Twisted conjugation braidings and link invariants [Elektronische Ressource] / vorgelegt von María Guadalupe Castillo Pérez

María Guadalupe Castillo Péerez

Sujets

• Mathematics

Informations

• Publié par: rheinische_friedrich-wilhelms-universitat_bonn
• Publié le: 01 janvier 2009
• Nombre de lectures: 22
• Langue: English

Extrait

Twisted conjugation braidings
and link invariants
Dissertation
zur
Erlangung des Doktorgrades (Dr. rer. nat.)
der
Mathematisch-Naturwissenschaftlichen Fakultat¨
der
Rheinischen Friedrich-Wilhelms-Universit¨at Bonn
vorgelegt von
Mar´ıa Guadalupe Castillo P´erez
aus
Mexico City
Bonn, Februar 2009AngefertigtmitGenehmigungderMathematisch-Naturwissenschaftlichen
Fakulta¨t der Rheinischen Friedrich-Wilhelms-Universitat¨ Bonn.
1. Referent: Prof. Dr. C.-F. Bod¨ igheimer
2. Referent: Prof. Dr. Catharina Stroppel
Promotionsdatum: 5. Juni 2009
Erscheinungsjahr: 2009Abstract
This work is about link invariants arising from enhanced Yang-Baxter opera-
tors. For each enhanced Yang-Baxter operatorR = (R,D,λ,β) and any braid
−ω(ξ) −n ⊗nBr(n) Turaev defined a link invariant T (ξ) = λ β trace(b (ξ)◦D ),R R
where ω : Br(n) →Z is a homomorphism and b is the representation of theR
Artin braid group Br(n) arising from the solution of the Yang-Baxter equati-
on R. Therefore, we first introduce new solutions of the Yang-Baxter equation
ϕ ⊗2 ⊗2 ϕ −1B : V → V , B (a⊗b) = abϕ(a) ⊗ϕ(a), for V =K[G], ϕ∈ Aut(G),
where G is any group. We call these solutions twisted conjugation braidings.
Then we give sufficient and necessary conditions for a map D to decide whe-
ϕther the quadruple (B ,D,λ,β) is an EYB-operator. Moreover, we prove that
ϕthe twisted conjugation braidings B can be enhanced using character theory.
These enhancements are called character enhancements. It turns out that for
ϕevery character enhancementD of the twisted conjugation bradingB the link
invariant is constantly 1, i.e., T (ξ) = 1 for all ξ∈Br(n). In general, we proveB
that the link invariant for all ξ ∈ Br(n) and for every enhancement D of the
ϕ −n ⊗ntwisted conjugation braiding B is a map T (ξ) =β trace(b ϕ)◦D .B B
Our main result is the following theorem.
Letγ be a fixed invertible element ofK and letD denote a linear map. Asumme
ϕthat D⊗D commutes with the twisted conjugation braiding B . Then
ϕ ±1 21. Sp ((B ) ◦(D⊗D)) =γD =⇒ D =γD2
ϕ ϕ −12. Sp (B ◦(D⊗D)) =γD ⇐⇒ Sp ((B ) ◦(D⊗D)) =γD2 2
Inthe lastpartofthis work,we provethat for finite groupsG the twistedconju-
ϕ ϕ lgation braidingB satisfies (B ) (a⊗b) =a⊗b, withl = 2lcm(ord(a),ord(b). n
m1FromthisfollowsthatthelinkinvariantisT (ξ) = ,forbraidsξ inBr(n),B β
ǫǫ l1with ξ =σ ...σ , and with ǫ ,...,ǫ ≡ 0 modl, where m = trace(D). We1 l 1σ ii l1
call such braids mod-l braids. Furthermore, it follows that the link invariant is n−1
m ǫ1T = for braids ξ ∈ Br(n) such that ξ = σ , with ǫ≡ 0 mod l. WeB β i
call these braids single-power braids. Moreover, we wrote a program in JAVA
programming language which computes the link invariants for the enhancement
∗ qD = γI, (γ ∈ K ) for braids ξ ∈ Br(p), (p prime) with ξ = (σ σ ...σ ) ,1 2 p−1
and with (p,q) = 1 for the cases G = Σ and G =Z/nZ. In the cases were wen
havecomputedthe link invariantsT “thepolynomialisconstant,”i.e.,T ∈K,B B
⊗2since the only braidings we consider are permutations of the basisK[G] .
1Contents
Introduction 1
1 The twisted shuffle Hopf algebra of a group 8
1.1 Schardt’s Hopf algebraH(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
ϕ1.2 The twisted conjugation braidingB . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Action of the braid group Br(k) on T G . . . . . . . . . . . . . . . . . . . . . . . . . 12k
ϕ1.4 Algebra structure onH (G). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
ϕ1.5 Coalgebra structure onH (G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
ϕ ϕ1.6 The antipode maps S andS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
L R
2 The Yang-Baxter Equation and knot invariants 28
2.1 Traces and partial traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Enhanced Yang-Baxter operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 Invariants of braids and links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4 Elementary properties of T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42R
2.5 The link invariants for the twisted conjugation braiding . . . . . . . . . . . . . . . . 44
3 Character enhancements 48
3.1 Character χ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 Character enhancements D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49χ
3.3 Constancy of the link invariant T (ξ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 51B
4 The projection enhancements 53
4.1 The idempotence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Examples of projection enhancements . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5 Link invariants for EYB-operators of the twisted conjugation braiding 58
5.1 Computations of link invariants for some braidsξ∈Br(n) . . . . . . . . . . . . . . . 58
6 Specific computations 64
ϕ6.1 Orders of B for symmetric groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
ϕ6.2 Orders of B for cyclic groups C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68p
6.3 Consideration of the matrix sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.4 Link invariants of torus knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
iA 77
A.1 Connection to quasi-cocommutative Hopf algebras. . . . . . . . . . . . . . . . . . . . 77
B 80
B.1 Connection to braided Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 80
B.1.1 Whitehouse’s solutions of the Yang-Baxter-equation . . . . . . . . . . . . . . 80
B.1.2 Woronowicz’s solutions of the Yang-Baxter equation . . . . . . . . . . . . . . 83
B.1.3 Braided Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
C 90
C.1 Tensor product of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
C.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
D 93
D.1 The computer program. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Bibliography 96
iiIntroduction
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 letK 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
⊗3(R⊗1)(1⊗R)(R⊗1) = (1⊗R)(R⊗1)(1⊗R) in Aut(V ). This equation first has appeared in
independentpapersofC.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]). Forexample,arelationshipbetweentheYang-Baxterequationandpolynomial
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 with R,
±1(T2a) Sp (R◦(D⊗D)) =λ βD,2
−1 ±1(T2b) Sp (R ◦(D⊗D)) =λ βD, where Sp : V → V denotes the partial trace on the second2 2
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 group G to introduce new
ϕ ⊗2 ⊗2 ϕ −1solutionsoftheYang-Baxter equationB :V →V .WedefineB (a⊗b)=abϕ(a) ⊗ϕ(a),for
ϕany group G and for V =K[G], and ϕ∈Aut(G). Throughout this work B will be called twisted
conjugation braiding and by a link we will understanda finite family of disjoint, smooth oriented or
3 3 ϕunoriented, closed curves inR , or equivalently S . An example of a solution B is the following.
ϕ −1Set G to be an abelian group. Then the twisted conjugation braiding B (a⊗b) = aba ⊗a.
ϕMoreover, observe that if G is commutative then B is the twist map.
1In Theorem 2.2.6 we completely characterize EYB-operators by a set of three equations. This
ϕallows us to show that the twisted conjugation braidingB 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
′ ϕelement ofK. Then, B =(B ,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
ϕconjugation braidingB 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,X
D(a) = χ(a,c)c,
c∈G
then the following three conditions are satisfied:
ϕ1. The quadruple B =(B ,D,λ