ,t ), where t
n 1 n i
may be ∅. It can be explicited, as follows, for t = ∨ (t ,··· ,t ), where t 6= ∅ for
n 1 n i
all i:
t t
1 n n−1
∆ (t):= ⊗ ··· ⊗ + ∅⊗i⊗t⊗∅⊗n−i−1 ,
n
i=0
X
m−1∅⊗i⊗t⊗∅⊗m−i−1 , if m<n
i=0
∆m(t):= Pm−1∅⊗i⊗t⊗∅⊗m−i−1+
+Pi=0i1+···+in+1=m−n∅⊗i1 ⊗t1⊗∅⊗i2 ⊗···⊗tn⊗∅⊗in+1 , if m>n
n−1 P
∆ (|):= ∅⊗i⊗|⊗∅n−i−1 ,
n
i=0
X
∆ (∅):=∅⊗n .
n
As in the preceding section one candefine the n-ungraftingof labelled trees by the
n-ungrafting of planar trees and keeping the labelling on the leaves.
INFINITE MAGMATIC BIALGEBRAS 5
Remark that the empty tree ∅ plays here the role of the unit, it can then be
denoted by 1:=∅.
3.1.2. Construction of the cofree connected infinite magmatic coalgebra.
Definition8. Theheight ofaplanartreeT isthemaximalnumberofinnervertices
one can meet when going through all the paths starting from the root to a leaf.
77 ··· wwwn
_ _ _ _ _
···
_ _ _ _ _
NNNppp
1 .
Example 9. The n-corolla is of height 1. The tree ????????(cid:127)?(cid:127)?(cid:127)(cid:127)??(cid:127)(cid:127) is of height 3.
Proposition 10. Let V be a vector space. The space Mag∞(V) endowed with
the n-ungrafting co-operations on labelled trees is a connected infinite magmatic
coalgebra. Moreover it is cofree over V among the connected infinite magmatic
coalgebras.
Proof. We could prove this proposition by dualising Proposition (4), but since we
did not give a proof of it we will write completely this proof.
The co-operations are co-unital by definition, so Mag∞ is an magmatic coalge-
bra.
Then we verify the connectedness of Mag∞(V). It comes naturally that:
F Mag∞(V) = Mag (V)⊕Mag (V)
1 0 1
F Mag∞(V) = Mag (V)⊕Mag (V)⊕ {n−corollas}
2 0 1 n
One canconclude by induction onthe number ofheights ofthe tree. Indeed, let us
consider the tree T ∈ Mag∞(V). It can be seen as the n-grafting of other trees,
each of them having at least a height less than the considered tree. Moreover we
have:
∆ (T) = ∆ ◦µ (T ⊗···⊗T )
n n m 1 n
0 , if m6=n
=
T ⊗···⊗ T , if m=n.
1 n
∈Fi 1(V) ∈Fi n(V)
where i ,··· ,i ≤n, so ∆(T)∈|{Fz} −1 ⊗n. So|w{ze}can conclude that:
1 n n n
F Mag∞(V)=⊕m=r{ trees with height m} .
r m=0
It is clear that ∪ F Mag∞(V)=Mag∞(V).
n n
6 E.BURGUNDER
To prove the cofreeness of the coalgebra, it is sufficient to prove the commuta-
tivity of the following diagram:
(1)
φ˜
C U_UU_UU_UU_U//UUMUφUUaUgU(UVUU)∞UUU=UU⊕** n(cid:15)(cid:15)(cid:15)(cid:15)≥0Magn⊗V⊗n
V .
The map φ˜ can be decomposed into its homogeneous components as follows:
(2) φ˜(c)=φ˜(c) +φ˜(c) +φ˜(c) +...
(1) (2) (3)
By induction on n, one can determine the homogenous components of φ˜. As the
map φ˜is a coalgebra morphism defined on C¯, one defines φ˜(1)=1 .
The commutativity of the diagram (1) gives the following equality:
(3) φ˜(c) =(|,φ(c)).
1
By definition of Mag (V):
2
φ˜(c) = ( ??(cid:127)(cid:127) ,a a )
2 1 2
We adopt the following notation ∆¯(cX)=Σc ⊗c . And we compute:
1 2
φ˜ ⊗φ˜ ◦∆¯(c) = φ˜ (c )⊗φ˜ (c )
1 1 1 1 1 2
= (|,φ(c ))⊗(|,φ(c )) thanks to (3)
1 2
But ∆◦φ˜(c) = (|,Pa )⊗(|,a )
2 P1 2
= (|,φ(c ))⊗(|,φ(c ))
1 2
P
Therefore,
φ˜(c) = ( ??(cid:127)(cid:127) ,φ(c )φ(c ))
2 1 2
P
AnytreeT determinesaco-operationthatwedenoteby∆T. IfT isthecorollawith
n leaves then ∆T is ∆n. Another example is to consider the tree T = ????????(cid:127)?(cid:127)?(cid:127)(cid:127)??(cid:127)(cid:127) ,
we have
∆T =(Id⊗3⊗∆ )◦(Id⊗∆ )◦∆ .
3 3 2
Analogously for a tree T of degree n:
i
φ˜ (c) = (t,a ···a )
n 1 n
∆¯Tiφ˜(c) = X(|,ai)⊗···⊗(|,ai)
1 n
Denote: ∆¯Ti(c) = Xci ⊗···⊗ci
1 n
φ˜⊗n◦∆¯Ti(c) = X(|,φ(ci))⊗···⊗(|,φ(ci)),
1 1 n
which gives us: X
φ˜i(c)= (T ,φ(ci)···φ(ci)) .
n i 1 n
X
INFINITE MAGMATIC BIALGEBRAS 7
Going through all the trees of degree n, we have:
φ˜ (c)= (T ,φ(ci)···φ(ci))
n i 1 n
Ti ofXdegreen
(though we denote φ˜i, T , we don’t assume that there must be an order on the
i
trees, this notation is only used to distinguish the trees with same degree.)
Therefore one has:
φ˜(c) = (|,φ(c))+ ( ??(cid:127)(cid:127) ,φ(c )φ(c ))+
1 2
( ??(cid:127)?(cid:127)?(cid:127)(cid:127)(cid:127)P,φ(c1)φ(c1)φ(c1))+ ( ?????(cid:127)?(cid:127)(cid:127) ,φ(c2)φ(c2)φ(c2))+
(cid:127) 1 2 3 (cid:127) 1 2 3
P / P
///(cid:15)(cid:15)(cid:15) ,φ(c3)φc3φ(c3) +...
(cid:15) 1 2 3
(cid:16) (cid:17)
P
Byconstructionφ˜isamorphismofinfinitemagmaticcoalgebraswhichisunique,
since we have no other choice to have the commutativity of diagram (1) and the
coalgebra morphism property. (cid:3)
4. Infinite magmatic bialgebra
Definition 11. An infinite magmatic bialgebra (H,µ ,∆ ) is a vector space H=
n n
H¯⊕K1 such that:
1) H admits an infinite magmatic algebra structure with n-ary operations de-
noted µ ,
n
2) H admits a infinite magmatic coalgebra structure with n-ary co-operations de-
noted ∆ ,
n
3) H satisfies the following compatibility relation called the “infinite magmatic
compatibilty”:
(4)
∆ ◦µ (x ⊗···⊗x )=x ⊗···⊗x + n−11⊗i⊗x⊗1⊗n−i−1 ,
n n 1 n 1 n i=0
∆ ◦µ (x ⊗···⊗x )=
m n 1 n
m−11⊗i⊗x⊗1⊗m−i−1P, if m<n
i=0
P+Pmi=−0i11+1·⊗··+ii⊗n+x1=⊗m1−⊗nm1−⊗ii−11⊗+x1⊗1⊗i2 ⊗···⊗xn⊗1⊗in+1 if m>n
where x:=µ ◦x ⊗P···⊗x and x ,··· ,x ∈H¯ .
n 1 n 1 n
A fundamental example in our context is the following:
Proposition 12. Let V be a vector space. The space (Mag∞(V),∨ ,∆ ), where
n n
the operations ∨ (resp. the co-operations ∆ ) are defined in 2.1.1 and 3.1.1, is an
n n
infinite magmatic connected bialgebra.
Proof. Any tree can be seen as the n-grafting of n trees, except the empty tree
and the tree reduced to the root. Therefore the m-ungrafting of a tree can be
viewed as the m-ungrafting of the n-grafting of n trees. This observation gives the
compatibility relation. (cid:3)
8 E.BURGUNDER
5. The main theorem
Definition13. Thecompleted infinite magmatic algebra, denotedbyMag∞(K)∧ ,
is defined by
Mag∞(K)∧ = Mag ,
n
n≥0
Y
wherethefirstgenerator|isdenotedbyt. Thisdefinitionallowsustodefineformal
power series of trees in Mag∞(K)∧.
Lemma 14. The following two formal power series, g and f, are inverse for com-
position in Mag∞(K)∧:
g(|):=|− ??(cid:127)(cid:127) − ////(cid:15)(cid:15)(cid:15) − ???/?//?/(cid:127)(cid:15)(cid:15)(cid:127)(cid:15)(cid:15)(cid:127)(cid:127)(cid:127) −··· , f(|):= T,
(cid:15)
X
where the sum is extended to all planar trees T.
Here the tree T stands for the element T(x) := T(x,...,x), where x = | the
generator. ThecompositionofT ◦T isdefinedasT ◦T (x):=T ◦T (x,...,x)=
1 2 1 2 1 2
T (T (x,...,x),...,T (x,...,x)).
1 2 2
Proof. First, we show that g◦f =|, that is to say:
T − ∨ (T ⊗T )−···− ∨ (T ⊗···⊗T )−···=| ,
2 1 2 n 1 n
X TX1,T2 T1,X···,Tn
equivalently:
∨ (T ⊗T )−···− ∨ (T ⊗···⊗T )−···= T −| .
2 1 2 n 1 n
TX1,T2 T1X,···,Tn X
It is immediate, as every tree can be seen as the n-grafting of n trees for a certain
n, except |.
Then one verifies that, as in the associative case, a right inverse is also a left
inverse. Let f−1 denote the left inverse of f. Then:
f−1 =f−1◦(f ◦g)=(f−1◦f)◦g =g .
Remark that we have associativity of composition even in the infinite magmatic
context. Therefore one has f ◦g =Id et g◦f =Id. (cid:3)
Definition15. Then-convolution ofninfinitemagmaticalgebramorphismsf ,··· ,f
1 n
is defined by:
⋆ (f ···f ):=µ ◦(f ⊗···⊗f )◦∆ .
n 1 n n 1 n n
Observe that these operations are unitary.
Lemma16. Let(H,µ ,∆ )beaconnectedinfinitemagmaticbialgebra. Thelinear
n n
map e:H→H defined as:
e:=J −⋆ ◦J⊗2−⋆ ◦J⊗3−···−⋆ ◦J⊗n−···
2 3 n
where J =Id−uc, u the unit of the operations, c the co-unit of the co-operations,
has the following properties:
(1) Im e=Prim H,
(2) for all x ,··· ,x ∈H¯ one has e◦µ (x ⊗···⊗x )=0,
1 n n 1 n
(3) the linear map e is an idempotent,
INFINITE MAGMATIC BIALGEBRAS 9
(4) for H = (Mag∞(V),µ ,∆ ) defined above, e is the identity on V =
n n
Mag (V) and trivial on the other components.
1
Proof. In this proof, we adopt the following notation: Id := IdH¯, and for all
x∈H¯, ∆¯ (x):= x ⊗···⊗x
n 1 n
(1) Proof of Im e=Prim H .
P
∆ (e(x)) = ∆ (x)− ∆ ◦µ ◦∆ (x)
n n n m n
m
X
= x ⊗···⊗x −∆ ◦µ (x ⊗···⊗x − ∆ ◦µ ◦∆ (x)
1 n n n 1 n n m n
m6=n
X =0
= 0 .
| {z }
(2) Proofthatforallx ,··· ,x ∈H¯ onehase◦µ (x ⊗···⊗x )=0. Indeed,
1 n n 1 n
e◦µ (x ⊗·⊗x ) = µ (x ⊗···⊗x )− µ ◦∆ ◦µ (x ⊗···⊗x )
n 1 n n 1 n m m n 1 n
m
X
= µ (x ⊗···⊗x )−µ ◦∆ ◦µ (x ⊗···⊗x )
n 1 n n n n 1 n
= 0 .
(3) Proof that e is an idempotent. We compute:
e(e(x)) = e(x)− e(µ ◦∆ (x))
m n
m
X
= e(x).
(4) Proof that for H=(Mag∞(V),µ ,∆ ) defined above, e is the identity on
n n
V =Mag (V) and trivial on the other components.
1
On Mag (V) = |⊗V we have: e(|⊗x) = |⊗x. All other trees can be
1
seen as the n-grafting of n trees for a certain n. Then it suffices to apply
the second property of the idempotent e to complete the proof.
(cid:3)
Theorem 17. If H be a connected infinite magmatic bialgebra over a field K of
any characteristic, then the following are equivalent:
(1) H is connected,
(2) H∼=Mag∞(Prim H).
Proof. It is convenient to introduce the following notation: for T ∈Y
n
⋆ (J):H−→H:x7→xn 7→⋆ (J)(xn)
T T
where T ∈ Mag(K): we label the tree T by J on each leaf, and endow each inner
vertexbyann-aryoperation⋆n. ForexampleconsideringthetreeT = ????????(cid:127)?(cid:127)?(cid:127)(cid:127)??(cid:127)(cid:127) ,
we have
⋆ (J)=⋆ (J)◦(Id⊗⋆ (J))◦(Id⊗3⊗⋆ (J)) ,
T 2 3 3
and valued on an element of H, one has:
⋆ (J)(x)=⋆ (J)◦(Id⊗⋆ (J))◦(Id⊗3⊗⋆ (J))(x⊗6) ,
T 2 3 3
Observe that
J⋆T1 ⊗···⊗J⋆Tn =J⋆(∨n(T1⊗···⊗Tn))
10 E.BURGUNDER
by definition. Let us denote V :=Prim H.
We prove the isomorphism by explicitly giving the two inverse maps.
We define: G:H¯ →Mag∞(V) as
G(x):=J(x)−⋆ ◦J⊗2(x)−⋆ ◦J⊗3(x)−···−⋆ ◦J⊗n(x)−··· ,
2 3 n
and F :Mag(V)→H¯ by
F(x):= ⋆ (J)(x) ,
T
where the sum is extended to all planaXr trees T.
Moreover,denote by t the generatorof Mag∞(K), t:=|, and by tn :=∨ ◦t⊗n.
n
We define g(t) := t−t2 −t3 −···−tn −··· , and f(t) := T, where the sum
is extended to all planar trees T. By lemma (14) these two preceding maps are
P
inverse, for composition.
These seriescanbe appliedto elements ofHomK(H,H) sending 1 on0 using ⋆n
as a product, thanks to the following morphism:
Mag(V)∞∧ −→ HomK(H,H)
t 7→ J
φ(t)= a T 7→ φ⋆(J)=Φ= a J⋆T
n n
φ◦ψ(t) 7→ (φ◦ψ)⋆(J)=Φ◦Ψ=φ⋆(J)◦ψ⋆(J)
P P
It is clear that e=g⋆(J).
Therefore composing the two power formal series F and G gives as a result:
F ◦G=f⋆◦g⋆(J)=(f ◦g)⋆(J)=Id⋆(J)=J
G◦F =g⋆◦f⋆(J)=(g◦f)⋆(J)=Id⋆(J)=J
The proof is complete since J =Id on H¯.
(cid:3)
Remark 18. There is a slightly different definition under which the result still
holds. Change the definition in:
An infinite magmatic co-augmented coalgebra is connected if it verifies the fol-
lowing property:
H= F H where F H:=K1
r≥0 r 0
and, by induction F H:=∩r+1 x∈H | ∆¯ (x)∈F H⊗n ,
S r n≥2 n r−1
The definition of the primitive elements(cid:8)being unchanged. Then, w(cid:9)e find that
Mag∞(K) is still connected for the following description:
F Mag∞ = {|}
1
F Mag∞ = {|, ??(cid:127)(cid:127) }
2
F Mag∞ = {the n corolla and all the trees with a root being a m-grafting
n
where m≤n and all the operations being in F Mag∞}
n
We observe that the space of primitive elements Prim H is the same as defined in
the precedent cas, that is to say Prim H={|}.
6. m-magmatic bialgebras
Insteadofconsideringinfinitemagmaticbialgebrasonemayconsiderm-magmatic
bialgebras, with m ≥ 2, where the number of operations and co-operations is re-
stricted to m. Explicitly, we would have:
