Generalised mixed distributive laws and Hopf-Borel type theorem for operads

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by  Emily Burgunder

GENERALISED MIXED DISTRIBUTIVE LAWS AND
HOPF-BOREL TYPE THEOREM FOR OPERADS
7
1 E. BURGUNDERANDB. DELCROIX-OGER
0
2

n
a
J Abstract. In [Lod08], Loday generalises Hopf-Borel theorem to oper-
ads. We extend here his result by loosening and reducing hypotheses
5
of this theorem to a class of rewriting rules generalising the classical

] notion of mixed distributive laws, that we call generalised mixed dis-
O tributive laws. This enables us to show that for any operads P and Q
C havingthesameunderlyingS-module,thereexistsageneralised mixed
distributive law λ such that any connected P coQ-bialgebra satisfying
.
h λ is free and cofree over its primitive elements. Our reasoning permits
t ustogeneratemanynewexamples,whilerecoveringtheknownonesby
a
m considering dual relations.
[
Introduction

1
v Distributive laws first appeared in 1969 in Beck’s article [Bec69]. It takes
3
its name from the distributivity of addition over multiplication studied dur-
2
3 ing childhood. Distributive laws give a confluent way to rewrite expressions
1 mixing different products. An example of distributive laws is the one mix-
0
ing the commutative productand the Lie bracket in a Poisson algebra. This
.
1 notion has then been studied by Burroni in [Bur73] for algebras and by
0
Markl in [Mar96] generalising this definition to operads and linking it to the
7
1 topological notion of Koszulness. Fox and Markl have adapted in [FM97]
: the notion of distributive laws to expressions mixing operations and coop-
v
i erations: they called the obtained rewriting rules mixed distributive laws.
X
A bialgebra, where the algebra structure is encoded by an operad A, the
ar coalgebra structure by a cooperad C∗ linked through a mixed distributive
law λ, satisfies, under some assumptions, a Hopf-Borel type theorem: any
connected Cc − A-bialgebra satisfying λ is free and cofree over its primi-
tive elements. Such a theorem is called a rigidity theorem. Particular cases
of rigidity theorems were proven for instance in [Bor53] for commutative
cocommutative Hopf bialgebras, in [LR06] for associative coassociative bial-
gebras, in [Bur10] for Zinbiel coassociative bialgebras, in [Foi07] for den-
driform codendriform algebras and in [Liv06] for PreLie coNAP bialgebras.
Date: January 6, 2017.
TheauthorsthankJ.Mill`esforourfruitfulconversationsandM.Livernetforheruseful
comments about our work. We also thank Y. Guiraud for suggesting the name quasi-set
operad. The work of thefirst author was supported by ANR CATHRE. The work of the
second authorwas supported by LabEx CIMI.
1
2 E.BURGUNDERANDB.DELCROIX-OGER
ThegeneralframeworkforthistheoremwasintroducedbyLodayin[Lod08].
Rigidity theorems were then further studied, for instance in [LMW15] and
applications can be found for example in [BCR15] to compute explicit bases
of algebras. While studying the general framework and its rewriting for
particular symmetric operads, it became clear to the authors that the three
hypothesesofthistheoremhadtobeclarifiedandsomefurtherclarifications
were needed in the proof.
In this article we prove that one only of the hypotheses formulated by
Lodayin[Lod08]isneeded,theothersbeingimplied. Thisviewpointenables
ustoprovideananswertoaconjectureofLoday: foragivenoperadP which
encodedthestructureofalgebraandofcoalgebraofaPc−P bialgebrasthere
exists mixed distributive laws between the structures such that a rigidity
theorem holds. We also explore the equivalence between these hypotheses
when the operads encoding the bialgebra structure admit compatible bases.
In such a framework, we are able to make explicit an inductive algorithm to
compute the idempotent and the mixed distributive laws. This framework
also includes the particular case where the coproducts are given as the dual
of the products, in which case, we can recover most of the known cases.
Moreoveritprovidesasuitableenvironmentforthedevelopmentofrewriting
and Groebner bases theories for P-algebras. Indeed the mixed distributive
laws allow to decide whether a polynomial is a divisor of another one.
This article is organised as follows. The first part focuses on the general
theory and then particularises to the framework wherebases for operads are
known. The second part develops the dual case and is illustrated through
some examples.
More precisely, in the first section, we recall Loday’s theorem with its
three hypotheses (H0), (H1), (H2iso): for any Cc −A bialgebra satisfying
mixed distributive laws (H0), one can compute a morphism ϕ : A(V) →
V
Cc(V), functorial in V (H1). If this morphism is bijective (H2iso) then
a rigidity theorem holds. But the core property is that the morphism ϕ
induces a family of S-modules isomorphisms ϕ : A(n) → C∗(n). This
n
property alone gives the generalised mixed distributive laws which we can
compute explicitly when orthogonal bases are known (see Subsection 1.4).
Then in the second part, the dual case is explored. The above isomor-
phisms can be obtained by duality with respect to a fixed basis. We explain
this construction and give a simple condition for it to satisfy the hypotheses
of rigidity theorem: its compatibility to the S-module structure. We then
list the examples in literature which can be obtained in that way before
illustrating the strength of this case with a bunch of brand new examples:
PreLie coPreLie, Perm coPerm, NAP coNAP, PAN coPAN, PAN coPerm,
Leibniz coAssociative, Poisson coAssociative, Leibniz coZinbiel, 2as co2as
or even Dipt coDipt bialgebras.
HOPF-BOREL TYPE THEOREM FOR OPERADS 3
1 General Case
WegiveinthissectionanenlargementofthehypothesesforLoday’srigid-
ity theorem on generalised bialgebras. We first recall Loday’s formulation
of rigidity theorem before studying the intertwining between hypotheses of
the theorem. We then study the case when bases are orthogonal, so that we
can compute explicitly mixed distributive laws and idempotent, which was
one problem raised in [Lod08]. We finally give a new formulation of rigidity
theorem.
1.1 Loday’s rigidity theorem. We recall from [Joy86] that a species
is a functor from the category of finite sets and bijections to the category
of vector spaces. It is possible to define a composition ◦ on species (see
[BLL98]) and an operad is then given by the data of a species P and a map
π :P ◦P → P. The reader can consult [LV12] for further explanations.
Let K be a field of characteristic 0. Let A,C be two algebraic operads. In
this article, we will only consider connected operads, i.e. such that A(0) =
C(0) = ∅ and A(1) = C(1) = K.id and such that A(n) and C(n) are finite.
Following [FM97] and [Lod08], we define the notion of mixed distributive
laws:
Definition 1.1.1. A compatibility relation λ :C⊗A → A⊗C is defined as
a sequence {λ(m,n)} of maps
λ(m,n): C(m)⊗A(n)→
⊕A(t1)⊗···⊗A(tm)⊗St1×···×StmK[SN]⊗Ss1×···×Ssn C(s1)⊗···⊗C(sn),
where the summation is taken over all the N ≥ 1 and s +·+s = t +
1 n 1
···+t = N. A compatibility relation is said to be distributive (also called
m
mixed distributive law) if it is compatible with the operad structures of A
and C.
Let us first recall the definition of a cooperad and a coalgebra over an
operad.
Definition 1.1.2 ([LV12]). A cooperad C∗ is the data of a family of S -
n
modules C∗(n) for every integer n and of two morphisms of S-modules
∆ : C∗ → C∗ ⊗ C∗ and ǫ : C∗ → I (counit) satisfying some coassociativ-
ity and counitality axioms. When C is an operad, C∗(n) = Hom(C(n),K) is
a cooperad, called the dual cooperad of C.
Definition 1.1.3 ([LV12]). A coalgebra over an operad C is a vector space
C equipped with a S -equivariant morphism γn : C(n)⊗C → C⊗n. This
n C
definition is equivalent to the definition of a coalgebra over the cooperad C∗,
which is the data of a map ∆nC : C → C∗(n)⊗Sn C⊗n. We denote the free
coalgebra over an operad C whose vector space of primitives is V by Cc(V).
WecallCc− A-bialgebraanybialgebrawhichisaA-algebra,Cc-coalgebra
λ
and such that products and coproducts satisfy the mixed distributive laws
λ.
4 E.BURGUNDERANDB.DELCROIX-OGER
The rigidity theorem as stated by Loday is:
Theorem 1.1.4 ([Lod08]). Let Cc− A be a bialgebra type which satisfies
λ
(H0): the compatibility relations λ are distributive,
(H1): the free A-algebra is naturally a Cc− A-bialgebra,
λ
(H2iso): the Cc-coalgebra map ϕ(V) :A(V) → Cc(V) is an isomorphism.
Then any connected Cc− A-bialgebra H is free and cofree over its primitive
λ
elements
A(PrimH)∼= H ∼= Cc(PrimH).
The following cofiltration F H can be defined on any Cc-coalgebra H:
n
F H = {x ∈ H|∀p> n,∀δ ∈ C(p),δ(x) = 0}.
n
When there will be no ambiguity, we will only write F .
n
Inwhatfollows, wewillusethefollowing generalisation ofmixeddistribu-
tive laws:
Definition 1.1.5. A (Generalised) mixed distributive law between operads
A and C is a family of mixed distributive laws
(1) λ : C(m)⊗A(n)⊗(F )n →
m,n 1
⊕A(t1)⊗···⊗A(tm)⊗St1×···×Stm K[Sn]⊗Sn (F1)n,
where the summation is taken over all t +·+t = n, such that λ is
1 m m,n
compatible with the structure of operad of C and functorial in primitive
elements.
It is clear that mixed distributive laws as defined in Definition 1.1.1 is a
generalised mixed distributive law.
Example 1.1.6. An example of such mixed distributive laws is presented
in Equation (10).
Remark 1.1.7. Mixed distributive laws can be either represented in a fi-
nite sum with coefficients varying with the cofiltration (second definition
introduced) or as an possibly infinite mixed distributive law with constant
coefficients(firstdefinitionintroduced). Theadvantageofconsideringgener-
alised mixed distributive laws with non constant coefficients is to ease their
computations on explicit examples. They are called ”generalised” mixed
distributive laws because this notion generalises the notion of mixed dis-
tributive laws and gives the firstexample of infinite mixed distributive laws.
The Cc − A-bialgebras are then defined as bialgebras which is a A -
λg
algebra, Cc-coalgebra, such that products and coproducts satisfy the gener-
alised mixed distributive laws defined in 1.1.5 and generated by its primi-
tive elements (which is also satisfied by mixed distributive laws, see Lemma
1.3.5).
HOPF-BOREL TYPE THEOREM FOR OPERADS 5
1.2 Conditions on ϕ. In [Lod08], a condition on ϕ for the existence of a
rigidity theoremisthatthemorphismϕ isanisomorphismforevery vector
V
space V. We first give an example of the necessity of the injectivity of ϕ
before studying the link between filtration and cofiltration in the associated
bialgebras. This link is capital because, as we will see later, the idempotent
isnothingbuttheprojection onprimitiveelements paralleltodecomposable
elements.
The following example will show that the lack of injectivity of ϕ induces
a counter-example in the rigidity theorem.
Example1.2.1. Considerthecoassociativeassociativebialgebras(Asc,As,λ)
where the mixed distributive law λ is the Hopf mixed distributive law. We
compute ϕ : As(V) → As(V)c from the mixed distributive law: it is given
by ϕ(1...n ⊗(v ,...v )) = σ(1)...σ(n)⊗ (v ,...v ). It is to be
1 n σ∈Sn 1 n
noted that the ϕ constructed is not a monomorphism (therefore not an iso-
P
morphism and nor a epimorphism), failing to verify all the hypotheses for
the existence of a rigidity theorem.
Let us consider Solomon-Tits algebra, see for example [NT06, HNT08,
BR10], the space of surjection endowed with the shuffle product and the
block coproduct:
Forx ∈ STr,wewritex = (x(1),...,x(n))andr = max{x(i),1 ≤ i≤ n}.
n
Let x ∈ STr, y ∈ STs x⋆y = f ◦(x(1),···x(n),y(1)+r,···y(m)+r),
n m
wherethesumisoverallthestufflesf ∈SH(r,s). Givenamapx : [n]−→ N
P
there exists a unique surjective map std(x) in ST such that x(i) < x(j) if,
and only if, std(x)(i) < std(x)(j), for 1 ≤ i,j ≤ n. The map std(x) is
called the standardisation of x (see for instance [NT06]). Let K = {j <
1
··· < j } ⊆ {1,...,r}, we define the co-restriction of x to K by x|K :=
l
std(x(s ),...,x(s )), for x−1(K) = {s < ··· < s }. We define ∆ (x)
1 q 1 q block
by:
r−1
∆ (x) = x|{1,...,i}⊗x|{i+1,...,r},
block
i=1
X
and we extend it by linearity to all K[ST] (see for instance [BR10]).
The mixed distributive law between the product and the coproduct is the
Hopf mixed distributive law, which can be checked by direct inspection.
ST is conilpotent as for any element in x ∈ STr ∆r+1 (x) = 0, where
n block
∆2 = ∆ and ∆k = (∆ ⊗id⊗k−1)◦∆k−1 .
block block block block block
By a direct computation, one can prove that the element 112 cannot be
constructed as a linear combination of products of primitive elements. And
ST is therefore not generated by its primitives though it does verify the
conilpotent and the existence of the mixed distributive law hypotheses.
Link between filtration and cofiltration
We firstdecribethefreebialgebra, wewillthen consider freealgebras and
coalgebras in this subsection.
6 E.BURGUNDERANDB.DELCROIX-OGER
Proposition 1.2.2. The data of a family of isomorphisms ϕ : A(V) →
V
Cc(V) functorial in V and sending natural graduation by product to natural
cograduation by coproduct, is equivalent to the data of a family of isomor-
phisms of S-modules ϕ : A(n)→ C∗(n).
n
Proof. As K is a field of characteristic 0, we can identify coinvariants and
invariants and the notion of free coalgebra over the operad C is given by the
Schur functor defined by Cc(V) = C∗(n)⊗S V⊗n.
n
The data of a family of isomorphisms of S-modules ϕ : A(n) → C∗(n)
n
L
defines a natural transformation between the Schur functor associated to
Operad A, A :V 7→ A(n)⊗V⊗n and the Schur functor associated to
n≥1
Cooperad C∗, Cc described above. This natural transformation is directly
equivalent to a familNy of isomorphisms ϕ : A(V) → Cc(V) functorial in V
V
sending graduation A(n)⊗V⊗n to cograduation C∗(n)⊗S V⊗n.
n
Let us now suppose the existence of a family of isomorphisms ϕ :
V
A(V) → Cc(V) functorial in V, sending graduation by product to cograd-
uation by coproduct. Then, ϕV(A(n) ⊗Sn V⊗n) = C∗(n) ⊗Sn V⊗n, for
any integer n. Hence ϕ defines a family of isomorphisms of S-modules
V
ϕ : A(n)→ C∗(n). (cid:3)
n
We will denote by ϕ : A → C the morphism of S-module induced by the
family of ϕ .
n
Remark 1.2.3. • Ifthereexists suchafamily ϕ , thetwo S-modules
n
P and Q induced by the operads are equals, or in other words, the
underlying species are the same.
• For Comm, ”admissible” ϕ are homotheties, but it is not always the
case (it depends on the chosen decomposition in irreducible repre-
sentations of the S-module, which is not unique).
We call homogeneous mixed distributive law the set of all rewritings of
compositions of a cooperation of arity n, δ ∈ C(n), with an operation of
arity n, µ ∈ A(n), applied to primitive elements in terms of the primitive
elements p . We denote this set by (λ ). Then we have:
i p
(λ )= {δ◦µ(p ,...,p )= ασ p ⊗...⊗p
p 1 n δ,µ σ(1) σ(n)
(2) σX∈Sn
|δ ∈ C(n),µ ∈ A(n),ασ ∈ K,n ≥ 1}.
δ,µ
The data of homogeneous mixed distributive law is then strictly equivalent
to endowing the free A-algebra over a vector space V with a structure of
C-coalgebra sendingthegraduation ofthealgebratothecograduation ofthe
coalgebra. Indeed, the following result follows:
Lemma 1.2.4. The data of ϕ is equivalent to the data of a homogeneous
mixed distributive law.
Proof. ⇐ Given an homogeneous mixed distributive law, we can con-
siderthemapfromA(V)toitselfwhereatthesourceA(V)isviewed
HOPF-BOREL TYPE THEOREM FOR OPERADS 7
as an algebra and at the target A(V) is seen as a coalgebra. It pro-
duces a family of morphism ψ , sending filtration by product to
V
cofiltration by coproduct, and as result a morphism ψ defined by
n
ψ (µ)(δ) = αid .
n δ,µ
⇒ Denoting by h,i the duality pairing between C(n) and its dual space
C∗(n), and xσ the action of an element σ of the symmetric group on
an element x of C(n) = A(n), we define the mixed distributive law
(λ ) from the morphisms ϕ by:
p
(3) δ◦µ(p ,...,p )= hδσ,ϕ (µ)ip ⊗...⊗p ,
1 n n σ(1) σ(n)
σX∈Sn
where δ ∈ C(n), µ ∈ A(n) and p ∈ V.
i
Notice that we have:
hδσ,ϕ (µ)i = δ,ϕ (µσ−1) .
n n
D E
The two constructions are inverse of each other.
(cid:3)
Remark 1.2.5. This formula should contain a sum over permutation of
primitive elements to take into account the action of the symmetric group,
and automorphism groups of operations, which does not appear in [Lod08].
For instance, in PreLie(3) detailed in 2.3, we have, denoting the generating
product and coproduct respectively by ∆ and µ:
(∆⊗id)◦∆◦µ◦(µ⊗id−id⊗µ)(p ,p ,p ) = p ⊗p ⊗p +p ⊗p ⊗p ,
1 2 3 1 2 3 1 3 2
for any triples of primitive elements p . The second term of the sum will
i
always appear if the first one appear because of the equality:
µ◦(µ⊗id−id⊗µ)(p ,p ,p ) = µ◦(µ⊗id−id⊗µ)(p ,p ,p ).
1 2 3 1 3 2
1.3 Equivalence between conditions. The reasoning in Loday’s the-
orem is to first consider a mixed distributive law thanks to which we can
compute a map sending A(V) to C(V). To use this theorem, we then have
to prove that the induced morphism is an isomorphism. We adopt here an-
other reasoning: we start from the data of such an isomorphism and prove
thatwegetbackthehypothesesofthetheorem,moreprecisely, weprovethe
existenceofassociated generalised mixeddistributivelaws. Wefirstdescribe
this law on free bialgebras before defining general bialgebras.
Remark 1.3.1. The Φ part of mixed distributive laws always occurs, oth-
1
erwise the coproduct would vanish as we considered only non unitary co-
products.
We will show in what follows the intertwining between conditions of this
theorem.
8 E.BURGUNDERANDB.DELCROIX-OGER
Proposition 1.3.2. Let C and A be two algebraic operads. The existence of
ageneralised mixed distributivelaw λ impliesthe existence of afamily of S -
n
module morphism ϕ : A(n)→ C∗(n), for any positive integer n. Moreover,
n
thebijectivityofallϕ enablesustodefine the associated (generalised) mixed
n
distributive law on primitive elements of free and cofree bialgebras.
Remark 1.3.3. If such bijections exist, then A and C have the same un-
derlying S -module.
n
Proof.
(λ) ⇒ ϕ: (λ ) are special types of mixed distributive laws and can be directly
p
computed from a mixed distributive law (λ).
ϕ ⇒ (λ): We definefrom ϕ the associated mixed distributive law (λ) on prim-
itive elements.
WedenoterespectivelybymanddtheisomorphismsA(V)≃ n≥1A(n)⊗Sn
V⊗n and Cc(V) ≃ n≥1C∗(n)⊗Sn V⊗n. We moreover denLote ϕ(Mn) by
H .
n
L
The compatibility relations are defined as the following composite, using
notations of Definition 1.1.3:
C(k)⊗A(n)⊗Sn V⊗n −i−d⊗−ϕ−⊗−i−d−⊗→n C(k)⊗C∗(n)⊗Sn V⊗n −id−⊗−d−−→1 C(k)⊗Hn
k k
γk (ϕ−1)⊗k
−→C H −−−−−→ M
li li
l1+..X.+lk=n,Oi=1 l1+..X.+lk=n,Oi=1
li≥1 li≥1
k
−(m−−−−1)−⊗→k A(li) ⊗Sn V⊗n.
!
l1+..X.+lk=n, Oi=1
li≥1
As the operations are functorial in V and using Equation (3), we ob-
tain that for any cooperation δ of arity k and any operation µ of arity n,
there exists operations of arity r , µσ, such that on any primitive elements
i i
p ,...,p :
1 n
δ◦µ(p ,...,p ) = µσ(p ,...,p )⊗...⊗µσ(p ,...,p ),
1 n 1 σ(1) σ(r1) k σ(n−rk+1) σ(n)
r1σ+∈.SX..+shrukffl=en
where S is the set of permutations of {1,...,n} such that σ( l r +
shuffle i=1 i
l
1) < ... < σ( r +r ) for any 0 ≤ l ≤ k−1.
i=1 i l+1 P
We moreover check thatthesecompatibility relations arecompatible with
P
compositions of cooperations: the following diagram commutes.
HOPF-BOREL TYPE THEOREM FOR OPERADS 9
µ ⊗id
C
(C(k )⊗...⊗C(k ))⊗C(l)⊗A(n) C(k)⊗A(n)
1 l
(id)⊗l ⊗λ λ
((C(k )⊗A(q ))⊗...⊗(C(k )⊗A(q ))) A(r )⊗A(r ),
Pqi=n 1 1 l l (λ)⊗l Pri=n 1 k
P P
where k = k +...+k .
1 l
Hence the compatibility relations satisfy the hypotheses of Definition
1.1.5. (cid:3)
Remark 1.3.4. • If the (generalised) mixed distributive laws λ are
given by an isomorphismϕ, wewill writeequivalently theassociated
bialgebras Cc− A-bialgebras or Cc− A-bialgebras.
ϕ λ
• Therecanbedifferentmixeddistributivelawsassociatedtothesame
coupleofoperads: seeforinstancethemixeddistributivelawsforthe
Dendriform coDendriform bialgebra computed by Foissy in [Foi07]
and the one referred below.
Projection
Let us now consider a (not necessarily free or cofree) connected Cc− A-
λ
bialgebra H. We denote by F the cofiltration on H and ϕ the S-module
n
morphism associated to the mixed distributive laws.
We first show the following lemmas:
Lemma 1.3.5. The bialgebra H is generated, as an algebra, by its primitive
elements.
Proof. Let us consider the minimal n such that there exists an x in F H,
n
x not in the subalgebra generated by primitive elements of H. We will
construct a y in the subalgebra generated by primitive elements of H such
that x−y belongs to F H.
n−1
For any basis (µ ,...,µ ) of A(n), there exists a basis (δ ,...,δ ) of C(n)
1 k 1 k
such that ϕ(µ ) = δ∗. We write aσ = δ∗(δσ), denoting by δσ the action
i i i,j i j j
of an element σ of the symmetric group on an element δ of the considered
j
S-module. Hence, we have for any primitive elements p ,...,p , according
1 n
to Equation (3):
(4) δ ◦µ (p ,...,p )= aσ p ⊗...⊗p .
j i 1 n i,j σ(1) σ(n)
σX∈Sn
Without loss of generality, we can consider the case:
(5) δ (x) = djp ⊗...⊗p
j τ τ(1) τ(n)
τX∈Sn
for a given tuple of primitive elements (p ,...,p ), as any coproduct can be
1 n
decomposed as a sum of such elements on disjoint tuples.
10 E.BURGUNDERANDB.DELCROIX-OGER
Let us now consider an element y = k αiµ (p ,...,p ).
i=1 τ∈Sn τ i τ(1) τ(n)
Then, we have, using Equation (4):
P P
k
(6) δ (y) = αiaσ p ⊗...⊗p .
j τ i,j τ◦σ(1) τ◦σ(n)
Xi=1τ,Xσ∈Sn
Then, using Equations (5) and (6), for all j ∈ {1,...,k}, δ (x) = δ (y) is
j j
equivalent to:
k
(7) dj = αi aσ .
τ τ◦σ−1 i,j
Xi=1σX∈Sn
Moreover, foranyj, δσ = k aσ δ , whichgives onx,foranyτ,σ ∈ S :
j i=1 i,j i n
P k
(8) dj = aσ di
τ i,j τ◦σ−1
i=1
X
Thanks to Equations (7) and (8), choosing αi = 1di for any i and σ
σ n! σ
gives y in the S subalgebra of H generated by primitive elements such that
x−y is in F H, hence x−y is S by minimality of n and so is x. (cid:3)
n−1
Lemma 1.3.6. The vector space F admits a supplementary space in F ,
n−1 n
which is M = {µ(p ,...,p )|µ ∈ A(n),p ∈F }.
n 1 n i 1
Proof. • First, we have by definition F ⊆ F .
n−1 n
• Let us now show that M ∩F = {0}, for all n. Let us consider
n n−1
an element µ(p ,...,p ) in M −{0}. As ϕ is injective, there is an
1 n n
element δ ∈ C(n)suchthatδ◦µ(p ,...,p ) 6= 0, thusµ(p ,...,p ) ∈/
1 n 1 n
F .
n−1
• Then, we have M ⊂ F , for all n. Thus, F ⊃ F ⊕M . This
n n n n−1 n
comes from Equation (3).
• Let us prove that F ⊂ F ⊕M . As an algebra, H = A(V)/(R).
n n−1 n
By Lemma 1.3.5, we have that V ⊆ F . Let us consider an element
1
x ∈F . Then we can write (not necessarily uniquely) x as:
n
j j
x = µ (p ,...,p ),
j 1 j
j≥1
X
j
where p ∈ F and µ ∈ A(j).
k 1 j
From Equation (3), we have M ⊂ F . From the bijectivity of
j j
ϕ, M ∩F = {0} (see the second point of this proof). Then, as
j j−1
x ∈F , we get the decomposition:
n
x = y+z,
where y ∈ F and z ∈ M .
n−1 n
(cid:3)