On the Linearized Artin Braid Representation

arXiv:hep-th/9210020v1 5 Oct 1992On the Linearized Artin Braid RepresentationF. ConstantinescuFachbereich MathematikJohann Wolfgang Goethe Universit¨at Frankfurt,Robert Mayer Str.

10, D-6000 Frankfurt a. M. 1F. ToppanPhysikalisches InstitutUniversit¨at BonnNussallee 12, D-5300 Bonn 1AbstractWe linearize the Artin representation of the braid group given by (right)automorphisms of a free group providing a linear faithful representation ofthe braid group.

This result is generalized to obtain linear representationsfor the coloured braid groupoid and pure braid group too. Applications tosome areas of two-dimensional physics are discussed.1

1IntroductionLet us consider a free group Fn of rank n with generators x1, x2, ..., xn. Artin[1, 2] proved that the braid group Bn with generators σ1, ..., σn−1 and definingrelationsσiσi+1σi=σi+1σiσi+1σiσj=σjσifor|i −j| ≥2(1)has a faithful representation as a group of right automorphisms ˆσi,i = 1, ..., n −1 of Fn given byˆσi :xi7→xi · xi+1 · xi−1xi+17→xixj7→xjforj ̸= i, i + 1(2)On the other hand Magnus [3] obtained a class of representations of freegroups which can be used to get in some cases representations of subgroupsof the automorphism group of Fn.In this paper we prove that the lin-earization of the Artin transformations (2) using the concept of the Magnusrepresentation leads to a faithful representation of the braid group which wasalready introduced in some different context in [4]:Let J Bn be the ring over the braid group Bn with integer coefficients.

Therepresentatives of the braid group generators are the matrices(ˆσi)jk=1(i−1,i−1) · σi ⊕ αiβiσi0!⊕1n−2−i,n−2−i · σi(3)withαi=σi(1 −θiθi+1θi−1)βi=σiθi(4)andθi=σ1−1...σi−1−1σi2σi−1...σ1(5)The fact that (3) provides a representation of Bn with values in the braidring J Bn can be seen from the following equalities in the braid ring:αiσi+2αi + βiαi+1σi+1=σi+2αiσi+22

αiσi+2βi=σi+2βiαi+1βiβi+1σi+1=σi+2βiβi+1σi+1σi+2αi=αi+1σi+1σi+2σi+1σi+2βi=βi+1σi+1σi+2(6)A full tower of linear representations of the braid group is produced by itera-tively replacing the old generators with the new representatives. The trivialdiagonal representation σi = q ∈R generates for instance the Burau repre-sentation which again, introduced in (3), produces a new representation andso on.

The representation (3) is reducible as a consequence of the fact thatthe element x1 · x2 · ... · xn ∈Fn is left invariant under Artin transformations.Using the braid relations αi can be reexpressed to αi = (1 −θi)σi [5].Another proof of the fact that (3) is a representation of the braid group usinghomology was indicated to the authors by R. Lawrence [5]; it is based on hervery interesting thesis [6]. It seems that ideas similar to ours can also befound in the book [7] and go back to Artin.Here we also extend the construction which leads to (3) to the case of thecoloured braid groupoid and pure braid group.The representation (3) and especially its iterations have applications whichare listed here: determination of braid and monodromy properties of gen-eralized hypergeometric integrals, characterization of braid and monodromyrepresentations through bilinear forms [8, 9], contour methods in two dimen-sional conformal quantum field theory (especially perturbation theory wherethe vertex structure is partially spoiled [9]) and quantum groups [10, 11].These applications will be briefly discussed in section 5 of this paper.

Formore details see [4, 10, 11].Concerning the organization of the paper, we put in section 2 some remarksabout the free differential calculus and the Magnus representation of freegroups and braid group. In section 3 we prove that (3) provides a linearfaithful braid valued representation of the braid group; in section 4 we ex-tend the results to the coloured and pure braid group cases.2The Magnus representationWe follow [12] with some minor modifications.Let Φ be an arbitrary homomorphism action on the free group Fn and let3

FnΦ denote the image of Fn under Φ. Let J FnΦ denote the group ring of FnΦwith integer coefficients.

Elements in J FnΦ are formal linear combinationsof the formP agg (g ∈FnΦ, ag ∈J ).The operations in J FnΦ are defined as follows:Xagg +Xbgg=X(ag + bg)g(Xagg) · (Xbgg)=Xg(Xhagh−1bh)g(7)There is a well defined mapping∂∂xj , j = 1, ..., n, called the free differential∂∂xj:J Fn →J Fn(8)given by∂∂xj(xµ1ǫ1...xµrǫr)=rXi=1ǫiδµi,jxµ1ǫ1...xµi12(ǫi−1)∂∂xj(Xagg)=Xag∂g∂xj(9)where g ∈Fn, ag ∈J , ǫi = ±1.We have the rulesi)∂xi∂xj = δi,jii)∂xi−1∂xj= −δi,jxi−1iii)∂(wv)∂xj= ∂w∂xj vt + w ∂v∂xjiv)chainrule(10)In iii) vt means the sum of the integer-coefficients in v and iv) can be for-mulated as follows: let v1, ..., vn be another system of generators in Fn. Theyare words vi(x1, ..., xn), i = 1, ..., n in Fn.

We have∂∂xjw(v1, ..., vn)=nXk=1∂w∂vk∂vk∂xjRemark the unusual form of ii) and iii).In section 4 we will consider the monodromy subgroup of the pure braid4

group, given by the generators θi of formula (5). Every free group is iso-morphic to the group generated by θi.

Via this identification we are able toconsider the multiplication operation of elements of a free group by elementsof the braid group Bn.We are going now to recall to the reader the Magnus representations of a freegroup and of the braid group. Let Sn be a free abelian semigroup with basiss1, ..., sn and let A(R, Sn) be the semigroup ring of Sn with respect to thering R. Let Φ be a homomorphism acting on the free group Fn; for w ∈Fnlet wΦ be the image of w under Φ.

Then the mapping w →(w)Φ,(w)Φ= wΦPnj=1( ∂w∂xj )Φsj01! (11)with the entries of (w)Φ in A(J FnΦ, Sn), is a representation of Fn called theMagnus Φ-representation.

It is faithful if Fn is abelian; otherwise its kernelis the commutator subgroup of kerΦ [12]. Now let A be any group of right(automorphisms) of Fn such thatxΦ=xαΦ(12)for each x ∈Fn,α ∈A.

Then the matrix||α||Φ= ∂(xiα)∂xj!Φ(13)with entries in J FnΦ defines a linear representation of A called again Magnusrepresentation. If the automorphism group A is the Artin automorphism (2),then we get the Magnus representation of the braid group.

If Φ is definedby xiΦ = q ∈R, 1 ≤i ≤n, we recover the Burau representation as aspecial case of the Magnus representation. The proof of the above assertionsis based on the chain rule of the free differential calculus and uses heavily thecondition (12).

In fact it turns out that this condition is rather restrictivebecause in the braid case it implies that xiΦ must be independent of i.In the next section we will drop out the condition (12) in the particularcase when A coincides with the Artin’s automorphism group and prove whatwe call the braid valued generalization of the Magnus representation for thebraid group.5

3Braid valued representation of the braidgroupLet ˜Fn be now a copy of Fn wih generators si. Certainly there will be a(generalized) Magnus representation of Fn given by w →(w), w ∈Fn and(w)= wPnj=1∂w∂xj sj01!

(14)where the entries of (w) are in the free ring of ˜Fn with respect to the ringJ Fn. For the generators this representation reduces to(xi)= xisi01!

(15)At the first glance this representation seems not to be very interesting, how-ever we will prove that it provides a representation of the braid group viaArtin automorphism.We identify the free group Fn with the monodromy subgroup of the purebraid group via the identification of the generators xi and θi. For the mon-odromy group the Artin transformations are equivalent to the adjoint actionof the generators σi:σi−1θkσi=ˆσi(θk)forany1 ≤i, k ≤n(16)whereˆσi(θi)=θi · θi+1 · θi−1ˆσi(θi+1)=θiˆσi(θj)=θjforj ̸= i, i + 1For given i the right hand side of (16) provides a new system of generatorsof the monodroy group.We go now from the free group Fn to the free group F ′n whose generators θ′jare the images under the Artin transformation ˆσi of the Fn generators:θ′j=σi−1θjσi,1 ≤i, j ≤n6

Let us introduce for a given i the jacobian map J of the free differentialcalculusJ:J Fn →J FnJ= ∂ˆσi(θj)∂θk! (17)which is the linearization of the Artin transformation.

We proceed in thesame way as before going from the free group F ′n to F ′′n with generatorsθ′′j=σl−1θ′jσl,1 ≤l, j ≤nThe linearization of the Artin relations (applied to θ′j) is given by the jacobianJ′ : J F ′n →J F ′n defined as (17) with the replacement θj →θ′j.It is possible to transport the linear mapping J′ to a linear mapping in J Fnby using the commutativity of the following diagramJ FnJ−→J Fn↓↓J F ′nJ′−→J F ′n(18)where the vertical arrow denotes the adjoint map adσ:adσ : β 7→σi−1βσi ∈J F ′n,β ∈J Fn(19)The above diagram makes possible to writeJ′=σi−1Jσi(20)We get, using the braid relations:(σj−1Jiσj)Jj=(σi−1Jjσi)Ji,|i −j| ≥2(21)and[σi+1−1(σi−1Ji+1σi)σi+1](σi+1−1Jiσi+1)Ji+1 =[σi−1(σi+1−1Jiσi+1)σi](σi−1Ji+1σi)Ji(22)7

(21) and (22) give, by inserting factors σ−1σ,σj−1σi−1σiJiσjJj=σi−1σj−1σjJjσiJi,|i −j| ≥2(23)andσi+1σiσi+1σi+1−1σi−1Ji+1σiσi+1σi+1−1Jiσi+1Ji+1 =σiσi+1σiσi−1σi+1−1Jiσi+1σiσi−1Ji+1σiJi(24)The relations (23,24) can be written now by using again the braid relationsas follows(σiJi)(σjJj)=(σjJj)(σiJi),|i −j| ≥2(25)(σiJi)(σi+1Ji+1)(σiJi)=(σi+1Ji+1)(σiJi)(σi+1Ji+1)(26)This shows that σi →σiJi is a linear representation of the braid group. Bycomputing the matrix elements ( ∂ˆσi(θj)∂θk ) of Ji following the rule of the freedifferential calculus, we get (3).

As an example take∂ˆσi(θi)∂θi=∂∂θi(θiθi+1θi−1) = ∂θi∂θi+ θi∂∂θi(θi+1θi−1) ==1 + θiθi+1∂∂θi(θi−1) = 1 −θiθi+1θi(27)Since the representation of the braid group through the Artin right autho-morphism is faithful and σi−1θjσi for given i is an isomorphism of Fn onF ′n, it follows that the braid valued representation (3) is a linear faithfulrepresentation.4The coloured case and the pure braid groupIn this section we apply the previous construction to the coloured case (rep-resentations of the coloured braid grupoid) and to the pure braid group.We introduce first the coloured generators σi(λ, µ) (where λ, µ, ... can belooked as colours attached to strings) of the coloured braid groupoid Bnc;they satisfy the relationsσi(λ, µ)σi+1(λ, ν)σi(µ, ν)=σi+1(µ, ν)σi(λ, ν)σi+1(λ, µ)(28)8

andσi(λ, µ)σj(ν, ρ)=σj(ν, ρ)σi(λ, µ),for|i −j| ≥2(29)The inverses are introduced through the relationσi(λ, µ)σi−1(µ, λ)=1(30)The monodromies θi are straightforwardly generalized to the coloured caseas θρi (..., λ, µ), where λ , µ are the last enclosed colours and ρ is the colourassociated to θi:θρi (..., λ, µ)=...σi−2−1(ρ, λ)σi−1(ρ, µ)σi−1(µ, ρ)σi−2(λ, ρ)...(31)The considerations of section 3 can be generalized to the coloured braidgrupoid and provide a matrix representation of the generators generalizing(3). The matrix Bci (λ, µ) is obtained from (3) by replacing σi →σi(λ, µ),θi →θρi (..., µ, λ):(Bci )jk(..., λ, µ) ≡(Bci )jk(λ, µ) =1(i−1,i−1) · σi(λ, µ) ⊕ αi(λ, µ)βi(λ, µ)σiλ, µ0!⊕1n−2−i,n−2−i · σi(λ, µ)(32)where αi(λ, µ), βi(λ, µ) can be easily read from (4).In the particular case in which the generators σi(λ, µ) are represented byqi ∈R, we get from the matrices Bci (λ, µ) a representation of the colouredbraid grupoid; when specialized to the generators of the pure braid group thisrepresentation coincides with the one discovered by Gassner [12].

It followsthat the Gassner matrices satisfy not only the pure braid relations but also10001 −ti+1ti0101 −ti+2ti010000110001 −ti+2ti+1010=1 −ti+2ti+1010000110001 −ti+2ti0101 −ti+1ti0100001(33)and iterated relations of this kind, (see also [13]).Equation (33) remembers the Yang-Baxter relation, but for the fact that it9

lives on a direct sum instead of a tensor product. The problem of promotinga braid group representation from acting on a direct sum to a braid grouprepresentation on a tensor product space is of particular interest in the theoryof quantum groups.

A particular nice example was provided by L. Kauffmanand H. Saleur [13] in connection with the quantum supergroup Uqgl(1, 1).The tensor product space is constructed in this case with the help of theexterior algebra over the direct sum space.We come now to the pure braid group Pn. It can be algebraically defined asthe subgroup of Bn having generatorsθk(i)=σi−1...σk−2−1σk−12σk−2...σi(34)for 1 ≤i < k ≤n and defined relations which appear for instance in [12], p.29.

Here we will consider the pure braid group Pn+1 with the extra generatorsθk(0)=σ0−1...σk−2−1σk−12σk−2...σ0,1 < k ≤nwhere σ0 is the extra generator of Bn+1 as compared to Bn.The sameprocedure as before allows us to construct a representation of Pn realizedwith matrices having entries in J Pn+1. It is convenient to express the n × nrepresentative matrices ˆθ(i)kof the generators in terms of their action on then generators yj of a (right-) module over the ring J Pn+1.

It turns out that:i) For j < i or j > k we getˆθ(i)k:yj 7→θk(i)yjthis relation being trivial.ii)ˆθ(i)k:yk 7→θk(i)(1 −θi(0)θk(0)θi(0))yi +θk(i)θi(0)ykiii) for i ≤j < kˆθ(i)k:yj 7→A(i,j,k)yj + B(i,j,k)yk + C(i,j,k)yiwhereA(i,j,k)=θi(0)θk(i)θi(0)10

B(i,j,k)=[1 −θi(0) −θj(0) + θj(0)θi(0)] ··θk(i)θi(0)C(i,j,k)=[1 −θj(0) + θj(0)θi(0)]θk(i) +θj(0)θk(i)θi(0) −θk(i)θi(0)θk(0)θi(0) −−θi(0)θk(i)θi(0)θj(0)θi(0)In ii) and iii) the inverse of θi(k) has been denoted by θi(k) for tipograficalreasons.5ApplicationsA geometric visualization of the representation (3) was given in [4]. We startby describing the connection of (3) with the realization of the braid groupon analytic functions with isolated branch points singularities (for details see[4]).

LetM>n={(z1, ..., zn) : |zi| > |zj|,ifi < j,−iπ < argzk ≤iπ} (35)be a simply connected subset of Cn. Let {fj, j ∈J} be a family of holomor-phic functions in M>n .

Singularities can appear only if two variables approacheach other, zi →zj. We continue this function on the universal covering ofM>n which isMn={(z1, ..., zn) :zi ̸= zjifi ̸= j}(36)Choosing a point P ∈M>n and denoting by γ a path in Mn starting atγ(0) = P and arriving at γ(1) = z ∈Mn, the expression fj(zγ) denotes the(unique) analytic continuation of fj from P ∈M>n to z ∈Mn along γ. Weintroduce an action ˜σi of the braid group Bn on functions fj by(˜σifj)(z)=fj(z, γi(z))(37)for a path γi(z) in Mn running from P first interchanging Pi and Pi+1 inpositive direction and then connecting the resulting point to(z1, ..., zi−1, zi+1, zi, zi+2, ..., zn) = ti(z) on a path in tiM>n .11

The interesting point is that the Artin relations (2) can be realized on ˜xiwhere(˜xifj)(z)=Zγifj(t, z1, ..., zn−1)dt(38)with γi a loop around zi. Now the claim is that the relations˜xj˜σi=Xk(B(i))jk˜xk(39)for i = 1, 2, ..., n are verified [4], where B(i) are the braid matrices (3).

Thisallows us to get a representation of the braid groups on integrals if the braidrepresentation on the integrands is known. The relation (39) can be under-stood as commuting the generators of the braid group through the integrals.Certainly this procedure can be iterated making possible a computation of thebraid and monodromy properties of generalized hypergeometric integrals (inparticular no “charge condition” is necessary).

The graded structure whichresults from this construction as well as details and specific computationsare contained in [4]. Let us remark that in the two-dimensional conformalperturbation theory braid properties of such generalized hypergeometric in-tegrals without charge condition are needed.

These problems were studied in[9]. Finally there are applications of the results and methods of this paper inthe theory of quantum groups.

A step in this direction was taken in [10, 11]where the vertex construction of Gomez and Sierra [14] was generalized andput in the framework of an abstract braid module. For some related ideas inthe supergroup case Uqgl(1, 1) see [13].Acknowledgements We would like to thank G. Burde for having givensome hints to us, M. L¨udde, H.F. de Grote and R. Flume for useful discus-sions.References[1] E. Artin “Theorie der Zopfe”, Hamb.

Abh. 4 (1925), 47-72[2] E. Artin “Theory of braids”, Ann.

of Math. 48 (1947), 101-12612

[3] Magnus “On a Theorem of Marshall Hall”, Ann. of Math.

40 (1939),764-768[4] F. Constantinescu and M. L¨udde, Preprint Bonn-HE-91, July 1991. [5] R. Lawrence, private communication.

[6] R. Lawrence, Thesis and Comm. Math.

Phys. 135 (1990), 141-191[7] G. Burde and H. Zieschang, “Knots”, de Gruyter, Berlin 1985[8] F. Constantinescu and M. L¨udde “Monodromy groups and bilinear in-variants”[9] P. Chaselon, F. Constantinescu and R. Flume, Phys.

Lett B 257 (1991)63-68[10] M. L¨udde and F. Toppan, “Matrix solutions of Artin’s braid relations”,Bonn preprint 92, to appear in Phys. Lett.

B (1992)[11] M. L¨udde, “Yang-Baxter deformations of semisimple Lie algebras”,Bonn preprint 92, to appear in Journ. Math.

Phys. (1992)[12] Birman, “Braids, Links and Mapping Class Group”, Ann.

of Math. Stud-ies 82, Princeton University Press (1975)[13] L. Kauffman and H. Saleur “Fermions and link invariants”, Int.

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1A (1992) 493-532[14] C.Gomez, R.Sierra, Nucl.Phys.B352 (1991) 791-82613

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