Twisted conjugation braidings and link invariants

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Twisted conjugation braidings and link invariants

Dissertation

zur

Erlangung des Doktorgrades (Dr. rer. nat.) der

Mathematisch-Naturwissenschaftlichen Fakult¨at der

Rheinischen Friedrich-Wilhelms-Universit¨at Bonn

vorgelegt von

Mar´ıa Guadalupe Castillo P´erez aus

Mexico City

Bonn, Februar 2009

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Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨at Bonn.

1. Referent: Prof. Dr. C.-F. B¨odigheimer 2. Referent: Prof. Dr. Catharina Stroppel

Promotionsdatum: 5. Juni 2009 Erscheinungsjahr: 2009

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Abstract

This work is about link invariants arising from enhanced Yang-Baxter operators. For each enhanced Yang-Baxter operatorR= (R, D, λ, β) and any braid Br(n) Turaev defined a link invariantTR(ξ) = λ^−ω(ξ)β⁻ⁿtrace(bR(ξ)◦D^⊗n), where ω :Br(n)→ Zis a homomorphism and bR is the representation of the Artin braid group Br(n) arising from the solution of the Yang-Baxter equati- onR.Therefore, we first introduce new solutions of the Yang-Baxter equation B^ϕ :V^⊗2→ V^⊗2, B^ϕ(a⊗b) =abϕ(a)⁻¹⊗ϕ(a), forV =K[G], ϕ∈Aut(G), whereGis any group. We call these solutionstwisted conjugation braidings.

Then we give sufficient and necessary conditions for a map D to decide whe- ther the quadruple (B^ϕ, D, λ, β) is an EYB-operator. Moreover, we prove that the twisted conjugation braidingsB^ϕ can be enhanced using character theory.

These enhancements are called character enhancements. It turns out that for every character enhancementDof the twisted conjugation bradingB^ϕthe link invariant is constantly 1, i.e.,TB(ξ) = 1 for allξ∈Br(n).In general, we prove that the link invariant for all ξ ∈Br(n) and for every enhancement D of the twisted conjugation braidingB^ϕis a mapTB(ξ) =β⁻ⁿtrace(bB^ϕ)◦D^⊗n . Our main result is the following theorem.

Letγbe a fixed invertible element ofKand letD denote a linear map. Asumme thatD⊗D commutes with the twisted conjugation braiding B^ϕ. Then

1. Sp2((B^ϕ)^±1◦(D⊗D)) =γD =⇒ D²=γD

2. Sp2(B^ϕ◦(D⊗D)) =γD ⇐⇒ Sp2((B^ϕ)⁻¹◦(D⊗D)) =γD

In the last part of this work, we prove that for finite groupsGthe twisted conjugation braidingB^ϕsatisfies (B^ϕ)^l(a⊗b) =a⊗b,withl= 2·lcm(ord(a), ord(b).

From this follows that the link invariant isTB(ξ) =

m1

β

n

, for braidsξinBr(n), withξ=σ_σ^ǫ¹_i1. . . σ^ǫ_i_l^l, and withǫ1, . . . , ǫl≡ 0 modl,where m1= trace(D). We call such braidsmod-lbraids. Furthermore, it follows that the link invariant is TB=

m1

β

n−1

for braidsξ ∈Br(n) such that ξ=σ_i^ǫ, withǫ ≡ 0 modl. We call these braidssingle-power braids. Moreover, we wrote a program in JAVA programming language which computes the link invariants for the enhancement D = γI,(γ ∈ K^∗) for braids ξ ∈Br(p), (p prime) with ξ = (σ1σ2. . . σp−1)^q, and with (p, q) = 1 for the cases G= Σn andG=Z/nZ. In the cases were we have computed the link invariantsTB“the polynomial is constant,” i.e.,TB∈K, since the only braidings we consider are permutations of the basisK[G]^⊗2.

1

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Introduction

In the 1988’s [14] Turaev defined a criteria called an enhancement. If satisfied, would produce a Markov trace and hence lead to a link invariant. To describe his criteria let K be a commutative ring with 1 and let V be a K− free module of finite rank m ≥ 0. A solution of the Yang- Baxter equation R is an invertible linear map R : V ⊗V → R⊗R which satisfies the equation (R⊗1)(1⊗R)(R⊗1) = (1⊗R)(R⊗1)(1⊗R) in Aut(V^⊗3).This equation first has appeared in independent papers of C. N. Yang and R. J. Baxter in the late 1960’s and early 1970’s, respectively.

This equation and its solutions play a fundamental role in statistical mechanics ([18]) and in knot theory ([7], [9], [10]). For example, a relationship between the Yang-Baxter equation and polynomial invariants of links can be found in [6]. In this paper, Jones introduced his famous polynomial of links via the study of certain finite dimensional von Neumann algebras. A remark of D. Evans mentioned in [6] points out that these algebras were earlier discovered by physicists who used them to study the Potts model of statistical mechanics.

For describing Turaev’s criteria we need to recall as well his definition of an enhanced Yang-Baxter operator. An enhanced Yang-Baxter operator (EYB) is a quadruple R = (R, D : V → V, λ ∈ K^∗, β ∈K^∗), where R is a solution of the Yang-Baxter equation and D is an endomorphism of V which satisfies

(T1) D⊗D commutes withR, (T2a) Sp₂(R◦(D⊗D)) =λ^±1βD,

(T2b) Sp2(R⁻¹◦(D⊗D)) =λ^±1βD, whereSp2 :V → V denotes the partial trace on the second factor. For the definition and properties of partial trace we refer the reader to Definition 2.1.1, Lemma 2.1.2 and Lemma 2.1.3.

In chapter 1 we use group rings V =K[G] and automorphisms of the groupG to introduce new solutions of the Yang-Baxter equationB^ϕ:V^⊗2→V^⊗2.We defineB^ϕ(a⊗b) =abϕ(a)⁻¹⊗ϕ(a),for any group Gand for V =K[G],and ϕ∈Aut(G). Throughout this workB^ϕ will be called twisted conjugation braidingand by a link we will understand a finite family of disjoint, smooth oriented or unoriented, closed curves in R³, or equivalently S³.An example of a solution B^ϕ is the following.

Set G to be an abelian group. Then the twisted conjugation braiding B^ϕ(a⊗b) = aba⁻¹ ⊗a.

Moreover, observe that if Gis commutative thenB^ϕ is the twist map.

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In Theorem 2.2.6 we completely characterize EYB-operators by a set of three equations. This allows us to show that the twisted conjugation braidingB^ϕ is an enhanced Yang-Baxter operator.

(We refer the reader to Theorem 2.2.6 for a precise formulation).

As a corollary of Theorem 2.2.6, we have:

Corollary 2.2.7 Let G be any finite group, V = K[G], and D = qId, where q is an invertible element of K. Then, B^′= (B^ϕ, D, λ= 1, β =q) is an EYB-operator.

Moreover, in Chapter 3 we prove in terms of characters of the group G×G that the twisted conjugation braiding B^ϕ is an enhanced Yang-Baxter operator. Indeed we have

Theorem 3.2.1 Let χ be a character defined from G×G into K^∗. Define the K-linear map D:K[G]→K[G],via its action on the basis elements a∈G,

D(a) =X

c∈G

χ(a, c)c,

then the following three conditions are satisfied:

1. The quadruple B= (B^ϕ, D, λ= 1, β =trace(D))is an EYB-operator, 2. B^ϕ◦(D⊗D) =D⊗D,

3. Sp₂(B^ϕ◦(D⊗D)) =trace(D)D

Coming back to the description of Turaev’s criteria. For each EYB operator R, Turaev defines in [14] a map T_R:`

Br(n)→K, as follows.

For a braid ξ ∈Br(n),

T_R(ξ) =λ^−ω(ξ)β⁻ⁿtrace(b_R(ξ)◦D^⊗n),

whereωis the homomorphism fromBr(n) to the additive group of integersZwhich sendsσ1, . . . , σn−1

into 1, and bRis the representation of the Artin braid groupBr(n), arising from the Yang-Baxter solution R:V^⊗2→V^⊗2. Namely, bR sends σi intoid^⊗(i−1)⊗R⊗id^(n−i−1).

The most important properties of the map T_R are given by the following theorem.

Theorem ((3.1.2), [14]) For any ξ, η,∈Br(n)

TR(η⁻¹ξη) =TR(ξσn) =TR(ξσ_n⁻¹) =TR(ξ).

Due to a theorem of J.W. Alexander (first part) and A. A. Markov, any oriented link is isotopic to the closure of some braid. The closures of two braids are isotopic (in the category of oriented links) if and only if these braids are equivalent with respect to the equivalence relation in `

nBr(n) generated by the Markov moves ξ 7→ η⁻¹ξη, ξ 7→ ξσ_n^±1, where ξ, η ∈ Br(n). Turaev’s theorem

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(Theorem 2.3.1) shows that for any enhanced Yang-Baxter operatorR= (R, D, λ, β),the mapping T_R:`

nBr(n)→Kinduces a mapping of the set of oriented isotopy classes of links into K. Motivated by Turaev’s work (mentioned above), we prove in Chapter 2 (Corollary 2.5.3) that the link invariant TB of any EYB-operatorB= (B^ϕ, D, λ, β) of the twisted conjugation braidingB^ϕ is given by the formula

T_B(ξ) =β⁻ⁿtrace(b_B^ϕ(ξ)◦D^⊗n) for any braid ξ ∈Br(n).

Moreover, in Chapter 3 we prove that the link invariant associated to any character enhancement Dχ of the twisted conjugation braidingB^ϕ is constantly 1, i.e.,T_B(ξ) = 1 for all ξ∈Br(n).(We refer the reader to Theorem 3.3.2 for a precise formulation).

Remark Theorem 3.3.2 shows that new link invariants will only arise from enhancements D of the twisted conjugation braiding B^ϕ that do not arise from a character χ:G×G→K.

The main result in this work is that any enhancementD of the twisted conjugation braidingB^ϕ is idempotent. Indeed we have the following theorem.

Theorem 4.1.1 (Idempotence) Letγ be fixed invertible element ofK, and letDdenote a linear map. Assume that D⊗Dcommutes with the twisted conjugation braiding B^ϕ.

1. IfSp₂(B^ϕ◦(D⊗D)) =γ · D,then D² =γ D.

2. IfSp2((B^ϕ)⁻¹◦(D⊗D)) =γ · D,then D²=γ D.

3. The following two statements are equivalent.

(a) Sp₂(B^ϕ◦(D⊗D)) =γ D, (b) Sp₂((B^ϕ)⁻¹◦(D⊗D)) =γ D.

Other important properties of the map T_R are given by the following result of Turaev (see [14]).

For the trivial knot we have

T_R() =β⁻¹trace(D).

If a link L=L1⊔L2 is the disjoint union of two linksL1 and L2 then T_R(L) =T_R(L₁)T_R(L₂),

i.e., the map T_R is multiplicative.

In particular, if L is the trivial n-component link, then T_R(L) =β⁻ⁿtrace(D)ⁿ.

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In this work, we compute the link invariants T_B for enhancements of the twisted conjugation braiding B^ϕ, for braids ξ in Br(n), with ξ = σ^ǫ_i₁¹. . . σ^ǫ_i^l

l, and with ǫ₁, . . . , ǫ_l ≡ 0 mod l. Such braids are called mod-l braids. We also compute the link invariants T_B for enhancements of the twisted conjugation braiding B^ϕ for braidsξ ∈Br(n) such thatξ=σ_i^ǫ,withǫ≡1 modl.We call these braids single-power braids. In Chapter 6, by using the program “Bhi orders” we compute the link invariants for the enhancement D = γI, (γ ∈ K^∗) for braids ξ ∈ Br(p), (p prime) with ξ = (σ₁σ₂. . . σ_p−1)^q,and with (p, q) = 1.

Our results are the following.

Remark In the cases were we have computed the link invariantsT_B,“the polynomial is constant”, i.e, T_B ∈ Kas we see in the following table (Table 6.13), since the only braidings we consider are permutations of the basis of K[G]^⊗2.

Table 1: Link invariants forG= Σ₅, ϕ(s) =s₂ss⁻¹₂ and D=γI

Knot Name (p, q) T_B

Hop link (2,2) 840

3₁ Trefoil knot (2, 3) 600

5₁ Solomon’s seal knot (2, 5) 720 7₁ 7 crossing torus knot (2, 7) 120 8₁₉ 8 crossing torus knot (3, 4) 1200

91 9 crossing torus knot (2, 9) 600 10₁₂₄ 10 crossing torus knot (3, 5) 600 11 crossing torus knot (2, 11) 120

Proposition 5.1.1 Asumme that D is an enhancement of the twisted conjugation braiding B^ϕ. Moreover, assume that (B^ϕ)^l◦(D⊗D) =D⊗D for some l∈N.Then

1. T_B(ξ) =

m1

β

n

, for all mod-lbraidsξ∈Br(n), wherem₁ =rank (D).

2. T_B(ξ) =

m1

β

n−1

, for all single-power braidsξ=σ^ǫ_i ∈Br(n), withǫ≡1modl.

In particular, for the enhancement D=qI, withq ∈K (invertible) 1. T_B(ξ) =|d|ⁿ, for all mod-lbraidsξ ∈Br(n), where, d=|G|

2. T_B(ξ) =|d|ⁿ⁻¹, for all single-power braids ξ=σ^ǫ_i ∈Br(n), withǫ≡1modl.

Examples of enhancements D of the twisted conjugation braiding B^ϕ, satisfying the hypothesis (B^ϕ)^l◦(D⊗D) =D⊗D,occur for example in the following situations.

Examples

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1. Let G be commutative group and set ϕ=id. Then the twisted conjugation braiding B^ϕ is the twist map, i.e. B^ϕ(a⊗b) =b⊗a. Therefore, (B^ϕ)² =id (see Proposition 5.1.3).

LetG=Z/3Z={1, x, x²},withx³ = 1 and assume thatϕis the automorphims which sends x7→x², x² 7→x.Then, (B^ϕ)³ =id (see Proposition 5.1.5).

Another example of enhancements Dof the twisted conjugation braidingB^ϕ, satisfying the condi- tion (B^ϕ)^l◦(D⊗D) =D⊗Dof previous Lemma is given by the following theorem.

Theorem 5.1.9 Let D :K[G] → K[G], defined as D(a) = P

∆_c∈(a, c)c. Assume that (D⊗D) commutes with the twisted conjugation braiding B^ϕ. Moreover, assume that there is no pair of elements aand c∈G such that∆(a, c) and∆(ϕ(a), ϕ(c)) vanish at the same time. Then

B^ϕ◦(D⊗D)◦B^ϕ =D⊗D In particular,

(B^ϕ)²⊗(D⊗D) =D⊗D= (D⊗D)◦(B^ϕ)².

Our work is organized as follows:

In Chapter 1, we introduce the twisted conjugation braiding (solution of the Yang-Baxter equation) B^ϕ. Moreover, motivated by the work of Sarah Schardt, (see [11]), we define an action of the Braid group Br(n) on K[G]^⊗n. With the help of this action, we give a slight generalization of Schardt’s Hopf algebra H(G). Namely, we define two Hopf algebra structures, (µ^ϕ_R,∆, ǫ, η) and (µ^ϕ_R,∆, ǫ, η), on the tensor algebra H^ϕ := ⊕_n≥0V^⊗n, compare with [11] Moreover, we prove that these Hopf algebras have invertible antipode maps S_L^ϕ and S_R^ϕ, respectively.

In Chapter 2, we recall the definition of the partial trace (Definition 2.1.1, Definition 2.1.4, see [3, 8]), and we prove that the partial trace does not depend on the choice of the basis (Lemma 2.1.2). Moreover, we recall Turaev’s work (see [14]) and we give the proof of Theorem 2.2.6 and Corollary 2.2.7.

In Chapter 3, we prove in terms of characters of the group G×G that the twisted conjugation braiding B^ϕ is an enhanced Yang-Baxter operator. Namely, we prove that if the map D:K[G]→ K[G] is defined asD(a) =P

c∈Gχ(a, c)c,for alla∈G,withχa character fromG×Ginto a fieldK. Then Dis an enhancement of the twisted braidingB^ϕ.Such enhancements will be calledcharacter enhancements and will be denoted by Dχ.Moreover, we prove that character enhancementsDχ of the twisted conjugation braiding B^ϕ satisfy the property

B^ϕ◦(D⊗D) =D⊗D.

At the end of this chapter we give the proof of Theorem 3.3.2.

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InChapter 4, we prove that any enhancementDof the twisted conjugation braidingB^ϕ satisifies D² =γ · D,whereγ is a fixed invertible element inK.In particular, ifDis invertible thenD=γI, i.e. we recover the enhancement Dgiven by Corollary 2.2.7.

In Chapter 5,we give the proof of Proposition 5.1.1 and give some examples of enhancements D of the twisted conjugation braidingB^ϕ,satisfying the hypothesis (B^ϕ)^l◦(D⊗D) =D⊗D.At the end of this chapter we give the proof of Theorem 5.1.9.

In Chapter 6, we prove thatord(B^ϕ) =ord(B^id) for all ϕ∈Inn(G). Moreover, we prove that if the least common mutiplem of the order of all elementsa∈Gexists, then the order of the twisted conjugation braiding B^ϕ is smaller than or equal to 2m. With the help of he computer program

“Bphi orders,” which is written in JAVA programming language, we compute at the end of this chapter the link invariantsT_B for the enhancementD=γI(γ ∈K^∗) for braidsξ ∈Br(p) (p prime) with ξ= (σ₁. . . σ_p−1)^q,and with (p, q) = 1 for the cases G= Σ_n and G=Z/nZ.

InAppendix A,we prove that the Hopf algebras (H^ϕ(G), µ^ϕ_L,∆, η, ǫ, S_L^ϕ) and (H^ϕ(G), µ^ϕ_R,∆, η, ǫ, S_R^ϕ) are neither quasi-commutative nor quasi-cocommutative, therefore they are not quantum groups.

In Appendix B, using Whitehouse and Worocnicz’s (see [15] and [17]) solutions of the YB- equation, we prove that the Hopf algebras (H^ϕ(G), µ^ϕ_L,∆, η, ǫ, S_L^ϕ) and (H^ϕ(G), mu^ϕ_R,∆, η, ǫ, S_R^ϕ) are not braided Hopf algebras.

In Appendix C,we recall the main properties of the tensor product of matrices.

In Appendix D, we explain how to use the program ”Bphi orders” which is written in JAVA programming language.

Acknowledgments First, I would like to thank CONACYT for giving me financial support during three years. Without this financial support, this work would not have been possible. More- over, I would like to thank my advisor Carl-Friedrich B¨odigheimer. He suggested this project, and I am grateful for all his help and useful suggestions in the development of this work. He always answered my questions with patience and always found time for our discussions . I learned and benefited a lot from all our meetings.Without all his support and help, this work would not have been possible. I also would like to thank Andres Angel, Christian Ausoni, Ryan Budney, Gerald Gaudens, Birgit Richter and Eduardo Santillan for a lot of discussions and helpful suggestions on this project. Moreover, I could learn and benefit a lot from exchanging several emails with Fred Cohen and Sarah Whitehouse. Furthermore, I would like to thank the second referee of this thesis, Catharina Stroppel, for reading a preliminary version of this thesis and for all her useful suggestions and remarks.

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While this project was carried out, I was supported by the Mathematical Institute of the University of Singapore, which enabled me to visit for one month in June 2007. I would like to thank the Graduiertenkolleg 1150 “Homotopy and Cohomology” , which gave me the opportunity to attend several conferences. Furthermore, I would like to thank to my brother student Balazs Visy for listening to my talks before I gave them, for his support during the last months of the development of this work and for reading a previous versions of this thesis. Furthermore, I would like to thank the family Misgeld for their support and hospitality during the last two months while I was writing the last chapters of this thesis. I would also like to thank the secretaries of the Mathematical Institute, Karen Bingel and Sabine George, for all their non mathematical support. Moreover, I would like to thank all my friends, in particular, I am grateful to Susso Schüller for all his help with the computer program, for listening to me, and for his encouragement and non mathematical support during all the years of my PhD studies. My thanks also go to my friends Jörn Müller, Philipp Rheinhard and Rui Wang for all the nice time and conversations I had with them. Furthermore, I would like to thank Oscar Loaiza Brito for all the nice moments that I had with him during 13 years. For he has encouraged me to apply to the Universitiy of Bonn, and for all his help, and for believing in me and in this work.

Finally my warmest thanks go to my parents Isaias Castillo Ortega and Mar´ıa Luisa Pérez Mart´ınez for giving the life, for believing in me, for their love and all their support through all the years of my studies. Without them and their encouragement I would not have been able to complete my studies and in particular this work . To my sisters and brothers Armando Aparicio Pérez, Santiago, Gregorio, Mar´ıa Luisa and Rosaura Castillo Pérez for all their support during all my studies, for their love, and for giving the best moments of my life. To my nieces and nephews Alexia Cortes Castillo, Rodrigo and Claudia Villavicencio Castillo , Armando Aparicio Aparicio, Alejandro Castillo, Raul Cortes Castillo and all my nieces and nephews who I still have to meet.

Last but not least to my dear and loved Ingo for all his support, for making it easy for me the last two years of my PhD studies. For being next to me in the most difficult times, and for giving me the force to go on, even when it seemed that there would not be an end. With his love and words he encouraged me not to give up.

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Chapter 1

The twisted shuffle Hopf algebra of a group

In the first section of this chapter we recall Schardt’s Hopf algebra H(G),(see[11]).In the second section, we define the twisted conjugation braiding B^ϕ (solution of the Yang-Baxter equation), which will play an important role throughout this work, since it will help us to describe some link invariants for some finite groups, as we will see in the next chapter of this thesis. In section 3, we give a slight generalization of Schardt’s Hopf algebra HG. The main part of this chapter is based on her work. We define two Hopf algebra structures on the tensor algebra H^ϕ(G). First, we define the two products µ^ϕ_L and µ^ϕ_R, respectively. We then define the twist mapstw^ϕ_Land tw_R^ϕ, respectively, and a coproduct ∆. Secondly, we prove that the coproduct ∆ is compatible with both products, and finally we show that the Hopf algebras (H^ϕ(G), µ^ϕ_L,∆, η, ǫ) and (H^ϕ(G), µ^ϕ_R,∆, η, ǫ) have antipode maps S_L^ϕ and S_R^ϕ, respectively. Moreover, in Apendix A and Appendix B, we prove that these Hopf algebras are neither quasi-commutative nor quasi-cocommutative; therefore they are not quantum groups. We will show as well using Whitehouse and Woroniwicz’s solutions of the YBE Ψ,Ψ^′; respectively Φ,Φ^′.(See [15], [17]), that they are not braided Hopf algebras.

1.1 Schardt’s Hopf algebra H(G)

In this section, we recall Schardt’s Hopf algebra, which has been introduced in [11], for two reasons.

First, because the main part of this chapter is based on her work and second, because it is an example of the Hopf algebra H^ϕ(G), which will be introduced later in this chapter. Thus, using her definition of the shuffle product on H(G), we compute the shuffle-products, coproduct and antipode maps, when we set Gto be the trivial group.

In [11], Schardt introduced the Hopf algebra H(G), associated to a group as follows: LetKbe any commutative ring with unit 1, and denote V =K[G] the ring group ofG. SetH(G) =L

n≥0V^⊗n.

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If we use the usual concatenation product

(x₁⊗ · · · ⊗xn)(y₁⊗ · · · ⊗ym) = (x₁⊗ · · · ⊗xn⊗y₁⊗ · · · ⊗ym) on H(G) we called it the tensor algebra, but Schardt defined a shuffle-productµ, as:

(x₁⊗ · · · ⊗x_l).(x_l+1⊗ · · · ⊗x_n) = X

σ∈(l,n−l)−shuffle

sgn(σ)(x^σ₁ ⊗ · · · ⊗x^σ_n) with

x^σ_j =





x_σ−1(j) if σ⁻¹(j), ∈ {l+ 1, . . . , n}

(x_σ⁻¹_(j))x_l+1...x_l+r ifσ⁻¹(j), ∈ {1, . . . , l}

andσ(l+r)< j < σ(l+r+ 1) and x_y =y⁻¹xy.

Moreover, she defined a coproduct ∆ and an antipode mapS, which are given as:

∆(x₁⊗ · · · ⊗x_n) = P_n

l=0(x₁, . . . , x_l)⊗(x_l+1, . . . , x_n)

S(x₁⊗. . . x_n) = (−1)^⌈ⁿ²^⌉n(x_n,(x_n−1)_x_n, . . . ,(x₂)_x₃_...x_n,(x₁)_x₂_...x_n)

(1.1.1) Furthermore, she proved that H(G)

1. is a graded differential algebra with the differential given by

∂=

n−1X

i=1

∂i

with

∂i(x1⊗ · · · ⊗xn) = (x1, . . . , xixi+1, . . . , xn).

2. S has finite order if the order of all elements of the group G have finite smallest common multiple. In particular,S is invertible for all finite groups.

3. H is neither commutative nor cocommutative.

Example

Set G={e}. Recall that K[G]∼=K and thatAut(G)∼={id}.

Denote by ǫk= 1⊗ · · · ⊗1 (ktimes) andǫl= 1⊗ · · · ⊗1 (l-times) the generators ofHk=K[G]^k and H_l=K[G]^l, respectively. Ifk=l= 1 the shuffle productǫ₁•ǫ₁ =ǫ₂−ǫ₂ vanishes. For any k and l= 1, the shuffle product is given by:

ǫ_k•ǫ₁=ǫ_k+1−ǫ_k+1+· · ·+ (−1)^kǫ_k+1=

0 forkodd ǫ_k+1 forkeven =

1 + (−1)^k 2

ǫ_k+1

Recursively, one can deduce that the shuffle product of ǫ_k and ǫ_l is given by:

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ǫ_k•ǫ_l= X

σ∈Sh(k,l)

sgn(σ)ǫ_k+l:=C_k,l . ǫ_k+l=

ǫ_k+l if k= 0 orl= 0 0 otherwise where

C_k,l=

1 + (−1)^k+l−1 2

C_k,l−1= Yl

i=1

1 + (−1)^k+i−1 2

C_k,0 (1.1.2)

C_k,0= 1, C_0,l = 1, C_0,0= 1 and ¹⁺⁽⁻¹⁾₂^k+i−1 =

0 for k+ieven 1 for k+iodd The antipode and the coproduct maps are given by:

∆(ǫ_k) = P

i+j=kǫ_i⊗ǫ_j

= ǫ₀⊗ǫk+ǫ₁⊗ǫ_k−1+· · ·+ǫk⊗ǫ₀ where by convention we set ǫ₀∈(K[G])^⊗0 =K,ǫ₀ = 1 inK

S(ǫ_k) = (−1)^⌈^k²^⌉ǫ_k

1.2 The twisted conjugation braiding B

^ϕ

In this section, we give a slight generalization of Schardt’s conjugation braiding, which has been introduced in [11]. More precisely, for a a group G (not necessarily commutative) she defines B :K[G]^⊗2→K[G]^⊗2 asa⊗b7→aba⁻¹⊗a.

Before we give the generalization of Schardt’s conjugation braidingB, we need to recall the following definition.

Definition 1.2.1. A solution of the Yang-Baxter equation is a linear mapR :V^⊗2 →V^⊗2 which satisfies

(R⊗idV)(idV ⊗R)(R⊗idV) = (idV ⊗R)(R⊗idV)(idV ⊗R) in Aut(V^⊗3),whereV is a finitely generated K-module of rank m≥0.

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Definition 1.2.2. Let G be a group, and let ϕ :G→ G be an automorphism. Define the twisted conjugation braidng B^ϕ :V^⊗2 →V^⊗2, where V =K[G]by:

B^ϕ(a⊗b) :=abϕ(a)⁻¹⊗ϕ(a).

It is easy to see that B^ϕ is invertible. Its inverse (B^ϕ)⁻¹:K[G]^⊗2→ K[G]^⊗2 is given by a⊗b7−→ϕ⁻¹(b)⊗ϕ⁻¹(b)⁻¹ab

for alla⊗bgenerator ofK[G]^⊗2. Figure 1.1 gives a graphic representation of the twisted conjugation bradingB^ϕ.

abϕ(a)⁻¹ ϕ(a)

a b a b

ϕ⁻¹(b)⁻¹ab ϕ⁻¹(b)⁻¹

B^ϕ (B^ϕ)⁻¹

Figure 1.1: The braidingB^ϕand its inverse (B^ϕ)⁻¹.

Proposition 1.2.3. B^ϕ satisfies the braiding equation in Aut(V^⊗3), i.e., B₁₂B₂₃B₁₂=B₂₃B₁₂B₂₃,

where B₁₂=B^ϕ⊗1 andB₂₃= 1⊗B^ϕ.

Proof Leta⊗b⊗cbe a generator ofV^⊗3 then:

B12(a⊗b⊗c) =abϕ(a)⁻¹⊗ϕ(a)⊗c and

B₂₃(a⊗b⊗c) =a⊗bcϕ(b)⁻¹⊗ϕ(b).

Therefore,

B₁₂B₂₃B₁₂(a⊗b⊗c) = B₁₂B₂₃(abϕ(a)⁻¹⊗ϕ(a)⊗c)

= B₁₂(abϕ(a)⁻¹⊗ϕ(a)cϕ²(a)⁻¹⊗ϕ²(a))

= abcϕ(ab)⁻¹⊗ϕ(ab)ϕ²(a)⁻¹⊗ϕ²(a)

= B₂₃(abcϕ(ab)⁻¹⊗ϕ(a)⊗ϕ(b))

= B₁₂B₂₃(a⊗bcϕ(b)⁻¹⊗ϕ(b))

= B23B12B23(a⊗b⊗c) From this follows that B^ϕ satisfies the braid equation.

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Remark 1.2.4.

1. Here, unless mentioned otherwise, we will understand by a braiding a solution of the Yang- Baxter equation.

2. Let ψ, ϕ :G→ G be homomorphism of the group G. Define B^ψ, B^ψ :K[G]^⊗2 → K[G]^⊗2 as above. Consider B =B^ψ◦B^ϕ, which is

a⊗b7−→ab ψϕ(a)ψ(ab)⁻¹⊗ψ(ab)ψϕ(a)⁻¹.

It is easy to see that B does not satisfy the Yang Baxter equation. But, up to an isomorphism C it is

C_ψ(ab) (B^ψϕ(a⊗b)) =B^ψ(B^ϕ(a⊗b)) with Cx(a⊗b) :=ax⁻¹⊗xb.

Therefore, in general composition of the Yang-Baxter equation is not a solution of the Yang- Baxter equation.

Lemma 1.2.5. Let V =K[G]^⊗l, letϕ=ϕ₁×ϕ₂× · · · ×ϕ_l,withϕi inAut(G)for all i∈ {1, . . . , l}.

Define B :V ⊗V →V ⊗V as:

a⊗b7→a₁b₁ϕ₁(a₁)⁻¹⊗a₂b₂ϕ₂(a₂)⁻¹⊗ · · · ⊗a_lb_lϕ_l(a_l)⁻¹⊗ϕ₁(a₁)⊗ϕ₂(a₂)⊗ · · · ⊗ϕ_l(a_l), for a⊗b generator of V ⊗V. (a= (a₁, . . . , a_l), b= (b₁, b₂, . . . , b_l)). Then B is a braiding on V. Proof It is similar to the proof of Proposition 1.2.3.

1.3 Action of the braid group Br(k) on T

_k

G

In this section, we define two actions of the braid group onK[G]^⊗k.

LetGdenote a groupG(not necessarily commutative). Letϕbe an automorphism of the groupG.

The following proposition gives two actions of the braid groupBr(k) onTkG, whereTkG=K[G]^⊗k. In the next section, we will use these actions to describe the two algebras and coalgebras structures on the tensor algebra H^ϕ(G) = L

k≥0TkG. Moreover, with the help of these actions we define twists maps and the antipode maps of the corresponding Hopf algebras.

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Definition 1.3.1. For each k>0 the braid group Br(k) is defined as:

Br(k) = < b₁, . . . , b_k−1 | ∀1≤i, j ≤k−1 : b_ib_j =b_jb_i for |i−j|>1 and bibi+1bi =bi+1bibi+1>

Proposition 1.3.2. For all k≥0, the braid group Br(k) acts on T_kG, this action is given by:

bi·(g₁, . . . , gi, g_i+1, . . . , gk) := (g₁, . . . , gig_i+1ϕ(gi)⁻¹, ϕ(gi), . . . , gk) and

(g₁, . . . , gi, g_i+1, . . . , g_k)·bi := (g₁, . . . , ϕ⁻¹(g_i+1), ϕ⁻¹(g_i+1)⁻¹gig_i+1, ϕ(gi), . . . , g_k) for all tuple (g₁, g₂, . . . , g_k)∈T_kG and each generator bi of Br(k).

Proof The action ofbi ∈Br(k) is an automorphism ofT_kG; an inverse is given by:

T_kG −→ T_kG

(g₁, . . . , g_i, g_i+1, . . . , g_k) 7−→ (g₁, . . . , ϕ⁻¹(g_i+1), ϕ⁻¹(g_i+1)⁻¹g_ig_i+1, . . . , g_k).

Now, it remains to prove the compatibility with the relations on the braid group.

Let bi, bj ∈Br_k with i < j, |i−j|>1. Then:

b_ib_j·(g₁. . . , g_i, g_i+1, . . . , g_j, g_j+1, . . . , g_k)

= (g₁, . . . , gig_i+1ϕ(gi)⁻¹, ϕ(gi), . . . , gjg_j+1ϕ(gj)⁻¹, ϕ(gj), . . . , gk)

=bjbi·(g1, . . . , gi, gi+1, . . . , gj, gj+1, . . . , gk) Now, if i < j, |i−j|= 1 and j=i+ 1, then

b_ib_i+1b_i·(g₁, . . . , g_k) = b_ib_i+1·(g₁, . . . , g_ig_i+1ϕ(g_i)⁻¹, ϕ(g_i), . . . , g_k)

= bi·(g₁, . . . , gig_i+1ϕ(gi)⁻¹, ϕ(gi)g_i+2ϕ²(gi)⁻¹, ϕ(gi), . . . , gk)

= (g₁, g₂, . . . , g_ig_i+1g_i+2ϕ(g_ig_i+1)⁻¹, ϕ(g_ig_i+1ϕ²(g_i)⁻¹, ϕ²(g_i), . . . , g_k) On the other hand:

b_i+1b_ib_i+1·(g₁, . . . , g_k) = b_i+1b_i·(g₁, . . . , g_i, g_i+1g_i+2ϕ(g_i+1)⁻¹, ϕ(g_i+1, . . . , g_k)

= b_i+1·(g₁, . . . , gig_i+1g_i+2ϕ(gig_i+1)⁻¹, ϕ(gi), ϕ(g_i+1, . . . , g_k) From this follows that b_ib_i+1b_i =b_i+1b_ib_i+1.

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bi

g₁

g₁ g_k

gk

g_i

ϕ(gi) g_i+1

gig_i+1ϕ(gi)⁻¹

Figure 1.2: Left braid action.

bi

g₁

g₁ g_i g_i+1 g_k

g_k ϕ⁻¹(g_i+1) ϕ⁻¹(g_i+1)⁻¹g_ig_i+1

Figure 1.3: Right braid action.

1.4 Algebra structure on H

^ϕ

(G).

With the help of proposition 1.3.2 we define in this section two algebra structures µ^ϕ_L respectively µ^ϕ_R on H^ϕ(G).

Definition 1.4.1. (Left Product) We define a left product: µ^ϕ_L:TlG⊗T_k−lG→TkG:

µ^ϕ_L(a⊗b) := P

σ∈(l,k−l)

−shuffle

sgn(σ)(b_σ(k). . . b_k−2b_k−1). . .(b_σ(l+2). . . blb_l+1)·(b_σ(l+1). . . b_l−1bl)·(a, b)

| {z }

=:S^ϕ_L(a,b;σ)l,k−l

for a∈TlG and for b∈T_k−lG.

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We define a unit:

η:K−→ H^ϕ(G)

17−→1∈T0G=K

Remark In view of the definition of the action of the braid groupBr(n) onTnV (see Proposition 1.3.2), we can describe µ^ϕ_L as in Figure 1.4.

Definition 1.4.2. (Right product) We define a right product: µ^ϕ_R:TlG⊗T_k−lG→TkG µ^ϕ_R: (a⊗b) := P

σ∈(l,k−l)−shuffle

sgn(σ)(a⊗b)·(blb_l+1. . . b_σ(l−1)·(b_l−1bl. . . b_{σ(l−1)−1}). . .(b₁b₂. . . b_σ(1)−1)

| {z }

=:S^ϕ_R(a,b;σ)l,k−l

for a∈TlG and for b∈T_k−lG.

We define a unit:

η:K→ H^ϕ(G) 17−→1∈T0G.

Note, that each of these products together with the unit η give a structure of graded algebra to H^ϕ(G).

Remark The algebra H^ϕ(G) is not commutative. Indeed we have that he following diagram T_lG⊗T_k−lG

TTTTTTTTT**

T TT TT TT T

µ^ϕ_L

//T_kG

T_k−lG⊗T_lG

µ^ϕ_L

OO

does not commute in general, whereT denotes the twist map, T_k(a⊗b) = (−1)^pqb⊗afora∈T_pG and b∈TqGand p+q=k.

Notation Let a= (g₁, . . . , g_l)∈T_lGand let b= (g_l+1, . . . , g_k) ∈T_k−lG. Denote by S_L,σ^ϕ (a, b) :=

S^ϕ_L(a, b, σ)_l,k−l.

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1.5 Coalgebra structure on H

^ϕ

(G)

In this section, we describe a coalgebra structure on (H^ϕ(G), µ^ϕ_L, η),and on (H^ϕ(G), µ^ϕr, η), respectively. Moreover, we define right and twist maps tw_R^ϕ, tw_L^ϕ and we prove that the coproduct is compatible with both products.

Definition 1.5.1. We define

∆ :TkG→(T G⊗T G)k= Mk

l=0

(TlG⊗T_k−lG)

∆(g1, . . . , gk) :=

Xk

l=0

(g1, . . . , gl)⊗(g_l+1, . . . , gk)

| {z }

=:∆l(g1,...,gk)

Define a counit ǫ:H^ϕ(G)→Kas T₀G ∋ 17−→1 (g₁, . . . , g_k)7−→0 for all k >0.

The above definition of ∆ together with the definition of the counit ǫ give a graded coalgebra structure to H^ϕ(G).

Remark H^ϕ(G) is not cocommutative. Indeed we have that the following diagram L_k

l=0(T_lG⊗T_k−lG)

TVVVVVVVVV**

VV VV VV VV V

T_kG

oo ∆

∆

Lk

l=0(T_k−lG⊗TlG) does not commute in general, whereT denotes the twist map.

Definition 1.5.2. (Right twist map) Let a = (g₁, . . . , g_l) ∈ T_lG and let b = (g_∈T_k−lG. We define the rigth twist map:

tw_R^ϕ :T_lG⊗T_k−lG→T_k−lG⊗T_lG

tw^ϕ_R(a⊗b) := (−1)^l(k−l)∆_k−l((a, b)·(b_lb_l+1. . . b_k−1)·(b_l−1b_l. . . b_k−2). . .(b₁b₂. . . b_k−l)

| {z }

t^ϕ_R(a,b)l,k−l

)

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Definition 1.5.3. (Left Twist map) Let a = (g₁, . . . , g_l) ∈ T_lG and let b = (g_l+1, . . . , g_k) ∈ T_k−lG. We define the left twist map:

tw^ϕ_L:T_lG⊗T_k−lG→T_k−lG⊗T_lG

tw^ϕ_L(a⊗b) : = (−1)^l(k−l)∆_k−l((b_k−l. . . b_k−2b_k−1). . .(b₂. . . b_lb_l+1)·(b₁. . . b_l−1b_l)·(a, b)

| {z }

t^ϕ_L(a,b)l,k−l

)

Using the action of the braid group Br(k) on T_kG, we see that the left twist map and the right twist map respectively, can be defined as

tw^ϕ_L(a⊗b) = (−1)^l(k−l)(ag_l+1ϕ(a)⁻¹, ϕ(a)g_l+2ϕ²(a)⁻¹, . . . , ϕ^k−l−2(g₁. . . g_l−1)gk)

⊗(ϕ^l−2(g₁), . . . , ϕ^k−l−2(g_l−1), ϕ^k−l−1(g_l)) This is graphically represented in Figure 1.5.

tw^ϕ_R(a⊗b) = (−1)^l(k−l)(ϕ^−(l+2)(g_l+1), ϕ^(l−1)(g_l+2), . . . , ϕ^{−(k−l−1)}(gk))

⊗(ϕ^{(−k−l−2)}(g₁, . . . ,), . . . , ϕ⁻²(gk. . . g_l+1)⁻¹g_l−1ϕ⁻¹(gk. . . , g_l+1gl), ϕ⁻¹(gl·b)) This is graphically represented in Figure 1.6.

Remark 1.5.4. tw^ϕ_R◦tw_L^ϕ=tw_L^ϕ◦tw^ϕ_R=id.

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. . .

. . . . . . . . .

. . . . . .

. . . . . . . . .

. . . . . . . . . . . . . . .

. . .

. . .. . .. . .. . .

g1

g₁

g_l gk

g_l−1 g_l+1 g_l+2

bl

b_l−1

b_l+1 b_σ(l+1)

b_σ(l+2)

b_k−2 b_k−1

e g= b_σ(k)

ϕ^k−l−1(g_l) ϕ^k−l−2(g_l−1) ϕ^k−r−2(gr_k). . .ϕ^k−l−2(g1)gkϕ^k−r−1(grk)⁻¹

eg

¯ g

¯

g=g_r_l+1. . . g_l+1ϕ(g_r_l+1. . . g_l)⁻¹

Figure 1.4: Graphic representation of the left-shuffle product.

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. . . . . .

. . .

. . . . . .

. . .

. . . . . .

. . .. . .. . .. . .

g₁ g_l−1 gl g_l+1 g_l+2 gk

b_l−1

b₁ b_l+1

b_l b_l

b₂

b_k−1

b_k−l b_k−2

g₁. . . g_l+1ϕ(g₁. . . gl)⁻¹

ϕ(g1. . . gl)g_l+2ϕ²(g1. . . gl)⁻¹

ϕ^k−l−2(g₁. . . g_l−1)gk ϕ^k−l−2(g_l−1)

ϕ^k−l−1(g_l)

Figure 1.5: Graphic representation of the left twist map.

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. . . . . .

. . .

. . . . . .

. . .

. . . . . .

. . .. . .. . .. . .

g₁ g_l−1 g_l g_l+1 g_l+2 gk

b_l+1

b_k−1 b_k−1 b_l−1 b_l

bl

b₁ b₂ b_k−2

ϕ^−(l+2)(g_l+1)

ϕ^−(l+1)(g_l+2)

ϕ⁻²(g_k. . . g_l+1)⁻¹g_l−1ϕ⁻¹(g_k. . . g_l) ϕ⁻¹(g_l. . . g_k)

Figure 1.6: Graphic representation of the right twist map.

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Proposition 1.5.5. ∆ is an algebra homomorphism for µ^ϕ_R and for µ^ϕ_L; i.e

∆◦µ^ϕ_L = (µ^ϕ_L⊗µ^ϕ_L)◦(id⊗tw_L^ϕ⊗id)◦(∆⊗∆)

∆◦µ^ϕ_R = (µ^ϕ_R⊗µ^ϕ_R)◦(id⊗tw_R^ϕ⊗id)◦(∆⊗∆), respectively.

Proof We only will prove the first equality, because the proof for the second equality is similar.

Leta= (a1, . . . , as)∈TsGandb= (b1, . . . , bt)∈TtG. Lets^′ ∈ {0, . . . s} andt^′ ∈ {0, . . . , t}. Letσ1

and σ₂ denote a fixed (s^′, t^′) and (s−s^′, t−t^′)- shuffles respectively. We have:

((µ^ϕ_L⊗µ^ϕ_L)◦(id⊗tw^ϕ_L⊗id)◦(∆⊗∆)(a⊗b))s^′,t^′,σ1,σ2:=

= (S₍^ϕL, σ₁)⊗S₍^ϕL, σ₂))◦(id⊗tw^ϕ_L⊗id)(∆_s(a₁, . . . , a_s)⊗∆_t(b₁, . . . , b_t))

= (S_L,σ^ϕ ₁ ⊗S_L,σ^ϕ ₂)((−1)^(s−s^′^)t^′((a₁, . . . , a_s^′)⊗t^ϕ_L(a, b)_s−s^′_,t^′)⊗(b_t^′₊₁, . . . , bt)

= (−1)^(s−s^′^)t^′S_L^ϕ(a, b, σ1)s^′,t^′⊗S_L^ϕ(a, b, σ2)_s−s^′_,t−t^′

Now, consider the permutation σ₀ ∈Σ_s+t which is given by:

{1, . . . , s+t} −→ {1, . . . , s+t}

i7−→









i if 1≤i≤s^′ i+t^′ ifs^′+ 1≤i≤s i−(s−s^′) ifs+ 1≤i≤s+t^′ i ifs+t^′+ 1≤i≤s+t

Clearly, sgn(σ₀) = (−1)^(s−s^′^)t^′. On the other hand, let σ₁^′ ∈ Σ_s+t denote the permutation that coincides with σ₁ in the first k+l positions, and the identity in the remained positions. Let σ^′₂ ∈ Σs+t denote the permutation that coincides with σ2 in the last s+t−(k+l) positions, and the identity in the remained positions. It is not difficult to see that σ^′ := σ₁^′ . σ₂^′ . σ₀ is a (s, t)-shuffle.

We have:

(∆◦µ^ϕ_L)s^′+t^′,σ^′(a⊗b) := ∆s^′+t^′ ◦S_L^ϕ(a, b, σ^′)

= (−)^(s−s^′^)t^′∆_s^′_+t^′(S_L^ϕ(a, b, σ^′₁σ₂^′))

= (−1)^(s−s^′^)t^′(S_L^ϕ(a, b, σ₁)⊗S_L^ϕ(a, b, σ₂)), supp(σ^′₁)⊆ {1, . . . , s^′+t^′}and supp(σ₂^′)⊆ {s^′+t^′+ 1, . . . , s+t}.

Twisted conjugation braidings and link invariants