INVARIANTS OF COLORED LINKS AND A PROPERTY OF

arXiv:hep-th/9207090v1 28 Jul 1992INVARIANTS OF COLORED LINKS AND A PROPERTY OFTHE CLEBSCH-GORDAN COEFFICIENTS OF Uq(g)1TETSUO DEGUCHI and TOMOTADA OHTSUKI †Department of Physics, Faculty of Science,University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan† Department of Mathematical Sciences,University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113, JapanAbstract.We show that multivariable colored link invariants are derived from the rootsof unity representations of Uq(g).We propose a property of the Clebsch-Gordancoefficients of Uq(g), which is important for defining the invariants of colored links. ForUq(sl2) we explicitly prove the property, and then construct invariants of colored linksand colored ribbon graphs, which generalize the multivariable Alexander polynomial.1IntroductionRecently a new family of invariants of colored oriented links and colored orientedribbon graphs is introduced, which gives generalizations of the multivariable Alexan-der polynomial.1,2,6 The new invariants are related to the roots of unity representa-tions of Uq(sl2) where (X±)N = 0 and (K)N = q2Np (p ∈C) and q = ǫ.5,6,7 Hereǫ = exp(πis/N), and the integers N and s are coprime.

We call the representationsnilpotent representations.The invariants of colored links have a property that they vanish for disconnectedlinks.2,6 Due to this property a proper regularization method is necessary for definitionof the invariants.In this paper we show an important property of the Clebsch-Gordan coefficients(CGC) of the nilpotent representations of Uq(sl2), which leads to the definition of thecolored link invariants. We consider the nilpotent reps of Uq(g) such that (X±i )N = 0,(Ki)N = q2Npi (pi ∈C ) and q = ǫ.

We give a conjecture that the property of CGCalso holds for the nilpotent reps of Uq(g) and we can define invariants of colored linksfor Uq(g) in the same way as Uq(sl2).2CGC of the nilpotent representationsWe introduce some symbols for a positive integer n and a complex parameter p.[n]q = qn −q−nq −q−1 ,[n]q! =nYk=1[k]q,[p; n]q!

=n−1Yk=0[p −k]q. (1)1 to appear in the Proceedings of the 21st International Conference on the Differential GeometryMethods in Theoretical Physics, 5-9 June, 1992, Tianjin, China

For n = 0 we assume [0]q! = [p; 0]q!

= 1.We introduce the nilpotent rep.V (p)5:π(X+)ab =q[2p −a]ǫ[a + 1]ǫ δa+1,b,π(X−)ab =q[2p −a + 1]ǫ[a]ǫ δa−1,b, π(K)ab = ǫ(2p−2a)δa,b. Here a, b = 0, 1, · · ·N −1.We note that N is related to the dimensions of the representation.

When N is even,ǫ is 2N-th primitive root of unity, while when N is odd, ǫ may be 2N-th or N-thprimitive root of unity.In Ref.1 the explicit matrix representations of the colored braid group wereintroduced. It was shown that the representations are equivalent to the R matricesof the nilpotent reps, which are derived from the universal R matrix R : Ra1,a2b1,b2 =πp1 ⊗πp2(R)a1a2b1b2 .

5We can show the fusion rule for the tensor product: V (p1)⊗V (p2) = Pp3 Np3p1p2V (p3),where Np3p1,p2 = 1 for p3 = p1 + p2 −n ( 0 ≤n ≤N −1 and n ∈Z); Np3p1,p2 = 0,otherwise. The CGC for V (pi) (i = 1, 2, 3) are given by 6C(p1, p2, p3; z1, z2, z3) = δ(z3, z1 + z2 −n)×q[2p1 + 2p2 −2n + 1]ǫq[n]ǫ![z1]ǫ![z2]ǫ!

[z3]ǫ!×Xν(−1)νǫ−ν(p1+p2+p3+1) × ǫ(n−n2)/2+(n−z2)p1+(n+z1)p2[ν]ǫ! [n −ν]ǫ!

[z1 −ν]ǫ! [z2 −n + ν]ǫ!×vuut[2p1 −n; z1 −ν]ǫ!

[2p1 −z1; n −ν]ǫ! [2p2 −n; z2 + ν −n]ǫ!

[2p2 −z2; ν]ǫ! [2p1 + 2p2 −n + 1; z1 + z2 + 1]ǫ!.

(2)Here 0 ≤zi ≤N −1, for i=1,2,3, and the sum over the integer ν in (2) is taken underthe condition: max {0, n −z2} ≤ν ≤min {n, z1}. The expression of the CGC wasproved through the infinite dimensional representations.

63A Property of CGCWe consider the following two sums;Anp1p2=Xz1[C(p1, p2, p3; z1, z2, z1 + z2 −n)]2 q2ρ(z1),Bnp1p2=Xz2[C(p1, p2, p3; z1, z2, z1 + z2 −n)]2 q−2ρ(z2). (3)For the nilpotent reps of Uq(sl2) we assume ρ(z) = p −z + Np/2 and q = ǫ.

Due tothe irreducibility of the representations the sum Anp1p2 (Bnp1p2) does not depend on z2(z1). We now introduce an important property of CGC.Proposition 3.1 The CGC of the nilpotent representations satisfyAnp1p2/Bnp1p2 = g(p1, p2) = f(p1)/f(p2), independent of n(p3 = p1 + p2 −n).

(4)

(For the nilpotent reps we have g(p1, p2) = [2p2; N −1]ǫ!/[2p1; N −1]ǫ! ).ProofWe set z2 = 0 for Anp1p2 and z1 = 0 for Bnp1p2, and we define Lnp1p2 and Rnp1p2 byAnp1p2 = Lnp1p2[2p3 + 1]/[2p3 + 1 + n; N]!

and Bnp1p2 = Rnp1p2[2p3 + 1]/[2p3 + 1 + n; N]! ,respectively.

Then we can showLnp1p2 = (−1)nN[2p2; N −1]ǫ!,Rnp1p2 = (−1)nN[2p1; N −1]ǫ!. (5)The first eq.

in (5) can be shown by using the following recurrence relation on n.Lnp1p2 = [2p1 + 2p2 −n −N + 2][2p2 + 1]ǫ2p1−nLn+1p1+1/2,p2+1/2 −[2p1 −n][2p2 + 1] Ln+1p1p2+1/2. (6)The second eq.

in (5) (for Rnp1p2) is derived from that of Lnp1p2 by exchanging p1 withp2, and by setting ǫ →ǫ−1.It is easy to see that the property (4) of CGC also holds for the finite dimensional(spin) representations of Uq(sl2) with q generic, where we have g(j1, j2) = [2j1]/[2j2].We can define invariants of colored links using the proposition 3.1. Let T be a(1, 1)- tangle.

We denote by ˆT the link obtained by closing the open strings of T. Wenote the following proposition. 2Proposition 3.2 Let T1 and T2 be two (1, 1)-tangles.

If ˆT1 is isotopic to ˆT2 as a linkin S3 by an isotopy which carries the closing component of ˆT1 to that of ˆT2. Then T1is isotopic to T2 as a (1, 1)-tangle.Let us introduce the functor φ(·) for the tangle diagrams.

2 We denote by φ(T, α)abthe value φ for the tangle with variables a and b on the closing component (or edge).It is easy to show that φ(T, α)ab = λδab. 2We put L = ˆT and s is the color of the closing component (or edge) of ˆT.

Fora colored link (L, α) and a color s of closing component (or edge), we define Φ byΦ(L, s, α) = λ where L, T, s are as above and φ(T, α)ab = λδab. We can show that Φ iswell-defined, i.e.

Φ(L, s, α) does not depend on a choice of T. 2 From the proposition3.1 we have the following.Proposition 3.3 2 For a link L and its color α = (p1, · · ·, pn), we haveΦ(L, s, α)([ps; N −1]ǫ! )−1 = Φ(L, s′, α)([ps′; N −1]ǫ!)−1.

(7)Definition 3.4 2 For a colored oriented link (L, α), we define an isotopy invariantˆΦ of (F, α) byˆΦ(L, α) = Φ(L, s, α)([ps; N −1]ǫ!)−1. (8)Thus we have constructed the multivariable invariants from the property (4) ofCGC.

In the same way we can define the multivariable invariants of colored ribbongraphs. 6

We now consider CGC of Uq(g), where g is a simple Lie algebra.AΛ3Λ1Λ2=X⃗z1[C(Λ1, Λ2, Λn; ⃗z1, ⃗z2, ⃗z1 + ⃗z2 −⃗n)]2 q2ρ(⃗z1),BΛ3Λ1Λ2=X⃗z2[C(Λ1, Λ2, Λ3; ⃗z1, ⃗z2, ⃗z1 + ⃗z2 −⃗n)]2 q−2ρ(⃗z2). (9)We assume that ρ is given by ”half the sum of positive roots”.

Finally, we proposethe following conjecture.Conjecture 3.5 (1) The CGC of the nilpotent representations Λi of Uq(g) satisfyAΛ3Λ1Λ2/BΛ3Λ1Λ2 = g(Λ1, Λ2), independent of Λ3. (10)(2) With a proper normalization of the quantum trace ( ρ(⃗z) →ρ(⃗z) + constant), wecan set g(Λ1, Λ2) = f(Λ1)/f(Λ2).

(3) f(Λ) is equivalent to the tangle invariant for the Hopf link.If the conjecture is true, we can construct multivariable invariants of colored linksand colored ribbon graphs from Uq(g) in the same way as Uq(sl2).It is easy to show that the property (10) holds for CGC of finite dimensionalrepresentations of Uq(g) with q generic.References[1] Y. Akutsu and T. Deguchi, A New Hierarchy of Colored Braid Group Represen-tations, Phys. Rev.

Lett. 67 (1991) 777.

[2] Y. Akutsu, T. Deguchi and T. Ohtsuki, Invariants of Colored Links, Journal ofKnot Theory and Its Ramifications 1 (1992) pp. 161-184.

[3] T. Deguchi and Y. Akutsu, A General Formula for Colored ZN Graded BraidMatrices and Fusion Braid Matrices, J. Phys. Soc.

Jpn. 60 (1991) 2559.

[4] T. Deguchi and Y. Akutsu, A New Hierarchy of Colored Vertex Models, J. Phys.Soc. Jpn.

60 (1991) 4053. [5] T. Deguchi and Y. Akutsu, Colored braid matrices from infinite dimensional rep-resentations of Uq(g), Mod.

Phys. Lett.

A7 (1992) 767. [6] T. Deguchi and Y. Akutsu, Colored Vertex Models, Colored IRF Models, andInvariants of Colored Framed Graphs, preprint January 1992.

[7] T. Ohtsuki, Colored ribbon Hopf algebras and universal invariants of framed links,preprint June 1992.

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