RE-NORMALIZED LINK INVARIANTS FROM THE UNROLLED QUANTUM GROUP. 1. Introduction

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1 RE-NORMALIZED LINK INARIANS FROM HE UNROLLED QUANUM GROUP NAHAN GEER 1 Introduction In the last several years, C Blanchet, F Costantino, B Patureau, N Reshetikhin, uraev and myself (in various collaborations) have developed a theory of renormalized quantum invariants of links and 3-manifolds which lead to QFs, see [2, 3, 7, 8, 9] his theory leads to quantum invariants arising from non-semi-simple categories which have objects with vanishing quantum dimensions In this note we will discuss the re-normalized link invariant arising from a particular quantum group which we call the unrolled quantum associated to sl(2), see [5] hese link invariants contain Kashaev s quantum dilogarithm invariants of knots, the Akutsu-Deguchi- Ohtsuki invariants of links and J Murakami s colored Alexander invariant (see [1, 10]) 2 Some algebra Fix a positive integer r and let q = e π 1 r be a 2r th -root of unity We use the notation q x = e π 1x r Here we give a slightly generalized version of quantum sl(2) called the unrolled quantum group Let Uq H sl(2) be the C(q)-algebra given by generators E,F,K,K 1,H and relations: HK = KH, HK 1 = K 1 H, [H, E] =2E, [H, F] = 2F, KK 1 = K 1 K =1, KEK 1 = q 2 E, KFK 1 = q 2 F, [E,F]= K K 1 q q 1 he algebra U H q sl(2) is a Hopf algebra where the coproduct, counit and antipode are defined by (E) =1 E + E K, ε(e) =0, S(E) = EK 1, (F )=K 1 F + F 1, ε(f )=0, S(F )= KF, (H) =H 1+1 H, ε(h) =0, S(H) = H, (K) =K K ε(k) =1, S(K) =K 1, (K 1 )=K 1 K 1 ε(k 1 )=1, S(K 1 )=K 1

2 2 N GEER Define Ū H q (sl(2)) to be the Hopf algebra U H q sl(2) modulo the relations E r = F r = 0 Let be a finite dimensional Ū H q (sl(2))-module An eigenvalue λ C of the operator H : is called a weight of and the associated eigenspace is called a weight space We call a weight module if splits as a direct sum of weight spaces and q H = K as operators on Let C be the category of finite dimensional weight Ū H q (sl(2))-modules he Hopf algebra structure of Ū H q (sl(2)) makes C into a tensor category he category C is not semi-simple but the simple modules are in one to one correspondence with the complex numbers C, one for each highest weight For C, let be the r-dimensional module with a vector v 0 such that Ev 0 = 0 and Kv 0 = q +r 1 v 0 Here the module has a basis of vectors {v i }, i =0,,r 1, where the vector v i = F i v 0 has an H-weight of + r 1 2i If C := (C \ Z) rz then is simple We will now recall that the category C is a ribbon category Let and W be objects of C Let {v i } be a basis of and {v i } be a dual basis of hen b :C, given by 1 v i v i d : C, given by f w f(w) are left duality morphisms of C We also have the following right duality maps: d : C, f v f(k 1 r v) b : C C, 1 K r 1 v i v i In [11], Ohtsuki defines an R-matrix operator defined on W by r 1 R = q H H/2 n=0 where q H H/2 is the operator given by {1} 2n {n}! qn(n 1)/2 E n F n (1) q H H/2 (v v )=q λλ /2 v v for weight vectors v and v of weights of λ and λ hus, the action of R on the tensor product of two objects of C is well defined and induces an endomorphism on such a tensor product Moreover, R gives rise to a braiding c,w : W W on C defined by v w τ(r(v w)) where τ is the permutation x y y x (see [10, 11]) Also, in [11] Ohtsuki defines twist and shows it is compatible with the braiding and right duality

3 RE-NORMALIZED LINK INARIANS 3 3 Basic topological definitions and the functor F In this section we recall some basic definitions and the Reshetikhin-uraev functor F For good references on this material see [11, 12] A knot is the image of a smooth embedding S 1 R 3 We say two knots K and K are isotopic if K is obtained from K by a continuous deformation which has no self-intersection, written K K Here we will consider maps on the set of knots into the complex numbers C which are constant on isotopy classes of a link: a map I : {Knots} C is an isotopy invariant of knots if I(K) =I(K )wheneverk K Adiagram is a smooth immersion S 1 R 2 with finite number of transversal double points all assigned to be an over or under crossing o define our knot invariants we will choose a diagram of a knot then use algebra to assign a value to this diagram and show that the value is independent of the choice of diagram Our main tool in doing this is Reidemeister moves: R1: R2: R3: he following theorem says it is enough to consider diagrams up to Reidemeister moves and isotopies in the plane heorem 1 {Knots}/isotopy of R 3 = {knot diagrams}/r1-r3, isotopy of R 2 In the rest of the paper we will consider links (and more generally tangles) which are multi-component versions of knots Next we describe how to compute the functor F for a link whose component are labeled by complex numbers Here a complex number is associated with the simple module Let L be an oriented (framed) link or tangle whose components are labeled by complex numbers Choose a nice diagram D of L which decompose into building blocks: Assign morphisms of C to each building block: d_v d _v b_v b _v ID_v C_v,v C 1_v,v

4 4 N GEER Use the link or tangle diagram to tell you how to put the morphism together to create a morphism which we call F (D) o illustrate this algorithm let us consider the Hopf link labeled with and W : H = W W d d W Id c W, Id W Id c,w Id W b b W then F (H) =(d d W ) (Id c W, Id W ) (Id c,w Id W ) (b b W ) It can be shown that F (D) only depends on L here are many details to check to confirm this but the main points are: 1) the properties of the R-matrix defined in Equation (1) imply that F satisfies the second and third Reidemeister moves, 2) the properties of the dualities imply that F is invariant for isotopies of the plane R 2 It should be emphasized that F is not invariant under the first Reidemeister move, this is why F is an invariant of framed links Let us point out two important features of the invariant F First,ifLis a link then F (L) is an endomorphism of the trivial module C, butend C (C) = C So we can associate a complex number to each link Second, let is a (1,1)-tangle whose open strings are colored by a simple module for C henf ( ) is an endomorphism of,whereend C ( )=CId hus, to each such we can associate a complex number Let us denote this complex number by < >, in other words: F ( )=< > Id Suppose L is a link which is the closure of a (1,1)-tangle as above hen F (L) =F = F (2) Here F ( )=qdim( ) is the quantum dimension Equation (2) implies that if the quantum dimension of is zero then the invariant F (L) is zero his observation combined with the following lemma implies that F vanishes on all links which have at least one label in the set C Lemma 2 If C then quantum dimension of is zero

5 RE-NORMALIZED LINK INARIANS 5 Proof he quantum dimension is given by: r 1 r 1 qdim( )= vi (K 1 r v i )= q (r 1)(+r 1 2i) (r 1)(+r 1) 1 q2r = q 1 q 2 i=0 i=0 Since q is a 2r-th root of unity then 1 q 2r = 0 and the lemma follows 4 he invariant F In this section we show how to renormalize the invariant F so that it is non-zero on links having labels in C he main idea is to replace the quantum dimension in Equation (2) with a non-zero function d : C C which will lead to a link invariant If is a simple module define S (, )= Asimplemodule = is ambidextrous if = for all (2,2)-tangles whose open strings are labeled by Lemma 3 If C \ 1 2 Z then is ambidextrous Proof Let be a (2,2)-tangle whose open strings are colored by and let f = F ( ) End( ) its corresponding endomorphism For C \ 1 2 Z it can be shown that is multiplicity free direct sum of simple modules hus, End( ) is commutative and so in particular the braiding commutes with f Implying f = c, c 1, f = c 1, fc,

6 6 N GEER and so = = = = = = where all arrows are colored with and = indicates the equality under F, ie G = H if F (G) =F (H) Let J = γ for γ C \ 1 2 Z If C =(C \ Z) rz then a direct computation shows S (J, ) = 0 Let L be the set of framed C-colored links with at least one color is in the set C Defined : C C by d() =d( )= S (,J) S (J, ) C heorem 4 he map F : L C given by L = closure of d( ) < > is a well defined invariant of closed colored ribbon graphs F (L) is independent of and In particular, Let us prove the following lemma which immediately implies the theorem Lemma Let L be a ribbon graph which is the closer of a (2, 2)-tangle ribbon graph, in other words, L = then U If = and U = β are where d(u) U = d( ) U Proof he lemma follows in two steps Step 1 Notice that since J is ambidectrous we have J J = J J for all ribbon graphs

7 RE-NORMALIZED LINK INARIANS 7 Step 2 Now by definition of the ribbon functor F we have J J U J = U J U Similarly, J U = S (U, J)S (J, ) U J = S (,J)S (J, U) U hen Step 1 implies that the left sides of the above equations are equal he lemma follows from the fact that d( )= S (,J) S (J, ) 5 Axiomatic definition of the invariant of graphs in S 3 In this section a set of defining relations is given for the invariant defined above o do this we extend the invariant to trivalent graphs he proof that these are defining relations is given in [3] Here we identify with for C As above, let r Z with r 2 and let q =e iπ/r Let L be the set of oriented trivalent framed graphs in S 3 whose edges are colored by element of C he following axioms define an invariant : L C Let, L (1) Let e be an edge of colored by If is obtained from by changing the orientation of e and its color to then ( )= ( ) In other words, = (N a) (2) Let, β, γ be the colors of a vertex v of If all the orientations of edges of v are incoming and + β + γ is not in H r then ( ) = 0, ie β =0if + β + γ/ H r (N b) γ (3) If denotes the connected sum of and along an edge colored by then ( )=d() 1 ( ) ( ): = δ β d() 1 (N c) β

8 8 N GEER (4) he invariant has the following normalizations: = d(), =1, =0, (N d) here we assume that the Θ graph is colored with any coloring which is not as in (N b) (5) If denotes the connected sum of and along a vertex with compatible incident colored edges then ( )= ( ) ( ): = (N e) (6) is zero on split graphs: ( ) = 0 (7) he following relations hold whenever all appearing colors are in C: = q 2 (r 1) 2 2, (N f) β γ = q γ2 2 β 2 +(r 1) 2 4 γ, (N g) β j 2 j 4 j 3 j 1 j 5 β =( 1) r 1 rq β, (N h) β = d(γ) γ, (N i) γ +β+h r = d(j 6 ) j j 1 j 2 j 3 2 j 4 j 4 j 5 j 6 j 6 (N j) j 1 j 5 j 6 j 1 +j 5 +H r Here the 6j-symbol j 1 j 2 j 3 j 4 j 5 j 6 = N r j 1 j 2 j 6 j 5 j 3 j 4 is given by: j 1 j 2 j 3 {B j 4 j 5 j 6 =( 1)r 1+B }! {B 123 }! 1 j3 + r 1 j3 + r 1 {B 246 }! {B 165 }! A r B 354 M ( 1) z A B156 + z B264 + B 345 z B453 + z j 5 + z + r z=m B 156 B 264 where A xyz = jx+jy+jz+3(r 1) 2, B xyz = jx+jy jz+r 1 2, m = max(0, j 3+j 6 j 2 j 5 and M =min(b 435, B 165 ) 2 ) B 462

9 RE-NORMALIZED LINK INARIANS 9 Remark 5 It can be shown that the above axioms determine the value of ( ) for any L (cf [6, Proposition 46] for a similar proof) In particular, the axioms can be used to reduce ( ) to a linear combination of 6j-symbols which are in turn determined by the above formula In [3] it is proved that these axioms are consistent References 1 Akutsu, Y, Deguchi,, and Ohtsuki,, Invariants of colored links J Knot heory Ramifications 1 (1992), no 2, C Blanchet, Costantino, F, Geer, N, and Patureau-Mirand, B, Non semi-simple QFs, Reidemeister torsion and Kashaev s invariants, arxiv: Costantino, F, Geer, N, and Patureau-Mirand, B, Quantum invariants of 3- manifolds via link surgery presentations and non-semi-simple categories, Accepted for publication in Journal of opology, arxiv: Costantino, F, Geer, N, and Patureau-Mirand, B, Relations between Witten- Reshetikhin-uraev and non semi-simple sl(2) 3-manifold invariants, accepted for publication in Algebraic & Geometric opology, arxiv: Costantino, F, Geer, N, and Patureau-Mirand, B, Some remarks on the unrolled quantum group of sl(2), accepted for publication in Journal of Pure and Applied Algebra, arxiv: N Geer, B Patureau-Mirand - Multivariable link invariants arising from sl(2 1) and the Alexander polynomial J Pure Appl Algebra 210 (2007), no 1, Geer, N, Patureau-Mirand, B, and uraev, Modified quantum dimensions and re-normalized link invariants Compos Math 145 (2009), no 1, Geer, N, Patureau-Mirand, B, and uraev, Modified 6j-Symbols and 3-Manifold invariants, Advances in Mathematics, 228 (2011), no 2, , arxiv: Geer, N and Reshetikhin N, On invariants of graphs related to quantum sl(2) at roots of unity, Letters in Mathematical Physics 88 (2009), no 1-3, Murakami, J, Colored Alexander Invariants and Cone-Manifolds Osaka J Math olume 45, Number 2 (2008), Ohtsuki - Quantum invariants A study of knots, 3 manifolds, and their sets Series on Knots and Everything, 29 World Scientific Publishing Co, Inc, River Edge, NJ, uraev,, Quantum invariants of knots and 3-manifolds de Gruyter Studies in Mathematics, 18 Walter de Gruyter & Co, Berlin, (1994) Mathematics & Statistics, Utah State University, Logan, Utah 84322, USA

arxiv: v1 [math.gt] 14 May 2014

arxiv: v1 [math.gt] 14 May 2014 NON SEMI-SIMPLE sl(2) QUANTUM INVARIANTS, SPIN CASE arxiv:1405.3490v1 [math.gt] 14 May 2014 CHRISTIAN BLANCHET, FRANCESCO COSTANTINO, NATHAN GEER, AND BERTRAND PATUREAU-MIRAND Abstract. Invariants of 3-manifolds