Cantidad de dígitos del divisor

While the product given by “disjoint unions” considered so far is very natural when considering forests, it is much less natural when considering spaces of trees. There, the more natural thing to do is to join trees together by their roots. Given a typed forest F, we then define the typed treeJ(F) by joining all the roots of F together. In other words, we setJ(F) = F/ ∼, where∼is the equivalence relation on nodes in NF given byx ∼ y if and only if either

x =y or bothx andybelong to the setF of nodes ofF. For example

When considering coloured or decorated trees as we do here, such an operation cannot in general be performed unambiguously since different trees may have roots of different colours. For example, if

(F,Fˆ)=

then we do not know how to define a colouring ofJ(F)which is compatible withFˆ. This justifies the definition of the subset Di(J)⊂Fas the set of all forests(F,Fˆ,n,o,e)such that Fˆ()∈ {0,i}for every rootofF. We also write D(J) = _i≥0 Di(J)and Dˆ i(J) ⊂ Di(J)for the set of forests such thateveryroot has colouri.

Example 4.6 Using as usual red for 1 and blue for 2, we have

∈ D1(J), ∈ ˆD1(J), ∈ ˆD2(J).

We can then extendJ to D(J)in a natural way as follows.

Definition 4.7 Forτ =(F,Fˆ,_n,_o,_e)∈ D(J), we define the decorated tree

J(τ)∈Fby

J(τ)=(J(F),[ ˆF],[n],[o],e) ,

where[n](x)=_y∈xn(y),[o](x)=_y∈xo(y), and[ ˆF](x)=sup_y∈xFˆ(y).

Example 4.8 The following coloured forests belong toD2(J)

τ1 = τ2 = τ1·τ2= J(τ1·τ2)=

The following coloured forests belong toDˆ 2(J)

τ1 = τ2 = τ1·τ2= J(τ1·τ2)=

It is clear that the Di’s are closed under multiplication and that one has

J(τ · ¯τ)=Jτ·J(τ)¯ , τ,τ¯ ∈ Di(J) (4.1) for everyi ≥0. Furthermore,J is idempotent and preserves our bigrading. The following fact is also easy to verify, whereK,Kˆi, i,ˆi and Pˆi were defined in Sect.3.5.

Lemma 4.9 For i ≥0, the sets Di(J)and Dˆi(J)are invariant underK,

i, Pˆi andJ. Furthermore,J commutes with bothKand Pˆi on Di(J)

and satisfies the identity ˆ

KiJ = ˆKiJKˆi , on Dˆ i(J). (4.2)

In particularKˆiJ is idempotent on Dˆi(J).

Proof The spaces Di(J) and Dˆi(J) are invariant under K, i and Pˆi because these operations never change the colours of the roots. The invariance underJ follows in a similar way.

The fact thatJ commutes withKis obvious. The reason why it commutes withPˆiis thatovanishes on colourless nodes by the definition ofF. Regarding (4.2), sinceKˆi = ˆPiiK, and all three operators are idempotent and commute with each other, we have

ˆ

KiJ =iPˆiJK, KˆiJKˆi =iPˆiJiK so that it suffices to show that

ˆ

PiJK= ˆPiJiK. (4.3)

For this, consider an elementτ ∈ ˆDi(J)and writeτ = μ·ν as in (3.37). By the definition of this decomposition and ofK, there existk ≥0 and labels

nj ∈Nd,oj ∈Zd ⊕Z(L)with j ∈ {1, . . . ,k}such that

Kτ =(Kμ)·x_n(i₁)_,o1. . .x_n(i_k)_,ok ,

wherexn(i,)o=(•,i,n,o,0). It follows that

iKτ =(Kμ)·x_n(i_,)₀ (4.4) withn =k_j=1nj. On the other hand, by (4.1), one has

JKτ =J(Kμ)·x_n(i_,)_o ,

withodefined from theoi similarly ton. Comparing this to (4.4), it follows thatJKτ differs fromJiKτ only by itso-decoration at the root of one of its connected components in the sense of Remark2.10. Since these are set

Finally, we show that the operation of joining roots is well adapted to the definitions given in the previous subsection. In particular, we assume from now on that theAi fori = 1,2 are given by Definition4.1. Our definitions guarantee that

• F1⊂ D1(J)

• F2⊂ ˆD2(J).

We then have the following, whereJ is extended to the relevant spaces as a triangular map.

Proposition 4.10 One has the identities

2J =(J ⊗J)2=(J ⊗J)2J , on D(J),

1J =(id⊗J)1=(id⊗J)1J , onF2.

Proof ExtendJ to coloured trees byJ(F,Fˆ)=(J(F),[ ˆF])with[ ˆF]as in Definition4.7. The first identity then follows from the following facts. By the definition of_A2, one has

A2(J(F,Fˆ))= {JFA : A∈A2(F,Fˆ)}, (4.5)

whereJFAis the subforest ofJF obtained by the image of the subforest

A of F under the quotient map. The map JF is furthermore injective on

A2(F,Fˆ), thus yielding a bijection between A2(J(F,Fˆ)) and A2(F,Fˆ).

Finally, as a consequence of the fact that each connected component of A

contains a root of F, there is a natural tree isomorphism betweenJFAand

JA. Combining this with an application of the Chu–Vandermonde identity on the roots allows to conclude.

The identity (4.5) fails to be true for A1 in general. However, if

(F,Fˆ,n,o,e) ∈ F2, then each of the roots of F is covered by Fˆ−1(2), so

that (4.5) with A2 replaced byA1 does hold in this case. Furthermore, one

then has a natural forest isomorphism betweenJFAandA(as a consequence of the fact that Adoes not contain any of the roots of F), so that the second

identity follows immediately.

We now use the “root joining” mapJ to define

ˆ

H2

def

= F2/ker(JKˆ2)%H2/ker(JPˆ2) . (4.6)

Note here thatJPˆ2 is well-defined on H2 by (4.2), so that the last identity

so the order in which the two operators appear here does not matter. We define also

ˆ

B2

def

=Vec(_C2)/ker(JK2)%B2/ker(J) , (4.7)

whereJ :_C2 →C2 is defined by(J(F),Fˆ), which makes sense since all

roots inF have the same (blue) colour. Finally, we define thetree productfori ≥0

Di(J)×Di(J)(τ,τ)¯ →ττ¯

def

=J(τ · ¯τ) (4.8)

Then we have the following complement to Corollary4.5

Proposition 4.11 Denoting byMˆ the tree product(4.8),

1. (Hˆ2,Mˆ, 2,12,1₂)is a Hopf algebra and a comodule bialgebra over the

Hopf algebra(H1,M, 1,11,11)with coaction1and counit11.

2. (Bˆ2,Mˆ, 2,12,1₂)is a Hopf algebra and a comodule bialgebra over the

Hopf algebra(B1,M, 1,11,1₁)with coaction1 and counit1₁.

Proof The Hopf algebra structure ofH2 turnsHˆ2 into a Hopf algebra as well

by the first part of Proposition4.10and (4.1), combined with [48, Thm 1 (iv)], which states that ifH is a Hopf algebra over a field andI a bi-ideal of Hsuch thatH/I is commutative, thenH/I is a Hopf algebra. For_Bˆ2, the same proof

holds.

The second assertion in Proposition 4.11 is in fact the same result, just written differently, as [8, Thm 8]. Indeed, our space_B2is isomorphic to the

Connes-Kreimer Hopf algebraHCK, andB1is isomorphic to an extension of

the extraction/contraction Hopf algebra H. The difference between our _B1

andHin [8] is that we allow extraction of arbitrary subforests, including with connected components reduced to single nodes; a subspace ofB1which turns

out to be exactly isomorphic to His the linear space generated by coloured forests(F,Fˆ)∈C1such thatNF ⊂ ˆF1.