The Long-Moody construction and polynomial functors
Arthur Souli{é}



The Long-Moody construction and polynomial functors
    Arthur Souli
    Accepted ???. Received ???; in original form December 30, 2023
==================================================================


In 1994, Long and Moody gave a construction on representations of
braid groups which associates a representation of 𝐁_n
with a representation of 𝐁_n+1. In this paper, we prove
that this construction is functorial and can be extended: it inspires
endofunctors, called Long-Moody functors, between the category of
functors from Quillen's bracket construction associated with the braid
groupoid to a module category. Then we study the effect of Long-Moody
functors on strong polynomial functors: we prove that they increase
by one the degree of very strong polynomiality.

Published in Annales de l'Institut Fourier, Volume 69 (2019) no. 4 p. 1799-1856.
This work was partially supported by the ANR Project ChroK, ANR-16-CE40-0003.
2010 Mathematics Subject Classification: 
 18D10, 18A25, 20C07, 20C99, 20J99, 20F36, 20F38, 57M07, 57N05.
Keywords: braid groups, functor categories, Long-Moody construction, polynomial functors.



§ INTRODUCTION


Linear representations of the Artin braid group on n strands 𝐁_n
is a rich subject which appears in diverse contexts in mathematics
(see for example <cit.> or <cit.> for
an overview). Even if braid groups are of wild representation type,
any new result which allows us to gain a better understanding of them
is a useful contribution.

In 1994, as a result of a collaboration with Moody in <cit.>,
Long gave a method to construct from a linear representation ρ:𝐁_n+1→ GL(V)
a new linear representation ℒℳ(ρ):𝐁_n→ GL(V^⊕ n)
of 𝐁_n (see <cit.>). Moreover, the
construction complicates in a sense the initial representation. For
example, applying it to a one dimensional representation of 𝐁_n+1,
the construction gives a mild variant of the unreduced Burau representation
of 𝐁_n. This method was in fact already implicitly present
in two previous articles of Long dated 1989 (see <cit.>).
In the article <cit.> dating from 2008, Bigelow and
Tian consider the Long-Moody construction from a matricial point of
view. They give alternative and purely algebraic proofs of some results
of <cit.>, and they slightly extend some of them. In a survey
on braid groups (see the Open Problem 7 in <cit.>),
Birman and Brendle underline the fact that the Long-Moody construction
should be studied in greater detail.

Our work focuses on the study of the Long-Moody construction ℒℳ
from a functorial point of view. More precisely, we consider the category
𝔘β associated with braid groups.
This category is an example of a general construction due to Quillen
(see <cit.>) on the braid groupoid β.
In particular, the category 𝔘β
has natural numbers ℕ as objects. For each natural number
n, the automorphism group Aut_𝔘β(n)
is the braid group 𝐁_n. Let 𝕂-𝔐𝔬𝔡
be the category of 𝕂-modules, with 𝕂 a commutative
ring, and 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
be the category of the functors from 𝔘β
to 𝕂-𝔐𝔬𝔡. An object M of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
gives by evaluation a family of representations of braid groups { M_n:𝐁_n→ GL(M(n))} _n∈ℕ,
which satisfies some compatibility properties (see Section <ref>).
Randal-Williams and Wahl use the category 𝔘β
in <cit.> to construct a general framework to
prove homological stability for braid groups with twisted coefficients.
Namely, they obtain the stability for twisted coefficients given by
objects of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).

In Proposition <ref>, we prove that a version of the
Long-Moody construction is functorial. We fix two families of morphisms
{ a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
and {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ,
satisfying some coherence properties (see Section <ref>).
Once this framework set, we show:
[Theorem A (Proposition <ref>) ]
There is a functor 𝐋𝐌_a,ς:𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)→𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡),
called the Long-Moody functor with respect to coherent families of
morphisms { a_n} _n∈ℕ and {ς_n} _n∈ℕ,
which satisfies for σ∈𝐁_n and M∈ Obj(𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡))

    𝐋𝐌_a,ς(M)(σ)=ℒℳ(M_n)(σ).


Among the objects in the category 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
the strong polynomial functors play a key role. This notion extends
the classical one of polynomial functors, which were first defined
by Eilenberg and Mac Lane in <cit.> for functors
on module categories, using cross effects. This definition can also
be applied to monoidal categories where the monoidal unit is a null
object. Djament and Vespa introduce in <cit.> the definition
of strong polynomial functors for symmetric monoidal categories with
the monoidal unit being an initial object. Here, the category 𝔘β
is neither symmetric, nor braided, but pre-braided in the sense of
<cit.>. However, we show that the notion of strong
polynomial functor extends to the wider context of pre-braided monoidal
categories (see Definition <ref>). We also introduce the
notion of very strong polynomial functor (see Definition <ref>).
Strong polynomial functors turn out inter alia to be very useful for
homological stability problems. For example, in <cit.>,
Randal-Williams and Wahl prove their homological stability results
for twisted coefficients given by a specific kind of strong polynomial
functors, namely coefficient systems of finite degree (see <cit.>).

We investigate the effects of Long-Moody functors on very strong polynomial
functors. We establish the following theorem, under some mild additional
conditions (introduced in Section <ref>)
on the families of morphisms { a_n} _n∈ℕ
and {ς_n} _n∈ℕ,
which are then said to be reliable.
[Theorem B (Corollary <ref>) ]
Let M be a very strong polynomial functor of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
of degree n and let { a_n} _n∈ℕ
and {ς_n} _n∈ℕ be coherent
reliable families of morphisms. Then, considering the Long-Moody functor
𝐋𝐌_a,ς with respect to the morphisms { a_n} _n∈ℕ
and {ς_n} _n∈ℕ, 𝐋𝐌_a,ς(M)
is a very strong polynomial functor of degree n+1.

Thus, iterating the Long-Moody functor on a very strong polynomial
functor of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
of degree d, we generate polynomial functors of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡),
of any degree bigger than d. For instance, Randal-Williams and
Wahl define in <cit.> a functor
𝔅𝔲𝔯_t:𝔘β→ℂ[t^±1]-𝔐𝔬𝔡
encoding the unreduced Burau representations. Similarly, we introduce
a functor 𝔗𝔜𝔐_t:𝔘β→ℂ[t^±1]-𝔐𝔬𝔡
corresponding to the representations considered by Tong, Yang and
Ma in <cit.>. These functors 𝔅𝔲𝔯_t and 𝔗𝔜𝔐_t
are very strong polynomial of degree one (see Proposition <ref>),
and moreover, we prove that the functor 𝔅𝔲𝔯_t is equivalent
to a functor obtained by applying the Long-Moody construction. Thus,
the Long-Moody functors will provide new examples of twisted coefficients
corresponding to the framework of <cit.>.

This construction is extended in the forthcoming work <cit.>
for other families of groups, such as automorphism groups of free
groups, braid groups of surfaces, mapping class groups of orientable
and non-orientable surfaces or mapping class groups of 3-manifolds.
The results proved here for (very) strong polynomial functors will
also hold in the adapted categorical framework for these different
families of groups.

The paper is organized as follows. Following <cit.>,
Section <ref> introduces the category 𝔘β
and gives first examples of objects of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).
Then, in Section <ref>, we introduce the
Long-Moody functors, prove Theorem A and give some of their properties.
In Section <ref>, we review the notion
of strong polynomial functors and extend the framework of <cit.>
to pre-braided monoidal categories. Finally, Section <ref>
is devoted to the proof of Theorem B and to some other properties
of these functors. In particular, we tackle the Open Problem 7
of <cit.>.

We will consider a commutative ring
𝕂 throughout this work. We denote by 𝕂-𝔐𝔬𝔡
the category of 𝕂-modules. We denote by 𝔊𝔯
the category of groups. We take the convention that the set of natural
numbers ℕ is the set of nonnegative integers { 0,1,2,…}.

Let ℭ𝔞𝔱 denote the category of small categories. Let
ℭ be an object of ℭ𝔞𝔱. We use the abbreviation
Obj(ℭ) to denote the objects of ℭ.
For 𝔇 a category, we denote by 𝐅𝐜𝐭(ℭ,𝔇)
the category of functors from ℭ to 𝔇.
If 0 is initial object in the category ℭ, then we
denote by ι_A:0→ A the unique morphism from 0
to A. The maximal subgroupoid 𝒢𝔯(ℭ)
is the subcategory of ℭ which has the same objects as
ℭ and of which the morphisms are the isomorphisms of
ℭ. We denote by 𝒢𝔯:ℭ𝔞𝔱→ℭ𝔞𝔱
the functor which associates to a category its maximal subgroupoid.



The author wishes to thank most sincerely his PhD advisor Christine
Vespa, and Geoffrey Powell, for their careful reading, corrections,
valuable help and expert advice. He would also especially like to
thank Aurlien Djament, Nariya Kawazumi, Martin Palmer, Vladimir Verchinine
and Nathalie Wahl for the attention they have paid to his work, their
comments, suggestions and helpful discussions. Additionally, he would
like to thank the anonymous referee for his reading of this paper.





§ THE CATEGORY 𝔘Β


The aim of this section is to describe the category 𝔘β
associated with braid groups that is central to this paper. On the
one hand, we recall some notions and properties about Quillen's construction
from a monoidal groupoid and pre-braided monoidal categories introduced
by Randal-Williams and Wahl in <cit.>. On the
other hand, we introduce examples of functors over the category 𝔘β.

We recall that the braid group on n≥2 strands denoted by 𝐁_n
is the group generated by σ_1, ..., σ_n-1 satisfying
the relations:


  * ∀ i∈{ 1,…,n-2}, σ_iσ_i+1σ_i=σ_i+1σ_iσ_i+1;

  * ∀ i,j∈{ 1,…,n-1} such that | i-j|≥2,
σ_iσ_j=σ_jσ_i.

𝐁_0 and 𝐁_1 both are the trivial group.
The family of braid groups is associated with the following groupoid.

The braid groupoid β
is the groupoid with objects the natural numbers n∈ℕ
and morphisms (for n,m∈ℕ):

    Hom_β(n,m)=𝐁_n   if 
    ∅   if .




The composition of morphisms ∘ in the groupoid β
corresponds to the group operation of the braid groups. So we will
abuse the notation throughout this work, identifying σ∘σ'=σσ'
for all elements σ and σ' of 𝐁_n with
n∈ℕ (with the convention that we read from the right
to the left for the group operation).





 §.§ Quillen's bracket construction associated with the groupoid β


This section focuses on the presentation and the study of Quillen's
bracket construction 𝔘β (see <cit.>)
on the braid groupoid β. It associates to β
a monoidal category whose unit is initial. The category 𝔘β
has further properties: Quillen's bracket construction on β
is a pre-braided monoidal category (see Section <ref>)
and β is its maximal subgroupoid. For an introduction
to (braided) strict monoidal categories, we refer to <cit.>.

A strict monoidal category will be denoted by (ℭ,♮,0),
where ℭ is the category, ♮ is the monoidal
product and 0 is the monoidal unit.





  §.§.§ Generalities


In <cit.>, Randal-Williams and Wahl study a construction
due to Quillen in <cit.>, for a monoidal category
S acting on a category X in the case S=X=𝔊 where
𝔊 is a groupoid. It is called Quillen's bracket construction.
Our study here is based on <cit.>
taking 𝔊=β.

<cit.> A
monoidal product ♮:β×β⟶β
is defined by the usual addition for the objects and laying two braids
side by side for the morphisms. The object 0 is the unit of this
monoidal product. The strict monoidal groupoid (β,♮,0)
is braided, its braiding is denoted by b_-,-^β.
Namely, the braiding is defined for all natural numbers n and m
such that n+m≥2 by:

    b_n,m^β=(σ_m∘⋯∘σ_2∘σ_1)∘⋯∘(σ_n+m-2∘⋯∘σ_n∘σ_n-1)∘(σ_n+m-1∘⋯∘σ_n+1∘σ_n)

where {σ_i} _i∈{ 1,…,n+m-1}
denote the Artin generators of the braid group 𝐁_n+m.


We consider the strict monoidal groupoid (β,♮,0)
throughout this section.

<cit.> Quillen's
bracket construction on the groupoid β, denoted
by 𝔘β, is the category defined by:



  * Objects: Obj(𝔘β)=Obj(β)=ℕ;

  * Morphisms: for n and n' two objects of β,
the morphisms from n to n' in the category 𝔘β
are given by:

    Hom_𝔘β(n,n')=βcolim[Hom_β(-♮ n,n')].

In other words, a morphism from n to n' in the category 𝔘β,
denoted by [n'-n,f]:n→ n', is an equivalence
class of pairs (n'-n,f) where n'-n is an object of
β, f:(n'-n)♮ n→ n'
is a morphism of β, in other words an element
of 𝐁_n'. The equivalence relation ∼ is defined
by (n'-n,f)∼(n'-n,f') if and only if there
exists an automorphism g∈ Aut_β(n'-n)
such that the following diagram commutes.

    (n'-n)♮ n[d]_g♮ id_n[r]^                f    n'
    (n'-n)♮ n[ur]_f'


  * For all objects n of 𝔘β, the identity
morphism in the category 𝔘β is given
by [0,id_n]:n→ n.

  * Let [n'-n,f]:n→ n' and [n”-n',g]:n'→ n”
be two morphisms in the category 𝔘β.
Then, the composition in the category 𝔘β
is defined by:

    [n”-n',g]∘[n'-n,f]=[n”-n,g∘(id_n'-n♮ f)].




The relationship between the automorphisms of the groupoid β
and those of its associated Quillen's construction 𝔘β
is actually clear. First, let us recall the following notion.

Let (𝔊,♮,0)
be a strict monoidal category. It has no zero divisors if for all
objects A and B of 𝔊, A♮ B≅0 if
and only if A≅ B≅0.


The braid groupoid (β,♮,0) has
no zero divisors. Moreover, by Definition <ref>,
Aut_β(0)={ id_0}. Hence, we
deduce the following property from <cit.>.

The groupoid β is the maximal subgroupoid of 𝔘β.


In addition, 𝔘β has the additional
useful property.

<cit.> The unit 0 of
the monoidal structure of the groupoid (β,♮,0)
is an initial object in the category 𝔘β.



Let n be a natural number and ϕ∈ Aut_β(n).
Then, as an element of Hom_𝔘β(n,n),
we will abuse the notation ϕ=[0,ϕ]. This comes
from the canonical functor:

    β   →   𝔘β
    ϕ∈ Aut_β(n)    ↦   [0,ϕ].



Finally, we are interested in a way to extend an object of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡)
to an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).
This amounts to studying the image of the restriction 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)→𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡).

Let M be an object
of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡).
Assume that for all n,n',n”∈ℕ such that n”≥ n'≥ n,
there exists an assignment M([n'-n,id_n']):M(n)→ M(n')
such that:

    M([n”-n',id_n”])∘ M([n'-n,id_n'])=M([n”-n,id_n”])

Then, we define a functor M:𝔘β→𝕂-𝔐𝔬𝔡
(assigning M([n'-n,σ])=M(σ)∘ M([n'-n,id_n'])
for all [n'-n,σ]∈ Hom_𝔘β(n,n'))
if and only if for all n,n'∈ℕ such that n'≥ n:

    M([n'-n,id_n'])∘ M(σ)=M(ψ♮σ)∘ M([n'-n,id_n'])

for all σ∈𝐁_n and all ψ∈𝐁_n'-n.



Note that for n'=n, M([n'-n,id_n'])=Id_M(n).


Let us assume that relation (<ref>)
is satisfied. We have to show that the assignment on morphisms is
well-defined with respect to 𝔘β. First,
let us prove that our assignment conforms with the defining equivalence
relation of 𝔘β (see Definition <ref>).
For n and n' natural numbers such that n'≥ n, let us consider
[n'-n,σ] and [n'-n,σ'] in Hom_𝔘β(n,n')
such that there exists ψ∈𝐁_n'-n so that σ'∘(ψ♮ id_n)=σ.
Since M is a functor over β, M([n'-n,σ])=M(σ')∘(M(ψ♮ id_n)∘ M([n'-n,id_n'])).
According to the relation (<ref>) and since M satisfies
the identity axiom, we deduce that M([n'-n,σ])=M(σ')∘ M(ψ♮ id_n)∘ M([n'-n,id_n'])=M([n'-n,σ']).

Now, we have to check the composition axiom. Let n, n' and n”
be natural numbers such that n”≥ n'≥ n, let ([n'-n,σ])
and ([n”-n',σ']) be morphisms respectively
in Hom_𝔘β(n,n') and in
Hom_𝔘β(n',n”). By our
assignment and composition in 𝔘β (see
Definition <ref>) we have that:

    M([n”-n',σ'])∘ M([n'-n,σ])=M(σ')∘(M([n”-n',id_n”])∘ M(σ))∘ M([n'-n,id_n']).

According to the relation (<ref>), we deduce that:

    M([n”-n',σ'])∘ M([n'-n,σ])     =     M(σ')∘(M([n”-n',id_n”])∘ M(σ))∘ M([n'-n,id_n']).
        =     M(σ')∘(M(id_n”-n'♮σ)∘ M([n”-n',id_n”]))∘ M([n'-n,id_n']).

Hence, it follows from relation (<ref>) that:

    M([n”-n',σ'])∘ M([n'-n,σ])=M(σ'∘(id_n”-n'♮σ))∘ M([n”-n,id_n])=M([n”-n',σ']∘[n'-n,σ]).


Conversely, assume that the functor M:𝔘β→𝕂-𝔐𝔬𝔡
is well-defined. In particular, composition axiom in 𝔘β
is satisfied and implies that for all n,n'∈ℕ such that
n'≥ n, for all σ∈𝐁_n:

    M([n'-n,id_n'])∘ M(σ)=M([n'-n,id_n'-n♮σ]).

It follows from the defining equivalence relation of 𝔘β
(see Definition (<ref>)) that for all ψ∈𝐁_n'-n:

    M([n'-n,id_n'])∘ M(σ)=M([n'-n,ψ♮σ]).

We deduce from the composition axiom that relation (<ref>)
is satisfied.


Let M and M' be objects
of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
and η:M→ M' a natural transformation in the category
𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡).
Then, η is a natural transformation in the category 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
if and only if for all n,n'∈ℕ such that n'≥ n:

    η_n'∘ M([n'-n,id_n'])=M'([n'-n,id_n'])∘η_n.




The natural transformation η extends to the category 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
if and only if for all n,n'∈ℕ such that n'≥ n,
for all [n'-n,σ]∈ Hom_𝔘β(n,n'):

    M'([n'-n,σ])∘η_n=η_n'∘ M([n'-n,σ]).

Since η is a natural transformation in the category 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡),
we already have η_n'∘ M(σ)=M'(σ)∘η_n'.
Hence, this implies that the necessary and sufficient relation to
satisfy is relation (<ref>).




  §.§.§ Pre-braided monoidal category


We present the notion of a pre-braided category, introduced by Randal-Williams
and Wahl in <cit.>. This is a generalization
of that of a braided monoidal category.

<cit.>
Let (ℭ,♮,0) be a strict monoidal category
such that the unit 0 is initial. We say that the monoidal category
(ℭ,♮,0) is pre-braided if:



  * The maximal subgroupoid 𝒢𝔯(ℭ,♮,0)
is a braided monoidal category, where the monoidal structure is induced
by that of (ℭ,♮,0).

  * For all objects A and B of ℭ, the braiding associated
with the maximal subgroupoid b_A,B^ℭ:A♮ B⟶ B♮ A
satisfies:

    b_A,B^ℭ∘(id_A♮ι_B)=ι_B♮ id_A:A⟶ B♮ A.

Recall that the notation ι_B:0→ B was introduced
in Notation <ref>.

Since the groupoid (β,♮,0) is
braided monoidal and it has no zero divisors, we deduce from <cit.>
the following properties.




 The category 𝔘β
is pre-braided monoidal. The monoidal structure (𝔘β,♮,0)
is defined on objects as that of (β,♮,0)
and defined on morphisms letting for [n'-n,f]∈ Hom_𝔘β(n,n')
and [m'-m,g]∈ Hom_𝔘β(m,m'):

    [m'-m,g]♮[n'-n,f]=[(m'-m)♮(n'-n),(g♮ f)∘(id_m'-m♮(b_m,n'-n^β)^-1♮ id_n)].

In particular, the canonical functor β→𝔘β
is monoidal.



The category (𝔘β,♮,0)
is pre-braided monoidal, but not braided. Indeed, as Figure 1 shows,
the pre-braiding defined on 𝔘β is not
a braiding: Figure 1 shows that b_1,2^β∘(ι_1♮ id_2)≠ id_2♮ι_1
whereas these two morphisms should be equal if b_-,-^β
were a braiding.





 §.§ Examples of functors associated with braid representations


Different families of representations of braid groups can be used
to form functors over the pre-braided category 𝔘β
to the category 𝕂-𝔐𝔬𝔡. Namely, considering
{ M_n:𝐁_n→𝕂-𝔐𝔬𝔡} _n∈ℕ
representations of braid groups, or equivalently an object M of
𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡),
we are interested in the situations where Proposition <ref>
applies so as to define an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).



  
Tong-Yang-Ma results

In 1996, in the article <cit.>, Tong, Yang and Ma investigated
the representations of 𝐁_n where the i-th generator
is sent to a matrix of the form Id_i-1⊕ T⊕ Id_n-i-1,
with T a m× m non-singular matrix and m≥2. In particular,
for m=2, they prove that there exist up to equivalence only two
non trivial representations of this type. We give here their result
and an interpretation of their work from a functorial point of view,
considering the representations over the ring of Laurent polynomials
in one variable ℂ[t^±1].

Let 𝔤𝔯 denote the full subcategory
of 𝔊𝔯 of finitely generated free groups. The free product
*:𝔤𝔯×𝔤𝔯→𝔤𝔯 defines
a monoidal structure over 𝔤𝔯, with 0 the unit, denoted
by (𝔤𝔯,*,0).

Let (ℕ,≤) denote the category of natural
numbers (natural means non-negative) considered as a poset. For all
natural numbers n, we denote by γ_n the unique element
of Hom_(ℕ,≤)(n,n+1). For all
natural numbers n and n' such that n'≥ n, we denote by
γ_n,n':n→ n' the unique element of Hom_(ℕ,≤)(n,n'),
composition of the morphisms γ_n'-1∘γ_n'-2∘⋯∘γ_n+1∘γ_n.
The addition defines a strict monoidal structure on (ℕ,≤),
denoted by ((ℕ,≤),+,0).



Let 𝐁_-:(ℕ,≤)→𝔊𝔯
and GL_-:(ℕ,≤)→𝔊𝔯
be the functors defined by:



  * Objects: for all natural numbers n, 𝐁_-(n)=𝐁_n
the braid group on n strands and GL_-(n)=GL_n(ℂ[t^±1])
the general linear group of degree n.

  * Morphisms: let n be a natural number. We define 𝐁_-(γ_n)=id_1♮-:𝐁_n↪𝐁_n+1
(where ♮ is the monoidal product introduced in Example <ref>).
We define GL_-(γ_n):GL_n(ℂ[t^±1])↪ GL_n+1(ℂ[t^±1])
assigning GL_-(γ_n)(φ)=id_1⊕φ
for all φ∈ GL_n(ℂ[t^±1]). 




For all natural numbers n≥2, for all i∈{ 1,…,n-1},
we denote by incl_i^n:𝐁_2≅ℤ↪𝐁_n
the inclusion morphism induced by:

    incl_i^n(σ_1)=σ_i.




<cit.> Let η:𝐁_-⟶ GL_-
be a natural transformation. Assume that for all natural numbers n≥2,
for all i∈{ 1,…,n-1}, the following diagram
is commutative:

    𝐁_n@->[rr]^η_n       GL_n(ℂ[t^±1])
    𝐁_2@->[rr]_η_2@->[u]^incl_i^n       GL_2(ℂ[t^±1]).@->[u]_id_i-1⊕-⊕ id_n-i-1

Here, two such natural transformations η and η' are said
to be equivalent if there exists a natural equivalence μ:GL_-⟶ GL_-
such that μ∘η=η' or if η'=η^* where -^*
denotes the dual representation. Then, η is equivalent to one
of the following natural transformations.



  * The trivial natural transformation, denoted by 𝔦𝔡: for
every generator σ_i of 𝐁_n, 𝔦𝔡_n(σ_i)=Id_GL_n(ℂ[t^±1]).

  * The unreduced Burau natural transformation, denoted by 𝔟𝔲𝔯:
for all generators σ_i of 𝐁_n,

    𝔟𝔲𝔯_n,t(σ_i)=Id_i-1⊕ B(t)⊕ Id_n-i-1,

with B(t)=[[   0   t;   1 1-t ]].

  * The natural transformation denoted by 𝔱𝔶𝔪: for every
generator σ_i of 𝐁_n if n≥2,

    𝔱𝔶𝔪_n,t(σ_i)=Id_i-1⊕ TYM(t)⊕ Id_n-i-1,

with TYM(t)=[[ 0 t; 1 0 ]]. We call it the Tong-Yang-Ma representation.



The unreduced Burau representation (see <cit.>
or <cit.> for more details about
this family of representations) is reducible but indecomposable, whereas
the Tong-Yang-Ma representation is irreducible (see <cit.>).
We can also consider a natural transformation using the family of
reduced Burau representations (see <cit.>
for more details about the associated family of representations):
these are irreducible subrepresentations of the unreduced Burau representations.

Let :(ℕ,≤)→𝔊𝔯
be the functor defined by:



  * Objects: for all natural numbers n, (n)=GL_n-1(ℂ[t^±1])
the general linear group of degree n-1.

  * Morphisms: let n be a natural number. We define (γ_n):GL_n-1(ℂ[t^±1])↪ GL_n(ℂ[t^±1])
assigning GL_-(γ_n)(φ)=id_1⊕φ
for all φ∈ GL_n-1(ℂ[t^±1]).




The reduced Burau natural transformation, denoted
by 𝔟𝔲𝔯:𝐁_-→
is defined by:



  * For n=2, one assigns 𝔟𝔲𝔯(σ_1)
to be the automorphism of ℂ[t^±1] defined
by the multiplication by -t.

  * For all natural numbers n≥3, we define for every Artin generator
σ_i of 𝐁_n with i∈{ 2,…,n-2}:

    𝔟𝔲𝔯_n,t(σ_i)=Id_i-2⊕B(t)⊕ Id_n-i-2

with

    B(t)=[[  1  0  0;  1 -t  t;  0  0  1 ]]

and

    𝔟𝔲𝔯_n,t(σ_1)=[[ -t  t;  0  1 ]]⊕ Id_n-3          ;        𝔟𝔲𝔯_n,t(σ_n-1)=Id_n-3⊕[[  1  0;  1 -t ]].




Let us use these natural transformations to form functors over the
category 𝔘β. Indeed, a natural transformation
η:𝐁_-→ GL_- (or )
provides in particular:



  * a functor 𝔑:β⟶ℂ[t^±1]-𝔐𝔬𝔡;

  * morphisms 𝔑([n'-n,id_n']):𝔑(n)→𝔑(n')
for all natural numbers n'≥ n, such that the relation (<ref>)
of Proposition <ref> is satisfied.

Therefore, according to Proposition <ref>,
it suffices to show that the relation (<ref>) is satisfied
to prove that 𝔑 is an object of 𝐅𝐜𝐭(𝔘β,ℂ[t^±1]-𝔐𝔬𝔡).

Recall that 0 is a null object in
the category of R-modules, and that the notation ι_G:0→ G
was introduced in Notation <ref>. Let n∈ℕ.
For all natural numbers n and n' such that n'≥ n, we define
ι_ℂ[t^±1]^⊕ n'-n⊕ id_ℂ[t^±1]^⊕ n:ℂ[t^±1]^⊕ n↪ℂ[t^±1]^⊕ n'
the embedding of ℂ[t^±1]^⊕ n as
the submodule of ℂ[t^±1]^⊕ n' given
by the n last copies of ℂ[t^±1].





  
Tong-Yang-Ma functor:

This example is based on the family introduced by Tong, Yang and Ma
(see Theorem <ref>). Let 𝔗𝔜𝔐_t:β→ℂ[t^±1]-𝔐𝔬𝔡
be the functor defined on objects by 𝔗𝔜𝔐_t(n)=ℂ[t^±1]^⊕ n
for all natural numbers n, and for all numbers n≥2, for every
Artin generator σ_i of 𝐁_n, by 𝔗𝔜𝔐_t(σ_i)=𝔱𝔶𝔪_n,t(σ_i)
for morphisms. For all natural numbers n and n' such that n'≥ n,
we assign 𝔗𝔜𝔐_t([n'-n,id_n']):ℂ[t^±1]^⊕ n↪ℂ[t^±1]^⊕ n'
to be the embedding ι_ℂ[t^±1]^⊕ n'-n⊕ id_ℂ[t^±1]^⊕ n
(where these morphisms are introduced in Notation <ref>).
For all natural numbers n”≥ n'≥ n, for all Artin generators
σ_i∈𝐁_n and all ψ_j∈𝐁_n'-n,
our assignments give:

    𝔗𝔜𝔐_t(ψ♮σ)∘𝔗𝔜𝔐_t([n'-n,id_n'])     =    (Id_j-1⊕ TYM(t)⊕ Id_(n'-n)-j-1⊕ Id_n'-n+i-1⊕ TYM(t)⊕ Id_n'-i-1)
          ∘(ι_ℂ[t^±1]^⊕ n'-n⊕ id_ℂ[t^±1]^⊕ n).

Remark that (Id_j-1⊕ TYM(t)⊕ Id_(n'-n)-j-1)∘ι_ℂ[t^±1]^⊕(n'-n)=ι_ℂ[t^±1]^⊕(n'-n).
Hence we deduce that

    𝔗𝔜𝔐_t(ψ♮σ)∘𝔗𝔜𝔐_t([n'-n,id_n'])=𝔗𝔜𝔐_t([n'-n,id_n'])∘𝔗𝔜𝔐_t(σ)

for all σ∈𝐁_n and all ψ∈𝐁_n'-n.
According to Proposition <ref>,
our assignment defines a functor 𝔗𝔜𝔐_t:𝔘β→ℂ[t^±1]-𝔐𝔬𝔡,
called the Tong-Yang-Ma functor.



  
Burau functors:

Other examples naturally arise from the Burau representations.

Let 𝔅𝔲𝔯_t:β⟶ℂ[t^±1]-𝔐𝔬𝔡
be the functor defined on objects by 𝔅𝔲𝔯_t(n)=ℂ[t^±1]^⊕ n
for all natural numbers n, and for all numbers n≥2, for every
Artin generator σ_i of 𝐁_n, by 𝔅𝔲𝔯_t(σ_i)=𝔟𝔲𝔯_n,t(σ_i)
for morphisms. For all natural numbers n and n' such that n'≥ n,
we assign 𝔅𝔲𝔯_t([n'-n,id_n']):ℂ[t^±1]^⊕ n↪ℂ[t^±1]^⊕ n'
to be the embedding ι_ℂ[t^±1]^⊕ n'-n⊕ id_ℂ[t^±1]^⊕ n
(where these morphisms are introduced in Notation <ref>). 

As for the functor 𝔗𝔜𝔐, the assignment for 𝔅𝔲𝔯
implies that for all natural numbers n”≥ n'≥ n, for all
σ∈𝐁_n and all ψ∈𝐁_n'-n, 𝔅𝔲𝔯_t([n'-n,id_n'])∘𝔅𝔲𝔯_t(σ)=𝔅𝔲𝔯_t(ψ♮σ)∘𝔅𝔲𝔯_t([n'-n,id_n']).
According to Proposition <ref>,
our assignment defines a functor 𝔅𝔲𝔯_t:𝔘β⟶ℂ[t^±1]-𝔐𝔬𝔡,
called the unreduced Burau functor. The dual version of the functor
𝔅𝔲𝔯_t was already considered by Randal-Williams and
Wahl in <cit.>.

Analogously, we can form a functor from the reduced Burau representations.
Let 𝔅𝔲𝔯_t:β⟶ℂ[t^±1]-𝔐𝔬𝔡
be the functor defined on objects by 𝔅𝔲𝔯_t(0)=0
and 𝔅𝔲𝔯_t(n)=ℂ[t^±1]^⊕ n-1
for all nonzero natural numbers n, and by 𝔅𝔲𝔯_t(σ_i)=𝔟𝔲𝔯_n,t(σ_i)
for morphisms for every Artin generator σ_i of 𝐁_n
for all numbers n≥2.

For all natural numbers n and n' such that n'≥ n, we assign
𝔅𝔲𝔯_t([n'-n,id_n']):ℂ[t^±1]^⊕ n-1↪ℂ[t^±1]^⊕ n'-1
to be the embedding ι_ℂ[t^±1]^⊕ n'-n⊕ id_ℂ[t^±1]^⊕ n-1
(where these morphisms are introduced in Notation <ref>).
Repeating mutadis mutandis the work done for the functor 𝔗𝔜𝔐,
the assignment for 𝔅𝔲𝔯_t implies that for
all natural numbers n”≥ n'≥ n, for all σ∈𝐁_n
and all ψ∈𝐁_n'-n, 𝔅𝔲𝔯_t([n'-n,id_n'])∘𝔅𝔲𝔯_t(σ)=𝔅𝔲𝔯_t(ψ♮σ)∘𝔅𝔲𝔯_t([n'-n,id_n']).
According to Proposition <ref>,
our assignment defines a functor 𝔅𝔲𝔯_t:𝔘β⟶ℂ[t^±1]-𝔐𝔬𝔡,
called the reduced Burau functor.



  
Lawrence-Krammer functor:

The family of Lawrence-Krammer representations was notably used to
prove that braid groups are linear (see <cit.>).
For this paragraph, we assign 𝕂=ℂ[t^±1][q^±1]
the ring of Laurent polynomials in two variables and consider the
functor GL_- of Definition <ref> with this
assignment. Let 𝔏𝔎:𝔘β→ℂ[t^±1][q^±1]-𝔐𝔬𝔡
be the assignment:



  * Objects: for all natural numbers n≥2, 𝔏𝔎(n)=1≤ j<k≤ n⊕V_j,k,
with for all 1≤ j<k≤ n, V_j,k is a free ℂ[t^±1][q^±1]-module
of rank one. Hence, 𝔏𝔎(n)≅(ℂ[t^±1][q^±1])^⊕ n(n-1)/2
as ℂ[t^±1][q^±1]-modules.
Moreover, one assigns 𝔏𝔎(1)=0 and 𝔏𝔎(0)=0.

  * Morphisms:



  * Automorphisms: for all natural numbers n, for every Artin generator
σ_i of 𝐁_n (with i∈{ 1,…,n-1}),
for all v_j,k∈ V_j,k (with 1≤ j<k≤ n),

    𝔏𝔎(σ_i)v_j,k=
    v_j,k   if ,
    
    tv_i,k+(t^2-t)v_i,i+1+(1-t)v_i+1,k   if ,
    
    v_i+1,k   if ,
    
    tv_j,i+(1-t)v_j,i+1-(t^2-t)qv_i,i+1   if ,
    
    v_j,i+1   if ,
    
    -qt^2v_i,i+1   if .


  * General morphisms: let n,n'∈ℕ, such that n'≥ n.
For all natural numbers j and k such that 1≤ j<k≤ n,
we define the embedding 𝔙_j,k^n,n':V_j,k∼⟶V_j+(n'-n),k+(n'-n)↪1≤ j<k≤ n'⊕V_j,k
of free ℂ[t^±1][q^±1]-modules.
Then we define 𝔏𝔎([n'-n,id_n']):1≤ j<k≤ n⊕V_j,k→1≤ j<k≤ n'⊕V_j,k
to be the embedding 1≤ j<k≤ n⊕𝔙_j,k^n,n'.


Since we consider a family of representations of 𝐁_n
(see <cit.>), the assignment 𝔏𝔎 defines an
object of 𝐅𝐜𝐭(β,ℂ[t^±1]-𝔐𝔬𝔡).

Let n, n' and n” be natural numbers such that n”≥ n'≥ n.
It follows directly from our definitions of 𝔏𝔎([n'-n,id_n']),
𝔏𝔎([n”-n',id_n”]) and 𝔏𝔎([n”-n,id_n”])
that relation (<ref>) of Proposition <ref>
is satisfied.

According to the definition of 𝔏𝔎(σ_l)
with σ_l an Artin generator of 𝐁_n'-n, for
all v_j,k∈ V_j,k with 1+(n'-n)≤ j<k≤ n',
𝔏𝔎(σ_l)v_j,k=v_j,k. Hence for
all ψ∈𝐁_n'-n:

    𝔏𝔎(ψ♮ id_n)∘𝔏𝔎([n'-n,id_n'])=𝔏𝔎([n'-n,id_n']).

Note also that for all l∈{ 1,…,n-1}, for all
v_j,k∈ V_j,k with 1+(n'-n)≤ j<k≤ n',
it follows from the assignment of 𝔏𝔎 that:

    𝔏𝔎(id_n'-n♮σ_l)(v_(n'-n)+j,(n'-n)+k)=𝔏𝔎(σ_n'-n+l)(v_(n'-n)+j,(n'-n)+k)=𝔏𝔎([n'-n,id_n'])(𝔏𝔎(σ_l)(v_j,k)).

Therefore, this implies that for all σ∈𝐁_n, 𝔏𝔎([n'-n,id_n'])∘𝔏𝔎(σ)=𝔏𝔎(id_n'-n♮σ)∘𝔏𝔎([n'-n,id_n']).
Hence, 𝔏𝔎 satisfies the relation (<ref>)
of Proposition <ref>. Hence,
the assignment defines a functor 𝔏𝔎:𝔘β→ℂ[t^±1][q^±1]-𝔐𝔬𝔡,
called the Lawrence-Krammer functor.



§ FUNCTORIALITY OF THE LONG-MOODY CONSTRUCTION


The principle of the Long-Moody construction, corresponding to Theorem
2.1 of <cit.>, is to build a linear representation of the
braid group 𝐁_n starting from a representation 𝐁_n+1.
We develop a functorial version of this construction, which leads
to the notion of Long-Moody functors (see Section <ref>).
Beforehand, we need to introduce various tools, which are consequences
of the relationships between braid groups and free groups (see Section
<ref>). Finally, in Section <ref>,
we investigate examples of functors which are recovered by Long-Moody
functors.



 §.§ Braid groups and free groups


This section recalls some relationships between braid groups and free
groups. We also develop tools which will be used throughout our work
of Sections <ref> and <ref>.

We consider the free group on n generators, which we denote by
𝐅_n=⟨ g_1,…,g_n⟩.

We denote by e_𝐅_n the unit element of the free group
on n generators 𝐅_n, for all natural numbers n.


Recall that the category of finitely generated free groups is monoidal
using free product of groups (see Notation <ref>). The
object 0 being null in the category 𝔤𝔯, recall that
ι_𝐅_n:0→𝐅_n denotes the unique
morphism from 0 to 𝐅_n as in Notation <ref>.

Let n be a natural number. We consider ι_𝐅_1*id_𝐅_n:𝐅_n↪𝐅_n+1.
This corresponds to the identification of 𝐅_n as the
subgroup of 𝐅_n+1 generated by the n last copies
of 𝐅_1 in 𝐅_n+1. Iterating this morphism,
we obtain for all natural numbers n'≥ n the morphism ι_𝐅_n'-n*id_𝐅_n:𝐅_n↪𝐅_n'.


Let {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ
be a family of group morphisms from the free group 𝐅_n
to the braid group 𝐁_n+1, for all natural numbers n.
We require these morphisms to satisfy the following crucial property.

For all elements g∈𝐅_n,
for all natural numbers n'≥ n, the following diagram is commutative
in the category 𝔘β:


    1♮ n@->[rrr]^ς_n(g)@->[d]_id_1♮[n'-n,id_n']          1♮ n@->[d]^id_1♮[n'-n,id_n']
    
    1♮ n'@->[rrr]_ς_n'(e_𝐅_n'-n*g)          1♮ n'.




Condition <ref> will be used to prove that
the Long-Moody functor is well defined on morphisms with respect to
the tensor product structure in Theorem <ref>. Moreover,
it will also be used in the proof of Propositions <ref>
and <ref>.



 Condition <ref>
is equivalent to assume that for all natural numbers n, for all
elements g∈𝐅_n, the morphisms {ς_n} _n∈ℕ
satisfy the following equality in 𝐁_n+2:

    ((b_1,1^β)^-1♮ id_n)∘(id_1♮ς_n(g))=ς_n+1(e_𝐅_1*g)∘((b_1,1^β)^-1♮ id_n).




Let n and n' be natural numbers such that n'≥ n. The equality
(<ref>) implies that for all 1≤ k≤ n'-n, the following
diagram in the category β is commutative :

    1♮ n'@->[rrrr]^id_n'-(n+k)♮ς_n+k-1(e_𝐅_k-1*g)@->[d]_id_n'-(n+k)♮(b_1,1^β)^-1♮ id_(k-1)+n             1♮ n'@->[d]^id_n'-(n+k)♮(b_1,1^β)^-1♮ id_(k-1)+n
    
    1♮ n'@->[rrrr]_id_n'-(n+k)♮ς_n+k(e_𝐅_k*g)             1♮ n'.

Hence composing squares, we obtain that the following diagram is commutative
in the category β:


    1♮⋯♮(1♮1)♮ n@->[rrr]^id_n'-n-1♮(b_1,1^β)^-1♮ id_n@->[d]_id_n'♮ς_n(g)          1♮⋯♮1♮(1♮ n)@->[d]_id_n'-1♮ς_n+1(e_𝐅_1*g)@->[rrr]^               id_n'-n-2♮(b_1,1^β)^-1♮ id_1+n         ⋯@->[rr]^(b_1,1^β)^-1♮ id_n'-1       1♮ n'@->[d]^ς_n'(e_𝐅_1*g)
    
    1♮⋯♮1♮ n@->[rrr]_id_n'-n-1♮(b_1,1^β)^-1♮ id_n          1♮⋯♮1♮(1♮ n)@->[rrr]_               id_n'-n-2♮(b_1,1^β)^-1♮ id_1+n         ⋯@->[rr]_(b_1,1^β)^-1♮ id_n'-1       1♮ n'.

By definition of the braiding (see Definition <ref>),
we deduce that the composition of horizontal arrows is the morphism
(b_1,n'-n^β)^-1♮ id_n
in β. Recall from Proposition <ref>
that id_1♮[n'-n,σ]=[n'-n,(id_1♮σ)∘((b_1,n'-n^β)^-1♮ id_n)].
Hence Condition <ref> is satisfied if we
assume that the equality (<ref>) is satisfied for all natural
numbers n.

Conversely, assume that Condition <ref> is
satisfied. Condition <ref> with n'=n+1
ensures that:

    [1,((b_1,1^β)^-1♮ id_n)∘(id_1♮ς_n(g))]=[1,ς_n'(e_𝐅_1*g)∘((b_1,1^β)^-1♮ id_n)].

Since Aut_𝔘β(1)=𝐁_1
is the trivial group, we deduce from the defining equivalence relation
of 𝔘β (see Definition <ref>)
the equality in 𝐁_n+2:

    ((b_1,1^β)^-1♮ id_n)∘(id_1♮ς_n(g))=ς_1+n(e_𝐅_1*g)∘((b_1,1^β)^-1♮ id_n).



It follows from Lemma <ref> that, for i≥2,
ς_n(g_i) is determined by ς_k(g_1) for
k≤ n by the equalities (<ref>).



The family ς_n,1, based on what
is called the pure braid local system in the literature (see <cit.>),
is defined by the following inductive assignment for all natural numbers
n≥1.

    ς_n,1:𝐅_n   ⟶   𝐁_n+1
    
    g_i   ⟼   σ_1^2   if 
    σ_1^-1∘σ_2^-1∘⋯∘σ_i-1^-1∘σ_i^2∘σ_i-1∘⋯∘σ_2∘σ_1   if

We assign ς_0,1 to be the trivial morphism.



The family of morphisms {ς_n,1} _n∈ℕ
satisfies Condition <ref>.



Relation (<ref>) is trivially satisfied for n=0. Let
n≥1 be a fixed natural number. By definition <ref>,
we have (b_1,1^β)^-1=σ_1^-1.
Moreover, for all i∈{ 2,…,n}, we have  ς_n+1(e_𝐅_1*g_i-1)=ς_n+1(g_i)
and

    id_1♮ς_n,1(g_i-1)=σ_2^-1∘⋯∘σ_i-1^-1∘σ_i^2∘σ_i-1∘⋯∘σ_2.

We deduce that:

    ((b_1,1^β)^-1♮ id_n)∘(id_1♮ς_n,1(g_i-1))∘(b_1,1^β♮ id_n)=ς_n,1(g_i).

Hence Relation (<ref>) of Lemma <ref>
is satisfied for all natural numbers.



Let us consider the trivial morphisms ς_n,*:𝐅_n→0_𝔊𝔯→𝐁_n+1
for all natural numbers n. The relation of Lemma <ref>
being easily checked, this family of morphisms {ς_n,*:𝐅_n→𝐁_n+1} _n∈ℕ
satisfies Condition <ref>.





  
Action of braid groups on automorphism groups of free groups:

There are several ways to consider the group 𝐁_n
as a subgroup of Aut(𝐅_n). For instance,
the geometric point of view of topology gives us an action of 𝐁_n
on the free group 𝐅_n (see for example <cit.>
or <cit.>) identifying 𝐁_n as the mapping
class group of a n-punctured disc Σ_0,1^n: fixing a
point y on the boundary of the disc Σ_0,1^n, each free
generator g_i can be taken as a loop of the disc based y turning
around punctures. Each element σ of 𝐁_n, as
an automorphism up to isotopy of the disc Σ_0,1^n, induces
a well-defined action on the fundamental group π_1(Σ_0,1^n)≅𝐅_n
called Artin representation (see Example <ref> for more details).


In the sequel, we fix a family of group actions of 𝐁_n
on the free group 𝐅_n: let { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
be a family of group morphisms from the braid group 𝐁_n
to the automorphism group Aut(𝐅_n). For the
work of Sections <ref> and <ref>,
we need the morphisms a_n:𝐁_n→ Aut(𝐅_n)
to satisfy more properties.

Let n and n' be natural
numbers such that n'≥ n. We require (ι_𝐅_n'-n*id_𝐅_n)∘(a_n(σ))=(a_n'(σ'♮σ))∘(ι_𝐅_n'-n*id_𝐅_n)
as morphisms 𝐅_n→𝐅_n' for all elements
σ of 𝐁_n and σ' of 𝐁_n'-n,
ie the following diagrams are commutative:


    𝐅_n@->[rr]^a_n(σ)@->[d]_ι_𝐅_n'-n*id_𝐅_n      𝐅_n@->[d]^ι_𝐅_n'-n*id_𝐅_n      𝐅_n@->[rr]^ι_𝐅_n'-n*id_𝐅_n@->[dr]_ι_𝐅_n'-n*id_𝐅_n      𝐅_n'
    𝐅_n'@->[rr]_a_n'(id_n'-n♮σ)      𝐅_n'         𝐅_n'.@->[ur]_a_n'(σ'♮ id_n)




Condition <ref> will be used to define
the Long-Moody functor on morphisms in Theorem <ref>.
Moreover, it will also be used for the proof of Propositions <ref>
and <ref>.


We will also require the families of morphisms {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ
and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
to satisfy the following compatibility relations.

Let n be a natural number.
We assume that the morphism given by the coproduct ς_n*(id_1♮-):𝐅_n*𝐁_n→𝐁_n+1
factors across the canonical surjection to 𝐅_na_n⋊𝐁_n.
In other words, the following diagram is commutative:

    𝐅_n@^(->[r]@->[dr]_ς_n   𝐅_na_n⋊𝐁_n@->[d]    𝐁_n@_(->[l]@->[dl]^id_1♮-
       𝐁_n+1.

where the morphism 𝐅_na_n⋊𝐁_n→𝐁_n+1
is induced by the morphism 𝐅_n*𝐁_n→𝐁_n+1
and the group morphism id_1♮-:𝐁_n→𝐁_n+1
is induced by the monoidal structure. This is equivalent to requiring
that, for all elements σ∈𝐁_n and g∈𝐅_n,
the following equality holds in 𝐁_n+1:

    (id_1♮σ)∘ς_n(g)=ς_n(a_n(σ)(g))∘(id_1♮σ).




Condition <ref> is essential in the
definition of the Long-Moody functor on objects in Theorem <ref>.


We fix a choice for these families of morphisms {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ
and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ.

The families {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ
and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
are said to be coherent if they satisfy conditions <ref>,
<ref> and <ref>.



A classical family is provided by the Artin representations
(see for example <cit.>). For n∈ℕ,
a_n,1:𝐁_n→ Aut(𝐅_n)
is defined for all elementary braids σ_i where i∈{ 1,…,n-1}
by:

    a_n,1(σ_i):𝐅_n   ⟶   𝐅_n
    
    g_j   ⟼   
    g_i+1   if 
    
    g_i+1^-1g_ig_i+1   if 
    
    g_j   if .

It clearly follows from their definitions that the morphisms { a_n,1:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
satisfy Condition <ref>.



The morphisms { a_n,1:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
together with the morphisms {ς_n,1:𝐅_n↪𝐁_n+1} _n∈ℕ
of Example <ref> satisfy Condition <ref>.



Let i be a fixed natural number in { 1,…,n-1}.
We prove that the equality (<ref>)
of Condition <ref> is satisfied for
all Artin generator σ_i and all generator g_j of the
free group (with j∈{ 1,…,n}). First, it follows
from the braid relation σ_iσ_i+1σ_i=σ_i+1σ_iσ_i+1
that:

    σ_1+i^-1∘σ_i^-1∘σ_1+i^-2∘σ_i^2∘σ_1+i^2∘σ_i∘σ_1+i=σ_i^-1∘σ_1+i^2∘σ_i,

and we deduce that:

    σ_1+i^-1∘ς_n,1(a_n,1(σ_i)(g_1+i))∘σ_1+i=ς_n,1(g_1+i).

Also, the braid relation σ_i+1∘σ_i∘σ_i+1=σ_i∘σ_i+1∘σ_i
implies that σ_i+1^-1∘σ_i^-1∘σ_i+1^2∘σ_i∘σ_i+1=σ_i^2
and a fortiori:

    σ_1+i^-1∘ς_n,1(a_n,1(σ_i)(g_i))∘σ_1+i=ς_n,1(g_i).

Finally, for a fixed ,
the commutation relation σ_iσ_j=σ_jσ_i
and the braid relation σ_iσ_i+1σ_i=σ_i+1σ_iσ_i+1
give directly:

    ς_n,1(g_j)=σ_1+i^-1∘ς_n,1(a_n,1(σ_i)(g_j))∘σ_1+i.



The families of morphisms { a_n,1:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
and {ς_n,1:𝐅_n→𝐁_n+1} _n∈ℕ
are coherent.



Consider the family of morphisms {ς_n,*:𝐅_n→𝐁_n+1} _n∈ℕ
of Example <ref> and any family of morphisms { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ.
Then Condition <ref> is always satisfied.
As a consequence, these families of morphisms {ς_n,*:𝐅_n→𝐁_n+1} _n∈ℕ
and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
are coherent if and only if the family of morphisms { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
satisfies Condition <ref>.





 §.§ The Long-Moody functors


In this section, we prove that the Long-Moody construction of <cit.>
induces a functor 

    𝐋𝐌:𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)→𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).

 We fix families of morphisms {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ
and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ,
which are assumed to be coherent (see Definition <ref>).

We first need to make some observations and introduce some tools.
Let F be an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
and n be a natural number. A fortiori, the 𝕂-module
F(n+1) is endowed with a left 𝕂[𝐁_n+1]-module
structure. Using the morphism ς_n:𝐅_n→𝐁_n+1,
F(n+1) is a 𝕂[𝐅_n]-module
by restriction.

Let us consider the augmentation ideal of the free group 𝐅_n,
denoted by ℐ_𝕂[𝐅_n].
Since it is a (right) 𝕂[𝐅_n]-module,
one can form the tensor product ℐ_𝕂[𝐅_n]𝕂[𝐅_n]F(n+1).
Also, for all natural numbers n and n' such that n'≥ n,
the morphism ι_𝐅_n'-n*id_𝐅_n:𝐅_n↪𝐅_n'
canonically induces a morphism ι_ℐ_𝕂[𝐅_n'-n]*id_ℐ_𝕂[𝐅_n]:ℐ_𝕂[𝐅_n]↪ℐ_𝕂[𝐅_n'].
In addition, the augmentation ideal ℐ_𝕂[𝐅_n]
is a 𝕂[𝐁_n]-module too:

The action a_n:𝐁_n→ Aut(𝐅_n)
canonically induces an action of 𝐁_n on ℐ_𝕂[𝐅_n]
denoted by a_n:𝐁_n→ Aut(ℐ_𝕂[𝐅_n])
(abusing the notation).



For any group morphism H→ Aut(G), the group
ring 𝕂[G] is canonically an H-module and
so is the augmentation ideal ℐ_G, as a submodule of
𝕂[G].


If the family of morphisms { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
is coherent with respect to the family of morphisms {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ,
the relation of Condition <ref> remains
true mutatis mutandis, for all natural numbers n and n', considering
the induced morphisms a_n:𝐁_n→ Aut(ℐ_𝕂[𝐅_n])
and ι_ℐ_𝕂[𝐅_n'-n]*id_ℐ_𝕂[𝐅_n]:ℐ_𝕂[𝐅_n]→ℐ_𝕂[𝐅_n'].


In the following theorem, we define an endofunctor of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
corresponding to the Long-Moody construction. It will be called the
Long-Moody functor with respect to {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ
and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ.

Recall that we have fixed coherent families
of morphisms {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ
and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ.
The following assignment defines a functor 𝐋𝐌_a,ς:𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)→𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).



  * Objects: for F∈ Obj(𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)),
𝐋𝐌_a,ς(F):𝔘β→𝕂-𝔐𝔬𝔡
is defined by:



  * Objects: ∀ n∈ℕ, 𝐋𝐌_a,ς(F)(n)=ℐ_𝕂[𝐅_n]𝕂[𝐅_n]F(n+1).

  * Morphisms: for n,n'∈ℕ, such that n'≥ n, and [n'-n,σ]∈ Hom_𝔘β(n,n'),
assign:

    𝐋𝐌_a,ς(F)([n'-n,σ])(i𝕂[𝐅_n]v)=a_n'(σ)(ι_ℐ_𝕂[𝐅_n'-n]*id_ℐ_𝕂[𝐅_n])(i)𝕂[𝐅_n']F(id_1♮[n'-n,σ])(v),

for all i∈ℐ_𝕂[𝐅_n]
and v∈ F(n+1).


  * Morphisms: let F and G be two objects of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡),
and η:F→ G be a natural transformation. We define
𝐋𝐌_a,ς(η):𝐋𝐌_a,ς(F)→𝐋𝐌_a,ς(G)
for all natural numbers n by:

    (𝐋𝐌_a,ς(η))_n=id_ℐ_𝕂[𝐅_n]𝕂[𝐅_n]η_n+1.


In particular, the Long-Moody functor 𝐋𝐌_a,ς
induces an endofunctor of the category 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡).



When there is no ambiguity, once the morphisms {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ
and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
are fixed, we omit them from the notation 𝐋𝐌_a,ς
for convenience (especially for proofs).



For this proof, n, n' and n” are natural numbers such that
n”≥ n'≥ n.


  * First let us show that the assignment of 𝐋𝐌 defines an
endofunctor of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡).
The two first points generalize the proof of <cit.>.
Let F, G and H be objects of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡).



  * We first check the compatibility of the
assignment 𝐋𝐌(F) with respect to the tensor
product. Consider σ∈𝐁_n g∈𝐅_n,
i∈ℐ_𝕂[𝐅_n] and v∈ F(n+1).
Since (id_1♮σ)∘ς_n(g)=ς_n(a_n(σ)(g))∘(id_1♮σ)
by Condition <ref>, we deduce that:

    𝐋𝐌(F)(σ)(i𝕂[𝐅_n]F(ς_n(g))(v))     =     a_n(σ)(i)𝕂[𝐅_n]F(id_1♮σ)(F(ς_n(g))(v))
        =     a_n(σ)(i)𝕂[𝐅_n](F(ς_n(a_n(σ)(g)))∘ F(id_1♮σ))(v)
        =     a_n(σ)(i· g)𝕂[𝐅_n]F(id_1♮σ)(v)
        =    𝐋𝐌(F)(σ)(i· g𝕂[𝐅_n](v)).


  * Let us prove that the assignment 𝐋𝐌(F)
defines an object of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡).
According to our assignment and since a_n and id_1♮-
are group morphisms, it follows from the definition that 𝐋𝐌(F)(id_𝐁_n)=id_𝐋𝐌(F)(n).
Hence, it remains to prove that the composition axiom is satisfied.
Let σ and σ' be two elements of 𝐁_n,
i∈ℐ_𝕂[𝐅_n] and v∈ F(n+1).
From the functoriality of F over β and the
compatibility of the monoidal structure ♮ with composition,
we deduce that F(id_1♮(σ'))∘ F(id_1♮(σ))=F(id_1♮(σ'∘σ)).
Since a_n is a group morphism, we have:

    (a_n(σ'∘σ))(i)=a_n(σ')(a_n(σ)(i)).

 Hence, it follows from the assignment of 𝐋𝐌 that:

    𝐋𝐌(F)(σ'∘σ)(i𝕂[𝐅_n]v)     =    (a_n(σ'∘σ))(i)𝕂[𝐅_n]F(id_1♮(σ'∘σ))(v)
        =     a_n(σ')(a_n(σ)(i))𝕂[𝐅_n](F(id_1♮(σ'))∘ F(id_1♮(σ)))(v)
        =    𝐋𝐌(F)(σ')∘𝐋𝐌(F)(σ)(i𝕂[𝐅_n]v).


  * It remains to check the consistency of our
definition of 𝐋𝐌 on morphisms of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡).
Let η:F→ G be a natural transformation. Hence, we
have that:

    G(id_1♮σ)∘η_n+1=η_n'+1∘ F(id_1♮σ).

Hence, it follows from the assignment of 𝐋𝐌 that:

    𝐋𝐌(G)(σ)∘𝐋𝐌(η)_n=𝐋𝐌(η)_n'∘𝐋𝐌(F)(σ)

Therefore 𝐋𝐌(η) is a morphism in the category
𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡).
Denoting by id_F:F→ F the identity natural transformation,
it is clear that 𝐋𝐌(id_F)=id_𝐋𝐌(F).
Finally, let us check the composition axiom. Let η:F→ G
and μ:G→ H be natural transformations. Let n be
a natural number, i∈ℐ_𝕂[𝐅_n]
and v∈ F(n). Now, since μ and η are morphisms
in the category 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡):

    𝐋𝐌(μ∘η)_n(i𝕂[𝐅_n]v)     =     i𝕂[𝐅_n](μ_n+1∘η_n+1)(v)=𝐋𝐌(μ)_n∘𝐋𝐌(η)_n(i𝕂[𝐅_n]v).



  * Let us prove that the assignment 𝐋𝐌 lifts to define an
endofunctor of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).
Let F, G and H be objects of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).



  * First, let us check the compatibility of the assignment 𝐋𝐌(F)
with respect to the tensor product. In fact, this compatibility being
already done for automorphisms (see <ref>), the
remaining point to prove is the compatibility of 𝐋𝐌(F)([n'-n,id_n']).
Let g∈𝐅_n, i∈ℐ_𝕂[𝐅_n]
and v∈ F(n+1). It follows from Condition <ref>
that in 𝐁_n+1:

    id_1♮[n'-n,id_n'-n♮ς_n(g)]=ς_n'(e_𝐅_n'-n*g)∘(id_1♮[n'-n,id_n']).

Since (ι_ℐ_𝕂[𝐅_n'-n]*id_ℐ_𝕂[𝐅_n])(i· g)=(e_ℐ_𝕂[𝐅_n'-n]*i)·(e_𝐅_n'-n*g),
we deduce that:

    𝐋𝐌(F)([n'-n,id_n'])(i𝕂[𝐅_n]F(ς_n(g))(v))
        =    (ι_ℐ_𝕂[𝐅_n'-n]*id_ℐ_𝕂[𝐅_n])(i)𝕂[𝐅_n']F(id_1♮[n'-n,id_n'])(F(ς_n(g))(v))
        =    (ι_ℐ_𝕂[𝐅_n'-n]*id_ℐ_𝕂[𝐅_n])(i· g)𝕂[𝐅_n']F(id_1♮[n'-n,id_n'])(v)
        =    𝐋𝐌(F)([n'-n,id_n'])(i· g𝕂[𝐅_n]v).


  * Let us prove that the assignment 𝐋𝐌(F) defines
an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
using Proposition <ref>. Recall
the compatibility of the monoidal structure ♮ with respect
to composition and that F is an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).
Consider [n'-n,σ]∈ Hom_𝔘β(n,n').
It follows from our assignment, that:

    𝐋𝐌(F)([n'-n,σ])=𝐋𝐌(F)(σ)∘𝐋𝐌(F)([n'-n,id_n']).

Moreover, the composition of morphisms introduced in Definition <ref>
implies that:

    𝐋𝐌(F)([n”-n,id_n”])     =    𝐋𝐌(F)([n”-n',id_n”])∘𝐋𝐌(F)([n'-n,id_n']).

Hence, the relation (<ref>) of Proposition <ref>
is satisfied. Let σ∈𝐁_n and ψ∈𝐁_n'-n.
Since (ι_n'-n*id_n)∘(a_n(σ))=(a_n'(ψ♮σ))∘(ι_n'-n*id_n)
by Condition <ref>, we deduce that:

    𝐋𝐌(F)(ψ♮σ)∘𝐋𝐌(F)([n'-n,id_n'])     =    𝐋𝐌(F)([n'-n,id_n'])∘𝐋𝐌(F)(σ).

Hence the relation (<ref>) of Proposition <ref>
is also satisfied. Therefore, according to Proposition <ref>,
since 𝐋𝐌(F) is an object of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡),
the assignment 𝐋𝐌(F) defines an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).

  * Finally, let us check the consistency of our assignment for 𝐋𝐌
on morphisms. Let η:F→ G be a natural transformation.
We already proved in <ref> that 𝐋𝐌(η)
is a morphism in the category 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡).
Since η is a natural transformation between objects of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡),
we have that:

    G(id_1♮[n'-n,id_n'])∘η_n+1=η_n'+1∘ F(id_1♮[n'-n,id_n']).

Hence, it follows from the assignment of 𝐋𝐌 that:

    𝐋𝐌(G)([n'-n,id_n'])∘𝐋𝐌(η)_n=𝐋𝐌(η)_n'∘𝐋𝐌(F)([n'-n,id_n']).

Hence the relation (<ref>) of Proposition <ref>
is satisfied, and we deduce from this last proposition that 𝐋𝐌(η)
is a morphism in the category 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).
The verification of the composition axiom repeats mutatis mutandis
the one of <ref>.



Recall the following fact on the augmentation ideal of the free group
𝐅_n where n∈ℕ.

<cit.>
The augmentation ideal ℐ_𝕂[𝐅_n]
is a free 𝕂[𝐅_n]-module with basis
the set {(g_i-1)| i∈{ 1,…,n}}.


This result allows us to prove the following properties.

The functor 𝐋𝐌_a,ς:𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)→𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
is reduced and exact. Moreover, it commutes with all colimits and
all finite limits.



Let 0_𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡):𝔘β→𝕂-𝔐𝔬𝔡
denote the null functor. It follows from the definition of the Long-Moody
functor that 𝐋𝐌(0_𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡))=0_𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).

Let n be a natural number. Since the augmentation ideal ℐ_𝕂[𝐅_n]
is a free 𝕂[𝐅_n]-module (as stated
in Proposition <ref>), it is therefore
a flat 𝕂[𝐅_n]-module. Then, the
result follows from the fact that the functor ℐ_𝕂[𝐅_n]𝕂[𝐅_n]-:𝕂-𝔐𝔬𝔡→𝕂-𝔐𝔬𝔡
is an exact functor, the naturality for morphisms following from the
definition of the Long-Moody functor (see Theorem <ref>).

Similarly, the fact that the functor 𝐋𝐌_a,ς
commutes with all colimits is a formal consequence of the commutation
with all colimits of the tensor products ℐ_𝕂[𝐅_n]𝕂[𝐅_n]-
for all natural numbers n. The commutation result for finite limits
is a property of exact functors (see for example <cit.>).



Let F be an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
and n a natural number. For all k∈{ 1,…,n},
we denote F(n+1)_k=𝕂[(g_k-1)]𝕂[𝐅_n]F(n+1)
with g_k a generator of 𝐅_n. We define an isomorphism

    _n,F:ℐ_𝕂[𝐅_n]𝕂[𝐅_n]F(n+1)    ⟶   [k=1]n⊕F(n+1)_k≅(F(n+1))^⊕ n
    (g_k-1)𝕂[𝐅_n]v    ⟼   (0,…,0,k-thv,0,…,0).

Thus, for η:F→ G a natural transformation,
with :

    ∀ n∈ℕ, _n((𝐋𝐌(η))_n)=η_n+1^⊕ n.

Hence, we can have a matricial point of view on this construction
(see <cit.>). Similarly, the study of Bigelow
and Tian in <cit.> is performed from a purely matricial
point of view.





  
Case of trivial ς:

Finally, let us consider the family of morphisms {ς_n,*:𝐅_n→𝐁_n+1} _n∈ℕ
of Example <ref>.

As stated in Example <ref>, we only need to consider a
family of morphisms { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
which satisfies Condition <ref> so
that the families {ς_n,*:𝐅_n→𝐁_n+1} _n∈ℕ
and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
are coherent.



We denote by 𝔛:𝔘β→𝕂-𝔐𝔬𝔡
the constant functor such that 𝔛(n)=𝕂
for all natural numbers n.


We have the following remarkable property.

Let F be an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
a family of morphisms which satisfies Condition <ref>.
Then, as objects of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡),
𝐋𝐌_a,ς_*(F)≅𝐋𝐌_a,ς_*(𝔛)𝕂⊗F(1♮-).



Remark <ref> shows that there is an isomorphism
of 𝕂-modules of the form:

    𝐋𝐌_a,ς_*(F)(n)@->[rr]^_n,F      (F(n+1))^⊕ n@->[rr]^(_n,𝔛𝕂⊗id_F(1♮ n))^-1                      𝐋𝐌_a,ς_*(𝔛)(n)𝕂⊗F(1♮ n)
    .

It is straightforward to check that this isomorphism is natural if
ς is trivial.




 §.§ Evaluation of the Long-Moody functor


A first step to understand the behaviour of a Long-Moody endofunctor
is to investigate its effect on the constant functor 𝔛.
This is indeed the most basic functor to study. Moreover, as Proposition
<ref> shows, the evaluation on this functor
is the fundamental information to understand a given Long-Moody endofunctor
when we consider the family of morphisms {ς_n,*:𝐅_n→𝐁_n+1} _n∈ℕ
of Example <ref>.

Fixing coherent families of morphisms {ς_n:𝐅_n→𝐁_n+1} _n∈ℕ
and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ,
we consider the Long-Moody functor 

    𝐋𝐌_a,ς:𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡)→𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡).

For a fixed natural number n, using the isomorphism _n
of Remark <ref>, we observe that 𝐋𝐌_a,ς(𝔛)(n)≅𝕂^⊕ n.

Let y be an invertible element of 𝕂.
Let y𝔛:β→𝕂-𝔐𝔬𝔡
be the functor defined for all natural numbers n by y𝔛(n)=𝕂
and such that:



  * if n=0 or n=1, then y𝔛(id)=id_𝕂;

  * if n≥2, for every Artin generator σ_i of 𝐁_n,
(y𝔛)(σ_i):𝕂→𝕂
is the multiplication by y.

For an object F of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡),
we denote the functor y𝔛𝕂⊗F:β→-𝔐𝔬𝔡
by yF. 





  §.§.§ Computations for 𝐋𝐌_1


Let us assume that 𝕂=ℂ[t^±1].
Let us consider the coherent families of morphisms {ς_n,1:𝐅_n↪𝐁_n+1} _n∈ℕ
(introduced in Example <ref>) and { a_n,1:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
(introduced in Example <ref>). We denote by 𝐋𝐌_1
the associated Long-Moody functor. We are interested in the behaviour
of the functor t^-1𝐋𝐌_1(t𝔛):β⟶ℂ[t^±1]-𝔐𝔬𝔡
on automorphisms of the category 𝔘β.
Indeed, adding a parameter t is necessary to recover functors specifically
associated with the category 𝔘β, such
as 𝔅𝔲𝔯_t (see Section <ref>).
Let us fix n a natural number and σ_i an Artin generator
of 𝐁_n.

Beforehand, let us understand the action a_n,1:𝐁_n⟶ Aut(ℐ_𝕂[𝐅_n])
induced by a_n,1:𝐁_n→ Aut(𝐅_n).
We compute:

    a_n,1(σ_i):ℐ_𝕂[𝐅_n]   ⟶   ℐ_𝕂[𝐅_n]
    
    g_j-1    ⟼   
    g_i+1-1    if 
    
    g_i+1^-1g_ig_i+1-1=[g_i-1]g_i+1+[g_i+1-1](1-g_i+1^-1g_ig_i+1)    if 
    
    g_j-1    if


Hence, we have the following result.

 As objects of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡),
t^-1𝐋𝐌_1(t𝔛)=𝔅𝔲𝔯_t^2.



Using the isomorphism _n of Remark <ref>,
we obtain that for σ_i an Artin generator of 𝐁_n:

    t^-1𝐋𝐌_1(t𝔛)(σ_i)=Id_i-1⊕[[     0   t^2;     1 1-t^2 ]]⊕ Id_n-i-1=𝔅𝔲𝔯_t^2(σ_i).





  
Recovering of the Lawrence-Krammer functor:

Let us first introduce the following result due to Long in <cit.>.
We assume that 𝕂=ℂ[t^±1][q^±1].
For this paragraph, we assume that 1+qt=0, q has a square root,
q^2≠1 and q^3≠1.

We denote by 𝔛':β⟶ℂ[t^±1][q^±1]-𝔐𝔬𝔡
the constant functor such that 𝔛'(n)=ℂ[t^±1][q^±1]
for all natural numbers n. Generally speaking, for F an object
of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡)
the representation of 𝐁_n induced by F will be denoted
by F_|𝐁_n.



<cit.>
Let n be a natural number such that n≥4. Then, the Lawrence-Krammer
representation 𝔏𝔎_|𝐁_n is a subrepresentation
of q^-1(𝐋𝐌_1(q(t^-1𝐋𝐌_1(t𝔛))))_|𝐁_n.


We first need to introduce new tools. Let n and m be two natural
numbers. Let w_n=(w_1,…,w_n)∈ℂ^n
such that w_i≠ w_j if . We consider
the configuration space: 

    Y_w_n,m={(z_1,…,z_m)| z_i∈ℂ, z_i≠ w_k for , z_i≠ z_j if } .

The two following results due to Long will be crucial to prove Proposition
<ref>.

<cit.> Let n
be a natural number and ρ:𝐁_n+1→ GL(V)
be a representation of 𝐁_n with V a ℂ[t^±1][q^±1]-module.
Then, the representation defined by Long in <cit.>,
which we denote by ℒℳ, is a group morphism:

    q^-1ℒℳ(qρ):𝐁_n→ GL(H^1(Y_w_n,1,E_ρ))

for E_ρ a flat vector bundle associated with ρ (see
<cit.>).



<cit.> For all natural
numbers m, there is an isomorphism of abelian groups:

    H^m+1(Y_w_n,m+1,E_𝔛_|𝐁_n)≅ H^1(Y_w_n,1,H^m(Y_w_n+1,m,E_𝔛_|𝐁_n)).

In particular, for m=1, H^2(Y_w_n,2,E_𝔛_|𝐁_n)≅ H^1(Y_w_n,1,H^1(Y_w_n+1,2,E_𝔛_|𝐁_n)).
By Proposition <ref>,
we can write as a representation:

    q^-1ℒℳ(q(t^-1ℒℳ(t𝔛))):𝐁_n→ GL(H^1(Y_w_n,1,E_t^-1ℒℳ(t𝔛))).

A fortiori by Lemma <ref>, q^-1ℒℳ(q(t^-1ℒℳ(t𝔛_|𝐁_n)))
is an action of 𝐁_n on H^2(Y_w_n,2,E_𝔛_|𝐁_n).
In particular, for m=2 and n≥4, according to <cit.>,
the representation of 𝐁_n factoring through the Iwahori–Hecke
algebra H_n(t) corresponding to the Young diagram
(n-2,2) is a subrepresentation of q^-1ℒℳ(q(t^-1ℒℳ(t𝔛_|𝐁_n))).
Moreover, this representation is equivalent to the Lawrence-Krammer
representation by <cit.>. By the definition of
the Long-Moody construction (see <cit.>), q^-1ℒℳ(q(t^-1ℒℳ(t𝔛_|𝐁_n)))
is the representation q^-1(τ_1𝐋𝐌_1)(q(t^-1𝐋𝐌_1(t𝔛)))_|𝐁_n.



We denote by 𝔏𝔎^≥4:β⟶(ℂ[t^±1])[q^±1]-𝔐𝔬𝔡
the subfunctor of the Lawrence-Krammer defined in Example <ref>
which is null on the objects such that n<4. The result of Proposition
<ref> implies that:

The functor 𝔏𝔎^≥4 is a subfunctor of q^-1(τ_1𝐋𝐌_1)(q(t^-1𝐋𝐌_1(t𝔛)))^≥4.





  §.§.§ Computations for other cases


Let us introduce examples of Long-Moody functors which arise using
other actions a_n:𝐁_n→ Aut(𝐅_n).



  
Wada representations

In 1992, Wada introduced in <cit.> a certain type
of family of representations of braid groups. We give here a functorial
approach to this work.

Let Aut_-:(ℕ,≤)→𝔊𝔯
be the functor defined by:



  * Objects: for all natural numbers n, Aut_-(n)=Aut(𝐅_n)
the automorphism group of the free group on n generators;

  * Morphisms: let n be a natural number. We define Aut_-(γ_n):Aut(𝐅_n)↪ Aut(𝐅_n+1)
assigning Aut_-(γ_n)(φ)=id_1*φ
for all φ∈ Aut(𝐅_n), using the monoidal
category (𝔤𝔯,*,0) recalled in Notation <ref>.




Let us consider two different non-trivial reduced
words W(g_1,g_2) and V(g_1,g_2)
on 𝐅_2, such that:



  * the assignments g_1↦ W(g_1,g_2) and g_2↦ V(g_1,g_2)
define a automorphism of 𝐅_2;

  * the assignment (W,V):𝐁_2⟶ Aut(𝐅_2):

    [(W,V)(σ_1)](g_j)=
    W(g_1,g_2)    if 
    
    V(g_1,g_2)    if

is a morphism.

Two morphisms (W,V):𝐁_2⟶ Aut(𝐅_2)
and (W',V'):𝐁_2→ Aut(𝐅_2)
are said to be swap-dual if W'(g_1,g_2)=V(g_2,g_1)
and V'(g_1,g_2)=W(g_2,g_1), backward-dual
if W'(g_1,g_2)=(W(g_1^-1,g_2^-1))^-1
and V'(g_1,g_2)=(V(g_1^-1,g_2^-1))^-1,
inverse if (W',V')=(W,V)^-1.



<cit.> Let W(g_1,g_2) and V(g_1,g_2)
be two words on 𝐅_2. A natural transformation 𝒲:𝐁_-→ Aut_-
is said to be of Wada-type if for all natural numbers n, for all
i∈{ 1,…,n-1}, the following diagram is commutative
(we recall that incl_i^n was introduced in Notation <ref>
and Aut_-(γ_2,i) in Definition <ref>):

    𝐁_n@->[rr]^𝒲_n       Aut(𝐅_n)
    𝐁_2@->[rr]_(W,V)@->[u]^incl_i^n       Aut(𝐅_2).@->[u]_Aut_-(γ_2,i)∗ id_𝐅_n-i-1




Note that therefore a Wada-type natural transformation is entirely
determined by the choice of (W,V).


Wada conjectured a classification of these type of representations.
This conjecture was proved by Ito in <cit.>.

<cit.> There are seven classes of Wada-type
natural transformation 𝒲 up to the swap-dual, backward-dual
and inverse equivalences, listed below.



  * (W,V)_1,m(g_1,g_2)=(g_2,g_2^mg_1g_2^-m)
where m∈ℤ;

  * (W,V)_2(g_1,g_2)=(g_1,g_2);

  * (W,V)_3(g_1,g_2)=(g_2,g_1^-1);

  * (W,V)_4(g_1,g_2)=(g_2,g_2^-1g_1^-1g_2);

  * (W,V)_5(g_1,g_2)=(g_2^-1,g_1^-1);

  * (W,V)_6(g_1,g_2)=(g_2^-1,g_2g_1g_2);

  * (W,V)_7(g_1,g_2)=(g_1g_2^-1g_1^-1,g_1g_2^2).




Note that the action given by the first Wada representation with m=1
is a generalization of the Artin representation.



The actions given by the k-th Wada-type
natural transformation will be denoted by a_n,k:𝐁_n↪ Aut(𝐅_n).
In particular, for k=1 with m=1, we recover the Artin representation
(see Example <ref>).


For all 1≤ k≤8, it clearly follows from their definitions
that the families of morphisms { a_n,k:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
satisfy Condition <ref>. Hence, for
1≤ k≤8, we consider a family of morphisms {ς_n,k:𝐅_n→𝐁_n+1}
assumed to be coherent with respect to the morphisms { a_n,k:𝐁_n↪ Aut(𝐅_n)} _n∈ℕ
(in the sense of Definition <ref>). Such morphisms
ς_n,k always exist because we could at least take the
family of morphisms {ς_n,*:𝐅_n→𝐁_n+1}
(see Example <ref>). We denote by 𝐋𝐌_k:𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡)→𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡)
the corresponding Long-Moody functor defined in Theorem <ref>
for k∈{ 1,…,8}.


Let us imitate the procedure of Section <ref>. We
assume that 𝕂=ℂ[t^±1]. Let n
be a fixed natural number. Let us consider the case of k=2. Using
the isomorphism _n of Remark <ref>,
we obtain the functor 𝐋𝐌_2(𝔛):β→ℂ[t^±1]-𝔐𝔬𝔡,
defined for σ_i∈𝐁_n by:

    𝐋𝐌_2(F)(σ_i)=(F(σ_i))^⊕ n.



For k=3, using _n, we compute that the functor t^-1𝐋𝐌_3(t𝔛):β→ℂ[t^±1]-𝔐𝔬𝔡
is defined for σ_i∈𝐁_n by:

    t^-1𝐋𝐌_3(t𝔛)(σ_i)     =     Id_i-1⊕[[           0 -ς_n,3(g_i);           1           0 ]]⊕ Id_n-i-1.

Hence, the functor t^-1𝐋𝐌_3(t𝔛)
is very similar to the one associated with the Tong-Yang-Ma representations
(recall Definition <ref>). We deduce that the identity
natural equivalence gives t^-1𝐋𝐌_3(t𝔛)≅𝔗𝔜𝔐_-ς_n,3(g_i)
as objects of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡).

For the actions given by the Wada-type natural transformation 4,
5, 6 and 7 in Theorem <ref>, the produced functors
t^-1𝐋𝐌_i(t𝔛):β⟶ℂ[t^±1]-𝔐𝔬𝔡
are mild variants of what is given by the case i=1.



§ STRONG POLYNOMIAL FUNCTORS


We deal here with the concept of a strong polynomial functor. This
type of functor will be the core of our work in Section <ref>.
We review (and actually extend) the definition and properties of a
strong polynomial functor due to Djament and Vespa in <cit.>
and also a particular case of coefficient systems of finite degree
used by Randal-Williams and Wahl in <cit.>.

In <cit.>, Djament and Vespa construct a framework
to define strong polynomial functors in the category 𝐅𝐜𝐭(𝔐,𝒜),
where 𝔐 is a symmetric monoidal category, the unit is
an initial object and 𝒜 is an abelian category. Here,
we generalize this definition for functors from pre-braided monoidal
categories having the same additional property. In particular, the
notion of strong polynomial functor will be defined for the category
𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).
The keypoint of this section is Proposition <ref>,
in so far as it constitutes the crucial property necessary and sufficient
to extend the definition of strong polynomial functor to the pre-braided
case.



 §.§ Strong polynomiality


We first introduce the translation functor, which plays the central
role in the definition of strong polynomiality.

Let (𝔐,♮,0) be
a strict monoidal small category, let 𝔇 be a category
and let x be an object of 𝔐. The monoidal structure
defines the endofunctor x♮-:𝔐⟶𝔐.
We define the translation by x functor τ_x:𝐅𝐜𝐭(𝔐,𝔇)→𝐅𝐜𝐭(𝔐,𝔇)
to be the endofunctor obtained by precomposition by the functor x♮-.


The following proposition establishes the commutation of two translation
functors associated with two objects of 𝔐. It is the
keystone property to define strong polynomial functors.

Let (𝔐,♮,0)
be a pre-braided strict monoidal small category and 𝔇
be a category. Let x and y be two objects of 𝔐.
Then, there exists a natural isomorphism between functors from 𝐅𝐜𝐭(𝔐,𝔇)
to 𝐅𝐜𝐭(𝔐,𝔇):

    τ_x∘τ_y≅τ_y∘τ_x.




First, because of the associativity of the monoidal product ♮
and the strictness of 𝔐, we have that τ_x∘τ_y=τ_x♮ y
and τ_y∘τ_x=τ_y♮ x. We denote by b_-,-^𝔐
the pre-braiding of 𝔐. The key point is the fact that
as b_-,-^𝔐 is a braiding on the maximal subgroupoid
of 𝔐 (see Definition <ref>), b_x,y^𝔐:x♮ y≅⟶y♮x
defines an isomorphism. Hence, precomposition by b_x,y^𝔐♮ id_𝔐
defines a natural transformation (b_x,y^𝔐♮ id_𝔐)^*:τ_x♮ y→τ_y♮ x.
It is an isomorphism since we analogously construct an inverse natural
transformation ((b_x,y^𝔐)^-1♮ id_𝔐)^*:τ_y♮ x→τ_x♮ y.


In Proposition <ref>, the natural isomorphism
is not unique: as the proof shows, we could have used the morphism
(b_y,x^𝔐)^-1♮ id_𝔐
instead to define an isomorphism between τ_x♮ y(F)
and τ_y♮ x(F). In fact, a category only
needs to be equipped with natural (in x and y) isomorphisms
x♮ y≅ y♮ x to satisfy the conclusion of Proposition
<ref>.


Let us move on to the introduction of the evanescence and difference
functors, which will characterize the (very) strong polynomiality
of a functor in 𝐅𝐜𝐭(𝔐,𝒜).
Recall that, if 𝔐 is a small category and 𝒜
is an abelian category, then the functor category 𝐅𝐜𝐭(𝔐,𝒜)
is an abelian category (see <cit.>).

From now until the end of Section <ref>,
we fix (𝔐,♮,0) a pre-braided strict
monoidal category such that the monoidal unit 0 is an initial object,
𝒜 an abelian category and x denotes an object of 𝔐.

For all objects F of 𝐅𝐜𝐭(𝔐,𝒜),
we denote by i_x(F):τ_0(F)→τ_x(F)
the natural transformation induced by the unique morphism ι_x:0→ x
of 𝔐. This induces i_x:Id_𝐅𝐜𝐭(𝔐,𝒜)→τ_x
a natural transformation of 𝐅𝐜𝐭(𝔐,𝒜).
Since the category 𝐅𝐜𝐭(𝔐,𝒜)
is abelian, the kernel and cokernel of the natural transformation
i_x exist. We define the functors κ_x=(i_x)
and δ_x=coker(i_x). The endofunctors
κ_x and δ_x of 𝐅𝐜𝐭(𝔐,𝒜)
are called respectively evanescence and difference functor associated
with x. 


The following proposition presents elementary properties of the translation,
evanescence and difference functors. They are either consequences
of the definitions, or direct generalizations of the framework considered
in <cit.> where 𝔐 is symmetric monoidal.

Let y be an object of 𝔐.
Then the translation functor τ_x is exact and we have the
following exact sequence in the category of endofunctors of 𝐅𝐜𝐭(𝔐,𝒜):

    0⟶κ_xΩ_x⟶Idi_x⟶τ_x_x⟶δ_x⟶0.

Moreover, for a short exact sequence 0⟶ F⟶ G⟶ H⟶0
in the category 𝐅𝐜𝐭(𝔐,𝒜),
there is a natural exact sequence in the category 𝐅𝐜𝐭(𝔐,𝒜):

    0⟶κ_x(F)⟶κ_x(G)⟶κ_x(H)⟶δ_x(F)⟶δ_x(G)⟶δ_x(H)⟶0.

In addition:



  * The translation endofunctor τ_x of 𝐅𝐜𝐭(𝔐,𝒜)
commutes with limits and colimits.

  * The difference endofunctors δ_x and δ_y of 𝐅𝐜𝐭(𝔐,𝒜)
commute up to natural isomorphism. They commute with colimits.

  * The endofunctors κ_x and κ_y of 𝐅𝐜𝐭(𝔐,𝒜)
commute up to natural isomorphism. They commute with limits.

  * The natural inclusion κ_x∘κ_x↪κ_x
is an isomorphism.

  * The translation endofunctor τ_x and the difference endofunctor
δ_y commute up to natural isomorphism.

  * The translation endofunctor τ_x and the endofunctor κ_y
commute up to natural isomorphism.

  * We have the following natural exact sequence in the category of endofunctors
of 𝐅𝐜𝐭(𝔐,𝒜):

    0⟶κ_y⟶κ_x♮ y⟶τ_xκ_y⟶δ_y⟶δ_x♮ y⟶τ_yδ_x⟶0.





In the symmetric monoidal case, this is <cit.>:
the numbered properties are formal consequences of the commutation
property of the translation endofunctors given by Proposition <ref>.
Hence, the proofs carry over mutatis mutandis to the pre-braided setting.

Using Proposition <ref>, we can define strong polynomial
functors.

We recursively define on n∈ℕ the category 𝒫ol_n^strong(𝔐,𝒜)
of strong polynomial functors of degree less than or equal to n
to be the full subcategory of 𝐅𝐜𝐭(𝔐,𝒜)
as follows:



  * If n<0, 𝒫ol_n^strong(𝔐,𝒜)={ 0};

  * if n≥0, the objects of 𝒫ol_n^strong(𝔐,𝒜)
are the functors F such that for all objects x of 𝔐,
the functor δ_x(F) is an object of 𝒫ol_n-1^strong(𝔐,𝒜).

For an object F of 𝐅𝐜𝐭(𝔐,𝒜)
which is strong polynomial of degree less than or equal to n∈ℕ,
the smallest d∈ℕ (d≤ n) for which F is an object
of 𝒫ol_d^strong(𝔐,𝒜)
is called the strong degree of F.



By Proposition <ref>, the category (𝔘β,♮,0)
is a pre-braided monoidal category such that 0 is initial object.
This example is the first one which led us to extend the definition
of <cit.>. Thus, we have a well-defined notion of strong polynomial
functor for the category 𝔘β.


The following three propositions are important properties of the framework
in <cit.> adapted to the pre-braided case. Their proofs follow
directly from those of their analogues in <cit.>.

<cit.> Let 𝔐' be another pre-braided
strict monoidal category such that such that its monoidal unit is
an initial object and α:𝔐→𝔐'
be a strong monoidal functor. Then, the precomposition by α
provides a functor  𝒫ol_n^strong(𝔐,𝒜)→𝒫ol_n^strong(𝔐',𝒜).



<cit.> The category 𝒫ol_n^strong(𝔐,𝒜)
is closed under the translation endofunctor τ_x, under quotient,
under extension and under colimits. Moreover, assuming that there
exists a set 𝔈 of objects of 𝔐 such that:

    ∀ m∈ Obj(𝔐),  ∃{ e_i} _i∈ I∈ Obj(𝔈) where  is finite,  m≅i∈ I♮e_i,

then, an object F of 𝐅𝐜𝐭(𝔐,𝒜)
belongs to 𝒫ol_n^strong(𝔐,𝒜)
if and only if δ_e(F) is an object of 𝒫ol_n-1^strong(𝔐,𝒜)
for all objects e of 𝔈.



 Let n be a natural number. Let F
be a strong polynomial functor of degree n in the category 𝐅𝐜𝐭(𝔐,𝒜).
Then a direct summand of F is necessarily an object of the category
𝒫ol_n^strong(𝔐,𝒜).



According to Proposition <ref>, the category 𝒫ol_n^strong(𝔐,𝒜)
is closed under quotients.


The category 𝒫ol_n^strong(𝔐,𝒜)
is not necessarily closed under subobjects. For example, we will see
in  Section <ref> that for 𝔐=𝔘β
and 𝒜=ℂ[t^±1]-𝔐𝔬𝔡,
the functor 𝔅𝔲𝔯_t is a subobject of τ_1𝔅𝔲𝔯_t
(see Proposition <ref>), 𝔅𝔲𝔯_t
is strong polynomial of degree 2 (see Proposition <ref>)
whereas τ_1𝔅𝔲𝔯_t is strong polynomial
of degree 1 (see Proposition <ref>). If we
assume that the unit 0 is also a terminal object of 𝔐,
then κ_x is the null endofunctor, δ_x is exact
and commutes with all limits. In this case, the category 𝒫ol_n^strong(𝔐,𝒜)
is closed under subobjects.



If we consider 𝔐=𝔘β,
then each object n (ie a natural number) is clearly 1^♮ n.
Hence, because of the last statement of Proposition <ref>,
when we will deal with strong polynomiality of objects in 𝐅𝐜𝐭(𝔘β,𝒜),
it will suffice to consider τ_1.



<cit.> Let F be an
object of 𝐅𝐜𝐭(𝔐,𝒜). Then,
the functor F is an object of 𝒫ol_0^strong(𝔐,𝒜)
if and only if it is the quotient of a constant functor of 𝐅𝐜𝐭(𝔐,𝒜).


Finally, let us point out the following property of the strong polynomial
degree with respect to the translation functor.

Let d and k be natural numbers
and F be an object of 𝐅𝐜𝐭(𝔘β,-𝔐𝔬𝔡)
such that τ_k(F) is an object of 𝒫ol_d^strong(𝔘β,-𝔐𝔬𝔡).
Then, F is an object of 𝒫ol_d+k(𝔘β,-𝔐𝔬𝔡).



We proceed by induction on the degree of polynomiality of τ_k(F).
First, assuming that τ_k(F) belongs to 𝒫ol_0^strong(𝔘β,-𝔐𝔬𝔡),
we deduce from the commutation property 6 of Proposition <ref>
that τ_k(δ_1F)=0. It follows from the definition
of τ_k(F) (see Definition <ref>) that
for all n≥2, δ_1(F)(n)=0. Hence

    timesδ_1⋯δ_1δ_1(F)≅0

and therefore F is an object of 𝒫ol_k(𝔘β,-𝔐𝔬𝔡).
Now, assume that τ_k(F) is a strong polynomial
functor of degree d≥0. Since (τ_k∘δ_1)(F)≅(δ_1∘τ_k)(F)
by the commutation property 6 of Proposition <ref>,
(τ_k∘δ_1)(F) is an object
of 𝒫ol_d-1^strong(𝔘β,-𝔐𝔬𝔡).
The inductive hypothesis implies that δ_1(F)
is an object of 𝒫ol_d+k^strong(𝔘β,-𝔐𝔬𝔡).


Let us consider the atomic functor 𝔄_n (with n>0),
which is strong polynomial of degree n (see Example <ref>).
Then τ_k(𝔄_n)≅𝔄_n-k^⊕ n
is strong polynomial of degree n-k, for k a natural number such
that k≤ n. This illustrates the fact that d+k is the best
boundary for the degree of polynomiality in Lemma <ref>.





 §.§ Very strong polynomial functors


Let us introduce a particular type of strong polynomial functor, related
to coefficient systems of finite degree (see Remark <ref>
below). We recall that we consider a pre-braided strict monoidal category
(𝔐,♮,0) such that the monoidal unit
0 is an initial object and an abelian category 𝒜.

We recursively define the category 𝒱𝒫ol_n(𝔐,𝒜)
of very strong polynomial functors of degree less than or equal to
n to be the full subcategory of 𝒫ol_n^strong(𝔐,𝒜)
as follows:



  * If n<0, 𝒱𝒫ol_n(𝔐,𝒜)={ 0};

  * if n≥0, a functor F∈𝒫ol_n^strong(𝔐,𝒜)
is an object of 𝒱𝒫ol_n(𝔐,𝒜)
if for all objects x of 𝔐, κ_x(F)=0
and the functor δ_x(F) is an object of 𝒱𝒫ol_n-1(𝔐,𝒜).



For an object F of 𝐅𝐜𝐭(𝔐,𝒜)
which is very strong polynomial of degree less than or equal to n∈ℕ,
the smallest d∈ℕ (d≤ n) for which F is an object
of 𝒱𝒫ol_d(𝔐,𝒜) is called
the very strong degree of F.

A certain type of functor, called a coefficient
system of finite degree, closely related to the strong polynomial
one, is used by Randal-Williams and Wahl in <cit.>
for their homological stability theorems, generalizing the concept
introduced by van der Kallen for general linear groups <cit.>.
Using the framework introduced by Randal-Williams and Wahl, a coefficient
system in every object x of 𝔐 of degree n at N=0
is a very strong polynomial functor.



As we force κ_x to
be null for all objects x of 𝔐, the category 𝒱𝒫ol_n(𝔐,𝒜)
is closed under kernel functors of the epimorphisms. In particular,
this category is closed under direct summands. However, 𝒱𝒫ol_n(𝔐,𝒜)
is not necessarily closed under subobjects. For instance, as for Remark
<ref>, we have that the functor 𝔅𝔲𝔯_t
is strong polynomial of degree 2 (see Proposition <ref>),
the functor τ_1𝔅𝔲𝔯_t is very strong
polynomial of degree 1 (see Proposition <ref>),
but 𝔅𝔲𝔯_t is a subobject of τ_1𝔅𝔲𝔯_t
(see Proposition <ref>).



The category 𝒱𝒫ol_n(𝔐,𝒜)
is closed under the translation endofunctor τ_x, under kernel
of epimorphism and under extension. Moreover, assuming that there
exists a set 𝔈 of objects of 𝔐 such that:

    ∀ m∈ Obj(𝔐),  ∃{ e_i} _i∈ I∈ Obj(𝔈) (where  is finite),  m≅i∈ I♮e_i,

then, an object F of 𝐅𝐜𝐭(𝔐,𝒜)
belongs to 𝒱𝒫ol_n(𝔐,𝒜)
if and only if κ_e(F)=0 and δ_e(F)
is an object of 𝒱𝒫ol_n-1(𝔐,𝒜)
for all objects e of 𝔈.



The first assertion follows from the fact that for all objects x
of 𝔐, the endofunctor τ_x commutes with the
endofunctors δ_x and κ_x (see Proposition <ref>).
For the second and third assertions, let us consider two short exact
sequences of 𝐅𝐜𝐭(𝔐,𝒜):
0⟶ G⟶ F_1⟶ F_2⟶0
and 0⟶ F_3⟶ H⟶ F_4⟶0
with F_i a very strong polynomial functor of degree n for
all i. Let x be an object of 𝔐. We use the exact
sequence (<ref>) of Proposition <ref>
to obtain the two following exact sequences in the category 𝐅𝐜𝐭(𝔐,𝒜):

    0⟶κ_x(G)⟶0⟶0⟶δ_x(G)⟶δ_x(F_1)⟶δ_x(F_2)⟶0;


    0⟶0⟶κ_x(H)⟶0⟶δ_x(F_3)⟶δ_x(H)⟶δ_x(F_4)⟶0.

Therefore, κ_x(G)=κ_x(H)=0
and the result follows directly by induction on the degree of polynomiality.
For the last point, we consider the long exact sequence (<ref>)
of Proposition <ref> applied to an object F of
𝒱𝒫ol_n(𝔐,𝒜) to obtain
the following exact sequence in the category 𝐅𝐜𝐭(𝔐,𝒜):

    0⟶κ_y(F)⟶κ_x♮ y(F)⟶τ_xκ_y(F)⟶δ_y(F)⟶δ_x♮ y(F)⟶τ_yδ_x(F)⟶0.

Hence, by induction on the length of objects as monoidal product of
{ e_i} _i∈ I, we deduce that κ_m(F)=0
for all objects m of 𝔐 if and only if κ_e(F)=0
for all objects e of 𝔈. Moreover, since 𝒱𝒫ol_n(𝔐,𝒜)
is closed under extension and by the translation endofunctor τ_y,
the result follows by induction on the degree of polynomiality n.


Let F be an object of 𝐅𝐜𝐭(𝔐,𝒜).
The functor F is an object of 𝒱𝒫ol_0(𝔐,𝒜)
if and only if it is isomorphic to τ_kF for all natural numbers
k.



The result follows using the long exact sequence (<ref>)
of Proposition <ref> applied to F.

The following example show that there exist strong polynomial functors
which are not very strong polynomial in any degree.

Let us consider the categories 𝔘β
and 𝕂-𝔐𝔬𝔡, and n a natural number.
Let 𝕂 be considered as an object of 𝕂-𝔐𝔬𝔡
and 0 be the trivial 𝕂-module. Let 𝔄_n
be an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡),
defined by:



  * Objects: ∀ m∈ℕ, 𝔄_n(m)=𝕂   if 

0    otherwise.

  * Morphisms: let [j-i,f] with f∈𝐁_n
be a morphism from i to j in the category 𝔘β.
Then: 

    𝔄_n(f)=
    id_𝕂   if 
    
    0    otherwise.


The functor 𝔄_n is called an atomic functor in 𝕂
of degree n. For coherence, we fix 𝔄_-1 to be the
null functor of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).
Then, it is clear that i_p(𝔄_n) is the
zero natural transformation. On the one hand, we deduce the following
natural equivalence κ_1(𝔄_n)≅𝔄_n
and a fortiori 𝔄_n is not a very strong polynomial
functor. On the other hand, it is worth noting the natural equivalence
δ_1(𝔄_n)≅τ_1(𝔄_n)
and the fact that τ_1(𝔄_n)≅𝔄_n-1.
Therefore, we recursively prove that 𝔄_n is a strong
polynomial functor of degree n.



Contrary to 𝒫ol_n^strong(𝔐,𝒜),
a quotient of an object F of 𝒱𝒫ol_n(𝔐,𝒜)
is not necessarily a very strong polynomial functor. For example,
for 𝔐=𝔘β and 𝒜=𝕂-𝔐𝔬𝔡,
let us consider the functor 𝔄_0 defined in Example
<ref>, which we proved to be a strong polynomial
functor of degree 0. Let 𝔄 be the constant object
of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
equal to 𝕂. Then, we define a natural transformation α:𝔄→𝔄_0
assigning:

    ∀ n∈ℕ,  α_n=
    id_𝕂   if 
    
    t_𝕂   otherwise.

Moreover, it is an epimorphism in the category 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
since for all natural numbers n, coker(α_n)=0_𝕂-𝔐𝔬𝔡.
We proved in Example <ref> that 𝔄_0
is not a very strong polynomial functor of degree 0 whereas 𝔄
is a very strong polynomial functor of degree 0 by Proposition
<ref>.


Finally, let us remark the following behaviour of the translation
functor with respect to very strong polynomial degree.

Let d and k be a natural numbers
and F be an object of 𝒱𝒫ol_d(𝔐,-𝔐𝔬𝔡).
Then the functor τ_k(F) is very strong polynomial
of degree equal to that of F.



We proceed by induction on the degree of polynomiality of F. First,
if we assume that F belongs to 𝒱𝒫ol_0(𝔐,-𝔐𝔬𝔡),
then according to Proposition <ref>, τ_k(F)≅ F
is a degree 0 very strong polynomial functor. Now, assume that
F is a very strong polynomial functor of degree n≥0. Using
the commutation properties 5 and 6 of Proposition <ref>,
we deduce that (κ_1∘τ_k)(F)≅(τ_k∘κ_1)(F)=0
and (δ_1∘τ_k)(F)≅(τ_k∘δ_1)(F).
Since the functor δ_1(F) is a degree n-1 very
strong polynomial functor, the result follows from the inductive hypothesis.


The previous proof does not work for strong polynomial functors since
the initial step fails. Indeed, considering the atomic functor 𝔄_1,
which is strong polynomial of degree 1 (see Example <ref>),
then τ_2(𝔄_0)=0.





 §.§ Examples of polynomial functors over 𝔘β


The different functors introduced in Section <ref>
are strong polynomial functors.



  
Very strong polynomial functors of degree one:

Let us first investigate the polynomiality of the functors 𝔅𝔲𝔯_t
and 𝔗𝔜𝔐_t.

The functors 𝔅𝔲𝔯_t and
𝔗𝔜𝔐_t are very strong polynomial functors of degree
1.

For the functor 𝔅𝔲𝔯_t, the proof is mutatis mutandis
the same as the one for the dual version considered in <cit.>.
We will thus focus on the case of the functor 𝔗𝔜𝔐_t.
Let n be a natural number. By Remark <ref>, it is
enough to consider the application i_1𝔗𝔜𝔐_𝔱([0,id_n])=ι_ℂ[t^±1]^⊕ n'-n⊕ id_ℂ[t^±1]^⊕ n.
This map is a monomorphism and its cokernel is ℂ[t^±1].
Hence κ_1𝔗𝔜𝔐_t is the null functor of 𝐅𝐜𝐭(𝔘β,ℂ[t^±1]-𝔐𝔬𝔡).
Let n' be a natural number such that n'≥ n and let [n'-n,σ]∈ Hom_𝔘β(n,n').
By naturality and the universal property of the cokernel, there exists
a unique endomorphism of ℂ[t^±1] such that
the following diagram commutes, where the lines are exact. It is exactly
the definition of δ_1𝔗𝔜𝔐_t([n'-n,σ]).

    0@->[r]    ℂ[t^±1]^⊕ n@->[rrr]^ι_ℂ[t^±1]⊕ id_ℂ[t^±1]^⊕ n@->[d]_𝔗𝔜𝔐([n'-n,σ])         ℂ[t^±1]^⊕ n+1@->[rr]^π_n+1@->[d]^τ_1(𝔗𝔜𝔐)([n'-n,σ])      ℂ[t^±1]@->[r]@.>[d]^∃!    0
    
    0@->[r]    ℂ[t^±1]^⊕ n'@->[rrr]_ι_ℂ[t^±1]⊕ id_ℂ[t^±1]^⊕ n'         ℂ[t^±1]^⊕ n'+1@->[rr]_π_n'+1      ℂ[t^±1]@->[r]     0.

For all (a,b)∈ℂ[t^±1]⊕ℂ[t^±1]^⊕ n=ℂ[t^±1]^⊕ n+1,
τ_1(𝔗𝔜𝔐_t)([n'-n,σ])(a,b)=(a,𝔗𝔜𝔐_t([n'-n,σ])(b)).
Therefore, (π_n'+1∘τ_1(𝔗𝔜𝔐_t)([n'-n,σ]))(a,b)=a=π_n+1(a,b).
Hence, id_ℂ[t^±1] also makes the diagram
commutative and thus δ_1𝔗𝔜𝔐_t([n'-n,σ])=id_ℂ[t^±1].
Hence, δ_1𝔗𝔜𝔐_t is the constant functor equal
to ℂ[t^±1]. A fortiori, because of Proposition
<ref>, δ_1𝔗𝔜𝔐_t is a
very strong polynomial functor of degree 0.






  
The particular case of 𝔅𝔲𝔯_t:

Let 𝒯_1:𝔘β⟶ℂ[t^±1]-𝔐𝔬𝔡
be the subobject of the constant functor 𝔛 (see Notation
<ref>) such that 𝒯_1(0)=0 and
𝒯_1(n)=ℂ[t^±1] for
all non-zero natural numbers n.



It follows from Definition <ref>
that δ_1𝒯_1≅𝔄_0 (where 𝔄_0
is introduced in Example <ref>). Therefore, 𝒯_1
is a strong polynomial functor of degree 1, but is not very strong
polynomial. Nevertheless, it is worth noting that κ_1𝒯_1=0.



The functor 𝔅𝔲𝔯
is a strong polynomial functor of degree 2. This functor is not
very strong polynomial. More precisely, we have the following short
exact sequence in 𝐅𝐜𝐭(𝔘β,ℂ[t^±1]-𝔐𝔬𝔡):

    0@->[r]    𝔅𝔲𝔯_t@->[r]    τ_1𝔅𝔲𝔯_t@->[r]    𝒯_1@->[r]     0
    .


The natural transformation i_1(𝔅𝔲𝔯_t)_n:𝔅𝔲𝔯_t(n)→τ_1𝔅𝔲𝔯_t(n)
(introduced in Definition <ref>) is defined to be ι_ℂ[t^±1]^⊕ n'-n⊕ id_ℂ[t^±1]^⊕ n-1.
Let n≥2 be a natural number. This map is a monomorphism (so
κ_1𝔅𝔲𝔯_t=0) and its cokernel is
ℂ[t^±1]. Repeating mutatis mutandis the
work done in the proof of Proposition <ref>,
we deduce that for all [n'-n,σ]∈ Hom_𝔘β(n,n')
(with n'≥ n≥2), δ_1𝔅𝔲𝔯_t([n'-n,σ])=Id_ℂ[t^±1].
In addition, since 𝔅𝔲𝔯_t(1)=0
and τ_1𝔅𝔲𝔯_t(1)=ℂ[t^±1],
we deduce that δ_1𝔅𝔲𝔯_t(1)=ℂ[t^±1]
and for all n'≥1, for all [n'-1,σ]∈ Hom_𝔘β(1,n'),
δ_1𝔅𝔲𝔯_t([n'-1,σ])=Id_ℂ[t^±1].
Hence, we prove that δ_1𝔅𝔲𝔯_t≅𝒯_1
where 𝒯_1 is introduced in Definition <ref>.
The results follow from the fact that δ_1𝒯_1≅𝔄_0
by Remark <ref>.



For formal reasons (see Proposition <ref>), 𝔅𝔲𝔯_t
is a subfunctor of τ_1𝔅𝔲𝔯_t. The following
proposition illustrates Remarks <ref>
and <ref>.

The functor τ_1𝔅𝔲𝔯_t
is a very strong polynomial functor of degree 1.



Repeating mutatis mutandis the work done in the proof of Proposition
<ref>, we prove that δ_1τ_1𝔅𝔲𝔯_t
is the constant functor equal to ℂ[t^±1]
(denoted by 𝔛 in Notation <ref>). Since 𝔛
is a constant functor, δ_1τ_1𝔅𝔲𝔯_t
is by Proposition <ref> a very strong polynomial
functor of degree 0.




  
A very strong polynomial functor of degree two:

We could have defined the unreduced Burau functor of Example <ref>
assigning ((ℂ[t^±1])[q^±1])^⊕ n
to each object n∈ℕ.

Abusing the notation, (ℂ[t^±1])[q^±1]:𝔘β→(ℂ[t^±1])[q^±1]-𝔐𝔬𝔡
denotes the constant functor at (ℂ[t^±1])[q^±1].
The functor 𝔅𝔲𝔯_tℂ[t^±1]⊗(ℂ[t^±1])[q^±1]
is denoted by 𝔅̌𝔲̌𝔯̌_̌ť:𝔘β→(ℂ[t^±1])[q^±1]-𝔐𝔬𝔡.



These functors (ℂ[t^±1])[q^±1]
and 𝔅̌𝔲̌𝔯̌_̌ť are also very strong polynomial
of degree one (the proof is exactly the same as the one for 𝔅𝔲𝔯_t
in Proposition <ref>).



Considering the modified version of the unreduced
Burau functor of Remark <ref>, then δ_1𝔏𝔎
is equivalent to 𝔅̌𝔲̌𝔯̌_̌ť.



We consider the application i_1𝔏𝔎([0,id_n]).
This map is a monomorphism and its cokernel is 2≤ l≤ n+1⊕V_1,l.
Let n and n' be two natural numbers such that n'≥ n. Let
[n'-n,σ]∈ Hom_𝔘β(n,n').
By naturality and because of the universal property of the cokernel,
there exists a unique endomorphism of (ℂ[t^±1])[q^±1]-modules
such that the following diagram commutes, where the lines are exact.
It is exactly the definition of δ_1𝔏𝔎([n'-n,σ]).

    0@->[r]    1≤ j<k≤ n⊕V_j,k@->[rrr]^𝔏𝔎([1,id_1+n])@->[d]_𝔏𝔎([n'-n,σ])         1≤ i<l≤ n+1⊕V_i,l@->[rrr]^oooπ_n@->[d]^τ_1(𝔏𝔎)([n'-n,σ])         2≤ l≤ n+1⊕V_1,l@->[r]@.>[d]^∃!    0
    
    0@->[r]    1≤ j'<k'≤ n'⊕V_j',k'@->[rrr]_𝔏𝔎([1,id_1+n'])         1≤ l'≤ n'+1⊕V_i',l'@->[rrr]_oooπ_n'         2≤ l'≤ n'+1⊕V_1,l'@->[r]     0.

Let i∈{ 1,…,n-1}, l∈{ 2,…,n+1}
and v_1,l be an element of V_1,l. Then we compute:

    τ_1𝔏𝔎(σ_i)v_1,l=𝔏𝔎(σ_1+i)(v_1,l)=
    v_1,l   if ,
    
    tv_1,i+1+(1-t)v_1,i+2-(t^2-t)qv_i+1,i+2   if ,
    
    v_1,i+2   if .

We deduce that in the canonical basis {𝐞_1,2,𝐞_1,3,…,𝐞_1,n+1}
of 2≤ l≤ n+1⊕V_1,l: 

    δ_1𝔏𝔎(σ_i)=Id_i-1⊕[[   0   t;   1 1-t ]]⊕ Id_n-i-1=𝔅̌𝔲̌𝔯̌_̌ť(σ_i).

So as to identify δ_1𝔏𝔎, it remains to consider
the action on morphisms of type [1,id_n+1]. According
to the definition of the Lawrence-Krammer functor, we have τ_1(𝔏𝔎)([1,id_n+1])=𝔏𝔎(σ_1^-1)∘𝔏𝔎([1,id_n+2])
and:

    𝔏𝔎(σ_1)(v_1,k)=
    v_2,k   if ,
    
    -qt^2v_1,2   if .

It follows that for all v_i,l∈ V_i,l with 1≤ i<l≤ n+1:

    π_n+1∘τ_1(𝔏𝔎)([1,id_n+1])(v_i,l)=
    v_1,l+1   if  and ,
    
    0    otherwise.

Hence, we deduce that for all 2≤ l≤ n+1, δ_1𝔏𝔎([1,id_n+1])(v_1,l)=v_1,l+1=𝔅̌𝔲̌𝔯̌_̌ť([1,id_n+1])(v_1,l).


The functor 𝔏𝔎 is a very strong
polynomial functor of degree 2.

Let n be a natural number. By Remark <ref>, we only
have to consider the application i_1𝔏𝔎([0,id_n]).
Since this map is a monomorphism with cokernel 1≤ i≤ n⊕V_i,n+1,
κ_1𝔏𝔎 is the null constant functor.
Since the functor 𝔅̌𝔲̌𝔯̌_̌ť is very strong polynomial
of degree one (following exactly the same proof as the one of Proposition
<ref>), we deduce from Lemma <ref>
that 𝔏𝔎 is very strong polynomial of degree two.






§ THE LONG-MOODY FUNCTOR APPLIED TO POLYNOMIAL FUNCTORS


Let us move on to the effect of the Long-Moody functors on (very)
strong polynomial functors. For this purpose, it is enough by Remark
<ref> to consider the cokernel of the map i_1𝐋𝐌.
First, we decompose the functor τ_1∘𝐋𝐌 (see Proposition
<ref>) so as to understand the behaviour
of the image of i_1𝐋𝐌 through this decomposition. This
allows us to prove a splitting decomposition of the difference functor
(see Theorem <ref>). This is the key point to prove
our main results, namely Corollary <ref> and Theorem
<ref>. Finally, we give some additional properties
of Long-Moody functors with respect to polynomial functors.

Let {ς_n:𝐅_n↪𝐁_n+1} _n∈ℕ
and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
be coherent families of morphisms (see Definition <ref>),
with associated Long-Moody functor 𝐋𝐌_a,ς (see
Theorem <ref>), which we fix for all the work of this
section (in particular, we omit the "a,ς" from the notation).



 §.§ Decomposition of the translation functor


We introduce two functors which will play a key role in the main result.
First, let us recall the following crucial property of the augmentation
ideal of a free product of groups, which follows by combining <cit.>
and <cit.>.

Let G and H be groups. Then, there
is a natural 𝕂[G∗ H]-module isomorphism:

    ℐ_𝕂[G∗ H]≅(ℐ_𝕂[G]𝕂[G]⊗𝕂[G∗ H])⊕(ℐ_𝕂[H]𝕂[H]⊗𝕂[G∗ H]).




In the statement of Proposition <ref>,
recall that the augmentation ideal ℐ_𝕂[G]
(respectively ℐ_𝕂[H]) is a free
right 𝕂[G]-module (respectively 𝕂[H]-module)
by Proposition <ref>. Moreover,
the group ring 𝕂[G∗ H] is a left 𝕂[G]-module
(respectively left 𝕂[H]-module) via the morphism
id_G∗ι_H:G→ G*H (respectively ι_G∗ id_H:H→ G*H
).



Let n and n' be natural numbers
such that n'≥ n. We consider the morphism id_𝐅_n*ι_𝐅_n'-n:𝐅_n↪𝐅_n'.
This corresponds to the identification of 𝐅_n as the
subgroup of 𝐅_n' generated by the n first copies
of 𝐅_1 in 𝐅_n'.

In addition, the group morphism id_𝐅_n*ι_𝐅_n'-n:𝐅_n↪𝐅_n'
canonically induces a 𝕂-module morphism id_ℐ_𝕂[𝐅_n]*ι_ℐ_𝕂[𝐅_n'-n]:ℐ_𝕂[𝐅_n]↪ℐ_𝕂[𝐅_n'].


For F an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡),
we consider the functor (τ_1∘𝐋𝐌)(F).
For all natural numbers n, by Proposition <ref>,
we have a 𝕂[𝐅_1+n]-module isomorphism:


    ℐ_𝕂[𝐅_1+n]𝕂[𝐅_1+n]F(n+2)
          ≅((ℐ_𝕂[𝐅_1]𝕂[𝐅_1]𝕂[𝐅_1+n])⊕(ℐ_𝕂[𝐅_n]𝕂[𝐅_n]𝕂[𝐅_1+n]))𝕂[𝐅_1+n]F(n+2).

Now, by Remark <ref>, the 𝕂[𝐅_n+1]-module
F(n+2) is a 𝕂[𝐅_1]-module
via 

    F(ς_1+n(id_𝐅_1∗ι_𝐅_n)):𝐅_1→ Aut_𝕂-𝔐𝔬𝔡(F(n+2))

 and 𝕂[𝐅_n]-module via 

    F(ς_1+n(ι_𝐅_1∗ id_𝐅_n)):𝐅_n→ Aut_𝕂-𝔐𝔬𝔡(F(n+2)).

Therefore, because of the distributivity of tensor product with respect
to the direct sum, we have the following proposition.

Let F∈ Obj(𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡))
and n be a natural number. Then, we have the following 𝕂-module
isomorphism:

    τ_1𝐋𝐌(F)(n)    ≅   (ℐ_𝕂[𝐅_1]𝕂[𝐅_1]F(n+2))⊕(ℐ_𝕂[𝐅_n]𝕂[𝐅_n]F(n+2)).




For all natural numbers n and F∈ Obj(𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)),
we denote by



  * υ(F)_n the monomorphism of 𝕂-modules
(id_ℐ_𝕂[𝐅_1]∗ι_ℐ_𝕂[𝐅_n])𝕂[𝐅_1+n]id_F(n+2):ℐ_𝕂[𝐅_1]𝕂[𝐅_1]F(n+2)↪τ_1𝐋𝐌(F)(n),

  * ξ(F)_n the monomorphism of 𝕂-modules
(ι_ℐ_𝕂[𝐅_1]*id_ℐ_𝕂[𝐅_n])𝕂[𝐅_1+n]id_F(n+2):ℐ_𝕂[𝐅_n]𝕂[𝐅_n]F(n+2)↪τ_1𝐋𝐌(F)(n),

associated with the direct sum of Proposition <ref>.


The aim of this section is in fact to show that this 𝕂-module
decomposition leads to a decomposition of τ_1𝐋𝐌 (see
Theorem <ref>) as a functor.



  §.§.§ Additional conditions


We need two additional conditions so as to make the decomposition
of Proposition <ref> functorial. First, we require
the morphisms { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
to satisfy the following property.

Let n and n' be natural
numbers such that n'≥ n. We require a_1+n'((b_1,n'-n^β)^-1♮ id_n)∘(ι_𝐅_n'-n*id_𝐅_n+1)∘(id_𝐅_1∗ι_𝐅_n)=id_𝐅_1∗ι_𝐅_n'.
In other words, the following diagram is commutative:


    𝐅_1@->[d]_id_𝐅_1*ι_𝐅_n@->[rrrr]^id_𝐅_1∗ι_𝐅_n'            𝐅_1+n'
    𝐅_1+n@->[rrrr]_ι_𝐅_n'-n*id_𝐅_1+n            𝐅_n'-n*𝐅_1+n≅𝐅_1+n'.@->[u]_a_1+n'((b_1,n'-n^β)^-1♮ id_n)

 



Condition <ref> will be used to define
an intermediary functor (see Proposition <ref>).


In addition, we will assume that the morphisms { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
satisfy the following condition.

Let n and n' be natural
numbers such that n'≥ n. We require a_n'(id_n'-n♮-):𝐁_n→ Aut(𝐅_n')
maps to the stabilizer of the homomorphism id_𝐅_n'-n*ι_𝐅_n:𝐅_n'-n⟶𝐅_n',
ie for all element σ of 𝐁_n the following diagram
is commutative:


    𝐅_n'-n@->[rr]^id_𝐅_n'-n*ι_𝐅_n@->[dr]_id_𝐅_n'-n*ι_𝐅_n      𝐅_n'
       𝐅_n'.@->[ur]_a_n'(id_n'-n♮σ)




Condition <ref> will be used in the
proof of Propositions <ref> and <ref>.



The relations of Conditions <ref>
and <ref> remain true mutatis mutandis,
for all natural numbers n, considering the induced morphisms a_n:𝐁_n→ Aut(ℐ_𝕂[𝐅_n])
and id_ℐ_𝕂[𝐅_n]*ι_ℐ_𝕂[𝐅_n'-n]:ℐ_𝕂[𝐅_n]↪ℐ_𝕂[𝐅_n'].



If the morphisms { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
also satisfy conditions <ref> and
<ref>, the coherent families of morphisms
{ς_n:𝐅_n↪𝐁_n+1} _n∈ℕ
and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
are said to be reliable.



The coherent families of morphisms { a_n,1:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
and {ς_n,1:𝐅_n↪𝐁_n+1} _n∈ℕ
of Examples <ref> and <ref> are reliable.



Recall from Definition <ref> that (b_1,n'-n^β)^-1=σ_1^-1∘σ_2^-1∘⋯∘σ_n'-n^-1.
We consider the element e_𝐅_n'-n*g_1*e_𝐅_n=g_n'-n+1∈𝐅_(n'-n)+1+n.
The definition of a_n,1 gives that a_1+n',1(σ_n'-n)(g_n'-n)=g_n'-n+1.
Therefore, we have that:

    a_1+n',1(σ_n'-n^-1)(g_n'-n+1)=g_n'-n.

Iterating this observation, we deduce that a_1+n'((b_1,n'-n^β)^-1♮ id_n)(g_n'-n+1)=g_1∈𝐅_1+n'.
Hence, the family of morphisms { a_n,1:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
satisfies Condition <ref>.

Similarly to Example <ref> earlier, for all g∈𝐅_n'-n
and each Artin generator σ_i∈𝐁_n, a_n'(id_n'-n♮σ_i)(g*e_𝐅_n)=g*e_𝐅_n.
Hence, the family of morphisms { a_n,1:𝐁_n→ Aut(𝐅_n)} _n∈ℕ
satisfies Condition <ref>.

From now until the end of Section <ref>,
we fix coherent reliable families of morphisms {ς_n:𝐅_n↪𝐁_n+1} _n∈ℕ
and { a_n:𝐁_n→ Aut(𝐅_n)} _n∈ℕ.



  §.§.§ The intermediary functors




  
The functor τ_2:

Let us consider the factor ℐ_𝕂[𝐅_1]𝕂[𝐅_1]F(n+2)
of τ_1𝐋𝐌(F)(n) in the decomposition
of Proposition <ref>.

For all objects F of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡),
for all natural numbers n, we denote ℐ_𝕂[𝐅_1]𝕂[𝐅_1]F(n+2)
by (F)(n).


Recall the monomorphisms {υ(F)_n:(F)(n)↪τ_1𝐋𝐌(F)(n)} _n∈ℕ
of Definition <ref>.

Let F be an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).
For all natural numbers n and n' such that n'≥ n, and
for all [n'-n,σ]∈ Hom_𝔘β(n,n'),
assign: 

    (F)([n'-n,σ])=id_ℐ_𝕂[𝐅_1]𝕂[𝐅_1]F(id_2♮[n'-n,σ]).

This defines a subfunctor (F):𝔘β→𝕂-𝔐𝔬𝔡
of τ_1𝐋𝐌(F), using the monomorphisms {υ(F)_n} _n∈ℕ.



Let us check that the assignment (F) is well
defined with respect to the tensor product. Let n and n' be
natural numbers such that n'≥ n, and [n'-n,σ]∈ Hom_𝔘β(n,n')
with σ∈𝐁_n'. Recall from Proposition <ref>
that id_2♮[n'-n,σ]=[n'-n,(id_2♮σ)∘((b_2,n'-n^β)^-1♮ id_n)].
On the one hand, by Condition <ref>,
we have: 

    (id_2♮σ)∘ς_1+n'(g_1)=ς_1+n'(a_1+n'(id_1♮σ)(g_1))∘(id_2♮σ).

Hence, it follows from Condition <ref>
that 

    (id_2♮σ)∘ς_1+n'(g_1)=ς_1+n'(g_1)∘(id_2♮σ).

On the other hand, Condition <ref>
gives that 

    g_1=a_2+n'((b_1,n'-n^β)^-1♮ id_n+1)(g_n'-n+1)

and by Condition <ref> we have 

    g_1=a_2+n'(id_1♮(b_1,n'-n^β)^-1♮ id_n)(g_1).

By the definition of the braiding b_-,-^β
(see Definition <ref>), we deduce that:

    ς_1+n'(g_1)     =    ς_1+n'(a_2+n'((b_2,n'-n^β)^-1♮ id_n)(g_n'-n+1)).

Then, it follows from the combination of Conditions <ref>
and <ref> that as morphisms in 𝔘β:

    [n'-n,ς_1+n'(g_1)∘((b_2,n'-n^β)^-1♮ id_n)]
    
    =    [n'-n,((b_2,n'-n^β)^-1♮ id_n)∘(id_n'-n♮ς_1+n(g_1))].

Hence, we deduce from the relations (<ref>) and (<ref>)
that:

    [n'-n,((id_2♮σ)∘((b_2,n'-n^β)^-1♮ id_n))∘(id_n'-n♮ς_1+n(g_1))]
    
    =    [n'-n,ς_1+n'(g_1)∘((id_2♮σ)∘((b_2,n'-n^β)^-1♮ id_n))].

A fortiori, F(id_2♮[n'-n,σ])∘ F(ς_1+n(g_1))=F(ς_1+n'(g_1))∘ F(id_2♮[n'-n,σ]).
Hence, our assignment is well defined with respect to the tensor product.

Let us prove that the subspaces (F)(n)
are stable under the action of 𝔘β.
Let i∈ℐ_𝕂[𝐅_1] and
v∈ F(n+2). We deduce from the definition of the monoidal
structure morphisms of 𝔘β (see Proposition
<ref>) and from the definition of the Long-Moody
functor (see Theorem <ref>) that, for all i∈ℐ_𝕂[𝐅_1]
and for all v∈ F(n+2):

    ((τ_1𝐋𝐌(F)([n'-n,σ]))∘υ(F)_n)(i𝕂[𝐅_1]v)
    
    =     a_1+n'(id_1♮σ)(a_1+n'((b_1,n'-n^β)^-1♮ id_n)(ι_ℐ_𝕂[𝐅_n'-n]*id_ℐ_𝕂[𝐅_1]*ι_ℐ_𝕂[𝐅_n])(i))
       𝕂[𝐅_n'+1]F(id_1♮ id_1♮[n'-n,σ])(v).

It follows from Condition <ref> that:

    a_1+n'((b_1,n'-n^β)^-1♮ id_n)(ι_ℐ_𝕂[𝐅_n'-n]*id_ℐ_𝕂[𝐅_1]*ι_ℐ_𝕂[𝐅_n])(i)=(id_ℐ_𝕂[𝐅_1]*ι_ℐ_𝕂[𝐅_n'])(i).

Since by Condition <ref>, a_1+n'(id_1♮σ)(id_ℐ_𝕂[𝐅_1]*ι_ℐ_𝕂[𝐅_n'])(i)=(id_ℐ_𝕂[𝐅_1]*ι_ℐ_𝕂[𝐅_n'])(i)
for all elements σ of 𝐁_n', we deduce that:

    (τ_1𝐋𝐌(F)([n'-n,σ])∘υ(F)_n)(i𝕂[𝐅_1]v)     =    (υ(F)_n'∘(F)([n'-n,σ]))(i𝕂[𝐅_m]v).

Therefore, the functorial structure of τ_1𝐋𝐌(F)
induces by restriction the one of (F).

Now, we can lift this link between (F)
of τ_1𝐋𝐌(F) to endofunctors of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).

Let F and G be two objects of
𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡),
and η:F→ G be a natural transformation. For all natural
numbers n, assign :

    ((η))_n=id_ℐ_𝕂[𝐅_1]𝕂[𝐅_1]η_n+2.

Then we define a subfunctor :𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)→𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
of τ_1𝐋𝐌 using the monomorphisms {υ(F)_n} _n∈ℕ. 



The consistency of our definition follows repeating mutatis mutandis
point (<ref>) of the proof of Theorem <ref>.
It directly follows from the definitions of ((η))_n,
υ(G)_n and τ_1∘𝐋𝐌 (see
Definition <ref>) that υ(G)_n∘()(η)_n=(τ_1∘𝐋𝐌)(η)_n∘υ(F)_n.

In fact, we have an easy description of the functor .

There is a natural equivalence
≅τ_2 where τ_2 is the translation functor
introduced in Definition <ref>.



Let F be an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).
By Proposition <ref>, for all natural
numbers n, we have an isomorphism:

    χ_n,F:ℐ_𝕂[𝐅_1]𝕂[𝐅_1]F(n+2)    ≅⟶    F(n+2).
    (g_1-1)𝕂[𝐅_n]v    ⟼    v

It follows from Definition <ref> and Proposition <ref>
that the isomorphisms {χ_n,F} _n∈ℕ
define the desired natural equivalence χ→τ_2.




  
The functor 𝐋𝐌∘τ_1:

Now, let us consider the part ℐ_𝕂[𝐅_n]𝕂[𝐅_n]F(n+2)
of τ_1∘𝐋𝐌(F)(n) in the
decomposition of Proposition <ref>. In fact,
we are going to prove that these modules assemble to form a functor
which identifies with 𝐋𝐌(τ_1F). We recall
from Theorem <ref> and Definition <ref>
the following fact.

The functor 𝐋𝐌∘τ_1:𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)→𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
is defined by:



  * for F∈ Obj(𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)),
∀ n∈ℕ, (𝐋𝐌∘τ_1)(F)(n)=ℐ_𝕂[𝐅_n]𝕂[𝐅_n]F(n+2),
where F(n+2) is a left 𝕂[𝐅_n]-module
using F(id_1♮ς_n(-)):𝐅_n→ Aut_𝕂-𝔐𝔬𝔡(F(n+2)).
For n,n'∈ℕ, such that n'≥ n, and [n'-n,σ]∈ Hom_𝔘β(n,n'):

    (𝐋𝐌∘τ_1)(F)([n'-n,σ])=a_n'(σ)(ι_ℐ_𝕂[𝐅_n'-n]*id_ℐ_𝕂[𝐅_n])𝕂[𝐅_n']F(id_1♮ id_1♮[n'-n,σ]).


  * Morphisms: let F and G be two objects of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡),
and η:F→ G be a natural transformation. The natural
transformation (𝐋𝐌∘τ_1)(η):(𝐋𝐌∘τ_1)(F)→(𝐋𝐌∘τ_1)(G)
for all natural numbers n is given by:

    ((𝐋𝐌∘τ_1)(η))_n=id_ℐ_𝕂[𝐅_n]𝕂[𝐅_n]η_n+2.





For all F∈ Obj(𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)),
the monomorphisms {ξ(F)_n} _n∈ℕ
(see Definition <ref>) allow to define a natural
transformation ξ'(F):(𝐋𝐌∘τ_1)(F)→(τ_1∘𝐋𝐌)(F)
where, for all natural numbers n: 

    ξ'(F)_n=(ι_ℐ_𝕂[𝐅_1]*id_ℐ_𝕂[𝐅_n])𝕂[𝐅_1+n]F((b_1,1^β)^-1♮ id_n).

This yields a natural transformation ξ':𝐋𝐌∘τ_1→τ_1∘𝐋𝐌.



Let n and n' be natural numbers such that n'≥ n, and [n'-n,σ]∈ Hom_𝔘β(n,n')
with σ∈𝐁_n'. Let i∈ℐ_𝕂[𝐅_n],
v∈ F(n+2) and g∈𝐅_n. By Condition
<ref> (using Lemma <ref>
with n'=n+1) the following equality holds in 𝐁_n+2:

    ((b_1,1^β)^-1♮ id_n)∘(id_1♮ς_n(g))=ς_1+n(e_𝐅_1*g)∘((b_1,1^β)^-1♮ id_n).

Recall that F(n+2) is a 𝕂[𝐅_n]-module
via F(ς_1+n∘(ι_𝐅_1∗ id_𝐅_n))
and τ_1F(n+1) is a 𝕂[𝐅_n]-module
via F(id_1♮(ς_n∘ id_𝐅_n)).
Then it follows that the assignment ξ'(F)_n is well-defined
with respect to the tensor product structures of (𝐋𝐌∘τ_1)(F)(n)
and (τ_1∘𝐋𝐌)(F)(n).
Moreover, we compute that:

    ((τ_1∘𝐋𝐌)(F)([n'-n,σ]))∘(ξ'(F)_n)(i𝕂[𝐅_n]v)
        =     a_1+n'(id_1♮σ)(a_1+n'((b_1,n'-n^β)^-1♮ id_n)(ι_ℐ_𝕂[𝐅_1+n'-n]*id_ℐ_𝕂[𝐅_n])(i))
          𝕂[𝐅_n'+1]F((b_1,1^β)^-1♮[n'-n,σ])(v).

It follows from Condition <ref> that:

    a_1+n'((b_1,n'-n^β)^-1♮ id_n)∘(ι_ℐ_𝕂[𝐅_1+n'-n]*id_ℐ_𝕂[𝐅_n])(i)=(ι_ℐ_𝕂[𝐅_1+n'-n]*id_ℐ_𝕂[𝐅_n])(i).

Again by Condition <ref>, we deduce
that:

    a_1+n'(id_1♮σ)∘(ι_ℐ_𝕂[𝐅_1+n'-n]*id_ℐ_𝕂[𝐅_n])(i)=ι_ℐ_𝕂[𝐅_1]*a_n'(σ)(ι_ℐ_𝕂[𝐅_n'-n]*id_ℐ_𝕂[𝐅_n])(i).

Hence, we deduce that:

    ((τ_1∘𝐋𝐌)(F)([n'-n,σ]))∘(ξ'(F)_n)=(ξ'(F)_n')∘((𝐋𝐌∘τ_1)(F)([n'-n,σ])).


Let η:F→ G be a natural transformation in the category
𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
and let n be a natural number. Since η is a natural transformation,
we have:

    G((b_1,1^β)^-1♮ id_n)∘η_n+2=η_n+2∘ F((b_1,1^β)^-1♮ id_n).

Hence, we deduce from the definitions of τ_1∘𝐋𝐌
(see Theorem <ref>) and of 𝐋𝐌∘τ_1
(see Remark <ref>) that:

    ξ'(G)_n∘(𝐋𝐌∘τ_1)(η)_n=(τ_1∘𝐋𝐌)(η)_n∘ξ'(F)_n.





  §.§.§ Splitting of the translation functor


Now, we can establish a decomposition result for the translation functor
applied to a Long-Moody functor.

There is a natural equivalence of
endofunctors of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡):

    τ_1∘𝐋𝐌≅τ_2⊕(𝐋𝐌∘τ_1).




Recall the natural transformations υ:→τ_1∘𝐋𝐌
(introduced in Proposition <ref>) and ξ':𝐋𝐌∘τ_1→τ_1∘𝐋𝐌
(defined in Proposition <ref>). The direct sum in
the category 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
(induced by the direct sum in the category 𝕂-𝔐𝔬𝔡)
allows us to define a natural transformation:

    υ⊕ξ':⊕(𝐋𝐌∘τ_1)⟶(τ_1∘𝐋𝐌)(F).

This is a natural equivalence since for all natural numbers n,
we have an isomorphism of 𝕂-modules according to Proposition
<ref>: (F)(n)⊕(𝐋𝐌∘τ_1)(F)(n)≅(τ_1∘𝐋𝐌)(F)(n).
We conclude using Proposition <ref>.




 §.§ Splitting of the difference functor


Recall the natural transformation i_1:Id_𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)→τ_1
of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).
Our aim is to study the cokernel of i_1∘𝐋𝐌. We recall
that for F an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡),
for all natural numbers n, (i_1𝐋𝐌)(F)_n=𝐋𝐌(F)([1,id_1+n])
(see Definition <ref>).

Explicitly for all elements i of ℐ_𝕂[𝐅_n],
for all elements v of F(n):

    (i_1𝐋𝐌)(F)_n(i𝕂[𝐅_n]v)=(ι_ℐ_𝕂[𝐅_1]*id_ℐ_𝕂[𝐅_n])(i)𝕂[𝐅_1+n]F(id_1♮ι_1♮ id_n)(v).






  
The natural transformation 𝐋𝐌∘ i_1:

Let us consider the exact sequence (<ref>) in the category
of endofunctors of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
of Proposition <ref>:

    0@->[r]    κ_1@->[r]^Ω_1    Id@->[r]^i_1   τ_1@->[r]^_1   δ_1@->[r]     0
    .

Since the Long-Moody functor is exact (see Proposition <ref>),
we have the following exact sequence:

    0@->[r]    𝐋𝐌∘κ_1@->[rr]^ooo𝐋𝐌(Ω_1)      𝐋𝐌@->[rr]^𝐋𝐌(i_1)      𝐋𝐌∘τ_1@->[rr]^𝐋𝐌(_1)      𝐋𝐌∘δ_1@->[r]     0
    .



From the definition of 𝐋𝐌 (see
Theorem <ref>), we deduce that for F an object of
𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡),
for all natural numbers n, for all elements i of ℐ_𝕂[𝐅_n],
for all elements v of F(n):

    𝐋𝐌(i_1)(F)_n(i𝕂[𝐅_n]v)=i𝕂[𝐅_n]F(ι_1♮ id_1♮ id_n)(v).

Recall the natural transformation ξ':𝐋𝐌∘τ_1→τ_1∘𝐋𝐌
introduced in <ref>.



As natural transformations from 𝐋𝐌
to τ_1∘𝐋𝐌, which are endofunctors of the category
𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡),
the following equality holds:

    ξ'∘(𝐋𝐌(i_1))=i_1𝐋𝐌.




Let F be an object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).
Let n be a natural number. Let i be an element of ℐ_𝕂[𝐅_n]
and let v be an element of F(n). Since (b_1,1^β)^-1∘(ι_1♮ id_1)=id_1♮ι_1
by Definition <ref>, we deduce from Proposition
<ref>, Remark <ref> and Remark
<ref>, that:

    (ξ'∘(𝐋𝐌(i_1)))(F)_n(i𝕂[𝐅_n]v)=(id_1*i)𝕂[𝐅_1+n]F(id_1♮ι_1♮ id_n)(v)=(i_1𝐋𝐌)(F)_n(i𝕂[𝐅_n]v).





  
Decomposition results:

Lemma <ref> leads to the following key results.

There is a
natural equivalence in the category 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡):

    δ_1∘𝐋𝐌≅τ_2⊕(𝐋𝐌∘δ_1).

Moreover, there is a natural isomorphism κ_1∘𝐋𝐌≅𝐋𝐌∘κ_1.



It follows from the definition of i_1 (see Proposition <ref>)
and from Lemma <ref> that the following diagram
is commutative and the row is an exact sequence:


    0@->[r]    κ_1∘𝐋𝐌@->[rr]^oooΩ_1𝐋𝐌      𝐋𝐌@->[rr]^i_1𝐋𝐌@=[d]       τ_1∘𝐋𝐌@->[rr]^_1𝐋𝐌      δ_1∘𝐋𝐌@->[r]     0
             𝐋𝐌@->[rr]^𝐋𝐌(i_1)      𝐋𝐌∘τ_1.@^(->[u]_by Lemma <ref>^ξ'

We denote by i_𝐋𝐌∘τ_1^⊕ the inclusion
morphism 𝐋𝐌∘τ_1↪τ_2⊕(𝐋𝐌∘τ_1).
The functor 𝐋𝐌∘κ_1 is also the kernel of the
natural transformation i_𝐋𝐌∘τ_1^⊕∘(𝐋𝐌∘ i_1),
as the inclusion morphism i_𝐋𝐌∘τ_1^⊕:𝐋𝐌∘τ_1↪τ_2⊕(𝐋𝐌∘τ_1)
is a monomorphism. Then, recalling the exact sequence (<ref>),
we obtain that the following diagram is commutative and that the two
rows are exact:


    0@->[r]    κ_1∘𝐋𝐌@->[rr]^oooΩ_1𝐋𝐌      𝐋𝐌@->[rr]^i_1𝐋𝐌@=[d]       τ_1∘𝐋𝐌@->[rr]^_1𝐋𝐌      δ_1∘𝐋𝐌@->[r]     0
    
    0@->[r]    𝐋𝐌∘κ_1@->[rr]_ooo𝐋𝐌(Ω_1)      𝐋𝐌@->[rr]_i_𝐋𝐌∘τ_1^⊕∘(𝐋𝐌(i_1))                  τ_2⊕(𝐋𝐌∘τ_1)@->[rr]_id_τ_2⊕(𝐋𝐌(_1))@->[u]_υ⊕ξ'^  by Proposition <ref>      τ_2⊕(𝐋𝐌∘δ_1)@->[r]     0

A fortiori, by definition of δ_1 (see Definition <ref>)
and the universal property of the cokernel, we deduce that:

    τ_2⊕(𝐋𝐌∘δ_1)≅δ_1∘𝐋𝐌.

Furthermore, by the unicity up to isomorphism of the kernel, we conclude
that κ_1∘𝐋𝐌≅𝐋𝐌∘κ_1.




 §.§ Increase of the polynomial degree


The results formulated in Theorem <ref> allow us
to understand the effect of the Long-Moody functors on (very) strong
polynomial functors.

Let F be a non-null object of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡).
If the functor F is strong polynomial of degree d, then:



  * the functor τ_2(F) belongs to 𝒫ol_d^strong(𝔘β,-𝔐𝔬𝔡);

  * the functor 𝐋𝐌(F) belongs to 𝒫ol_d+1^strong(𝔘β,-𝔐𝔬𝔡).




We prove these two results by induction on the degree of polynomiality.
For the first result, it follows from the commutation property 5
of Proposition <ref> for τ_2. For the second
result, let us first consider F a strong polynomial functor of
degree 0. By Theorem <ref>, we obtain that δ_1𝐋𝐌(F)≅τ_2(F).
Therefore 𝐋𝐌(F) is a strong polynomial functor
of degree less than or equal to 1. Now, assume that F is a strong
polynomial functor of degree n≥0. By Theorem <ref>:
δ_1𝐋𝐌(F)≅𝐋𝐌(δ_1F)⊕τ_2(F).
By the inductive hypothesis and the result on τ_2, we deduce
that 𝐋𝐌(F) is a strong polynomial functor of
degree less than or equal to n+1.


For all natural numbers d, the endofunctor 𝐋𝐌
of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
restricts to a functor:

    𝐋𝐌:𝒫ol_d^strong(𝔘β,-𝔐𝔬𝔡)⟶𝒫ol_d+1^strong(𝔘β,-).




Let d be a natural number and F be an
object of 𝒫ol_d^strong(𝔘β,-𝔐𝔬𝔡)
such that the strong polynomial degree of τ_2(F)
is equal to d. Then, the functor 𝐋𝐌(F) is
a strong polynomial functor of degree equal to d+1.



Let d be a natural number and F be
an object of 𝒱𝒫ol_d(𝔘β,-𝔐𝔬𝔡)
of degree equal to d. Then, the functor 𝐋𝐌(F)
is a very strong polynomial functor of degree equal to d+1.



Using Lemma <ref>, it follows from Corollary
<ref> that 𝐋𝐌(F) is a strong
polynomial functor of degree equal to n+1.  Since the functor 𝐋𝐌
commutes with the evanescence functor κ_1 by Theorem <ref>,
we deduce that (κ_1∘𝐋𝐌)(F)≅(𝐋𝐌∘κ_1)(F)=0.
Moreover, using Theorem <ref>, we have:

    (κ_1∘(δ_1∘𝐋𝐌))(F)≅(κ_1∘τ_2)(F)⊕(κ_1∘(𝐋𝐌∘δ_1))(F).

Therefore, the fact that τ_2 commutes with the evanescence
functor κ_1 (see the commutation property 6 of Proposition
<ref>) and Theorem <ref>
together imply that:

    (κ_1∘(δ_1∘𝐋𝐌))(F)≅(τ_2∘κ_1)(F)⊕(𝐋𝐌∘(κ_1∘δ_1))(F).

The result then follows from the fact that F is an object of 𝒱𝒫ol_n(𝔘β,-𝔐𝔬𝔡)
and τ_2 is a reduced endofunctor of the category 𝐅𝐜𝐭(𝔘β,-𝔐𝔬𝔡).


By Proposition <ref>, 𝔛 is a
very strong polynomial functor of degree 0. Now applying the Long-Moody
functor 𝐋𝐌_1, we proved in Proposition <ref>
that t^-1𝐋𝐌_1(t𝔛) is naturally
equivalent to 𝔅𝔲𝔯_t^2, which is very strong polynomial
of degree 1 by Proposition <ref>.





 §.§ Other properties of the Long-Moody functors


We have proven in the previous  section that a Long-Moody functor
sends (very) strong polynomial functors to (very) strong polynomial
functors. We can also prove that a (very) strong polynomial functor
in the essential image of a Long-Moody functor is necessarily the
image of another strong polynomial functor.

Let d be a natural number. Let
F be a strong polynomial functor of degree d in the category
𝐅𝐜𝐭(𝔘β,-𝔐𝔬𝔡).
Assume that there exists an object G of the category 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
such that 𝐋𝐌(G)=F. Then, the functor G is
a strong polynomial functor of degree less than or equal to d+1
in the category 𝐅𝐜𝐭(𝔘β,-𝔐𝔬𝔡).



It follows from Theorem <ref> that:

    δ_1F≅τ_2(G)⊕(𝐋𝐌∘δ_1)(G).

According to Corollary <ref>, the functor τ_2(G)
is an object of the category 𝒫ol_d-1^strong(𝔘β,-𝔐𝔬𝔡),
and because of Lemma <ref> the functor
G is an object of the category 𝒫ol_d+1^strong(𝔘β,-𝔐𝔬𝔡).


The Long-Moody functor 𝐋𝐌:𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡)⟶𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡)
is not essentially surjective.



Let l be a natural number. Let E_l:𝔘β⟶𝕂-𝔐𝔬𝔡
be the functor which factorizes through the category ℕ,
such that E_l(n)=𝕂^⊕ n^l for all
natural numbers n and for all [n'-n,σ]∈ Hom_𝔘β(n,n')
(with n, n' natural numbers such that n'≥ n), E_l([n'-n,σ])=ι_ℂ[t^±1]^⊕ n'^l-n^l⊕ id_ℂ[t^±1]^⊕ n^l.
In particular, for all natural numbers n, for every Artin generator
σ_i of 𝐁_n, E_l(σ_i)=id_𝕂^⊕ n^l.
It inductively follows from this definition and direct computations
that E_l is a very strong polynomial functor of degree l.

Let us assume that 𝐋𝐌 is essentially surjective. Hence,
there exists an object F of 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡)
such that 𝐋𝐌(F)≅ E_l. Because of the
definition of 𝐋𝐌(F) on morphisms (see Theorem
<ref>), this implies that for all natural numbers n
and for all σ∈𝐁_n, a_n(σ)=id_n.
Also, if 𝐋𝐌 is essentially surjective, there exists an
object T of the category 𝐅𝐜𝐭(β,𝕂-𝔐𝔬𝔡)
such that we can recover the Burau functor from 𝐋𝐌(T),
ie something like α𝐋𝐌(T) (see Notation
<ref>) with α∈𝕂. We deduce from the definition
of 𝐋𝐌(T) on objects and morphisms that for
all n≥1, T(n)=𝕂 and for all generator
σ_i of 𝐁_n:

    𝐋𝐌(T)(σ_i)=T(σ_i)· Id_n.

Then necessarily, for all i∈{ 1,…,n}, T(σ_i)=δ
such that δ^2=t and we consider δ^-1𝐋𝐌(T).
We deduce that there exists a natural transformation ω:δ^-1𝐋𝐌(T)≅→𝔅𝔲𝔯_t.
This contradicts the fact that for all σ∈𝐁_n,
a_n(σ)=id_n.


The proof of Proposition <ref> shows in
particular that a Long-Moody functor 𝐋𝐌 is not essentially
surjective on very strong polynomial functors in any degree.


In <cit.>, Birman
and Brendle ask “whether all finite dimensional unitary matrix representations
of 𝐁_n arise in a manner which is related to the construction”
recalled in Theorem <ref>. Since the Tong-Yang-Ma and
unreduced Burau representations recalled in Theorem <ref>
are unitary representations, the proof of Proposition <ref>
shows that any Long-Moody functor (and especially the one based on
the version of the construction of Theorem <ref>) cannot
provide all the functors encoding unitary representations. Therefore,
we refine the problem asking whether all functors encoding families
of finite dimensional unitary representations of braid groups lie
in the image of a Long-Moody functor.

Another question is to ask whether we can directly obtain the reduced
Burau functor 𝔅𝔲𝔯_t by a Long-Moody functor.
Recall that for all natural numbers n, 𝔅𝔲𝔯_t(n)=ℂ[t^±1]^⊕ n-1
and 𝐋𝐌(F)(n)≅(F(n+1))^⊕ n
for any Long-Moody functor 𝐋𝐌 and any object F of 𝐅𝐜𝐭(𝔘β,𝕂-𝔐𝔬𝔡)
(see Remark <ref>). Therefore, for dimensional considerations
on the objects, it is clear that we have to consider a modified version
of the Long-Moody construction. This modification would be to take
the tensor product with ℐ_𝐅_n-1 on 𝐅_n-1,
the 𝕂-module F(n+1) being a 𝕂[𝐅_n-1]-module
using a morphism 𝐅_n-1→(𝐅_n-1a_n'⋊𝐁_n+1)→𝐁_n+1
for all natural numbers n, where a_n':𝐁_n+1→ Aut(𝐅_n-1)
is a group morphism.


plain



IRMA, Universit de Strasbourg, 7 rue Ren Descartes, 67084
Strasbourg Cedex, France

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