On the representation theory of braid groups

This work presents an approach towards the representation theory of the braid groups $B_n$. We focus on finite-dimensional representations over the field of Laurent series which can be obtained from representations of infinitesimal braids, with the help of Drinfeld associators. We set a dictionary between representation-theoretic properties of these two structures, and tools to describe the representations thus obtained. We give an explanation for the frequent apparition of unitary structures on classical representations. We introduce new objects -- varieties of braided extensions, infinitesimal quotients -- which are useful in this setting, and analyse several of their properties. Finally, we review the most classical representations of the braid groups, show how they can be obtained by our methods and how this setting enrich our understanding of them.


Motivations
Artin braid group on n strands B n is one of the finitely generated, infinite and torsion-free group which appears most often in mathematics. Concurrently, and in particular in the past thirty years, lots of linear (finite-dimensional, characteristic-zero) representations appeared in the most diverse contexts. Although it seems illusory to aim at a complete classification of its representations, there is thus a real need for some understanding of its representation theory. In particular it seems useful to start unifying as much as possible these approaches on some common ground -even if it entails a restriction on the range of representations which can be considered.
The main difficulty in the investigation of the representations of this kind of groups originates in the fact that they usually belong to families depending on transcendant parameters -contrary to the finite groups, whose representation theory can be studied over the field of algebraic numbers. Rather than study representations on a pure (algebraically closed) field it is thus useful to consider on this field additional structures, for instance to assume that it is (the algebraic closure of) the quotient field of some local ring. Let be an arbitrary field of characteristic 0 and K = ((h)) be the field of formal series with coefficients in . The field K is of infinite transcendance degree over , and the field of matrix coefficients of a linear representation of B n over K or its algebraic closure K is a finitely generated extension of .
We choose to investigate representations of B n over ((h)). It is well-known that such representations can be obtained from the monodromy of flat vector bundles over X n /S n , where X n = {(z 1 , . . . , z n ) ∈ n | z i = z j ⇔ i = j}. Indeed, let π : B n → S n be the projection of the braid group onto the symmetric group, P n = Ker π the pure braids group ; one has P n = π 1 (X n ), B n = π 1 (X n /S n ). The algebraic variety X n has good properties with respect to P n : X n is an Eilenberg-MacLane space and is the complement of hypersurfaces in some projective space. Associated to it we then find a holonomy Lie algebra T n , and it can be shown (see Kohno [14]) that the representations sufficiently close to the trivial action on the same vector space are monodromies of representations of the holonomy Lie algebra. The analogous object for B n is the Hopf algebra B n defined as the semidirect product of the group algebra of S n and the envelopping algebra UT n . The geometric properties of X n imply that the defining relations of T n are homogeneous with respect to its natural generators. This allows us to introduce a scalar parameter h inside any representation of T n or B n . The computation of the monodromy in terms of K.T. Chen's iterated integrals thus yields representations of B n over the local ring of formal (and even convergent) power series with complex coefficients.
This linear structure associated to the group structure of B n is far more tractable than its group algebra, and the monodromy operation commutes with direct sums and tensor products. It follows that one may hope to consider B n as the Lie algebra of B n , in the sense that the representation theory of connected Lie groups essentially relies on its Lie algebra ; in particular we would like to have at disposal a convenient dictionnary which avoids the explicit computation of the monodromy, in order to decide whether the representation is irreducible, to decompose tensor products, or decide invariance with respect to bilinear forms.
Transcendant monodromy however presents some major drawbacks. First of all it can only lead to representations over (finitely generated extensions of) the field of complex numbers . But above all it depends on the choice of a base point. This lack of symmetry raises problems in the study of the dual representations and the invariance with respect to bilinear forms. In order to solve these problems, we use Drinfeld associators. In his article [11], Drinfeld expressed the universal monodromy with respect to some "base point at infinity" in a very simple way, in terms of a formal power series in two non-commuting variables Φ KZ (x, y). He then determined which were the algebraic equations that such a series Φ(x, y), with coefficients in any field of characteristic 0, had to fulfill so that the same formulas of universal monodromy satisfy the braid relations. Solutions of these equations are called (Drinfeld) associators, and Drinfeld proved in the same article that there exists an associator with coefficients in the field É of rational numbers.
The use of associators enables us to remove the two a priori obstacles to our purpose -a contrario, the main obstruction to the generalization of several aspects of this work to other similar groups, such as generalized Artin groups, is due to the fact that analogues of Drinfeld associators have not been defined and extensively studied yet (see however [25] and the references there for Artin groups of type I 2 (m) and B). Drinfeld associators and its analogues may be seen as non-commutative versions of Chevalley's formal exponentiation (see [7]).

Outline of the results
In the first place (part 2) we recall the basic notions on braids, infinitesimal braids and associators, which will be of use to us in the sequel. The third part achieves the program of setting a dictionnary between representations of B n over and representations of B n over K. Beyong linear representations, we give a uniform explanation to the apparition of unitary strutures on "monodromy" representations. Let us assume ⊂ . Once we have obtained representations of B n over the local ring [[h]] of formal power series, representations over a pure field can be deduced by at least two means : either by forgetting the local structure on ((h)) -whose algebraic closure is isomorphic to -or by specialization in h ∈ . This last method may be used only in case the matrix coefficients of the representation are convergent power series. We show in appendix A that, up to a field automorphism, this situation can always be assumed. We then get unitary representations by this method.
In view of understanding representation-theoretic aspects, a first advantage of B n over B n is that B n is a semidirect product, whereas the short exact sequence 1 → P n → B n → S n → 1 is not split. We thus define in part 4, for every representation of S n , the variety of all representations of B n of which it is the restriction. This yields a systematic approach to the search for (irreducible) representations of B n . We analyse in this part how much information on the corresponding representations of B n is contained in this variety. In particular, we show how its factorization through classical quotients of the braid groups can be detected from the original data. Moreover, a third standard manipulation of the local ring [[h]], namely reduction modulo h, leads to representations of B n over which are iterated extensions of representations of S n -in fact, of the irreducible components of the original representation.
At the other end of the fundamental exact sequence, another operation is the restriction to the pure braid group of the representation (part 5). The replacement of P n by the holonomy Lie algebra T n enlights noteworthy phenomenons for a large class of representations -in particular the "generic" irreducibility of tensor products, or the irreducibility of the restriction to various subgroups of B n . We study the consequences of these properties on the "infinitesimal quotients" of B n , i.e. Hopf quotients of B n whose structure explains in particular the decomposition of tensor products. We moreover indicate (appendix B) how it is possible to get, in particularly auspicious situations, an explicit (matrix) description of representations of B n from their infinitesimal version.
The last part in this work (part 6) reviews the most classical constructions of representations of the braid group : the Iwahori-Hecke and Birman-Wenzl-Murakami algebras, Yang-Baxter representations, and Long's generalization of Magnus induction. We show how our approach enforces the understanding we have on them. In particular we insist on the unitary structures which appear on the representations and also indicate the general properties of part 5 that they satisfy.

Beyond
We underline here a few directions of research which are suggested by the uniform approach settled here. We part them in two types. The first one is concerned with the analysis of standard algebraic or geometric structures, shown here to be relevant for the representation theory of the braid group.
Among them are the infinitesimal quotients defined in section 5.6, and the reductive Lie algebras associated to them. The work of decomposing the Lie algebra associated to representations of the Temperley-Lieb algebra and the Iwahori-Hecke algebra was done by the author in [17,26]. The next step would be to decompose the Lie algebra associated to the Birman-Wenzl-Murakami algebra of type A -the part corresponding to the Krammer representation is already done in [27]. Another kind of structures are the varieties of braided extensions V s (M ) defined in section 4.1. The analysis of these varieties and the study of their relationship with the space of extensions is another interesting task. Apart from the examples studied in part 4, the analysis was carried out for irreducible M (see [19,18]).
A special question of particular interest is whether all irreducible representations of all the generic cyclotomic Hecke algebras considered in section 6.1.1 can be deduced in some way from our approach -answering this question is a mainly computational but still delicate matter. A third kind of structure is given by the connection of "essentially pure" representations with the Deligne-Simpson problem (section 5.4), namely a description of which roots of the Kac-Moody algebras defined by W. Crawley-Boevey corresponds to (irreducible, essentially pure) representations of the braid group.
The second one is concerned with questions which are consequences of this work. A first one is to determinate the field of matrix coefficients for the representations Φ(ρ) studied here from the infinitesimal datas, and at least its transcendance degree. A second one is to find a criterium on ρ such that Φ(ρ) is faithful. For the time being, the approaches to faithfulness questions are based on thorough studies of special representations, such as the Burau representation or the representation of the Birman-Wenzl-Murakami algebra which was intensively studied and shown to be faithful, algebraically by D. Krammer and geometrically by S. Bigelow. Now that the linearity of the braid group is known, we need to find a general criterium for faithfulness. It is our hope that such a criterium could be expressed in terms of the infinitesimal datas -weak evidence in this direction can be found in section 4.2. However, while the delicate questions of whether the Hecke algebra representation and the Burau representation for n = 4 are faithful remain unsettled, it seems prematurate to state even vague conjectures. The third one is about the "inverse problem", that is when does a representation R of B n "comes from" some representation ρ of B n . This third question is first not to answer this very vague formulation, but to specify what is meant. Clearly, some representations of B n cannot be isomorphic to some Φ(ρ), because their field of matrix coefficients is an algebraic extension of É -however, it may still be possible to get them by specialization (e.g. the representations of the Iwahori-Hecke algebra at roots of unity). Some other representations cannot be equal to some Φ(ρ), but are twists of some Φ(ρ) by elements of Gal(K/ ) or other field morphisms (see the 4-dimensional example in section 6.1.1, or take any Φ(ρ) and replace h by h 2 ). We then wonder how many (semisimple) representations of the braid group it is possible to get from representations of the form Φ(ρ) by an alternating use of these operations, namely manipulations of the field of matrix coefficients and representationtheoretic operations such as taking direct sums, sub-modules and quotients.
General Notations. By convention, all rings are commutative with unit, all algebras are associative, all representations are finite dimensional, and all Hopf algebras have antipodes. All the fields occuring in the text are of characteristic 0. Unless otherwise stated, n (the number of strands) and N (the dimension of the representation) are integers with n ≥ 2, N ≥ 1, is an arbitrary field of characteristic 0. Whenever g is a Lie algebra, Ug is its universal envelopping algebra. Whenever A is a ring, we shall denote by A × its set of invertible elements, by M N (A) the set of N × N matrices and, if G is a group, by A G the group algebra of G over A. We let m , r and µ r for r ≥ 1 be the algebraic varieties whose A-points are A × , A r and the set of r-th roots of 1 in A. We use the notation diag(a 1 , . . . , a N ) to designate diagonal matrices in M N (A) whose diagonal coefficients are a 1 , . . . , a N . We let K be the algebraic closure of the field K, and denote by Gal(L/K) the Galois group of an extension of L over K. Finally, if R is a representation, either of a group or of a Hopf algebra, we shall denote by R ∨ the dual representation, and by ½ the trivial representation.

Braids
Let n ≥ 2 be an integer. In the sequel, we shall denote by S n the symmetric group on n letters, and by s i the transposition (i i+1) for 1 ≤ i < n. The relations between the s i 's give a presentation, found by Moore in the end of the 19-th century, of the symmetric group : The braid group on n strands B n is defined by generators σ 1 , . . . , σ n−1 and relations We let π be the (non-split) projection B n → S n given by π(σ i ) = s i . The kernel P n of π is by definition the pure (or colored) braid group, and is generated by the elements Other fundamental elements of B n are γ n = (σ 1 . . . σ n−1 ) n , δ n = σ n−1 . . . σ 2 σ 2 1 σ 2 . . . σ n−1 . It is readily checked that γ n and δ n belong to P n . By convention we let S 1 = P 1 = B 1 = {e} (the trivial group), and we mention the obvious isomorphisms P 2 ≃ ≃ B 2 .
The following algebraic facts are known. See the book by Birman [4] and the references there for the most classical ones. The groups P n and B n are torsion free. For n ≥ 3 the center of B n is the same as the center of P n , is infinite cyclic and generated by γ n . Each B n (resp. P n ) embeds into B n+1 (resp. P n+1 ) by σ i → σ i (resp. ξ ij → ξ ij ) ; this will be refered to as the "usual inclusion" B n ⊂ B n+1 . Hence each γ r and δ r , for 2 ≤ r ≤ n, will be implicitly considered as an element of B n . The elements ξ 1,n , ξ 2,n , . . . , ξ n−1,n generate a free normal subgroup F n in P n , whose centralizer in P n is the center of P n , and the subgroup generated by σ 2 1 , . . . , σ 2 n−1 is "locally free" : this means that the relations between these elements are generated by the obvious commutation relations σ 2 i σ 2 j = σ 2 j σ 2 i for |i − j| ≥ 2 (see [9]). The elements δ 2 , . . . , δ n generate a free abelian subgroup in P n . It is readily checked that γ n = γ n−1 δ n−1 = δ n−1 γ n−1 , and δ n = σ n−1 δ n−1 σ n−1 . The action by conjugation of B n+1 on its normal subgroup P n+1 restricts to an action of B n ⊂ B n+1 which leaves the free subgroup generated by ξ 1,n+1 , . . . , ξ n,n+1 invariant. Hence induces an action (so-called Artin action) on a free group on n generators ; this action is known to be faithful. By the usual inclusion P n−1 ⊂ P n we have P n = P n−1 ⋉ < ξ 1,n , . . . , ξ n−1,n >, hence P n is an iterated extension of free groups.
We proceed by recalling some well-known quotients of B n . First note that all σ i 's belong to the same conjugacy class in B n . Likewise, all ξ i,j belong to the same conjugacy class in B n -but not in P n . The quotient of B n by its subgroup of commutators [B n , B n ] is infinite cyclic, generated by the image of any σ i . A canonical isomorphism between B n /[B n , B n ] and is given by the so-called length function l ; one has l(σ i ) = 1 for 1 ≤ i ≤ n. The quotient of B n by the subgroup of commutators of P n will be denoted by S n and refered to as the enhanced symmetric group. This probably non-standard terminology is a translation -and a specialization to type A -of Tits notion of groupes de Coxeterétendus (see [33]). The kernel of the projection S n → S n is free abelian of rank n(n − 1)/2, with free generators given by the images of the ξ ij 's in S n . Finally, let Hurw n be Hurwitz' group, also called the braid group of the sphere ; it is defined as the quotient of B n by its normal subgroup generated by δ n .

Infinitesimal braids
The Lie algebra of infinitesimal pure braids on n strands T n is defined over É by generators It is endowed with the natural grading defined by all the generators t ij being of degree 1. The sum of all these generators is easily seen to be central in T n , and even to generate its center. A remarkable set of homogeneous elements of degree 1 in T n is the following. We define, for thus Y 1 = 0 and Y 2 = t 12 . Considering T r for r < n as embedded in T n in the natural way, it is clear that each Y r belongs to T r and commutes to T r−1 . It follows that the Y r 's commute. The elements t ij and Y r are to be seen as infinitesimal analogues (or, better, "residues") of the elements ξ ij and δ r of P n . Likewise, the Lie subalgebra generated by t 1,n , . . . , t n−1,n is shown to be free and an ideal of T n , which is a semi-direct product of T n−1 and this Lie algebra. It is shown in [8] that the centralizer of F n is the center of T n -in the same way that the centralizer of F n in P n is the center of P n . There exists a natural action of S n on T n (hence also on UT n ) which preserves the grading : it is defined by s.t ij = t s(i)s(j) for s ∈ S n . The semi-direct product ÉS n ⋉ UT n is a Hopf algebra denoted by B n , and we call it the algebra of infinitesimal braids. The sum of the elements t ij remains central in B n . This algebra is naturally graded, with deg t ij = 1 and deg s = 0 for s ∈ S n . We denote by B n its completion with respect to this grading. Let L be the (associative) algebra defined by generators s, Y, Y ′ , t and relations The algebra L is graded, with Y, Y ′ and t of degree 1, and s of degree 0. Consequences of the defining relations are that Y + Y ′ is central in L and that [Y . It is easily checked that, for every n ≥ 3, there exists a graded algebra morphism L → B n given by This morphism naturally extends to the completions with respect to the grading, whence a morphism from the completion L of L to B n .

Associators
In its fundamental article [11], Drinfeld defined, for every commutative É-algebra , and every λ ∈ , the set ××Ó λ ( ) of all formal series Φ in two non-commuting variables A and B, which satisfy the following relations : where ××Ó λ ( ) is Drinfeld's M µ ( ) (see [11], §5) with µ = 2λ. The equation (3) is called the hexagonal relation, the equation (4) the pentagonal relation -these names are motivated by MacLane coherence conditions for monoidal categories. In relation (1), the symbol⊗ denotes the completed tensor product associated to the identification of the algebra ≪ A, B ≫ of formal series in non-commuting variables A and B with the completed envelopping bigebra of the free Lie algebra on two generators. The equation (4)
The state of knowledge about these associators is roughly as follows. Drinfeld defined an explicit associator Φ KZ ∈ ××Ó iπ ( ) and proved by nonconstructive means, not only that ××Ó 1 (É) = ∅, but also that ××Ó • } denotes the set of even associators -in particular, one can choose α = 0 in the expansion of proposition 1. The associator Φ KZ is not even, as show its first terms : where ζ is the Riemann zeta function. More generally, Le and Murakami [15] gave an explicit formula for its coefficients, which involves multiple zeta values. Deep transcendance conjectures on these values lead many people to think that all the algebraic relations over É between the multiple zeta values may be equivalent to those implied by the equations (1)(2)(3)(4). There now exists a rather clear algebraic picture of what these algebraic relations should be ; assuming these relations to be the only ones, it is then possible to guess explicit -but still complicated -formulas for rational even associators. In particular, Jean Ecalle announced the existence of a conjectural canonical even rational associator. In this paper we will not anticipate on the developpments in this field, and we will choose arbitrary associators in accordance with Drinfeld existence theorems. It should however be kept in mind that, although the formulas which involve even associators are non-explicit in the actual state of knowledge, it should not be so in a (rather) near future. In addition to explicit formulas, another unknown fact which could be of use to us concerns the convergence of these associators, when is a complete topological field. It can be shown that Φ KZ (A, B) is not universally convergent, since the coefficient of A n−1 B (resp. AB n−1 ) is ζ(n) ≥ 1. However, one easily shows, from Le and Murakami formulas, that Φ KZ (A, B) converges at least for ||A|| ≤ 1 4 and ||B|| ≤ 1 4 , whenever A and B lies in some Banach algebra. For general associators, we did not hear of any result concerning convergence. We do not know, in particular, whether there exists convergent Φ ∈ ××Ó 1 (Ê). One only knows that these cannot be universally convergent (see [25]). In order to repair this problem, we prove in appendix A several approximation results which help us making the series converge. Further developments in the field may (or may not) make these tools less necessary.

Drinfeld isomorphisms
In the same paper [11], Drinfeld states that, for any Φ ∈ ××Ó λ ( ), there exists an homomorphism from B n to the completion B n of B n , given by In particular, σ 1 → s 1 e λt 12 and σ 2 → Φ(t 23 , t 12 )s 2 e λt 23 Φ(t 12 , t 23 ). Note that equation (4) is not needed for the case n = 3. In order to lighten notations we identify here, whenever Φ is fixed, B n with its image in B n .
Let us notice that the image under Drinfeld isomorphism of σ n−1 ∈ B n is the image in by the morphism L → B n that we already defined. Using the expression of Φ(A, B) up to the order 3, with parameters λ and α, we now get the expression of σ up to the order 3.
Proof. One has Since e λt equals 1 + λt plus terms of order at least 2, Y ]], we get the conclusion.
These morphisms commute with the "addition of a strand" in the sense that the usual inclusions B n ֒→ B n+1 , B n ֒→ B n+1 along with the Drinfeld morphism B r → B r for r = n, n + 1 associated to the same Φ ∈ ××Ó λ ( ) form a commutative square. If n = 4 there is in addition a section of the usual inclusion B 3 ֒→ B 4 defined by σ 3 → σ 1 , as well as a section of the usual inclusion B 3 ֒→ B 4 . This last one is defined by mapping t i,4 to t k,l where {i, k, l} = {1, 2, 3} and S 4 to S 3 by s 1 , s 3 → s 1 and s 2 → s 2 . The corresponding square involving Drinfeld morphisms also commutes, as can be easily checked. We study finite dimensional representations of a finitely generated group G over K. We first deal with the general case before specializing to G = B n and to representations obtained through associators. In case has cardinality at most the continuum, the algebraic closure of ((h)) is isomorphic to , so these representations can be seen (in a highly non-canonical way) as ordinary representations of G. However their field of matrix coefficients, as finitely generated extensions of , have more structure. In particular their transcendance degree r has important meaning, namely that such a representation can be considered as a r -dimensional family of representations of G over .

Lifting properties
Our aim here is to establish properties of a representation of G over A or K from the study of its coefficients in h. In order to distinguish these coefficients from the matrix coefficients of the representation, and because this appellation has some geometrical meaning, we call them "infinitesimal datas". It turns out that the -algebra generated by these infinitesimal datas already contains an important part of the representation-theoretic information about the representation (see propositions 5, 7 and 8). In order to prove this, we first need to draw a parenthesis about idempotents in GL N (A

Conjugation of idempotents and Hensel lemma
Let S ∈ GL N (A) be a symmetry, i.e. an element such that S 2 = 1. Because S ∈ GL N ( ) is also a symmetry with the same trace, S is conjugated to S in GL N (K). This conjugation also holds in GL N (A) because of the following identity (1 + SS)S = S + S = S(SS + 1).
Since 1 + SS = 2, 1+SS is invertible and its inverse belongs to GL N (A). Because symmetries and projectors are linked by linear (rational) relations, the same result holds for idempotents. These two facts are particular instances of the following non-commutative version of Hensel lemma. Proof. For any n ≥ 0, we let I n = n r=0 a n−r a r ∈ A [[h]]. In particular I 0 = 1, I 1 = a + a. We have I n a − a n+1 = aI n − a n+1 . Let us write Q(X) = d r=0 c r X r with c r ∈ , and define P = d−1 r=0 c r+1 I r . Since it follows that P a = aP , and one easily checks P = Q ′ (a).

General setting
Let A be a -algebra with unit and G a group. We are interested in triples (R, ρ, N ) such that N is a positive integer, R : G → GL N (A) and ρ : A → M N ( ) are representations of G and A related by the following conditions : In the sequel, if the integer N is implicit we eventually drop the last index and consider pairs (R, ρ) satisfying (i) and (ii). The first condition implies that the coefficients in h of R (its infinitesimal datas) belong to the image of ρ ; it also implies that the images of elements of the group algebra KG which belong to M N (A) have their reduction modulo h inside ρ(A). Condition (ii) forces the reverse inclusion to be true. We sometimes have to be more specific about this last condition. Let m = hA be the maximal ideal of A. Let us assume that we are given a family S of generators for thealgebra A, and a function f : A → KG. Then consider the following condition on the pair (R, ρ).
(ii) f ∀j ∈ J R(f(a j )) ∈ ρ(a j ) + M N (m) It is clear that (i) and (ii) f imply (i) and (ii). Conversely, if (R, ρ) satisfy (i) and (ii), then (R, ρ) satisfy (i) and (ii) f for some f. The specification of the function f is useful in the study of intertwinners. For any triples (R 1 , ρ 1 , N 1 ) and (R 2 , ρ 2 , N 2 ), we shall denote by Hom (ρ 1 , ρ 2 ) the set of intertwinners of ρ 1 and ρ 2 , and We then have the following result.

Proof. We identify Hom
We assume by contradiction that there exists e ∈ Hom A (R 1 , R 2 ) which do not belong to Hom (ρ 1 , ρ 2 ) ⊗ A. Since we already proved the reverse inclusion, and since A contains [h], we may assume e ∈ Hom (ρ 1 , ρ 2 ). Then for every Thus eR 1 (b) = R 2 (b)e implies eρ 1 (a) = ρ 2 (a)e, a contradiction. Finally, the identity proves the last assertion.

Indecomposability and irreducibility
Since every couple (R, ρ) satisfying (i) and (ii) satisfies (i) and (ii) f for some f, proposition 5 proves in particular that End A (R) = End (ρ) ⊗ A and End K (R) = End (ρ) ⊗ K. Given such a pair (R, ρ), we would like to lift indecomposability, i.e. to relate the potential indecomposability of R and ρ. Here we have to be careful because, if ρ is indecomposable, then End (ρ) is a local -algebra and, if End (ρ) contains a non-invertible element u = 0, then submodules. We then have Proof. If ρ is decomposable, then R is decomposable because of condition (i). Now assume that R is decomposable. This means that there exists a non-trivial idempotent p ∈ End A (R), then conjugated in GL N (A) to its constant term p by proposition 4. Thus p is a non-trivial idempotent in End (ρ) by proposition 5, and ρ is decomposable.
By definition, a representation R over the local ring A is called (absolutely) irreducible if and only if it is (absolutely) irreducible over its quotient field K. Proposition 7. Let (R, ρ) be a pair satisfying (i) and (ii). Then ρ is irreducible iff R is irreducible.

and this concludes the proof.
When ρ is irreducible and is algebraically closed, in which case ρ is absolutely irreducible, it is not clear a priori that R is absolutely irreducible, because ((h)) is not algebraically closed. However, this is true : Proposition 8. Let (R, ρ) be a pair satisfying (i) and (ii). Then ρ is absolutely irreducible iff R is absolutely irreducible.

Proof. Let us recall that ρ is absolutely irreducible if and only if it is surjective, and that R is absolutely irreducible if and only if
Because of condition (i), this implies that m belongs to the image of ρ. Conversely, let us assume that ρ is surjective. There exists then a 1 , . . . , a N 2 ∈ A such that ρ(a 1 ), This family then forms a K-basis of M N (K), thereby proving that R is absolutely irreducible.

Representations of B n
Let Φ ∈ ××Ó λ ( ) with λ ∈ × . This associator Φ defines a morphism from the group algebra B n to B n , which is a Hopf algebra morphism for the canonical structures on B n and B n . On the other hand, if ρ : We study general properties of these functors Φ, from the category of finite-dimensional B n -modules to the category of representations of G over A.

Linear representations
Since the Drinfeld morphism B N → B n preserves coproducts and antipodes, the functor Φ commutes with tensor products and duals : for any representations ρ 1 , ρ 2 of B n , Let ρ : B n → M N ( ) be a representation of B n , R = Φ(ρ). Then the pair (R, ρ) satisfies the properties (i) and (ii) of the previous section, for A = B n and G = B n . Moreover, it satifies (ii) f for a universal function f independant of the given pair. Indeed, let us choose as generators of B n the set . Consequences of the above results can be condensed in the following theorem.
Then Φ preserves indecomposability, irreducibility and absolute irreducibility. It commutes with direct sums, tensor products and duals. Moreover, for any given representations ρ 1 : B n → M N 1 ( ) and ρ 2 : B n → M N 2 ( ), we have In addition, note that Φ is exact, since the functor ⊗ A from the category of finite dimensional vector spaces over to the category of free A-modules is exact. Finally, Φ obviously commutes with the natural restrictions corresponding to the removal of the last strand -with a slight imprecision in the notations, we may write : The determination of the field of matrix coefficients of Φ(ρ) is a far more delicate matter. A coarse lower bound for its transcendance degree over is given by the trivial case n = 2 : since B 2 is isomorphic to , we only have to determine the field L of matrix coefficients of exp(hX) when X ∈ M N ( ) is known. At least if is algebraically closed, a straightforward use of Jordan canonical form shows that, if the spectrum Sp(X) of X equals {a 1 , . . . , a r }, then L = (e ha 1 , . . . , e har ) if X is semisimple and L = (h, e ha 1 , . . . , e har ) otherwise. In particular, writing dim É S for the dimension of the É-vector space spanned by S ⊂ , its transcendance degree over is equal to dim É Sp(X) if X is semisimple and dim É Sp(X) + 1 otherwise.
do not depend on Φ and factorizes through S n . In the general case, it seems difficult to get a non-trivial upper bound for the transcendance degree ; a thiner lower bound is given by taking into account the spectrum of the ρ( Unfortunately, these eigenvalues are usually rational linear combinations of Sp(ρ(t 12 )) -see appendix B -so this may not improve the lower bound at all.
A last feature of this construction, and another consequence of the lifting properties is the following. Let A be one of the envelopping algebras UT n , UF n , UI n , where we denote by F n (resp. I n ) the Lie subalgebra of T n generated by the t i,n (resp. t i,i+1 ) for 1 ≤ i ≤ n − 1, and G be the corresponding group P n , F n or I n , where we denote by F n (resp. I n ) the subgroup of P n generated by ξ i,n (resp. σ 2 i ) for 1 ≤ i ≤ n − 1. Let ρ : B n → M N ( ) be a representation and consider R = Φ(ρ). It can be easily checked that whence an immediate consequence of the lifting properties is the following. Proposition 9. Let λ ∈ × , Φ ∈ ××Ó λ ( ) and ρ : B n → M N ( ). Let (H, g) be one of the following pairs : (P n , T n ), (F n , F n ), (I n , I n ). Then Res H Φ(ρ) is indecomposable (resp. irreducible, absolutely irreducible) if and only if Res g ρ is so.

Orthogonal, symplectic and unitary representations
An immediate corollary of theorem 1 is that the functors Φ for Φ ∈ ××Ó λ ( ), λ ∈ × , preserve semisimplicity. It is thus a natural question to ask whether B n -invariant bilinear or sesquilinear form can be detected at the infinitesimal level. This is all the more challenging that it appeared in the past twenty years, starting from the Squier form on the Burau representation, that the most classical representations of the braid group are naturally endowed with a "unitary" structure. In order to answer this question, we first need some notations.
Let L be a field of characteristic 0, N a positive integer and β a non-degenerate bilinear form on L N . To any x ∈ M N (L) we associate its transpose x † with respect to β : β(xu, v) = β(u, xv). As usual, If β is symmetric, we denote by O N (L, β) the group of isometries in GL N (L). If β is skew-symmetric, we denote it by SP N (L, β). If L is embedded in some larger field L ′ , then β is trivially extended to a non-degenerate bilinear form β ′ on (L ′ ) N . For the sake of simplicity, we write O N (L ′ , β) = O N (L ′ , β ′ ) and SP N (L ′ , β) = SP N (L ′ , β ′ ). Now assume that L admits an involutive non-trivial field automorphism ǫ. Given a non-degenerate symmetric bilinear form β on L N , we make ǫ act on M N (L) coefficientwise and define If β is implicit or is the standard bilinear form on L N we simply write U ǫ N (L). In particular, the ordinary unitary group U N is defined as U z →z N ( ). Recall that the Squier form of the Burau representation (see [30]) sends B n to U ǫ n (Ê(q)) where ǫ ∈ Gal(Ê(q)/Ê) is defined by ǫ(q) = q −1 , and ordinary unitary representations are obtained by specializing at q ∈ of modulus 1. On K, we will consider the field automorphism ǫ defined by f (h) → f (−h), i.e. the only automorphism in Gal(K/ ) continuous for the h-adic topology such that ǫ(h) = −h.
The conditions on the infinitesimal datas in order for our construction to provide orthogonal, symplectic or unitary representations of B n are the following ones.
Under these conditions, we have the following properties.
Proof. It is sufficient to show that Φ(ρ)(σ i ) is isometric for all 1 ≤ i < n. For this, we use its explicit expression. Since O N ( , β) ⊂ O N (K, β) and SP N ( , β) ⊂ SP N (K, β), ρ(s i ) is isometric. The same assertion for exp(λhρ(t ij )) is standard under these hypothesis. Now we have to show that Φ(hx, hy) for x, y ∈ M N ( ) is isometric as soon as x † = −x, y † = −y. This follows from the fact that Φ is the exponential of a Lie series Ψ, and that the set of skew-symmetric elements forms a Lie subalgebra of M N ( ) = gl n ( ). Hence Ψ(hx, hy) † = −Ψ(hx, hy) and Φ(hx, hy) † = Φ(hx, hy) −1 .
. It remains to show that Φ(hx, hy) belong to U ǫ N (K, β) as soon as x and y are selfadjoint. We know that Φ is the exponential of a Lie series Ψ. Let us introduce the outer Lie algebra automorphism τ : In case is a subfield of Ê, the field of coefficients of Φ(ρ) is a finitely generated extension of contained in Ê((h)), not necessarily in the field of convergent power series Ê({h}). We show in the appendix how this representation may be twisted by field isomorphisms in order to get convergent coefficients. It is then possible to get ordinary unitary representations of B n from representations of B n into U ǫ N (K), by specialization in h small and real -we refer to appendix A for more details on this topic.

Variety of braided extensions
One of the major discomforts in the study of braid group representations is that the extension 1 → P n → B n → S n → 1 of the symmetric group is not split. In particular we can not use the well-known representation theory of the symmetric group and study a restriction to the symmetric group of a given representation of B n .
Things change drastically with the infinitesimal version B n , which is a semi-direct product. It then appears an important algebraic variety associated to any S n -module M . Its set of -points is formed by all representations of B n whose restriction to S n is M . For reasons that we shall make clear later, we call this variety the variety of braided extensions of the module M .

Definition and elementary properties
Let M be a finite-dimensional S n -module. We define In order to have at disposal a more tractable definition of the same object, we introduce the following algebraic functions on End (M ) : Proof. Let S be the set introduced in the statement. If ρ ∈ V(M ) and τ = ρ(t 12 ), we have C 1 (τ ) = ρ([t 12 , t 13 + t 23 ]), C 2 (τ ) = ρ([t 12 , t 34 ]) so τ ∈ S and the map is well-defined. Since t 12 and S n generate B n it is injective. Surjectivity is a consequence of Frobenius reciprocity law Indeed, let e {i,j} for 1 ≤ i = j ≤ n be a basis of the induction of the trivial representation of S 2,n−2 , such that the action of S n is given by s.e {i,j} = e {s(i),s(j)} . It is readily checked that the canonical isomorphism given by Frobenius reciprocity law is such that this map are disjoint provided that M 1 is not isomorphic to M 2 . Indeed, let χ be the character of S n associated to the S n -module M . Then the trace of Φ(ρ)(σ) for σ ∈ B n belongs to A and its reduction modulo h is χ • π(σ), hence χ is determined by Φ(ρ) ∈ Hom(B n , GL N (K)) and so is M up to isomorphism. In accordance with the general conventions about variety of representations, we consider absolutely irreducible representations instead of irreducible ones. In case λ ∈ × , the elements in V(M ) which correspond to these are precisely the absolutely irreducible representations of B n , i.e. elements ρ ∈ V(M ) which are surjective. The tensor product of representations of B n admits as infinitesimal counterparts the mor- with zero trace. These maps are injective on the -points, and closed immersions at the algebraic level. The dual corresponds to the involution τ → −τ on V(M ). Also note that τ ∈ V(M ) has trace α ∈ iff the corresponding representation of B n deduced from some Φ ∈ ××Ó λ ( ) is such that (one whence) every σ i act with determinant exp(λhα). In view of the transcendance degree of the field of matrix coefficients, another important information encoded in V(M ) is the dimension of the É-vector space spanned by the eigenvalues of τ ∈ V(M ) -i.e. the roots of its characteristic polynomial. As an illustration, let us look for monic polynomials P ∈ É[X] of degree N > 1, such that the É-linear span of its roots has dimension N . First P cannot admit any rational root, for the coefficient of X N −1 in P would provide a rational linear relation. For the same reason, this coefficient cannot be zero. Finally, P of course has to be prime with its derivative P ′ . These conditions are not yet sufficient (see P = (X − 1) 4 − 2). They nevertheless imply that this É-linear span is at least 2-dimensional. Let α 1 , . . . , α N ∈ É be the roots of P and L = É(α 1 , . . . , α N ). A sufficient additional condition is that G = Gal(L/É) acts 2-transitively on the roots. It is indeed a classical fact that the permutation ÉG-module with basis {e α i | 1 ≤ i ≤ N } and action g.e α i = e g(α i ) then splits into two components, the invariant vector e α 1 + . . . + e α N and a (N − 1)-dimensional irreducible supplement. The kernel of its natural projection onto the É-linear span of the roots do not contain the invariant vector (the coefficient of X N −1 is non-zero) and do not contain its supplement because the image has to be of dimension at least 2. Then this morphism is bijective and the É-linear span of the roots has dimension N .
Other standard properties that can be read on V(M ) are dealt with in the next section.
The study of V(M ) = V s (M ) for irreducible M was carried out in [18] and the result is exposed in [19]. For example, if M is the irreducible 2-dimensional representation of S 3 , there exists matrix models of M such that (1 2) acts as the diagonal matrix diag(1, −1), and the S 2,n−2 -invariance condition implies that τ = diag(a, b) for a, b ∈ . We then have We recall from this earlier work that, even when M is irreducible, the variety V s (M ) may admit several irreducible components. As a concrete example for non-irreducible M , we now study the case where M is the sum of the 2-dimensional irreducible representation of S 3 and of the trivial one. Assuming that contains square roots of 3, we use Young orthogonal models to write down matrices for s 1 , s 2 and for the general form of τ ∈ End S 2,1 (M ).
The representation is absolutely irreducible iff b = 0 and c = 0. Note that two such τ 's define isomorphic representations iff the products bc are the same. An easy calculation shows that, In view of the field of matrix coefficients for the associated representations of B n we note that, provided is large enough, the parameters can be chosen so that these three eigenvalues are linearly independant over É. For instance, if ⊃ É( ) it is possible to get four eigenvalues linearly independant over É (take x = 1, y = 1 + √ 3, u = 1, v = 2). Moreover, 2t 12 + 2t 13 + 2t 23 acts by the scalar 3(x + y), which is not an integer linear combination of the eigenvalues of τ . It follows that the field of matrix coefficients of R = Φ(ρ) for Φ ∈ ××Ó λ ( ), λ ∈ × is larger than the field generated by the eigenvalues of R(σ 1 ), since R(γ 3 ) is then the exponential of 3λh(x + y). Let

Quotients of the braid group
Let M be a S n -module, τ ∈ V s (M ) and ρ : B n → End (M ) be the corresponding representation with τ = ρ(t 12 ). We choose once and for all Φ ∈ ××Ó λ ( ) with λ ∈ × and let R = Φ(ρ). We look for conditions on τ in order to characterize factorizations of R through the usual quotients Hurw n , = B n /[B n , B n ] and S n . We also consider the map π × l from B n to S n × . We let T be the sum of the ρ(t ij ) for 1 ≤ i < j ≤ n and recall that R designates the reduction modulo h of R. We assume that n ≥ 3 and state the results as a list of lemmas.
Proof. Indeed, this element T commutes to ρ(B n ), hence is scalar because ρ is absolutely irreducible. It is then zero if and only if its trace is zero. But since all the ρ(t ij )'s are conjugated in B n , it follows that tr(T ) = 0 iff tr(τ ) = 0. The last assertion is a direct consequence of R(γ n ) = exp(2λhT ).
Proof. The first part comes from R(δ n ) = exp(2λhρ(Y n )). If ρ(Y n ) = 0, it of course commutes with ρ(S n ). Conversely, for n ≥ 3, if ρ(Y n ) commutes to S n , it also commutes to ρ(B n ) since it commutes to ρ(t 12 ). It is then a scalar, and the equality tr(ρ(Y n )) = (n − 1)tr(τ ) concludes the proof.
Note that the hypothesis that ρ is absolutely irreducible was useful only for lemma 3. A characterization of the absolutely irreducible representations R which factorize through is very easy, since M must then be 1-dimensional. Lemma 4 gives a characterization of a more subtle situation, namely when R factorizes through the non-surjective map l × π : B n → × S n . Note that such an R in particular factorizes through S n .

General construction
Let A be a discrete valuation subring of [[h]] and K be its quotient field. Let R : B n → GL N (A) be a representation of the braid group. As usual, we let R be its reduction modulo h, and we assume that R(σ 2 1 ) = 1, i.e. that R factorizes through S n . This condition is of course satisfied if R = Φ(ρ) for some ρ : B n → M N ( ). More precisely, if ρ ∈ V s (M ), R corresponds to the S n -module M . By conjugation in GL N (K) we can get other representations S : B n → GL N (A). We denote this relationship between R and S by R ∼ S. It means that R and S are isomorphic as representations over K, but correspond to different B n -invariant lattices in K N . It is a classical fact that the simple modules appearing in the Jordan-Hölder decomposition of R and S are the same.
Proposition 12. If R is irreducible and R is not irreducible, then there exists S ∼ R such that S do not factorize through the symmetric group.
Proof. Under these hypothesis it is known (see [2,3]) that there exists S ∼ R such that S is not semisimple. Since all representations of the symmetric group are semisimple, the conclusion follows.
More generally, let M be a S n -module, and let (M i ) i∈I be its simple submodules. Let τ ∈ V s (M ), Φ ∈ ××Ó λ ( ) with λ ∈ × such that Φ(ρ)(B n ) ⊂ GL N (A). The relationship between the reductions modulo h of Φ(ρ) given by different choices of lattices in K N on the one hand and B n -extensions of the modules M i on the other hand enter a general setting extensively studied by Bellaïche and Graftieaux in [2,3]. This is the justification for the name we have chosen of "variety of braided extensions" of the given S n -module M . A case of particular interest is when M = M 1 ⊕ M 2 , with M 1 and M 2 irreducible. Then any τ ∈ V s (M ) allows us to define, following the original idea of Ribet [28] in the framework of algebraic number theory, elements in Ext Bn (M 1 , M 2 ) and Ext Bn (M 2 , M 1 ). We now explain this in more detail. Assume then, on the lattices hM 1 ⊕ M 2 and M 1 ⊕ hM 2 , the reduction of Φ(ρ)(σ i ) modulo h reads i,i+1 s (2) i respectively. Note that these expressions do not depend on the choice of Φ ∈ ××Ó λ ( ).
Moreover, since B n is generated by the s i 's and t i,i+1 's, the fact that τ belongs to V s (M ) implies that there exists i and j such that τ (12) i,i+1 and τ (21) j,j+1 are non zero, hence that the elements obtained in Ext Bn (M 1 , M 2 ) and Ext Bn (M 2 , M 1 ) are non trivial. The most elementary case is the following fact : Proposition 13. Let B and C be two non-trivial S n -module. Then Proof. We assume Hom S 2,n−2 (C, B) = 0. Since all S 2,n−2 -modules are semisimple, we also have Hom S 2,n−2 (B, C) = 0. The S 2,n−2 -endomorphisms of B ⊕ C decompose as follows : Then every τ ∈ V(B ⊕ C) leaves B and C invariant, hence can not belong to V s (B ⊕ C). Let us now consider an element of Ext Bn (C, B), i.e. a representation of B n on the vector space B ⊕ C which can be written in matrix form Since the image of σ 2 1 has the form 1 x 0 1 , this means that σ B x = xσ C for every σ ∈ B 2 × B n−2 , hence that x ∈ Hom S 2,n−2 (C, B). By hypothesis we then have x = 0, hence σ 2 1 = 1. Since the braids σ 2 i for 1 ≤ i ≤ n are conjugated in B n , it follows that σ 2 i = 1 for all 1 ≤ i ≤ n and the representation factorizes through S n , hence is semi-simple and Ext Bn (C, B) = 0. Similarly, Ext Bn (B, C) = 0.
In general, elements in Ext Bn (C, B) can then be constructed from elements in V s (B ⊕ C). Note that V s (B ⊕ C) is not a -vector space in general but only a cone, contrary to Ext Bn (B, C). A natural question, which we leave open, is whether these elements generate Ext Bn (B, C) as a -vector space.

Example : extensions between hooks
Let ½ be the trivial representation of S n , E its permutation representation over . Then E = ½ ⊕ U , where U is an irreducible representation of S n corresponding to the partition [n − 1, 1] of n. Let (e 1 , . . . , e n ) be a basis for E such that s.e i = e s(i) for s ∈ S n . The variety V s (E) was studied in [18]. It is shown there that V s (E) is of the form × τ for some non-zero τ ∈ V s (E), and that the corresponding irreducible representation of B n deduced from any associator Φ ∈ ××Ó λ ( ) is given by with q = e αh , α depending on λ and on the choice of τ ∈ V s (E). By the general procedure described above, we get the following non-split indecomposable representations of B n over : with the following notations : r ∈ {k, k + 1}, (f 1 , . . . , f n ) and (g 1 , . . . , g n ) are some basis of E, v = f 1 + . . . + f n , w = g 1 + . . . + g n . The p-th exterior power Λ p U of U is irreducible under S n and corresponds to the hook diagram [n − p, 1 p ]. One has Λ p V = Λ p−1 U ⊕ Λ p U . From these non-trivial elements in Ext Bn (½, U ) and Ext Bn (U, ½) we then get non-trivial elements in Ext Bn (Λ p−1 U, Λ p U ) and Ext Bn (Λ p U, Λ p−1 U ). For the sake of concision, we introduce the Koszul differential on the exterior algebra Λ • E, defined by If f is of the form f i 1 ∧ . . . ∧ f im for some m such that {i 1 , . . . , i m } ∩ {k, k + 1} = ∅, then the action of σ k in the first case is given by In order to describe the action of σ k in the second case, we introduce a permutation action of S n on the vector space Λ • E, defined by s.(g i 1 ∧ . . . ∧ g im ) = g s(i 1 ) ∧ . . . ∧ g s(im) for s ∈ S n . We then have σ k .g = s k .g + α n (g k + g k+1 ) − 2α n 2 w ∧ dg for all g ∈ Λ • E. Note that, since the restriction to S 2,n−2 of Λ p U and Λ q U have no isomorphic irreducible component for |p−q| ≥ 2, it follows from proposition 13 that Ext Bn (Λ p U, Λ q U ) = 0 in these cases. The cohomology groups of B n in its representation U were computed by Vassiliev in [34]. In particular, he there showed that Ext Bn (½, U ) = H 1 (B n , U ) ≃ for n ≥ 3.

Restriction to pure braids
Keeping in mind the short exact sequence 1 → P n → B n → S n → 1, we now focus on the restriction of representations of B n to its normal subgroup of finite index P n . Being considered as an iterated extension of free groups, P n do not have a well-understood representation theory to build on. On the contrary, it seems hopeless to classify its representationswhatever this could mean. Nevertheless, as it is a normal subgroup of finite index, most irreducible representations of B n are already irreducible under the action of P n . Moreover, its infinitesimal structure is a graded Lie algebra, that is a more elementary structure than the infinitesimal braid algebra B n . It follows that studying the restriction to T n helps in understanding the decomposition of semisimple representations into irreducible components. Since T n is generated by its elements of degree 1, it is first a natural task to investigate the image of these under a representation ρ of B n , and their role in irreducibility properties.
We show that the restriction of ρ to this subspace of homogeneous degree 1 elements of T n is injective, unless the associated B n -representation factorizes through one of the classical quotients of the braid group. We then focus on representation-theoretic properties which are tannakian, i.e. such that the collection of representations satisfying these forms a tannakian subcategory of the category of representations. For instance, semisimplicity is a tannakian property. In order to enlarge this setting, we define a notion of generically tannakian properties, making use of the natural m ( )-action on T n . Two generically tannakian properties naturally appear. One of them is essential purity. Essentially pure irreducible representations of B n restrict to irreducible representations of T n . As a matter of fact, they are already irreducible under the action of several Lie subalgebras of T n . This means that the associated representations of B n are irreducible under the action of several subgroups. The other one is agregation. This notion was first defined in [20], and is a generalization of the fact that tensor products of irreducible representations of a free group are "usually" irreducible.
Another interesting property of a representation ρ is when the eigenvalues of ρ(t 12 ) form a simplex over É. We conclude this part by studying the incidence of all these properties on the Hopf quotients of B n induced by ρ, and mention the connection with the algebraic hull of the pure braid group in the corresponding representations. For convenience, an associator Φ ∈ ××Ó λ ( ), λ ∈ × , integers n ≥ 2, N ≥ 1 are chosen once and for all.

Linear independance
Let ρ : B n → M N ( ) and R = Φ(ρ), for some Φ ∈ ××Ó λ ( ) and λ ∈ × . We showed in the last part that it is possible to characterize the potential factorization of R through B n /Z(B n ), S n × , Hurw n or S n in terms of ρ. We show here that, if R do not factorize through one of these groups, then the -vector space spanned by ρ(t ij ) for 1 ≤ i < j ≤ n is n(n − 1)/2-dimensional. In particular, since B 2 = S 2 , we assume n ≥ 3. We first need to recall several elementary facts about representations of the symmetric group. Le E be the n(n − 1)/2-dimensional S n -module with basis e ij = e {i,j} and action s.e ij = e s(i),s(j) . It is the induced representation of the trivial representation of the Young subgroup S 2,n−2 . Littlewood-Richardson rule shows that provided n ≥ 4 it is the sum of three irreducible non-isomorphic components respectively corresponding to the partitions [n], [n − 1, 1] and [n − 2, 2] whose dimensions are 1, n − 1 and n(n − 3)/2. If n = 3 there are only two irreducible components, corresponding to [n] and [n − 1, 1]. In both cases, the first component is generated by the sum y of all the e ij 's for 1 ≤ i < j ≤ n, and the direct sum of the first two components admits as basis the y i 's defined for 1 ≤ i ≤ n as y i = j =i e ij .

Proposition 14.
If R do not factorize through B n /Z(B n ), Hurw n or S n , then the ρ(t ij )'s are linearly independant.
Proof. We let L be the sub-S n -module of End(M ) linearly generated by the ρ(t ij )'s. We use the notations above and let u i = ϕ(y i ). There is a natural surjective S n -morphism ϕ : E → L defined by ϕ(e ij ) = t ij . Consequences of the characterizations in section 4.2 are the following ones. Lemma 2 implies ϕ(y) = 0. Lemma 3 implies that u n cannot commute to every s ∈ S n , hence there exists i, j such that u i − u j = 0. Lemma 4 implies that L has dimension at least 2. It follows that Ker ϕ, as a S n -subrepresentation of E, cannot contain the invariant vector of E and may contain at most one of the remaining irreducible components. One of them is generated by the elements y i − y j , hence the kernel cannot contain this one. It follows that, if ϕ were not surjective L would be of dimension n, generated by u 1 , . . . , u n , and n ≥ 4. We assume this and show that it would contradict lemma 5 -that is, the family u 1 , . . . , u n would be commutative.

Tannakian and generically tannakian properties
Let A be a graded Hopf algebra over . We shall study here special properties that representations of A may have, in order to apply this setting to the case where A = UT n or A = B n . For a given property (P ), we study in particular their stability by the standard operations in representation theory, namely taking sub-modules and quotients, dual and tensor products, and extension of the base field. Definition 2. A property (P) is said to be tannakian if it is stable by taking sub-modules and quotients, dual and tensor products, and extension of the base field. In other words, if 1) If ρ satifies (P), then so does ρ ∨ , and all its quotients and sub-modules.
If a property (P ) is tannakian, the full sub-category of the (left) A-modules that satisfy (P) inherits the tannakian structure, hence corresponds to a pro-algebraic group scheme over . In particular, the property of factorizing through some Hopf quotient of A is the basic example of a tannakian property.
In this definition, the genericity has to be understood with respect to the Zariski topology on We end this section by studying properties of UT n and B n which are tannakian. All the proofs are standard. The first property under consideration is semisimplicity. We first need a lemma. Lemma 6. Let G be a finite group acting on a Lie algebra g by automorphisms. A representation of U G g = G ⋉ Ug is semi-simple if and only if its restriction to Ug is semi-simple.
Proof. Let ρ : U G g → M N ( ) be a representation of U G g, and ρ ′ be its restriction to Ug. We first assume that ρ is irreducible. Let us choose an irreducible subrepresentation of ρ ′ and let U be the underlying subspace of N . Then any vector space ρ(s)U for s ∈ G is ρ ′ -invariant and irreducible as a representation of Ug, and the sum of all of these is a non-zero ρ-invariant subspace of N . Since ρ is irreducible, this subspace is N and ρ ′ is semisimple. If ρ is semisimple, the same conclusion of course holds. Conversely, we assume that ρ ′ is semisimple. Let ψ be a sub-representation of ρ and U be the underlying subspace. Let i : U ֒→ N be the canonical inclusion. Since ρ ′ is semisimple, i admits a left inverse j as a Ug-module. Let J = ( ψ(s) • j • ρ(s) −1 )/(#G), with the sum being taken over all elements of the finite group G. One has J • i = id U and J is a morphism of U G g-modules. Then ρ is semisimple and the lemma is proved. Proof. For every algebra A, every sub-quotient of a semisimple module is semisimple. Condition 3) is moreover satified because is perfect, being of characteristic 0. In case A = UT n , the dual is semisimple and condition 2) is satified because A is an envelopping algebra. The general case follows from lemma 6.
We call a representation ρ of B n unitary if is unitary with respect to some non-degenerate symmetric bilinear form.

Proposition 16. The property of being unitary is tannakian.
Proof. Since any unitary representation is semisimple, condition 1) is true. Condition 2) holds because, if ρ, ρ ′ are unitary with respect to β, β ′ , then ρ ⊗ ρ ′ is unitary with respect to β ⊗ β ′ , which is symmetric and non degenerate. Condition 3) is clear.

Agregating representations
A less standard property is the property of agregation. This is our first example of a generically tannakian property. We recall from [20] its definition and its main application. It is sufficient to show this for UT n . See [20] for a proof. A consequence of this property is the following : Proposition 18. (see [20]) Let ρ be a representation of U T n . If ρ is agregating and irreducible, then it is absolutely irreducible. If ρ 1 , . . . , ρ r are agregating representations of UT n , then ρ 1 α 1 ⊗ . . . ⊗ ρ r αr is irreducible (resp. indecomposable) for generic values of α 1 , . . . , α r ∈ m ( ) if and only if all the ρ i 's are so.

Essential purity
In this section, we let s = (1 2), t = t 12 . As an associative algebra with unit, B 2 is generated by s and t, with relations s 2 = 1, st = ts.
Definition 5. A representation ρ of B 2 is said to be essentially pure if ρ(s) is a polynomial in ρ(t). A representation of B n is called essentially pure if its restriction to B 2 is so.
If ρ(t) is diagonalizable, it means that ρ(s) acts by a scalar on each sub-vector space Ker(ρ(t) − α) for α ∈ . Note that a representation ρ of B n is essentially pure if and only if any ρ((i j)) is a polynomial in ρ(t ij ).
This property has special incidences on the restrictions of a representation of B n or B n . The notations F n , I n , F n and I n were defined in section 3.2.1.

Proposition 19.
Let ρ be a representation of B n , and Φ ∈ ××Ó λ ( ) with λ ∈ × . If ρ is essentially pure, then the irreducible components of ρ (resp. Φ(ρ)) are irreducible under the action of T n , F n and I n (resp. P n , F n and I n ).
Proof. If ρ is essentially pure, the image ρ((i j)) of each transposition is a polynomial in ρ(t ij ). We can suppose that ρ is irreducible. Since the transpositions generate S n , its restriction to T n is irreducible. Since the transpositions of the special form (i n) (resp. (i i+1)) generate S n , its restriction to F n (resp. I n ) is also irreducible. The consequences for Φ(ρ) follow from proposition 7.
Since F n is a free group, it follows from this that every essentially pure representation of the braid group B n yields a solution of the Deligne-Simpson problem on n-tuples, in the particular case where the first n − 1 conjugacy classes are the same. Recall that this problem concerns the classification of n-tuples (C 1 , . . . , C n ) of conjugacy classes in GL N (K) such that there exists a n-tuple (X 1 , . . . , X n ) ∈ GL N (K) n satisfying X i ∈ C i for 1 ≤ i ≤ n, K N is irreducible under X 1 , . . . , X n , and X 1 . . . X n = 1. Here X i = R(ξ i,n ) for 1 ≤ i ≤ n − 1 and X n = R((ξ 1,n . . . ξ n−1,n ) −1 ). Conversely, a simple solution of this problem in this particular case could be of great benefit for the representation theory of the braid group. An infinitesimal variant of this problem was considered and solved by specialists in the field, namely the classification of n-tuples (c 1 , . . . , c n ) of conjugacy classes in M N ( ) such that there exists a n-tuple (x 1 , . . . , x n ) ∈ M N ( ) n satisfying x i ∈ c i for 1 ≤ i ≤ n, N is irreducible under x 1 , . . . , x n and x 1 + . . . + x n = 0. Here x i = ρ(t i,n ) for 1 ≤ i ≤ n − 1 and x n = −ρ(t 1n + . . . + t n−1,n ). Let us mention the solution recently given by Crawley-Boevey [10] in terms of representations of quivers and the associated Kac-Moody algebra. The solutions given by essentially pure irreducible representations of B n correspond to a special kind of quivers, and it would be most interesting to decide which roots of the associated Kac-Moody algebra are concerned by this construction. Let us moreover notice that the so-called Katz rigidity index is very easy to compute from a given essentially pure B n -representation, and gives an interesting invariant for such representations -for instance, the Burau representation leads to a rigid tuple in this sense, and it is the only one doing so among representations of the Iwahori-Hecke algebra of type A (see [22]). Lemma 7. The representation ρ : B n → M N ( ) is essentially pure iff it is essentially pure over .
Proof. Because of lemma 7, we may assume that is algebraically closed. Let s i = ρ i (s) and t i = ρ i (t). Let d i be the semisimple component of the endomorphism t i . Then t α = ρ α (t) is the sum of the elements α i 1 ⊗ . . . ⊗ t i ⊗ . . . 1 for 1 ≤ i ≤ r, and the sum d α of the elements and there exists a polynomial Q α ∈ [X] depending on α 1 , . . . , α r such that d α = Q α (t α ).
Since the representations ρ i are essentially pure, there exists for all 1 ≤ i ≤ r polynomials Since d α is semisimple and is algebraically closed, it follows that s α = P (d α ) for some P ∈ [X], hence s α = (P • Q α )(t α ) and ρ α is essentially pure.

It easily follows :
Proposition 21. The property of being essentially pure is generically tannakian.

Simplicial representations
We study here N -dimensional representations of B n such that the image of σ 1 or, equivalently, of σ 2 1 , is semisimple and its field of matrix coefficients is a purely transcendantal extension of . As we saw before, if R = Φ(ρ) for some Φ ∈ ××Ó λ ( ) with λ ∈ × and is algebraically closed, this means that the eigenvalues of ρ(t 12 ) have no É-linear relations between them -in particular, this condition only depends on the restriction to T n , hence fits our preoccupations in this part. Since a rescaling of R by multiplication of some scalar (equivalently, a rescaling of ρ by addition of some scalar) do not change the representation in a significant way, the convenient setting at the infinitesimal level is not linear algebra but affine geometry over É. In this framework, a simplex will refer to the structure of É-affine vector space of . Definition 6. A representation ρ : B n → M N ( ) is called simplicial if ρ(t 12 ) is diagonalizable over and its spectrum forms a simplex.
We introduce in appendix B a representation-theoretic criterium ensuring that the interior of this simplex contains the eigenvalues of 2T /n(n − 1), with T being the sum of the ρ(t ij )'s -thus illustrating the relevance of this geometric definition.
For any Hopf algebra A and representation ρ of A, the derived representations of ρ are the representations of the form ρ ⊗p ⊗ (ρ ∨ ) ⊗q for p, q ≥ 0, their subrepresentations and quotients. Besides its influence on the field of matrix coefficient, the most interesting aspect of simpliciality is that it makes essential purity extend to derived representations.
Proposition 22. Let ρ be simplicial and essentially pure. Then all its derived representations are essentially pure.
Proof. Let S ⊂ be the spectrum of ρ(t 12 ). Essential purity means that there exists a settheoretic map ε : Let r ≥ 0. We first show that ρ ⊗r is essentially pure. We let t = t 12 , s = (1 2) as before.
Let v ∈ ( N ) ⊗r be a pure tensor formed by tensoring several eigenvectors for t : n 1 with eigenvalue α 1 , . . . , n r with eigenvalue α r . One has, Since n i = r and the α ′ i s are affine independant, the value of ρ ⊗r (t 12 ) on v determines the n i 's as barycentric coordinates, hence the value of ρ ⊗r (s) on v. Since the tensors like v form a basis of ( N ) ⊗r which are eigenvectors for ρ ⊗r (t 12 ) it follows that ρ ⊗r (s) is a polynomial of ρ ⊗r (t 12 ) and ρ ⊗r is essentially pure. The same more generally holds forρ = ρ ⊗p ⊗ (ρ * ) ⊗q for p, q ≥ 0 : the eigenvalues ofρ(t) are then of the form (n i − m i )α i , with n i = p, m i = q, and uniquely determine the (n i − m i )'s, hence the parity of the (n i + m i )'s and the corresponding action ε(α i ) n i +m i ofρ(s). Since essential purity is generically tannakian and in particular satisfies condition 1) the conclusion follows.

Infinitesimal quotients and algebraic hulls
Let ρ : B n → M N ( ) be a representation and H ρ = ρ(T n ) the Lie subalgebra of gl N ( ) generated by the ρ(t ij )'s. The sum T of the ρ(t ij )'s belong to the center of H ρ . The symmetric group acts on H ρ by automorphisms and the Lie algebra morphism T n → H ρ extends to a surjective Hopf algebras morphism B n → Q ρ where Q ρ = S n ⋉ UH ρ is called the infinitesimal quotient of B n associated to ρ. Note that, if ρ can be decomposed as ρ 1 ⊕ . . . ⊕ ρ r , then H ρ naturally embeds into H ρ 1 ⊕ . . . ⊕ H ρr by an S n -equivariant Lie algebra morphism, hence Q ρ embeds into Q ρ 1 ⊕ . . . ⊕ Q ρr as a Hopf algebra.
The representation-theoretic conditions introduced above, when satisfied by ρ, have the following consequences on H ρ . Proposition 23. Let ρ be essentially pure and semisimple. Then H ρ is a reductive Lie algebra and Z(H ρ ) = ρ(Z(T n )) = T . In particular, if ρ(t 12 ) ∈ gl N ( ) has non-zero trace, then H ρ has 1-dimensional center. Moreover, if ρ is agregating then H ρ is split over .
Proof. The representation ρ induces a faithful representation ρ ′ of H ρ on N . Since ρ is essentially pure, every simple component of ρ is irreducible under the action of T n , hence ρ ′ is faithful and semisimple, and H ρ is reductive.
One always has ρ(Z(T n )) = T . Since H ρ is reductive, one has H ρ = Z(H ρ ) × [H ρ , H ρ ] and ρ([T n , T n ]) = [H ρ , H ρ ]. If T 1 n denotes the subspace generated by the t ′ ij s inside T n one has T n = T 1 n × [T n , T n ] as vector spaces and S n -modules, and H ρ = Z(H ρ ) × [H ρ , H ρ ] is also a decomposition as S n -modules because ρ is essentially pure hence S n acts trivially on Z(H ρ ). We denote by p the corresponding S n -equivariant projection H ρ ։ Z(H ρ ). It follows that Z(H ρ ) ⊂ p • ρ(T 1 n ) ρ(Sn) hence it is a quotient of the S n -module T 1 n . Because of the decomposition of this latter semisimple S n -module one has p • ρ(T 1 n ) ρ(Sn) = p • ρ((T 1 n ) Sn ) = p( T ). Hence T ⊂ Z(H ρ ⊂ p( T ) with dim T ≥ dim(p( T )) and the conclusion follows. Since the trace of ρ(T ) is n(n−1) 2 -times the trace of ρ(t 12 ), if the latter is non-zero then T has dimension 1.
If ρ is agregating, there exists x ∈ T n such that u = ρ(x) is a regular element in gl N ( ). It is then a regular element of the Lie algebra H ρ in the traditional sense and a Cartan subalgebra of H ρ is given by C ρ = {y ∈ H ρ | ∃m (ad u) m y = 0}. But since H ρ is reductive, then C ρ is commutative. It follows that C ρ ⊂ M N ( ) is a set of polynomials in u, hence all elements in C ρ are diagonalizable and H ρ is split over .
As before in this part, we are interested here in whether the irreducible components of a representation of Q ρ are already irreducible under the action of H ρ . Let ρ = ρ 1 ⊕ . . . ⊕ ρ r be a semisimple, essentially pure and simplicial representation of B n . It easily follows from propositions 22 and 23 that the decomposition of any tensor product of ρ 1 , . . . , ρ r and their duals only depends on the decomposition of the reductive Lie algebra H ρ into simple factors. For applications of this situation to the most classical representations of the braid group, see [17,26,27].
Recall that F n is a normal subgroup of P n such that P n /F n ≃ P n−1 and the centralizer of F n in P n is Z(P n ). Similarly, F n is an ideal of T n , T n /F n ≃ T n−1 and its centralizer in T n is the center of T n . We letȞ ρ = ρ(F n ).

Proposition 24.
Let H andȞ be the Zariski closure of R(P n ) and R(F n ) inside GL N (K), and h,ȟ be their Lie algebras. Then H ρ ⊗ K ⊂ h andȞ ρ ⊗ K ⊂ȟ.
Proof. It easily follows from the definition of Φ that Φ(ξ ij ) = Φ ij e 2λt ij Φ −1 ij with Φ ij ∈ exp T n , hence Φ(ξ ij ) = exp ψ ij where ψ ij ∈ T n and ψ ij = 2λt ij plus higher terms. In particular R(ξ ij ) = exp hψ ij with ψ ij ∈ ρ(T n ) ⊗ A and ψ ij = 2λρ(t ij ). It follows from the lemma that ψ ij ∈ h. Now the K-Lie algebra generated by the ψ ij 's is contained in H ρ ⊗ K, and has dimension at least dim H ρ since the ψ ij 's generate H ρ . It follows H ρ ⊗ K ⊂ h.
Because F n is an ideal of T n one has ψ ij ∈ F n and one getsȞ ρ ⊗ K ⊂ȟ by the same argument.
In fact we proved more, namely that the same result holds when P n and F n are replaced by the subgroups generated by the ξ k ij 's, and by the ξ k in 's, for some fixed integer k. As a consequence of this, we get Proposition 25. Let G be a finite index subgroup of B n , Γ the Zariski closure of R(G) and h the Lie algebra of Γ. Then H ρ ⊗ K ⊂ h. The same holds for any subgroup of B n containing ξ k ij , 1 ≤ i < j ≤ n, for some fixed non-zero integer k.
Proof. If G has finite index, there exists k ∈ \ {0} such that G contains the subgroup P (k) n of B n generated by the ξ k ij , 1 ≤ i < j ≤ n. Then R(G) ⊃ R(P (k) n ) and the Lie algebra of the Zariski closure of R(P (k) n ) contains H ρ ⊗ K by the above remark hence H ρ ⊗ K ⊂ h.
The last subgoups of interest are the right-angled group C (k) n generated by the elements σ 2k 1 , . . . , σ 2k n−1 , for some non-zero integer k.
Proposition 26. For all k ∈ \ {0}, let Γ k be the Zariski closure of R(C k n ) and h k its Lie algebra. If H ρ is generated by the elements i ) = exp ψ i with ψ i ∈ T n and ψ i = 2kλt i,i+1 plus higher terms. Hence R(σ 2k i ) = exp hψ i with ψ i = 2kλρ(t i,i+1 ) and ψ i ∈ h k . Let J be the Lie algebra generated by the ψ i . One has J ⊂ h k , J ⊂ H ρ ⊗ K and J has dimension dim H ρ ⊗ K since the elements ψ i generate H ρ .

Basic constructions
In this last part we review the basic constructions of representations of the braid groups which appeared in the last decades. We show how these can be dealt with in our terms, and how our approach -in particular our uniform approach to unitary representations -may simplify or improve either the constructions or the proof of their main properties. Incidentally, we shall get three different proofs of the unitarizability of the Burau representation : as a representation of the Iwahori-Hecke algebra (see 6.1.1), of the Birman-Wenzl-Murakami algebra (see 6.2.2) and as a product of Long's induction (see 6.3). For the sake of simplicity, we let A = [[h]], K = ((h)), and Φ ∈ ××Ó λ ( ), λ ∈ × is chosen once and for all.

Small representations
Let R : B n → GL N ( ) be an irreducible representation of B n . To any such representation are attached several integer invariants. Besides the dimension N , the most classical ones are the cardinality of the spectrum and the rank of R(σ 1 ) − 1. A systematic step-by-step search for irreducible representations of B n can thus be carried out by imposing small values on these parameters. First note that, letting R = Φ(ρ), τ = ρ(t 12 ), R(σ 1 ) ∈ GL N (K) is semisimple if and only if τ ∈ M N ( ) is semisimple. We now review what happens when one of these parameters is "small".

Small spectrum
The first parameter that we consider is the cardinality of the spectrum of R(σ 1 ). We assume that R(σ 1 ) is diagonalizable and let S = {a 1 , . . . , a r } ⊂ K × be this spectrum. Then R factorizes through the K-algebra H n (a 1 , . . . , a r ) defined as the quotient of KB n by the relation (σ 1 − a 1 )(σ 1 − a 2 ) . . . (σ 1 − a r ) = 0. If S has cardinality 1, then H n (a 1 ) = K and R factorizes through a morphism B n → → GL 1 (K). If R = Φ(ρ) the irreducible representation ρ must factorize through a morphism B n → S 2 ⋉ . We may thus assume r ≥ 2.
Iwahori-Hecke algebras. If S = {a 1 , a 2 } has cardinality 2, the algebra H n (a 1 , a 2 ) is known as the Iwahori-Hecke algebra of type A n−1 with parameters a 1 , a 2 . It is well-known that this algebra is isomorphic to KS n as soon as a 1 /a 2 ∈ K is not a root of unity and is a square. In particular, if a 1 and a 2 are algebraically independant over É and a 1 /a 2 is a square, then H n (a 1 , a 2 ) ≃ KS n and, in case R = Φ(ρ), the spectrum of τ = ρ(t 12 ) has to be of cardinality 2. We assume that Sp(τ ) = {u, v} ⊂ with u = v. Then s = ρ(s 1 ) acts as a scalar on each eigenspace of τ , otherwise R(σ 1 ) would admit three distinct eigenvalues ; these scalars are distinct otherwise ρ would be reducible. We can thus assume that τ (x) = ux (resp. τ (x) = vx) iff s(x) = x (resp. s(x) = −x), up to tensoring by the sign representation of S n . It follows that there exists α, β ∈ such that τ = α + βs.
Conversely, let M be an irreducible representation of S n . For every α, β ∈ an element τ ∈ V(M ) can be defined by τ = α + βs, with s denoting the action of s 1 on M . If we let ρ be the corresponding representation of B n , then Φ(ρ)(σ 1 ) is diagonalizable over K with two eigenvalues {e λh(α+β) , −e λh(α−β) }, which are algebraically independant as soon as β = 0, and it follows that R factorizes through the Hecke algebra. The cases when M corresponds to the partition [n − 1, 1] or [2, 1 n−2 ] lead to the reduced Burau representation. Note that the existence of a unitary structure on the Hecke algebra representations (for parameters of modulus 1 and not roots of unity) can be seen in the following way. If ⊂ Ê, there is a canonical euclidian structure on M for which the S n -action is orthogonal. In particular, each transposition is orthogonal and involutive, hence selfadjoint, and τ is selfadjoint. This gives then rise to a unitary representation of B n . Moreover, these representations are simplicial, essentially pure if β = 0, and the elements Y n are sent to so-called Jucys-Murphy elements of the symmetric group. It follows that these representations are also agregating. We determined the algebraic hull of B n in these representations in [26].
Cubic Hecke algebras . For generic values of a, b, c -in particular when a, b, c ∈ K are algebraically independant over -the algebras H n (a, b, c) for n ≤ 4 and, to the best of our knowledge conjecturally, n = 5, are finite-dimensional and semi-simple. This is related to the fact that they appear as deformations of group algebras of exceptional finite complex reflections groups. We first study in some detail the case n = 3.
The finite complex reflection group called G 4 in Shephard and Todd classification (see [29]) admits the presentation < S, T | S 3 = 1, T 3 = 1, ST S = T ST >. It is obviously a quotient of B 3 through the map σ 1 → S, σ 2 → T , and has order 24. Let j be a 3-root of 1. This finite group admits three 1-dimensional irreducible representations X ω for ω ∈ {1, j, j 2 } defined by S, T → ω, three 2-dimensional irreducible representations U ω characterized by the fact that S and T act with eigenvalues {1, j, j 2 } \ {ω}, and one 3-dimensional irreducible representation V . On the representation V , S and T act with eigenvalues {1, j, j 2 }. It follows from [5] that KG 4 is isomorphic to its "cyclotomic Hecke algebra" A first description of V was obtained in [5]. We obtained a symmetric model in [18] : σ 1 acts as the diagonal matrix with entries (a, b, c), and σ 2 acts as the following matrix :  The representations X ω , U ω arise from Hecke algebra representations, namely the trivial and reduced Burau representations. It follows from [6] that all these representations, including V , can be obtained by monodromy over the configuration space associated to G 4 from the corresponding G 4 -representations. A natural question is to ask whether these representations arise as Φ(ρ) for some ρ. For all irreducible representations except V , this is a consequence of the study of Hecke algebra representations. It is also true for V , due to our work on V s ([2, 1] ⊕ [3]) in section 4.1. It follows that every finite-dimensional representation can be obtained as a direct sum of representations of the form Φ(ρ), each twisted by some element in Gal(K/ ). Note however that some non-irreducible representations cannot be obtained directly as Φ(ρ) for some ρ. This is in particular the case for the representation U a ⊕ U b ⊕ U c : the determinant of σ 1 should be equal to (abc) 2 and have −1 as constant term at the same time.
For every n ≥ 2, there exists a finite-dimensional quotient of H n (a, b, c) known as the Birman-Wenzl-Murakami algebra. From the knowledge of the representation theory of G 4 , it is easy to determine from the infinitesimal datas whether the corresponding monodromy representation factorize through this algebra or not, and all representations of this algebra can be obtained by our method -we refer to [17] for details on these two points, and only mention that the corresponding representations of B n are simplicial, essentially pure and agregating.
Important representations of the Birman-Wenzl-Murakami algebra are the Krammer representations -essentially the only known faithful representation of the braid groups. The algebraic hull of the braid groups inside them was computed in [27].
Quartic relations and beyond. In the same vein , H 3 (a, b, c, d) is a deformation of the group algebra of the finite complex reflection group called G 8 . It is finite-dimensional, and isomorphic to KG 8 as soon as a, b, c, d ∈ K are algebraically independant over and abcd admits a square root in K. For a detailed study of H 3 (1, u, v, w) we refer to the work of Broué and Malle [5]. The irreducible representations of G 8 of dimension less than 4 are not of interest for us, since they can be obtained by specialization from the cyclotomic Hecke algebra of G 4 . The rest is formed by two 4-dimensional irreducible representations. The corresponding two irreducible representations of H 3 (a, b, c, d) can be distinguished by the scalar action of (σ 1 σ 2 ) 3 , and can be deduced from each other through the action of Gal(L/ (a, b, c, d)), where L ⊂ K is the minimal (quadratic) extension of (a, b, c, d) containing √ abcd. These representations can be obtained by our method. Indeed, we proved in section 4.1 that there exists τ ∈ V s ([2, 1] + [3] + [1 3 ]) such that the eigenvalues of τ can be chosen linearly independant over É, hence the corresponding eigenvalues of σ 1 are algebraically independent. Moreover, the determinant of σ 1 has all its square roots inside [[h]] -hence its companion representation is deduced by using some element in Gal(K/ ). This example shows however that we definitely have to take into account these Galois actions : there is no way to get this other representation directly as Φ(ρ) for some ρ. The algebra H 3 (a, b, c, d, e) is again a deformation of the group algebra of a finite complex reflections group called G 16 . However, this process does not go further : the quotient of B 3 by the relation σ r 1 = 1 for r ≥ 6 is infinite.

Small dimension
Small-dimensional irreducible representations of B n were studied by Formanek and Sysoeva (see [13,31,32]). In the infinitesimal setting, we denote by M a S n -module, and ask how small can the dimension of M be with V s (M ) = ∅. If M decomposes as a sum of 1-dimensional representations, then M should itself be of dimension 1 under this condition (see proposition 13). This happens in particular if M is of dimension at most n − 2. If M is of dimension at most n − 1, then the action of S n is the standard (n − 1)-dimensional representation, or its tensorization by the sign representation and elements in V s (M ) correspond to the reduced Burau representation. If M has dimension n then M is, up to tensorization by the sign representation, the sum of the standard representation plus a 1-dimensional representation. This 1-dimensional representation must be the trivial one by proposition 13. The analysis of V s ([n − 1, 1] ⊕ ½) carried out in [18] shows that we get the irreducible representation of S n described in section 4.3.2. These observations are infinitesimal analogues of the results of Formanek and Sysoeva. The other representations obtained in their work do not fits our approach, since the eigenvalues of R(σ 1 ) are then algebraic over : thay can be obtained only after specialization of the field of matrix coefficients -for instance, one of them is a (n − 2)-dimensional representation of the Hecke algebra with parameters roots of unity.

Small rank
A central ingredient in Formanek's and Sysoeva's approach is the analysis of the irreducible representations R such that R(σ 1 ) − 1 has rank 1. Then R(σ 1 ) − 1 is either semisimple or nilpotent. Let R = Φ(ρ), τ = ρ(t 12 ) for ρ : B n → M N ( ). In case R(σ 1 ) − 1 is semisimple it factorizes through the Hecke algebra of type A, and this general situation was already studied. In this particular case, since τ is then a linear combination of ρ(s 1 ) and 1, one easily shows that R(σ 1 ) − 1 has rank one iff R is the reduced Burau representation. If it is nilpotent then R(σ 1 ) − 1 ∈ ρ(s 1 ) − 1 + hM N (A) implies that ρ is the trivial action of S n on N . It thus contradicts our assumption that R is irreducible for N ≥ 2. Besides this, it also implies that R(σ 1 ) − 1 ∈ λρ(s 1 )τ h + h 2 M N (A), hence that τ is nilpotent and of rank 1. A richer infinitesimal analogue of this analysis is then to ask what happens when τ is a nilpotent transvection. The answer is that any such representation R will factorize through S n . Since it is equivalent to the fact that ρ([t 12 , t 23 ]) = 0, this is an immediate consequence of the following result.

Preliminaries
All the material here is standard. We recall that has characteristic zero. Let g be a semisimple Lie algebra over , and K g its Killing form. Since g is semisimple, this bilinear symmetric invariant form is nondegenerate. Let (e λ ) λ∈Λ be a basis for g, and (e λ ) λ∈Λ the dual basis of g with respect to K g : K g (e λ , e µ ) = δ λ,µ . The Casimir element C = e λ e λ ∈ Ug do not depend on the choice of the basis (e λ ) and is central in Ug. We let ∆ be the coproduct of the Hopf algebra Ug, and introduce c = ∆(C) − C ⊗ 1 − 1 ⊗ C ∈ (Ug) ⊗2 , τ = c ⊗ 1 ∈ (Ug) ⊗n . With respect to the action of S n on (Ug) ⊗n by permutation of the factors, τ is S 2 × S n−2invariant (because Ug is cocommutative), and we define τ ij = τ ji for 1 ≤ i < j ≤ n by the characteristic property τ s(1),s(2) = s.τ for s ∈ S n . It is well known that c commutes to ∆(g) ⊂ (Ug) ⊗2 and that there exists a S n -equivariant algebra morphism UT n → (Ug) ⊗n defined by t ij → τ ij .
As a consequence of this, there is an algebra morphism B n → S n ⋉ (Ug) ⊗n . Since g is semisimple, g n is semisimple and every representation of (Ug) ⊗n = U(g n ) is completely reducible. Moreover, every irreducible representation of g n has the form V 1 ⊗ . . . ⊗ V n where the V i 's are irreducible g-modules. By lemma 6, every representation of S n ⋉ (Ug) ⊗n is completely reducible. A special kind of irreducible representation of S n ⋉ (Ug) ⊗n are of the type V ⊗ . . . ⊗ V for V an irreducible representation of g. Since c commutes to ∆(g), the diagonal action of g on any (Ug) ⊗n -module (resp. S n ⋉ (Ug) ⊗n -module) commutes with the action of T n (resp. B n ).

Semisimplicity and unitarity
Let us consider n self-dual irreducible g-modules V 1 , . . . , V n , for which we fix an isomorphism where ½ designates the trivial representation of g and the last map is the canonical contraction. The form β i is either symmetric or skewsymmetric, depending on whether the g-module V i is orthogonal (½ ֒→ S 2 V i ) or symplectic (½ ֒→ Λ 2 V i ). We define a non-degenerate bilinear form β on V 1 ⊗ . . . ⊗ V n as the product of the forms β i . It is easily seen that g 1 ⊗ . . . ⊗ g n ∈ (Ug) ⊗n is selfadjoint with respect to β as soon as there are an even number of g i ∈ g while the other ones belong to . In particular, each τ ij is selfadjoint with respect to β and, if V 1 = . . . = V n , the action of S n is obviously isometric. This implies that the action of T n (resp. B n ) is semisimple.
If the form β is symmetric and V 1 = . . . = V n = V , then the action of B n is unitary. This happens for instance if V is the adjoint representation of g, in which case each β i is a multiple of the Killing form and is symmetric, or if n is even. However the most interesting situation is when ⊂ Ê, but then β is not in general positive definite.
In one case however we can get unitary representations of B n on V ⊗n for any field of characteristic zero. Let α be the canonical symmetric nondegenerate bilinear form on m and g = so m ( ), V = m . The forms β i coincide with α, and β is the canonical bilinear form on V ⊗n = ( m ) ⊗n = mn . It follows that the action of B n and the corresponding action of B n are unitary. This explains and proves the existence of a unitary structure on the representations of the Birman-Wenzl-Murakami algebra, since they appear in this way. More generally, if each representation V i of g is faithful and = Ê, this situation happens exactly when the Lie algebra g defined over Ê is compact ; indeed, β i can be chosen to be positive definite iff the image of g in V i is a compact Lie algebra.

Other properties
By its very definition, the eigenvalues of τ on V ⊗n are closely related to the eigenvalues of the Casimir operator on V ⊗ V . In particular, if the decomposition of V ⊗ V as a g-module is multiplicity-free and the Casimir operator acts with distinct eigenvalues on each irreducible component, then the action of B n on V ⊗n is essentially pure. More generally, the eigenvalues of the action of each Y r are related to the eigenvalues of the action of the Casimir operator on V ⊗r . We note the following.
Proof. By extension of the base field, we can assume = , hence there exists a basis (e λ ) λ∈Λ of g which is self-dual with respect for the Killing form and c = 2 e λ ⊗ e λ . Then [τ 12 , τ 23 ] is the sum over all λ, µ ∈ Λ of the elements e λ ⊗ [e λ , e µ ] ⊗ e µ and the conclusion follows.
In particular such a representation will in general not factorize through the enhanced symmetric group. In case g = sl 2 ( ), every representation is self-dual, so the actions of T n will be semisimple. We recall from [20] the following.

Artin action
We denote by f 1 , . . . , f n given free generators of the free group L n . The well-known faithful Artin action of B n on L n is given by making σ i for 1 ≤ i < n act on L n by i+1 f i f i+1 and f j → f j for j ∈ {i, i + 1}. A theorem of Artin says that this identifies B n with the subgroup of automorphisms of L n which preserve the set of conjugacy classes of {f 1 , . . . , f n } and the product f 1 . . . f n . The geometric meaning of this is that B n can be identified with a set of diffeomorphisms of the disc minus n punctures, whose fundamental group is a free group on n generators. This action enables one to form the semi-direct product B n ⋉ L n , which embeds in B n+1 through the inclusion B n ⊂ B n+1 and the map f i → ξ i,n+1 , thus identifying L n and F n+1 . In particular, every representation of B n+1 restricts to a representation of B n ⋉ L n ≃ B n ⋉ F n+1 ⊂ B n+1 . Also note that L n ⋊ B n can be mapped onto × B n by sending each f i to 1. In particular, every (irreducible) representation of B n and any non-zero scalar α yield a (irreducible) representation of B n ⋉ L n . In general every representation R of B n ⋉ L n can be rescaled by any non-zero scalar α in order to get a new representation R α , letting R α (σ) = R(σ) for σ ∈ B n , R α (f r ) = αR(f r ). We define here an infinitesimal analogue of Artin action.
Let L n be the free Lie algebra on the abstract generators g 1 , . . . , g n , considered as the Lie algebra of the pro-nilpotent completion of L n . We make s ∈ S n act as the automorphism defined by g i → g s(i) for 1 ≤ i ≤ n, and t ij as the derivation defined by This leads to a well-defined action by automorphisms of B n on UL n , in the sense that grouplike elements in B n act as automorphisms and primitive elements in B n act as derivations, because the relation [t ij , t ik + t kj ](g i ) = 0 means [[g i , g j ], g k ] − [[g i , g k ], g j ] − [g i , [g j , g k ]] = 0, which is a version of the Jacobi identity. As a consequence of this, the semi-direct product B n ⋉ UL n has a well-defined Hopf algebra structure, and the map g k → t k,n+1 together with the natural inclusion B n ⊂ B n+1 induce a Hopf algebra inclusion B n ⋉ UL n ⊂ B n+1 , thus identifying L n with F n+1 . The natural map L n ⋊ B n → × B n corresponds to the following. Let x be an indeterminate, and make B n act trivially on the 1-dimensional space x (i.e. s.x = x for s ∈ S n , t ij .x = 0). Now, x can be considered as a (commutative) 1-dimensional Lie algebra and the Lie algebra morphism L n → x defined by g i → x is B n -equivariant, hence leads to the awaited mapping UL n ⋊ B n → [x] × B n . Here we let [x] be the ring of polynomials in x and we identify it with the universal envelopping algebra of x.
We next apply the associator Φ. The image of ξ i,n+1 ∈ P n+1 in B n+1 under the associated Drinfeld morphism belongs to B n ⋉ UF n+1 . Indeed, ξ i,n+1 is a conjugate of σ 2 n by elements of B n , and the image of σ 2 n is Φ(t n,n+1 , Y n+1 )e 2λt n,n+1 Φ(Y n+1 , t n,n+1 ) hence belongs to UF n+1 . As a result, we get the following commutative diagram with all arrows injective.

Long induction
In [16], D.D. Long generalizes the classical Magnus construction to get (unitary) 1-parameter families of representations of B n from a (unitary) representation of F n ⋉ B n . In particular, it is possible to deduce from given representations of B n+1 , or even representations of B n , richer representations of B n . We refer to [16] for the geometric interpretation of this construction, and recall the algebraic construction. Let I be the augmentation ideal of L n , and R a representation of L n ⋉ B n on somevector space V of finite dimension m. There are then well-defined actions of B n on I and V , and the associated representation R + of B n on the nm-dimensional -vector space I ⊗ Ln V is given by making B n act simultaneously on both factors. From the above observations, it follows that every representation R of B n+1 (resp. B n ), by restriction to B n ⋉ F n and rescaling (resp. by the morphism B n ⋉ F n → B n × ) leads to a family of B n -representation (R α ) + . Using geometric means and a theorem of Deligne and Mostow, it is shown in [16] that, if = and R is unitary, then for generic α ∈ of modulus 1 the representation (R α ) + is unitary with respect to some non-degenerate hermitian form. This hermitian form is not explicitely given at the algebraic level, and comes from a Poincaré duality pairing.
We now give an infinitesimal analogue of this construction. Apart from linearizing the situation, it has the advantage that the infinitesimal bilinear form is explicitely described. Let ρ be a representation of UL n ⋊ B n , and ρ α the representation deduced from ρ by the B n -invariant twisting g i → g i + α. We denote by V the underlying -vector space. A linear B n -action ρ + is defined on V n by making s ∈ S n act as (2) , . . . , s.v s −1 (n) ) and, denoting x where v = (v 1 , . . . , v n ),ṽ = (ṽ 1 , . . . ,ṽ n ). It is easily checked that ρ + is a representation of B n . From the previous observations, we then get representations (ρ α ) + of B n from any representation ρ of B n+1 or B n .

Infinitesimal forms
We assume that a representation ρ of B n ⋉ UL n on a -vector space V is fixed, and that V is endowed with a non-degenerate bilinear form ( | ) such that ρ(s) is isometric for every s ∈ S n . By abuse of notation, we shall write tv = ρ(t)v (resp. t + v = ρ + (t)v) for every t ∈ B n . We introduce on V the twisted bilinear forms β i (v, v ′ ) = (g i v|v ′ ). We shall denote by v = (v 1 , . . . , v n ) and v ′ = (v ′ 1 , . . . v ′ n ) elements of V n . We define on V n the bilinear form Last, we decompose each t + ij into two endomorphisms, We need a technical lemma.
is proved. The last assertions are consequences of the following easy Appendix A : Formal and real unitarity Let be one of the topological fields Ê, É p , or one of their finite extensions. In particular, it is endowed with a natural non-trivial valuation. A power series f in [[h]] is called convergent if it has a non-zero radius of convergence with respect to the given valuation. We let {h} be the ring of convergent power series, and by ({h}) its quotient field -that is, the set of Laurent series f ∈ ((h)) such that h r f is a convergent power series for some integer r. We let K = ((h)), K * = ({h}) and, forgetting the given topology on , we endow these fields with the ordinary h-adic topology. The purpose of this appendix is to prove approximation results in the framework of representation theory. More precisely, we work in the following scope. Let R : G → GL N (K) be a representation of a finitely generated group G. Because of this finite generation hypothesis, the image of G in fact lies in GL N (L), for L a subfield of K which is finitely generated over . Let us assume next that there exists a subfield L * of K * which is isomorphic to L. The representation R * deduced from R by this isomorphism is convergent, and the two representations R and R * are essentially equivalent -they are conjugated by an element in Gal(K/É). Here we prove more. Assuming that L contains (h), we prove the existence of (h) ⊂ L * ⊂ K * such that L * is isomorphic to L as an extension of (h) -this means that this isomorphism is the identity on (h). We also say that L is isomorphic to L * over (h). Then R and R * are conjugated by an element in Gal(K/ (h)).
We next deal with a more subtle question, involving unitarity. We let now = Ê, and ǫ be the automorphism of K defined by f (h) → f (−h). This is the only continuous automorphism of K such that ǫ(h) = −h. It leaves K * invariant -we say that K * is an ǫ-invariant subfield of K. Using the notations of section 3.2.2, we denote by U ǫ N (K) the group {x ∈ GL N (K) | x −1 = t ǫ(x)}, and assume that we are given a representation R : G → U ǫ N (K). Again because G is finitely generated, R(G) lies in U ǫ N (L) for L some finitely generated subfield of K, which may be assumed to contain (h) and to be ǫ-invariant. We prove that there exists a finitely generated subfield L * of K * , containing (h) and ǫinvariant, such that U ǫ N (L) ≃ U ǫ N (L * ). We then explain how, after a convenient rescaling of the local parameter h, we get from this a convergent representation such that specialization in h leads to representations of G into the ordinary unitary group U N .
We first prove the following result.
Theorem 2. Let L be a subfield of K which contains (h) and is finitely generated over . Then there exists a subfield L * of K * which contains (h) and is isomorphic to L over (h).
Moreover, for any finite family a 1 , . . . , a t ∈ L and m ≥ 0, then the isomorphism L → L * can be chosen such that each a i is congruent to its image modulo h m .
Proof. Let (h) ⊂ L 0 ⊂ L be a maximal purely transcendantal extension of (h) contained in L. Since L is finitely generated over , its transcendance degree over (h) is finite, whence there exists f 1 , . . . , f r ∈ K algebraically independant over (h) such that L 0 = (h)(f 1 , . . . , f r ). The field L is by definition an algebraic extension of L 0 , and a finite extension because L is finitely generated over . The field K * is an extension of (h) of infinite transcendance degree -for instance, if = Ê, or É p , the family {e h d } d>0 is algebraically independant over (h) and its elements belong to K * . On the other hand, if g 1 , . . . , g r ∈ K * are algebraically independant over (h), so are g 1 + P 1 , . . . , g r + P r for P 1 , . . . , P r arbitrary elements of (h). It follows from these two facts that a family g 1 , . . . , g r of convergent power series, algebraically independant over (h), can be chosen as close to f 1 , . . . , f r as we want with respect to the h-adic topology. Let us now introduce the abstract fieldL 0 = (h)(Y 1 , . . . , Y r ), where Y 1 , . . . , Y r are indeterminates. Every family g = (g 1 , . . . , g r ) as above yield an embedding ofL 0 into K * , through Y i → g i . We denote by L g 0 the image subfield of K * . The extensionsL 0 , L 0 and L g 0 of (h) are by definition isomorphic, and the above considerations show that there exists a family g 1 , . . . , g r such that L g 0 ⊂ K * , so this proves the result in case L is a purely transcendantal extension. In order to prove the general case, we make use of the additional fact that (g 1 , . . . , g r ) may be chosen arbitrarily near to (f 1 , . . . , f r ). All families g 1 , . . . , g r chosen below are assumed to be algebraically independant over (h).
Since L is finite and separable as an extension of L 0 , the primitive element theorem yield α ∈ K such that L = L 0 (α). Because L 0 contains (h), we can and will assume α ∈ h [[h]]. Let P ∈ L 0 [X] be a minimal polynomial of α over L 0 . By not requiring P to be monic, we may assume P ∈ [h, f 1 , . . . , f r ]. The field L is then isomorphic to L 0 [X]/(P ). Since L 0 is isomorphic toL 0 over (h), there existsP ∈L 0 [X] such that P (resp. its derivative P ′ ) is deduced fromP (resp.P ′ ) through the substitution Y i → f i . Similarly, the substitution Y i → g i leads to polynomials P g and P ′ g in L g 0 [X], such that L ≃ L 0 [X]/(P ) ≃L 0 [X]/(P ) ≃ L g 0 [X]/(P g ) as extensions of (h). In order to map L g 0 [X]/(P g ) into K * , it is then sufficient to find convergent g 1 , . . . , g r such that P g admit a root in K * . Because of Artin's theorem (see Artin [1] th. 1.2), it is sufficient for this that P g admits a root in h [[h]]. Since P is a minimal polynomial for α over L 0 , and L 0 has characteristic 0, P (α) = 0 and P ′ (α) = 0. Since P ′ (α) ∈ [[h]], there exists than s ≥ 0, β ∈ \ {0} such that P ′ (α) = h s β modulo h s+1 . Choosing g 1 , . . . , g r ∈ {h} close enough to f 1 , . . . , f r so that P g (α) = 0 modulo h 2s and P ′ g (α) = h s β modulo h s+1 , we have P g (α) ∈ P ′ g (α) 2  [X]/(P g ) ≃ L g 0 (γ) ⊂ K * as extensions of (h). Now consider a family a 1 , . . . , a t as in the statement, let f 1 , . . . , f r and α be as above. There exists polynomials Q 1 , . . . , Q t with coefficients in (h) such that a i = Q i (f 1 , . . . , f r , α).
By continuity of these polynomials, there exists M 0 ∈ AE such that f i ≡ g i and α ≡ β modulo h M 0 implies a i ≡ Q i (g 1 , . . . , g r , β) modulo h m . By requiring the g 1 , . . . , g r chosen above to be congruent modulo h M 1 for some M 1 ≥ M 0 , one may assume that s ≥ M 0 hence γ ≡ α modulo h M 0 . Then Artin's approximation theorem says that the corresponding root β of P g in K * can be chosen arbitrarily close to γ, and the conclusion follows.
Recall that we denote by ǫ the automorphism of K defined by f (h) → f (−h). We will need the following proposition only for = Ê, although it can be proved in a more general context. Proposition 29. Let L be an ǫ-invariant finitely generated subfield of K containing (h). Then there exists an ǫ-invariant subfield L * of K * containing (h) and a field isomorphism Ω : L → L * such that ǫ • Ω = Ω • ǫ.
Moreover, for any finite family a 1 , . . . , a t ∈ L and m ≥ 0, then this isomorphism L → L * can be chosen such that each a i is congruent to its image modulo h m .
Proof. Let L ǫ = {x ∈ L | ǫ(x) = x} be the set of elements of L which are fixed by ǫ. It obviously contains (h 2 ) and is contained in K ǫ = ((h 2 )). Then L is a quadratic extension of L ǫ , L = L ǫ ⊕ hL ǫ as a L ǫ -vector space and L ≃ L ǫ [X]/(X 2 − h 2 ) as a field. We let Φ be the canonical field isomorphism ((h 2 )) → K defined by f (h 2 ) → f (h), and let Λ = Φ(L ǫ ). Note that Φ and Φ −1 send convergent series to convergent series.
We have L ≃ Λ + = Λ[X]/(X 2 − h) and the action of ǫ on Λ + is the non-trivial element of the Galois group Gal(Λ + /Λ). The theorem claims that there exists a subfield Λ * of K * , which contains (h) and is isomorphic to Λ over (h). Thus there exists a field isomorphism between the field extensions Λ + /Λ and Λ * + /Λ * ; it sends non-trivial elements in Gal(Λ/Λ + ) to non-trivial elements in Gal(Λ * /Λ * + ). We finally let L * − = Φ −1 (Λ * ) and L * = L * − (h) ⊂ K * . The composite of this isomorphisms is a field isomorphism Ω : L → L * , and it is easily checked that Ω • ǫ = ǫ • Ω. We summarize the situation by the following commutative diagram in the category of field extensions. All vertical unlabelled arrows represent the only non-trivial element in the Galois group of the corresponding quadratic extensions.
For the last part, a i ∈ L = L ǫ ⊕ hL ǫ can be decomposed as a i = a + i + ha − i . Then the isomorphism Λ → Λ * can be chosen such that the a ± i 's equal their images modulo h m and the conclusion follows.
We finally use this proposition in the set-up of the introduction. Let R : G → U ǫ N (K) be a representation of the finitely generated group G into the formal unitary group U ǫ N (K) with = Ê. Our purpose is to deduce from this non-trivial representations of G into the unitary group U N . Let L be the smallest ǫ-invariant subfield of K = Ê((h)) containing Ê(h) and the coefficients of R(g) for g ∈ G. Because G is finitely generated as a group, and ǫ has finite order, L is finitely generated over Ê and we can apply the proposition to this field. Let L * ⊂ K * = Ê({h}) and Ω : L → L * be the field and field isomorphism given by the proposition.
We extend Ω : L → L * coefficientwise to a group isomorphism GL N (L) → GL N (L * ). Since L contains the coefficients of R, R factorizes through U ǫ N (L) ⊂ GL N (L), and we have U ǫ N (L * ) = {x ∈ GL N (L * ) | x −1 = t ǫ(x)} = {Ω(y) | y ∈ GL N (L), Ω(y) −1 = t ǫ • Ω(y)} = {Ω(y) | y ∈ GL N (L), Ω(y −1 ) = Ω( t ǫ(y))} = Ω(U ǫ N (L)). hence Ω • ρ : G → U ǫ N (M ) is a representation of G into U ǫ N (Ê({h})). Let c be the automorphism of ((h)) defined by f (h) → f (ih), and η the automorphism f → f induced by the complex conjugation of the coefficients. These two automorphisms leave ({h}) invariant, and it is readily checked that ǫ and η coincide on c(Ê((h))) = Ê((ih)). In particular, if we let J = c(L * ) ⊂ ({h}), then c • Ω • ρ : G → U ǫ N (K) = U η N (K) ⊂ U η N ( ({h})) hence, by specialization in h real and close to 0, we get morphisms G → U N , i.e. unitary representations of G in the ordinary sense. Now assume that the natural coloring of Γ is known, as well as the representations R 1 , . . . , R r . Assuming this coloring known, we do not need the algebraic closeness assumption anymore. For a complete description of R, the only missing piece is R(σ n−1 ). Since σ n−1 commutes to B n−2 , it suffices to determine the action of σ n−1 on each vector space Hom B n−2 (Ř, R), forŘ a vertex of level n − 2 in Γ. The equation δ n = σ n−1 δ n−1 σ n−1 can be rewritten as (σ n−1 δ n−1 ) 2 = (δ n−1 σ n−1 ) 2 = γ n . Depending on the complexity of Γ, this yields to a sometimes very tractable set of equations on each space Hom B n−2 (Ř, R). Applications of this method can be found in [18]. Its interest heavily depends on the knowledge we have on Γ, the roots of unity involved in its coloring, and the spectrum of σ 1 .
As we have seen before, these datas are known if R = Φ(ρ) with Φ ∈ ××Ó λ ( ), λ = 0 and ρ an essentially pure irreducible representation of B n . This combinatorial approach has thus the advantage of avoiding the (intricated) calculation of Φ(x, y) for x = ρ(t i,i+1 ), y = ρ(Y i ) and nevertheless getting an explicit (matrix) description of R.