We show how to define invariants of graphs related to quantum
$\mathfrak{sl}(2)$ when the graph has more then one connected component and
components are colored by blocks of representations with zero quantum
dimensions

N. Geer

N. Reshetikhin

Nathan Geer

Nicolai Reshetikhin

English

arXiv

Geer, Nathan

Reshetikhin, Nicolai

Max Planck Society eDoc Server

On invariants of graphs related to quantum sl(2) at roots of unity

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On Invariants of Graphs Related to Quantum $${\mathfrak {sl}(2)}$$ at Roots of Unity

We show how to define invariants of graphs related to quantum sl 2 when the graph has more then one connected component and components are colored by blocks of representations with zero quantum dimensions

Geer, N.

Reshetikhin, N.

NARCIS 

UvA-DARE

We show how to define invariants of graphs related to quantum sl
                2 when the graph has more then one connected component and components are colored by blocks of representations with zero quantum dimensions

eScholarship - University of California

UC BerkeleyUC Berkeley Previously Published WorksTitleOn Invariants of Graphs Related to Quantum    $${\mathfrak {sl}(2)}$$ at Roots of UnityPermalinkhttps://escholarship.org/uc/item/9660n0g4JournalLetters in Mathematical Physics: A Journal for the Rapid Dissemination of Short Contributions in the Field of Mathematical Physics, 88(1)ISSN1573-0530AuthorsGeer, NathanReshetikhin, NicolaiPublication Date2009-06-01DOI10.1007/s11005-009-0320-9 Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaDOI 10.1007/s11005-009-0320-9Lett Math Phys (2009) 88:321–331On Invariants of Graphs Related to Quantum sl(2)at Roots of UnityNATHAN GEER1,2 and NICOLAI RESHETIKHIN3,41Max-Planck Institute für Mathematik, Vivatgasse 7, 53111 Bonn, Germany.2School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA.e-mail: geer@math.gatech.edu3Department of Mathematics, University of California, Berkeley, 970 Evans Hall #3840,Berkeley, CA 94720-3840, USA.4KDV Institute for Mathematics, Universiteit van Amsterdam, Plantage Muidergracht 24,1018 TV, Amsterdam, The Netherlands. e-mail: reshetik@math.berkeley.eduReceived: 14 March 2009 / Accepted: 4 April 2009Published online: 13 May 2009 – © The Author(s) 2009. This article is published with openaccess at Springerlink.comAbstract. We show how to define invariants of graphs related to quantum sl2 when thegraph has more then one connected component and components are colored by blocks ofrepresentations with zero quantum dimensions.Mathematics Subject Classification (2000). 17B37, 57R56, 46A63.Keywords. invariants of knots, quantized universal enveloping algebras,topological quantum field theory.0. IntroductionThe invariants of oriented linked framed graphs colored by finite dimensional rep-resentations of quantized universal enveloping algebras were constructed in [4] fornon-zero quantum dimensions. In this construction the edges of the graph arecolored by finite dimensional representations. At each vertex a total order-ing of adjacent edges which agrees with counter-clockwise cyclic order definedby the framing is fixed. A vertex with adjacent edges colored by representa-tions V1, . . . , Vn (according to the total ordering) is colored by an element fromHom(C, V ε11 ⊗ . . .⊗V εnn ) of all Uq(sl2)-invariant vectors in the tensor product. HereV + =V, V − =V ∗ where V ∗ is the left dual to V , ε =+ if the edges is incoming andε =− if the edges is outgoing.When the quantum dimension is zero, it is easy to define such invariants forknots when the center of Uq(sl2) acts by scalars on the representation coloringthe knot. One should cut the knot (its diagram) at any place and then to computethe invariant of the corresponding (1,1) tangle. Because the center acts trivially at thecoloring representation, this invariant will be equal to a operator of multiplication by322 NATHAN GEER AND NICOLAI RESHETIKHINa scalar. It is easy to show that the scalar is the invariant of a knot (does not depend onwhere the knot was cut). But this technique does not work for links.Recently invariants of links colored by such representations were introduced in[2] under the assumption that these representations satisfy an additional property,i.e. they are ambidextrous (see [2] or Section 3.1 for the precise definition). We willabbreviate this name here to ambi-modules.The goal of this note is to present in details how the construction involvingambi-modules works for sl2. We prove that all generic irreducible representationsof◦U ε (sl2) are ambi-modules.We also conjecture but we do not prove it here that all generic irreducible rep-resentations of the◦U ε (g) are ambi-modules for any simple Lie algebra.We would like to thank V. Turaev and B. Patureau for discussions. This workwas supported by the Danish National Research Foundation through the NielsBohr initiative, the work of N.R. was also supported by the NSF grant DMS-0601912. The work of N.G. was supported by the NSF grant DMS-0706725. Bothauthors are grateful to the Aarhus University for the hospitality.1. The Split Form of Quantum sl2 at a Root of Unity1.1. QUANTUM sl2Let m, l be positive odd integers, t = exp( iπm2l ), and ε = t2.The algebra Uε(sl2) is the unital C-algebra generated by invertible K and byE, F with defining relations:K E = ε2 E K , K F = ε−2 F K , [E, F]= K − K−1ε − ε−1 .Recall that it is a Hopf algebra with the comultiplication acting on generatorsasK = K ⊗ K , E = E ⊗ K +1⊗ E, F = F ⊗1+ K −1 ⊗ FThe elements K l , Fl , El generate a central Hopf subalgebra, and El , Fl generatecentral Hopf ideal. Denote by Uε(sl2)′ its quotient by this central ideal.1.2. THE SPLIT FORM OF Uε(sl2)′DEFINITION 1. The split form of Uε(sl2)′ is the algebra◦U ε (sl2) generated byPi , i ∈Z and by E, F, x, x−1 such that x is central andPi Pj = δi j Pi , Pi E = E Pi−2, Pi F = F Pi+2,∑i∈ZPi =1,E F − F E =∑j∈Zx2ε j − x−2ε− jε − ε−1 Pj .ON INVARIANTS OF GRAPHS RELATED TO QUANTUM SL(2) 323The algebra◦U ε (sl2) is a Hopf algebra (in the category of its finite-dimensionalrepresentations) where the comultiplication , counit ε and antipode S morphismsare given by:(E)=∑j∈ZE ⊗ x2ε j Pj +1⊗ E, (F)= F ⊗1+∑j∈Zx−2ε− j Pj ⊗ F,(Pk)=∑i, j∈Z; i+ j=kPi ⊗ Pj , (x)= x ⊗ x,ε(E)= ε(F)=0, ε(Pi )= δi,0, ε(x)=1,S(E)=−E∑j∈Zx−2ε− j Pj , S(F)=−∑j∈Zx2ε j Pj F, S(Pi )=P−i , S(x)=x−1.It is clear that the map ι :Uε(sl2)→◦U ε (sl2) given by K →∑ j∈Z x2ε j Pj , E → Eand F → F is an injective morphism of Hopf algebras.Notice that K l →∑ j∈Z x2lε jl Pj and that K 2l → x4l .1.3. THE R- MATRIXFor n ∈N we set [n]= εn−ε−nε−ε−1 and [n]!= [n][n −1] . . . [1]. DefineR0 =∑i, j∈Zt i j x i Pj ⊗ x j Pi ,R1 =l−1∑n=0(ε − ε−1)n[n]! En ⊗ Fn,and R = R0 R1 ∈◦U ε (sl2)⊗2.Now we will show that R defines a quasitriangular structure on◦U ε (sl2).LEMMA 2. The element R0 satisfies the following identities.(1) (⊗ Id)(R0)= (R0)13(R0)23,(2) (Id ⊗)(R0)= (R0)13(R0)12,(3) R0(E ⊗1)= (E ⊗ ι(K ))R0, R0(1⊗ E)= (ι(K )⊗ E)R0,(4) R0(F ⊗1)= (F ⊗ ι(K )−1)R0, R0(1⊗ F)= (ι(K )−1 ⊗ F)R0.Proof. We will prove the first identity in Part (3), the other identities follow sim-ilarly. We haveR0(E ⊗1)=∑i, j∈Zt i j x i Pj E ⊗ x j Pi ==∑i, j∈Zt i j x i E Pj−2 ⊗ x j Pi =324 NATHAN GEER AND NICOLAI RESHETIKHIN=∑i, j∈Zt i j x i E Pj ⊗ x j x2εi Pi == (E ⊗ ι(K ))R0.THEOREM 3. The pair (◦U ε (sl2), R) is a quasitriangular Hopf algebra.Proof. Lemma 2 implies that R0 has the same commutator relation with E andF as the element exp( h4 (H ⊗ H)) in the C[[h]]-algebra Uh(sl2). Thus, we conclude(⊗ Id)(R)= R13 R23, (Id ⊗)(R)= R13 R12,andop(a)= R(a)R−1, for a ∈ ◦U ε (sl2)where op = τ ◦ and τ is the permutation a ⊗b →b ⊗a.Let R =∑ si ⊗ ti and define u =∑ S(ti )si . Then following [1] we haveε(u)=1, (u)= (R21 R)−1 (u ⊗u), S2(a)=uau−1 (1)for all a ∈ ◦U ε (sl2).A direct computation shows that for all a ∈ ◦U ε (sl2) we haveS2(a)= ι(K )aι(K )−1. (2)Let θ̃ = uι(K )−1 = ι(K )−1u. Comparing this formula with the formula for the S2involving u we see that θ̃ is central.LEMMA 4. The element θ̃ satisfies the relations:ε(θ̃)=1, (θ̃)= (R21 R)−1 (θ̃ ⊗ θ̃ ), S(θ̃)= ι(K )2l θ̃ .Proof. The first two relations follow from (1) and the definition of the counitand coproduct of ι(K ). The last relation can be proven by direct computation ofS(uι(K )−1) and of uι(K )−1. This computation was done essentially in [3]. One canalso prove the last identity by computing how S(θ̃) and θ̃ act on generic irreduc-ible modules. It is easy to see that on these modules the identity holds. On theother hand our algebra is finitely generated and is finite dimensional over the cen-ter. This implies that the identity S(θ̃ )= ι(K )2l θ̃ holds not only for generic pointsbut also for special points (where x acts as 4lth root of unity).ON INVARIANTS OF GRAPHS RELATED TO QUANTUM SL(2) 3251.4. THE AUTOMORPHISM φPROPOSITION 5. (1) The mapping φ : ◦U ε (sl2)→◦U ε (sl2) acting on generators asφ(Pi )= Pi+2, φ(x)= xε, φ(E)= E, φ(F)= Fextends uniquely to an algebra automorphism.(2) The set of fixed points form a Hopf subalgebra generated by x∑i∈Z t i Pi , x2l ,E, F .Proof. The first part is clear from defining relations, the second follows immedi-ately from the formula for φ.Note that despite the fact that fixed points of φ form a Hopf subalgebra, it isnot a Hopf algebra automorphism.The automorphism φ acts on the R-matrix as follows:(φ ⊗ id)(R)= (1⊗ x−2)R, (id ⊗φ)(R)= (x−2 ⊗1)(R)PROPOSITION 6. The automorphism φ acts on θ̃ as follows:φ(θ̃)= x4θ̃Proof. Let R = ∑i si ⊗ ti , then R−1 =∑i si ⊗ S(ti ). Combining the way φ actson the R-matrix with the fact that it is an algebra automorphism we obtain (φ ⊗φ)(R−1) = (x2 ⊗ x2)R−1. Therefore φ(∑i S(ti )si ) = x4∑i S(ti )si . Thus, φ(u) = x4u.Since φ(ι(K ))= ι(K ), this proves the proposition.Define ε =∑ j∈Z(−1) j Pj . It is central and unipotent. It is also easy to see thatφ acts trivially on it:φ(ε)= ε1.5. THE EXTENDED R- MATRIXConsider the algebra Aε =◦U ε (sl2) ⊗C C[τ, τ−1]. The element τ is central in thisalgebra.Remark 7. The motivation for this construction is the formal power series versionof the algebra◦U ε (sl2), when instead of having Laurent polynomials in x we haveformal powers series z:x = exp(imzπ2l), τ = exp(imz2π4l).326 NATHAN GEER AND NICOLAI RESHETIKHINAssume that the action of the comultiplication on τ is symmetric, i.e. op(τ )=(τ). Defineρ =(τ)τ−1 ⊗ τ−1.Extend the action of the automorphism φ on Aε as:φ(τ)= εx2τ, (φ ⊗ id)(ρ)= (1⊗ x2)ρ.Define the extended R-matrix asR=ρR.It is easy to see that R defines a quasitriangular structure on Aε . It is also easyto see that(id ⊗φ)(R)= (φ ⊗ id)(R)=R.Defineθ1 = τ−2ι(K )l θ̃ , θ2 = τ−2ι(K )lεθ̃ .Each of these elements satisfies identities:ε(θ)=1, (θ)= (R21R)−1 (θ ⊗ θ), S(θ)= θ, φ(θ)= θ.2. The Category Cε2.1. SIMPLE MODULESHere we will focus on simple modules over the◦U ε (sl2), Uε(sl2)′, and Aε . We willsay a ∈C∗ is generic if a4l =1.The following statement is a variation on a well known fact:PROPOSITION 8. For each generic a ∈C∗ and k ∈Z there exists unique simple ◦U ε(sl2)-module V (a, k) with highest weight vector vk such thatEvk =0, xvk =avk Pjvk = δ j,kvk, V (a, k)=l−1∑i=0CFivk .It is easy to compute the action of generators on the weight basis vk−2 j = F jvk :xvk−2i =avk−2iPjvk−2i = δ j,k−2ivk−2i , 0≥ i ≥ l −1,Fvk−2i =vk−2i−2, 1≤ i ≤ l −1Evk−2i =(a2εk+i+1 −a−2ε−k−1−iε − ε−1)(εi − ε−iε − ε−1)vk−2i+2, 0≤ i ≤ l −1.ON INVARIANTS OF GRAPHS RELATED TO QUANTUM SL(2) 327The homomorphism ι : Uε(sl2)′ →◦U ε (sl2) defines on V (a, k) the structure of aUε(sl2)′-module.PROPOSITION 9. The Uε(sl2)′-modules V (a, k) are irreducible, and in addition,V (a, k) and V (aε, k +1) are isomorphic as U ′ε-modules.In particular central element ε acts on V (a, k) as (−1)k .Note that the modules V (a, k) and V (aε, k +1) are not isomorphic as ◦U ε (sl2)-modules.For z ∈ C we can always define the Aε-module structure on the representationV (eimzπ2l , k) of◦U ε (sl2) by defining τv = exp( imz2π4l )v for any v ∈ V (eimzπ2l , k). It isclear that this defines an irreducible Aε-module.2.2. THE CATEGORY CεObjects of the category Cε are finite dimensional◦U ε (sl2)-modules (V, πV : Aε →End(V )) on which x acts as a multiplication by a scalar πV (x)v = exp( imzπ2l )v.Here v ∈ V , m, l are as above and z ∈ C. The central element τ on such moduleacts by multiplication on exp( imz2π4l ).Morphisms between two such modules are all◦U ε (sl2)-invariant linear maps.This category is monoidal because it is a category of finite dimensional modulesover a Hopf algebra. It is a rigid monoidal category with the left dual modulesdefined as usual (V ∗, πV ∗ =π∗V ◦ S) where π∗V (a) is the dual linear map to πV (a)andwith usual injection and evaluation morphisms:iV :C→ V ⊗ V ∗, given by 1 →∑ei ⊗ ei,eV : V ∗ ⊗ V →C, given by f ⊗w → f (w)It is easy to see that Cε is a braided category with the commutativity morphismc ={cV,W } where cV,W : V ⊗ W → W ⊗ V given by v ⊗w → τ(R(v ⊗w)). It is also aribbon category with the ribbon morphisms (twists): θV : V → V, v → θ−1v.In this category the braiding and the ribbon structure agree with isomorphismsof modules induced by φ.The objects of this category are semisimple for generic z, i.e. when mz it is notan integer.The linear mapping v→θ−1uv is a an isomorphism of representations V → V ∗∗.Recall that the quantum (functorial) dimension of V is defined as the compositionmapping C→V ⊗V ∗ →V ∗∗ ⊗V ∗ →C, or, as trV (θ−1u)= trV (τ 2ι(K )1−l). It is clearthat the quantum dimension of any generic representation is zero.328 NATHAN GEER AND NICOLAI RESHETIKHIN3. Invariants of Links3.1. AMBI- ELEMENTS IN A RIBBON CATEGORYHere we will recall some results and definitions from [2].Recall that in a ribbon category there is a natural notion of a trace of an endo-morphism of an object. If f : V → V , its trace istrV ( f )= eV ∗ ◦ (µV ⊗ idV ∗)◦ ( f ⊗ idV ∗)◦ iVwhere µV : V → V ∗∗ is the isomorphism between V and V ∗∗ determined by thebraiding and the ribbon structure. For the category of◦U ε (sl2), µV =πV (ι(K )1−l).We will use the following notations. If V is simple and f : V → V is a morphism,by definition f =c( f )idV for some c( f )∈κ where κ is the base field for our cate-gory. We will assume κ =C and will use the graphical notation for c( f ) shown onFigure 1.Define S′(U, V )= (idV ⊗ trU )(cU,V cV,U ) where cV,U : V ⊗U →U ⊗ V is the com-mutativity constraint in the category. This element can be written graphically as onFigure 2.For any f : V → V and simple V we have the identity shown on Figure 3.Recall that an object V is called ambidextrous if (idV ⊗ trV )( f )= (trV ⊗idV )( f )for any f : V ⊗2 → V ⊗2 (see also Figure 4). Here we will call such an object anambi-object.The following observation is one of the key tools from [2]. Let U be a simpleambi-object. Then for any two objects V1 and V2 and any morphism f : V1 ⊗ V2 →V1 ⊗ V2 the following identity holds:S′(U∗, V ∗1 )S′(V2,U )c((trV1 ⊗idV2)( f ))= S′(V1,U )S′(U∗, V ∗2 )c((idV1 ⊗ trV2)( f ))(3)Figure 1.Figure 2.ON INVARIANTS OF GRAPHS RELATED TO QUANTUM SL(2) 329Figure 3.Figure 4.Figure 5.Figure 6. T̂V is the closer of TV .It follows from the ambi-identity for the tangle from Figure 5. Here notice thatS′(V ∗, W ∗)= S′(V, W ).We have the following corollary of the identity (3).PROPOSITION 10. • If J is a simple ambi-object of the category C, then allobjects U ∈ A(J ) = {W |S′(W, J ) = 0, S′(J, W ) = 0, W is simple} are also ambi-objects.• Let TV be a (1,1)-tangle colored by elements of Cε , whose open component is col-ored by an element V ∈ A(J ). Then the expression in Figure 6 is an invariant ofthe link obtained by the closure of TV . In this expression U is any ambi-elementfrom A(J ).330 NATHAN GEER AND NICOLAI RESHETIKHIN3.2. AMBI- MODULES OVER◦U ε (sl2)3.2.1. As it was pointed out at the end of the previous section all modules V (a, k)have zero quantum dimension. Here we will prove that they are all ambi-objectsin the category◦U ε (sl2)−mod.LEMMA 11. The module V (a, k) is an ambi-object for all k ∈Z and a8l =1.Proof. It is easy to prove that for all a ∈C\ {a8l =1}, and k ∈ZV (a, k)⊗ V (a, k)∼= V (2a,2k)⊕ V (2a,2k −2)⊕· · ·⊕ V (2a,2k −2l +2).Since this decomposition is a direct sum of non-isomorphic simple modules thealgebra End(V (a, k) ⊗ V (a, k)) is commutative. Now let f ∈ End(V ⊗ V ), then(idV ⊗ trV )( f )= (trV ⊗idV )(c−1V,V ◦ f ◦ cV,V ). But cV,V commutes with End(V ⊗ V )and so c−1V,V ◦ f ◦ cV,V = f . Here we set V = V (a, k). This finishes the proof.The condition a8l =1 can be relaxed to a4l =1.LEMMA 12. Let V (a, k) and V (b,m) be two irreducible◦U ε (sl2)-modules describedin the previous section. ThenS′(V (a, k), V (b,m))= ε(k+1−l)(m+1−l)b2k−2l+2a2m−2l+2(−1)m+l−1 ××[b2l −b−2lεm+1−lb2 − (εm+1−lb2)−1]Proof. The proof is a straightforward computation. We want to compute theaction of (idV ⊗ trU )((πV ⊗πU )(σ (R)R)) on V . Let us apply to the highest weightvector. ThenS′((V (a, k), V (b,m))vm = (idV (b,m) ⊗ trV (a,k))((πV (b,m) ⊗πV (a,k))(σ (R0)R0))vm == εk(m+1−l)b2ka2m+2−2ll−1∑i=0ε−2i(m+1−l)b−4ivmThen one should sum up the geometric progression.Now, note that S′(V (a, k), V (b,m)) = 0 unless b4l = 1 when (εm+1−lb2)2 = 1.In the later case the representation V (b,m) is reducible. We assume that this is notthe case.Thus, we have proven the following theorem.THEOREM 13. All irreducible representations V (a, k) of◦U ε (sl2) with a4l = 1 areambi-modules.ON INVARIANTS OF GRAPHS RELATED TO QUANTUM SL(2) 331Open Access This article is distributed under the terms of the Creative Commons Attri-bution Noncommercial License which permits any noncommercial use, distribution, andreproduction in any medium, provided the original author(s) and source are credited.References1. Drinfeld, V.: Almost cocommutative Hopf algebras. Algebra i Analiz 1(2), 30–46 (1989)(Russian) [translation in Leningrad Math. J. 1(2), 321–342 (1990)]2. Geer, N., Patureau-Mirand, B., Turaev, V.: Modified quantum dimensions and re-nor-malized link invariants. Compos. Math. 145(01), 196–212 (2009). math.QA/0711.42293. Ohtsuki, T.: Quantum invariants. A study of knots and 3-manifolds. Series on Knotsand Everything, vol. 29. World Scientific Publishing Co., Inc., River Edge (2002)4. Reshetikhin, N., Turaev, V.: Ribbon graphs and their invariants derived from quantumgroups. Commun. Math. Phys. 127(1), 1–26 (1990)

On Invariants of Graphs Related to Quantum 
                
                  
                
                $${\mathfrak {sl}(2)}$$
               at Roots of Unity

International Migration, Integration and Social Cohesion online publications

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)On invariants of graphs related to quantum sl(2) at roots of unityGeer, N.; Reshetikhin, N.DOI10.1007/s11005-009-0320-9Publication date2009Document VersionFinal published versionPublished inLetters in  Mathematical PhysicsLink to publicationCitation for published version (APA):Geer, N., & Reshetikhin, N. (2009). On invariants of graphs related to quantum sl(2) at rootsof unity. Letters in Mathematical Physics, 88(1-3), 321-331. https://doi.org/10.1007/s11005-009-0320-9General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:14 Apr 2023DOI 10.1007/s11005-009-0320-9Lett Math Phys (2009) 88:321–331On Invariants of Graphs Related to Quantum sl(2)at Roots of UnityNATHAN GEER1,2 and NICOLAI RESHETIKHIN3,41Max-Planck Institute für Mathematik, Vivatgasse 7, 53111 Bonn, Germany.2School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA.e-mail: geer@math.gatech.edu3Department of Mathematics, University of California, Berkeley, 970 Evans Hall #3840,Berkeley, CA 94720-3840, USA.4KDV Institute for Mathematics, Universiteit van Amsterdam, Plantage Muidergracht 24,1018 TV, Amsterdam, The Netherlands. e-mail: reshetik@math.berkeley.eduReceived: 14 March 2009 / Accepted: 4 April 2009Published online: 13 May 2009 – © The Author(s) 2009. This article is published with openaccess at Springerlink.comAbstract. We show how to define invariants of graphs related to quantum sl2 when thegraph has more then one connected component and components are colored by blocks ofrepresentations with zero quantum dimensions.Mathematics Subject Classification (2000). 17B37, 57R56, 46A63.Keywords. invariants of knots, quantized universal enveloping algebras,topological quantum field theory.0. IntroductionThe invariants of oriented linked framed graphs colored by finite dimensional rep-resentations of quantized universal enveloping algebras were constructed in [4] fornon-zero quantum dimensions. In this construction the edges of the graph arecolored by finite dimensional representations. At each vertex a total order-ing of adjacent edges which agrees with counter-clockwise cyclic order definedby the framing is fixed. A vertex with adjacent edges colored by representa-tions V1, . . . , Vn (according to the total ordering) is colored by an element fromHom(C, V ε11 ⊗ . . .⊗V εnn ) of all Uq(sl2)-invariant vectors in the tensor product. HereV + =V, V − =V ∗ where V ∗ is the left dual to V , ε =+ if the edges is incoming andε =− if the edges is outgoing.When the quantum dimension is zero, it is easy to define such invariants forknots when the center of Uq(sl2) acts by scalars on the representation coloringthe knot. One should cut the knot (its diagram) at any place and then to computethe invariant of the corresponding (1,1) tangle. Because the center acts trivially at thecoloring representation, this invariant will be equal to a operator of multiplication by322 NATHAN GEER AND NICOLAI RESHETIKHINa scalar. It is easy to show that the scalar is the invariant of a knot (does not depend onwhere the knot was cut). But this technique does not work for links.Recently invariants of links colored by such representations were introduced in[2] under the assumption that these representations satisfy an additional property,i.e. they are ambidextrous (see [2] or Section 3.1 for the precise definition). We willabbreviate this name here to ambi-modules.The goal of this note is to present in details how the construction involvingambi-modules works for sl2. We prove that all generic irreducible representationsof◦U ε (sl2) are ambi-modules.We also conjecture but we do not prove it here that all generic irreducible rep-resentations of the◦U ε (g) are ambi-modules for any simple Lie algebra.We would like to thank V. Turaev and B. Patureau for discussions. This workwas supported by the Danish National Research Foundation through the NielsBohr initiative, the work of N.R. was also supported by the NSF grant DMS-0601912. The work of N.G. was supported by the NSF grant DMS-0706725. Bothauthors are grateful to the Aarhus University for the hospitality.1. The Split Form of Quantum sl2 at a Root of Unity1.1. QUANTUM sl2Let m, l be positive odd integers, t = exp( iπm2l ), and ε = t2.The algebra Uε(sl2) is the unital C-algebra generated by invertible K and byE, F with defining relations:K E = ε2 E K , K F = ε−2 F K , [E, F]= K − K−1ε − ε−1 .Recall that it is a Hopf algebra with the comultiplication acting on generatorsasK = K ⊗ K , E = E ⊗ K +1⊗ E, F = F ⊗1+ K −1 ⊗ FThe elements K l , Fl , El generate a central Hopf subalgebra, and El , Fl generatecentral Hopf ideal. Denote by Uε(sl2)′ its quotient by this central ideal.1.2. THE SPLIT FORM OF Uε(sl2)′DEFINITION 1. The split form of Uε(sl2)′ is the algebra◦U ε (sl2) generated byPi , i ∈Z and by E, F, x, x−1 such that x is central andPi Pj = δi j Pi , Pi E = E Pi−2, Pi F = F Pi+2,∑i∈ZPi =1,E F − F E =∑j∈Zx2ε j − x−2ε− jε − ε−1 Pj .ON INVARIANTS OF GRAPHS RELATED TO QUANTUM SL(2) 323The algebra◦U ε (sl2) is a Hopf algebra (in the category of its finite-dimensionalrepresentations) where the comultiplication , counit ε and antipode S morphismsare given by:(E)=∑j∈ZE ⊗ x2ε j Pj +1⊗ E, (F)= F ⊗1+∑j∈Zx−2ε− j Pj ⊗ F,(Pk)=∑i, j∈Z; i+ j=kPi ⊗ Pj , (x)= x ⊗ x,ε(E)= ε(F)=0, ε(Pi )= δi,0, ε(x)=1,S(E)=−E∑j∈Zx−2ε− j Pj , S(F)=−∑j∈Zx2ε j Pj F, S(Pi )=P−i , S(x)=x−1.It is clear that the map ι :Uε(sl2)→◦U ε (sl2) given by K →∑ j∈Z x2ε j Pj , E → Eand F → F is an injective morphism of Hopf algebras.Notice that K l →∑ j∈Z x2lε jl Pj and that K 2l → x4l .1.3. THE R- MATRIXFor n ∈N we set [n]= εn−ε−nε−ε−1 and [n]!= [n][n −1] . . . [1]. DefineR0 =∑i, j∈Zt i j x i Pj ⊗ x j Pi ,R1 =l−1∑n=0(ε − ε−1)n[n]! En ⊗ Fn,and R = R0 R1 ∈◦U ε (sl2)⊗2.Now we will show that R defines a quasitriangular structure on◦U ε (sl2).LEMMA 2. The element R0 satisfies the following identities.(1) (⊗ Id)(R0)= (R0)13(R0)23,(2) (Id ⊗)(R0)= (R0)13(R0)12,(3) R0(E ⊗1)= (E ⊗ ι(K ))R0, R0(1⊗ E)= (ι(K )⊗ E)R0,(4) R0(F ⊗1)= (F ⊗ ι(K )−1)R0, R0(1⊗ F)= (ι(K )−1 ⊗ F)R0.Proof. We will prove the first identity in Part (3), the other identities follow sim-ilarly. We haveR0(E ⊗1)=∑i, j∈Zt i j x i Pj E ⊗ x j Pi ==∑i, j∈Zt i j x i E Pj−2 ⊗ x j Pi =324 NATHAN GEER AND NICOLAI RESHETIKHIN=∑i, j∈Zt i j x i E Pj ⊗ x j x2εi Pi == (E ⊗ ι(K ))R0.THEOREM 3. The pair (◦U ε (sl2), R) is a quasitriangular Hopf algebra.Proof. Lemma 2 implies that R0 has the same commutator relation with E andF as the element exp( h4 (H ⊗ H)) in the C[[h]]-algebra Uh(sl2). Thus, we conclude(⊗ Id)(R)= R13 R23, (Id ⊗)(R)= R13 R12,andop(a)= R(a)R−1, for a ∈ ◦U ε (sl2)where op = τ ◦ and τ is the permutation a ⊗b →b ⊗a.Let R =∑ si ⊗ ti and define u =∑ S(ti )si . Then following [1] we haveε(u)=1, (u)= (R21 R)−1 (u ⊗u), S2(a)=uau−1 (1)for all a ∈ ◦U ε (sl2).A direct computation shows that for all a ∈ ◦U ε (sl2) we haveS2(a)= ι(K )aι(K )−1. (2)Let θ̃ = uι(K )−1 = ι(K )−1u. Comparing this formula with the formula for the S2involving u we see that θ̃ is central.LEMMA 4. The element θ̃ satisfies the relations:ε(θ̃)=1, (θ̃)= (R21 R)−1 (θ̃ ⊗ θ̃ ), S(θ̃)= ι(K )2l θ̃ .Proof. The first two relations follow from (1) and the definition of the counitand coproduct of ι(K ). The last relation can be proven by direct computation ofS(uι(K )−1) and of uι(K )−1. This computation was done essentially in [3]. One canalso prove the last identity by computing how S(θ̃) and θ̃ act on generic irreduc-ible modules. It is easy to see that on these modules the identity holds. On theother hand our algebra is finitely generated and is finite dimensional over the cen-ter. This implies that the identity S(θ̃ )= ι(K )2l θ̃ holds not only for generic pointsbut also for special points (where x acts as 4lth root of unity).ON INVARIANTS OF GRAPHS RELATED TO QUANTUM SL(2) 3251.4. THE AUTOMORPHISM φPROPOSITION 5. (1) The mapping φ : ◦U ε (sl2)→◦U ε (sl2) acting on generators asφ(Pi )= Pi+2, φ(x)= xε, φ(E)= E, φ(F)= Fextends uniquely to an algebra automorphism.(2) The set of fixed points form a Hopf subalgebra generated by x∑i∈Z t i Pi , x2l ,E, F .Proof. The first part is clear from defining relations, the second follows immedi-ately from the formula for φ.Note that despite the fact that fixed points of φ form a Hopf subalgebra, it isnot a Hopf algebra automorphism.The automorphism φ acts on the R-matrix as follows:(φ ⊗ id)(R)= (1⊗ x−2)R, (id ⊗φ)(R)= (x−2 ⊗1)(R)PROPOSITION 6. The automorphism φ acts on θ̃ as follows:φ(θ̃)= x4θ̃Proof. Let R = ∑i si ⊗ ti , then R−1 =∑i si ⊗ S(ti ). Combining the way φ actson the R-matrix with the fact that it is an algebra automorphism we obtain (φ ⊗φ)(R−1) = (x2 ⊗ x2)R−1. Therefore φ(∑i S(ti )si ) = x4∑i S(ti )si . Thus, φ(u) = x4u.Since φ(ι(K ))= ι(K ), this proves the proposition.Define ε =∑ j∈Z(−1) j Pj . It is central and unipotent. It is also easy to see thatφ acts trivially on it:φ(ε)= ε1.5. THE EXTENDED R- MATRIXConsider the algebra Aε =◦U ε (sl2) ⊗C C[τ, τ−1]. The element τ is central in thisalgebra.Remark 7. The motivation for this construction is the formal power series versionof the algebra◦U ε (sl2), when instead of having Laurent polynomials in x we haveformal powers series z:x = exp(imzπ2l), τ = exp(imz2π4l).326 NATHAN GEER AND NICOLAI RESHETIKHINAssume that the action of the comultiplication on τ is symmetric, i.e. op(τ )=(τ). Defineρ =(τ)τ−1 ⊗ τ−1.Extend the action of the automorphism φ on Aε as:φ(τ)= εx2τ, (φ ⊗ id)(ρ)= (1⊗ x2)ρ.Define the extended R-matrix asR=ρR.It is easy to see that R defines a quasitriangular structure on Aε . It is also easyto see that(id ⊗φ)(R)= (φ ⊗ id)(R)=R.Defineθ1 = τ−2ι(K )l θ̃ , θ2 = τ−2ι(K )lεθ̃ .Each of these elements satisfies identities:ε(θ)=1, (θ)= (R21R)−1 (θ ⊗ θ), S(θ)= θ, φ(θ)= θ.2. The Category Cε2.1. SIMPLE MODULESHere we will focus on simple modules over the◦U ε (sl2), Uε(sl2)′, and Aε . We willsay a ∈C∗ is generic if a4l =1.The following statement is a variation on a well known fact:PROPOSITION 8. For each generic a ∈C∗ and k ∈Z there exists unique simple ◦U ε(sl2)-module V (a, k) with highest weight vector vk such thatEvk =0, xvk =avk Pjvk = δ j,kvk, V (a, k)=l−1∑i=0CFivk .It is easy to compute the action of generators on the weight basis vk−2 j = F jvk :xvk−2i =avk−2iPjvk−2i = δ j,k−2ivk−2i , 0≥ i ≥ l −1,Fvk−2i =vk−2i−2, 1≤ i ≤ l −1Evk−2i =(a2εk+i+1 −a−2ε−k−1−iε − ε−1)(εi − ε−iε − ε−1)vk−2i+2, 0≤ i ≤ l −1.ON INVARIANTS OF GRAPHS RELATED TO QUANTUM SL(2) 327The homomorphism ι : Uε(sl2)′ →◦U ε (sl2) defines on V (a, k) the structure of aUε(sl2)′-module.PROPOSITION 9. The Uε(sl2)′-modules V (a, k) are irreducible, and in addition,V (a, k) and V (aε, k +1) are isomorphic as U ′ε-modules.In particular central element ε acts on V (a, k) as (−1)k .Note that the modules V (a, k) and V (aε, k +1) are not isomorphic as ◦U ε (sl2)-modules.For z ∈ C we can always define the Aε-module structure on the representationV (eimzπ2l , k) of◦U ε (sl2) by defining τv = exp( imz2π4l )v for any v ∈ V (eimzπ2l , k). It isclear that this defines an irreducible Aε-module.2.2. THE CATEGORY CεObjects of the category Cε are finite dimensional◦U ε (sl2)-modules (V, πV : Aε →End(V )) on which x acts as a multiplication by a scalar πV (x)v = exp( imzπ2l )v.Here v ∈ V , m, l are as above and z ∈ C. The central element τ on such moduleacts by multiplication on exp( imz2π4l ).Morphisms between two such modules are all◦U ε (sl2)-invariant linear maps.This category is monoidal because it is a category of finite dimensional modulesover a Hopf algebra. It is a rigid monoidal category with the left dual modulesdefined as usual (V ∗, πV ∗ =π∗V ◦ S) where π∗V (a) is the dual linear map to πV (a)andwith usual injection and evaluation morphisms:iV :C→ V ⊗ V ∗, given by 1 →∑ei ⊗ ei,eV : V ∗ ⊗ V →C, given by f ⊗w → f (w)It is easy to see that Cε is a braided category with the commutativity morphismc ={cV,W } where cV,W : V ⊗ W → W ⊗ V given by v ⊗w → τ(R(v ⊗w)). It is also aribbon category with the ribbon morphisms (twists): θV : V → V, v → θ−1v.In this category the braiding and the ribbon structure agree with isomorphismsof modules induced by φ.The objects of this category are semisimple for generic z, i.e. when mz it is notan integer.The linear mapping v→θ−1uv is a an isomorphism of representations V → V ∗∗.Recall that the quantum (functorial) dimension of V is defined as the compositionmapping C→V ⊗V ∗ →V ∗∗ ⊗V ∗ →C, or, as trV (θ−1u)= trV (τ 2ι(K )1−l). It is clearthat the quantum dimension of any generic representation is zero.328 NATHAN GEER AND NICOLAI RESHETIKHIN3. Invariants of Links3.1. AMBI- ELEMENTS IN A RIBBON CATEGORYHere we will recall some results and definitions from [2].Recall that in a ribbon category there is a natural notion of a trace of an endo-morphism of an object. If f : V → V , its trace istrV ( f )= eV ∗ ◦ (µV ⊗ idV ∗)◦ ( f ⊗ idV ∗)◦ iVwhere µV : V → V ∗∗ is the isomorphism between V and V ∗∗ determined by thebraiding and the ribbon structure. For the category of◦U ε (sl2), µV =πV (ι(K )1−l).We will use the following notations. If V is simple and f : V → V is a morphism,by definition f =c( f )idV for some c( f )∈κ where κ is the base field for our cate-gory. We will assume κ =C and will use the graphical notation for c( f ) shown onFigure 1.Define S′(U, V )= (idV ⊗ trU )(cU,V cV,U ) where cV,U : V ⊗U →U ⊗ V is the com-mutativity constraint in the category. This element can be written graphically as onFigure 2.For any f : V → V and simple V we have the identity shown on Figure 3.Recall that an object V is called ambidextrous if (idV ⊗ trV )( f )= (trV ⊗idV )( f )for any f : V ⊗2 → V ⊗2 (see also Figure 4). Here we will call such an object anambi-object.The following observation is one of the key tools from [2]. Let U be a simpleambi-object. Then for any two objects V1 and V2 and any morphism f : V1 ⊗ V2 →V1 ⊗ V2 the following identity holds:S′(U∗, V ∗1 )S′(V2,U )c((trV1 ⊗idV2)( f ))= S′(V1,U )S′(U∗, V ∗2 )c((idV1 ⊗ trV2)( f ))(3)Figure 1.Figure 2.ON INVARIANTS OF GRAPHS RELATED TO QUANTUM SL(2) 329Figure 3.Figure 4.Figure 5.Figure 6. T̂V is the closer of TV .It follows from the ambi-identity for the tangle from Figure 5. Here notice thatS′(V ∗, W ∗)= S′(V, W ).We have the following corollary of the identity (3).PROPOSITION 10. • If J is a simple ambi-object of the category C, then allobjects U ∈ A(J ) = {W |S′(W, J ) = 0, S′(J, W ) = 0, W is simple} are also ambi-objects.• Let TV be a (1,1)-tangle colored by elements of Cε , whose open component is col-ored by an element V ∈ A(J ). Then the expression in Figure 6 is an invariant ofthe link obtained by the closure of TV . In this expression U is any ambi-elementfrom A(J ).330 NATHAN GEER AND NICOLAI RESHETIKHIN3.2. AMBI- MODULES OVER◦U ε (sl2)3.2.1. As it was pointed out at the end of the previous section all modules V (a, k)have zero quantum dimension. Here we will prove that they are all ambi-objectsin the category◦U ε (sl2)−mod.LEMMA 11. The module V (a, k) is an ambi-object for all k ∈Z and a8l =1.Proof. It is easy to prove that for all a ∈C\ {a8l =1}, and k ∈ZV (a, k)⊗ V (a, k)∼= V (2a,2k)⊕ V (2a,2k −2)⊕· · ·⊕ V (2a,2k −2l +2).Since this decomposition is a direct sum of non-isomorphic simple modules thealgebra End(V (a, k) ⊗ V (a, k)) is commutative. Now let f ∈ End(V ⊗ V ), then(idV ⊗ trV )( f )= (trV ⊗idV )(c−1V,V ◦ f ◦ cV,V ). But cV,V commutes with End(V ⊗ V )and so c−1V,V ◦ f ◦ cV,V = f . Here we set V = V (a, k). This finishes the proof.The condition a8l =1 can be relaxed to a4l =1.LEMMA 12. Let V (a, k) and V (b,m) be two irreducible◦U ε (sl2)-modules describedin the previous section. ThenS′(V (a, k), V (b,m))= ε(k+1−l)(m+1−l)b2k−2l+2a2m−2l+2(−1)m+l−1 ××[b2l −b−2lεm+1−lb2 − (εm+1−lb2)−1]Proof. The proof is a straightforward computation. We want to compute theaction of (idV ⊗ trU )((πV ⊗πU )(σ (R)R)) on V . Let us apply to the highest weightvector. ThenS′((V (a, k), V (b,m))vm = (idV (b,m) ⊗ trV (a,k))((πV (b,m) ⊗πV (a,k))(σ (R0)R0))vm == εk(m+1−l)b2ka2m+2−2ll−1∑i=0ε−2i(m+1−l)b−4ivmThen one should sum up the geometric progression.Now, note that S′(V (a, k), V (b,m)) = 0 unless b4l = 1 when (εm+1−lb2)2 = 1.In the later case the representation V (b,m) is reducible. We assume that this is notthe case.Thus, we have proven the following theorem.THEOREM 13. All irreducible representations V (a, k) of◦U ε (sl2) with a4l = 1 areambi-modules.ON INVARIANTS OF GRAPHS RELATED TO QUANTUM SL(2) 331Open Access This article is distributed under the terms of the Creative Commons Attri-bution Noncommercial License which permits any noncommercial use, distribution, andreproduction in any medium, provided the original author(s) and source are credited.References1. Drinfeld, V.: Almost cocommutative Hopf algebras. Algebra i Analiz 1(2), 30–46 (1989)(Russian) [translation in Leningrad Math. J. 1(2), 321–342 (1990)]2. Geer, N., Patureau-Mirand, B., Turaev, V.: Modified quantum dimensions and re-nor-malized link invariants. Compos. Math. 145(01), 196–212 (2009). math.QA/0711.42293. Ohtsuki, T.: Quantum invariants. A study of knots and 3-manifolds. Series on Knotsand Everything, vol. 29. World Scientific Publishing Co., Inc., River Edge (2002)4. Reshetikhin, N., Turaev, V.: Ribbon graphs and their invariants derived from quantumgroups. Commun. Math. Phys. 127(1), 1–26 (1990)

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)On invariants of graphs related to quantum sl(2) at roots of unityGeer, N.; Reshetikhin, N.DOI10.1007/s11005-009-0320-9Publication date2009Document VersionFinal published versionPublished inLetters in  Mathematical PhysicsLink to publicationCitation for published version (APA):Geer, N., & Reshetikhin, N. (2009). On invariants of graphs related to quantum sl(2) at rootsof unity. Letters in Mathematical Physics, 88(1-3), 321-331. https://doi.org/10.1007/s11005-009-0320-9General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:25 Jul 2022DOI 10.1007/s11005-009-0320-9Lett Math Phys (2009) 88:321–331On Invariants of Graphs Related to Quantum sl(2)at Roots of UnityNATHAN GEER1,2 and NICOLAI RESHETIKHIN3,41Max-Planck Institute für Mathematik, Vivatgasse 7, 53111 Bonn, Germany.2School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA.e-mail: geer@math.gatech.edu3Department of Mathematics, University of California, Berkeley, 970 Evans Hall #3840,Berkeley, CA 94720-3840, USA.4KDV Institute for Mathematics, Universiteit van Amsterdam, Plantage Muidergracht 24,1018 TV, Amsterdam, The Netherlands. e-mail: reshetik@math.berkeley.eduReceived: 14 March 2009 / Accepted: 4 April 2009Published online: 13 May 2009 – © The Author(s) 2009. This article is published with openaccess at Springerlink.comAbstract. We show how to define invariants of graphs related to quantum sl2 when thegraph has more then one connected component and components are colored by blocks ofrepresentations with zero quantum dimensions.Mathematics Subject Classification (2000). 17B37, 57R56, 46A63.Keywords. invariants of knots, quantized universal enveloping algebras,topological quantum field theory.0. IntroductionThe invariants of oriented linked framed graphs colored by finite dimensional rep-resentations of quantized universal enveloping algebras were constructed in [4] fornon-zero quantum dimensions. In this construction the edges of the graph arecolored by finite dimensional representations. At each vertex a total order-ing of adjacent edges which agrees with counter-clockwise cyclic order definedby the framing is fixed. A vertex with adjacent edges colored by representa-tions V1, . . . , Vn (according to the total ordering) is colored by an element fromHom(C, V ε11 ⊗ . . .⊗V εnn ) of all Uq(sl2)-invariant vectors in the tensor product. HereV + =V, V − =V ∗ where V ∗ is the left dual to V , ε =+ if the edges is incoming andε =− if the edges is outgoing.When the quantum dimension is zero, it is easy to define such invariants forknots when the center of Uq(sl2) acts by scalars on the representation coloringthe knot. One should cut the knot (its diagram) at any place and then to computethe invariant of the corresponding (1,1) tangle. Because the center acts trivially at thecoloring representation, this invariant will be equal to a operator of multiplication by322 NATHAN GEER AND NICOLAI RESHETIKHINa scalar. It is easy to show that the scalar is the invariant of a knot (does not depend onwhere the knot was cut). But this technique does not work for links.Recently invariants of links colored by such representations were introduced in[2] under the assumption that these representations satisfy an additional property,i.e. they are ambidextrous (see [2] or Section 3.1 for the precise definition). We willabbreviate this name here to ambi-modules.The goal of this note is to present in details how the construction involvingambi-modules works for sl2. We prove that all generic irreducible representationsof◦U ε (sl2) are ambi-modules.We also conjecture but we do not prove it here that all generic irreducible rep-resentations of the◦U ε (g) are ambi-modules for any simple Lie algebra.We would like to thank V. Turaev and B. Patureau for discussions. This workwas supported by the Danish National Research Foundation through the NielsBohr initiative, the work of N.R. was also supported by the NSF grant DMS-0601912. The work of N.G. was supported by the NSF grant DMS-0706725. Bothauthors are grateful to the Aarhus University for the hospitality.1. The Split Form of Quantum sl2 at a Root of Unity1.1. QUANTUM sl2Let m, l be positive odd integers, t = exp( iπm2l ), and ε = t2.The algebra Uε(sl2) is the unital C-algebra generated by invertible K and byE, F with defining relations:K E = ε2 E K , K F = ε−2 F K , [E, F]= K − K−1ε − ε−1 .Recall that it is a Hopf algebra with the comultiplication acting on generatorsasK = K ⊗ K , E = E ⊗ K +1⊗ E, F = F ⊗1+ K −1 ⊗ FThe elements K l , Fl , El generate a central Hopf subalgebra, and El , Fl generatecentral Hopf ideal. Denote by Uε(sl2)′ its quotient by this central ideal.1.2. THE SPLIT FORM OF Uε(sl2)′DEFINITION 1. The split form of Uε(sl2)′ is the algebra◦U ε (sl2) generated byPi , i ∈Z and by E, F, x, x−1 such that x is central andPi Pj = δi j Pi , Pi E = E Pi−2, Pi F = F Pi+2,∑i∈ZPi =1,E F − F E =∑j∈Zx2ε j − x−2ε− jε − ε−1 Pj .ON INVARIANTS OF GRAPHS RELATED TO QUANTUM SL(2) 323The algebra◦U ε (sl2) is a Hopf algebra (in the category of its finite-dimensionalrepresentations) where the comultiplication , counit ε and antipode S morphismsare given by:(E)=∑j∈ZE ⊗ x2ε j Pj +1⊗ E, (F)= F ⊗1+∑j∈Zx−2ε− j Pj ⊗ F,(Pk)=∑i, j∈Z; i+ j=kPi ⊗ Pj , (x)= x ⊗ x,ε(E)= ε(F)=0, ε(Pi )= δi,0, ε(x)=1,S(E)=−E∑j∈Zx−2ε− j Pj , S(F)=−∑j∈Zx2ε j Pj F, S(Pi )=P−i , S(x)=x−1.It is clear that the map ι :Uε(sl2)→◦U ε (sl2) given by K →∑ j∈Z x2ε j Pj , E → Eand F → F is an injective morphism of Hopf algebras.Notice that K l →∑ j∈Z x2lε jl Pj and that K 2l → x4l .1.3. THE R- MATRIXFor n ∈N we set [n]= εn−ε−nε−ε−1 and [n]!= [n][n −1] . . . [1]. DefineR0 =∑i, j∈Zt i j x i Pj ⊗ x j Pi ,R1 =l−1∑n=0(ε − ε−1)n[n]! En ⊗ Fn,and R = R0 R1 ∈◦U ε (sl2)⊗2.Now we will show that R defines a quasitriangular structure on◦U ε (sl2).LEMMA 2. The element R0 satisfies the following identities.(1) (⊗ Id)(R0)= (R0)13(R0)23,(2) (Id ⊗)(R0)= (R0)13(R0)12,(3) R0(E ⊗1)= (E ⊗ ι(K ))R0, R0(1⊗ E)= (ι(K )⊗ E)R0,(4) R0(F ⊗1)= (F ⊗ ι(K )−1)R0, R0(1⊗ F)= (ι(K )−1 ⊗ F)R0.Proof. We will prove the first identity in Part (3), the other identities follow sim-ilarly. We haveR0(E ⊗1)=∑i, j∈Zt i j x i Pj E ⊗ x j Pi ==∑i, j∈Zt i j x i E Pj−2 ⊗ x j Pi =324 NATHAN GEER AND NICOLAI RESHETIKHIN=∑i, j∈Zt i j x i E Pj ⊗ x j x2εi Pi == (E ⊗ ι(K ))R0.THEOREM 3. The pair (◦U ε (sl2), R) is a quasitriangular Hopf algebra.Proof. Lemma 2 implies that R0 has the same commutator relation with E andF as the element exp( h4 (H ⊗ H)) in the C[[h]]-algebra Uh(sl2). Thus, we conclude(⊗ Id)(R)= R13 R23, (Id ⊗)(R)= R13 R12,andop(a)= R(a)R−1, for a ∈ ◦U ε (sl2)where op = τ ◦ and τ is the permutation a ⊗b →b ⊗a.Let R =∑ si ⊗ ti and define u =∑ S(ti )si . Then following [1] we haveε(u)=1, (u)= (R21 R)−1 (u ⊗u), S2(a)=uau−1 (1)for all a ∈ ◦U ε (sl2).A direct computation shows that for all a ∈ ◦U ε (sl2) we haveS2(a)= ι(K )aι(K )−1. (2)Let θ̃ = uι(K )−1 = ι(K )−1u. Comparing this formula with the formula for the S2involving u we see that θ̃ is central.LEMMA 4. The element θ̃ satisfies the relations:ε(θ̃)=1, (θ̃)= (R21 R)−1 (θ̃ ⊗ θ̃ ), S(θ̃)= ι(K )2l θ̃ .Proof. The first two relations follow from (1) and the definition of the counitand coproduct of ι(K ). The last relation can be proven by direct computation ofS(uι(K )−1) and of uι(K )−1. This computation was done essentially in [3]. One canalso prove the last identity by computing how S(θ̃) and θ̃ act on generic irreduc-ible modules. It is easy to see that on these modules the identity holds. On theother hand our algebra is finitely generated and is finite dimensional over the cen-ter. This implies that the identity S(θ̃ )= ι(K )2l θ̃ holds not only for generic pointsbut also for special points (where x acts as 4lth root of unity).ON INVARIANTS OF GRAPHS RELATED TO QUANTUM SL(2) 3251.4. THE AUTOMORPHISM φPROPOSITION 5. (1) The mapping φ : ◦U ε (sl2)→◦U ε (sl2) acting on generators asφ(Pi )= Pi+2, φ(x)= xε, φ(E)= E, φ(F)= Fextends uniquely to an algebra automorphism.(2) The set of fixed points form a Hopf subalgebra generated by x∑i∈Z t i Pi , x2l ,E, F .Proof. The first part is clear from defining relations, the second follows immedi-ately from the formula for φ.Note that despite the fact that fixed points of φ form a Hopf subalgebra, it isnot a Hopf algebra automorphism.The automorphism φ acts on the R-matrix as follows:(φ ⊗ id)(R)= (1⊗ x−2)R, (id ⊗φ)(R)= (x−2 ⊗1)(R)PROPOSITION 6. The automorphism φ acts on θ̃ as follows:φ(θ̃)= x4θ̃Proof. Let R = ∑i si ⊗ ti , then R−1 =∑i si ⊗ S(ti ). Combining the way φ actson the R-matrix with the fact that it is an algebra automorphism we obtain (φ ⊗φ)(R−1) = (x2 ⊗ x2)R−1. Therefore φ(∑i S(ti )si ) = x4∑i S(ti )si . Thus, φ(u) = x4u.Since φ(ι(K ))= ι(K ), this proves the proposition.Define ε =∑ j∈Z(−1) j Pj . It is central and unipotent. It is also easy to see thatφ acts trivially on it:φ(ε)= ε1.5. THE EXTENDED R- MATRIXConsider the algebra Aε =◦U ε (sl2) ⊗C C[τ, τ−1]. The element τ is central in thisalgebra.Remark 7. The motivation for this construction is the formal power series versionof the algebra◦U ε (sl2), when instead of having Laurent polynomials in x we haveformal powers series z:x = exp(imzπ2l), τ = exp(imz2π4l).326 NATHAN GEER AND NICOLAI RESHETIKHINAssume that the action of the comultiplication on τ is symmetric, i.e. op(τ )=(τ). Defineρ =(τ)τ−1 ⊗ τ−1.Extend the action of the automorphism φ on Aε as:φ(τ)= εx2τ, (φ ⊗ id)(ρ)= (1⊗ x2)ρ.Define the extended R-matrix asR=ρR.It is easy to see that R defines a quasitriangular structure on Aε . It is also easyto see that(id ⊗φ)(R)= (φ ⊗ id)(R)=R.Defineθ1 = τ−2ι(K )l θ̃ , θ2 = τ−2ι(K )lεθ̃ .Each of these elements satisfies identities:ε(θ)=1, (θ)= (R21R)−1 (θ ⊗ θ), S(θ)= θ, φ(θ)= θ.2. The Category Cε2.1. SIMPLE MODULESHere we will focus on simple modules over the◦U ε (sl2), Uε(sl2)′, and Aε . We willsay a ∈C∗ is generic if a4l =1.The following statement is a variation on a well known fact:PROPOSITION 8. For each generic a ∈C∗ and k ∈Z there exists unique simple ◦U ε(sl2)-module V (a, k) with highest weight vector vk such thatEvk =0, xvk =avk Pjvk = δ j,kvk, V (a, k)=l−1∑i=0CFivk .It is easy to compute the action of generators on the weight basis vk−2 j = F jvk :xvk−2i =avk−2iPjvk−2i = δ j,k−2ivk−2i , 0≥ i ≥ l −1,Fvk−2i =vk−2i−2, 1≤ i ≤ l −1Evk−2i =(a2εk+i+1 −a−2ε−k−1−iε − ε−1)(εi − ε−iε − ε−1)vk−2i+2, 0≤ i ≤ l −1.ON INVARIANTS OF GRAPHS RELATED TO QUANTUM SL(2) 327The homomorphism ι : Uε(sl2)′ →◦U ε (sl2) defines on V (a, k) the structure of aUε(sl2)′-module.PROPOSITION 9. The Uε(sl2)′-modules V (a, k) are irreducible, and in addition,V (a, k) and V (aε, k +1) are isomorphic as U ′ε-modules.In particular central element ε acts on V (a, k) as (−1)k .Note that the modules V (a, k) and V (aε, k +1) are not isomorphic as ◦U ε (sl2)-modules.For z ∈ C we can always define the Aε-module structure on the representationV (eimzπ2l , k) of◦U ε (sl2) by defining τv = exp( imz2π4l )v for any v ∈ V (eimzπ2l , k). It isclear that this defines an irreducible Aε-module.2.2. THE CATEGORY CεObjects of the category Cε are finite dimensional◦U ε (sl2)-modules (V, πV : Aε →End(V )) on which x acts as a multiplication by a scalar πV (x)v = exp( imzπ2l )v.Here v ∈ V , m, l are as above and z ∈ C. The central element τ on such moduleacts by multiplication on exp( imz2π4l ).Morphisms between two such modules are all◦U ε (sl2)-invariant linear maps.This category is monoidal because it is a category of finite dimensional modulesover a Hopf algebra. It is a rigid monoidal category with the left dual modulesdefined as usual (V ∗, πV ∗ =π∗V ◦ S) where π∗V (a) is the dual linear map to πV (a)andwith usual injection and evaluation morphisms:iV :C→ V ⊗ V ∗, given by 1 →∑ei ⊗ ei,eV : V ∗ ⊗ V →C, given by f ⊗w → f (w)It is easy to see that Cε is a braided category with the commutativity morphismc ={cV,W } where cV,W : V ⊗ W → W ⊗ V given by v ⊗w → τ(R(v ⊗w)). It is also aribbon category with the ribbon morphisms (twists): θV : V → V, v → θ−1v.In this category the braiding and the ribbon structure agree with isomorphismsof modules induced by φ.The objects of this category are semisimple for generic z, i.e. when mz it is notan integer.The linear mapping v→θ−1uv is a an isomorphism of representations V → V ∗∗.Recall that the quantum (functorial) dimension of V is defined as the compositionmapping C→V ⊗V ∗ →V ∗∗ ⊗V ∗ →C, or, as trV (θ−1u)= trV (τ 2ι(K )1−l). It is clearthat the quantum dimension of any generic representation is zero.328 NATHAN GEER AND NICOLAI RESHETIKHIN3. Invariants of Links3.1. AMBI- ELEMENTS IN A RIBBON CATEGORYHere we will recall some results and definitions from [2].Recall that in a ribbon category there is a natural notion of a trace of an endo-morphism of an object. If f : V → V , its trace istrV ( f )= eV ∗ ◦ (µV ⊗ idV ∗)◦ ( f ⊗ idV ∗)◦ iVwhere µV : V → V ∗∗ is the isomorphism between V and V ∗∗ determined by thebraiding and the ribbon structure. For the category of◦U ε (sl2), µV =πV (ι(K )1−l).We will use the following notations. If V is simple and f : V → V is a morphism,by definition f =c( f )idV for some c( f )∈κ where κ is the base field for our cate-gory. We will assume κ =C and will use the graphical notation for c( f ) shown onFigure 1.Define S′(U, V )= (idV ⊗ trU )(cU,V cV,U ) where cV,U : V ⊗U →U ⊗ V is the com-mutativity constraint in the category. This element can be written graphically as onFigure 2.For any f : V → V and simple V we have the identity shown on Figure 3.Recall that an object V is called ambidextrous if (idV ⊗ trV )( f )= (trV ⊗idV )( f )for any f : V ⊗2 → V ⊗2 (see also Figure 4). Here we will call such an object anambi-object.The following observation is one of the key tools from [2]. Let U be a simpleambi-object. Then for any two objects V1 and V2 and any morphism f : V1 ⊗ V2 →V1 ⊗ V2 the following identity holds:S′(U∗, V ∗1 )S′(V2,U )c((trV1 ⊗idV2)( f ))= S′(V1,U )S′(U∗, V ∗2 )c((idV1 ⊗ trV2)( f ))(3)Figure 1.Figure 2.ON INVARIANTS OF GRAPHS RELATED TO QUANTUM SL(2) 329Figure 3.Figure 4.Figure 5.Figure 6. T̂V is the closer of TV .It follows from the ambi-identity for the tangle from Figure 5. Here notice thatS′(V ∗, W ∗)= S′(V, W ).We have the following corollary of the identity (3).PROPOSITION 10. • If J is a simple ambi-object of the category C, then allobjects U ∈ A(J ) = {W |S′(W, J ) = 0, S′(J, W ) = 0, W is simple} are also ambi-objects.• Let TV be a (1,1)-tangle colored by elements of Cε , whose open component is col-ored by an element V ∈ A(J ). Then the expression in Figure 6 is an invariant ofthe link obtained by the closure of TV . In this expression U is any ambi-elementfrom A(J ).330 NATHAN GEER AND NICOLAI RESHETIKHIN3.2. AMBI- MODULES OVER◦U ε (sl2)3.2.1. As it was pointed out at the end of the previous section all modules V (a, k)have zero quantum dimension. Here we will prove that they are all ambi-objectsin the category◦U ε (sl2)−mod.LEMMA 11. The module V (a, k) is an ambi-object for all k ∈Z and a8l =1.Proof. It is easy to prove that for all a ∈C\ {a8l =1}, and k ∈ZV (a, k)⊗ V (a, k)∼= V (2a,2k)⊕ V (2a,2k −2)⊕· · ·⊕ V (2a,2k −2l +2).Since this decomposition is a direct sum of non-isomorphic simple modules thealgebra End(V (a, k) ⊗ V (a, k)) is commutative. Now let f ∈ End(V ⊗ V ), then(idV ⊗ trV )( f )= (trV ⊗idV )(c−1V,V ◦ f ◦ cV,V ). But cV,V commutes with End(V ⊗ V )and so c−1V,V ◦ f ◦ cV,V = f . Here we set V = V (a, k). This finishes the proof.The condition a8l =1 can be relaxed to a4l =1.LEMMA 12. Let V (a, k) and V (b,m) be two irreducible◦U ε (sl2)-modules describedin the previous section. ThenS′(V (a, k), V (b,m))= ε(k+1−l)(m+1−l)b2k−2l+2a2m−2l+2(−1)m+l−1 ××[b2l −b−2lεm+1−lb2 − (εm+1−lb2)−1]Proof. The proof is a straightforward computation. We want to compute theaction of (idV ⊗ trU )((πV ⊗πU )(σ (R)R)) on V . Let us apply to the highest weightvector. ThenS′((V (a, k), V (b,m))vm = (idV (b,m) ⊗ trV (a,k))((πV (b,m) ⊗πV (a,k))(σ (R0)R0))vm == εk(m+1−l)b2ka2m+2−2ll−1∑i=0ε−2i(m+1−l)b−4ivmThen one should sum up the geometric progression.Now, note that S′(V (a, k), V (b,m)) = 0 unless b4l = 1 when (εm+1−lb2)2 = 1.In the later case the representation V (b,m) is reducible. We assume that this is notthe case.Thus, we have proven the following theorem.THEOREM 13. All irreducible representations V (a, k) of◦U ε (sl2) with a4l = 1 areambi-modules.ON INVARIANTS OF GRAPHS RELATED TO QUANTUM SL(2) 331Open Access This article is distributed under the terms of the Creative Commons Attri-bution Noncommercial License which permits any noncommercial use, distribution, andreproduction in any medium, provided the original author(s) and source are credited.References1. Drinfeld, V.: Almost cocommutative Hopf algebras. Algebra i Analiz 1(2), 30–46 (1989)(Russian) [translation in Leningrad Math. J. 1(2), 321–342 (1990)]2. Geer, N., Patureau-Mirand, B., Turaev, V.: Modified quantum dimensions and re-nor-malized link invariants. Compos. Math. 145(01), 196–212 (2009). math.QA/0711.42293. Ohtsuki, T.: Quantum invariants. A study of knots and 3-manifolds. Series on Knotsand Everything, vol. 29. World Scientific Publishing Co., Inc., River Edge (2002)4. Reshetikhin, N., Turaev, V.: Ribbon graphs and their invariants derived from quantumgroups. Commun. Math. Phys. 127(1), 1–26 (1990)

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On invariants of graphs related to quantum $\mathfrak{sl}(2)$ at roots of unity

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On invariants of graphs related to quantum sl(2)\mathfrak{sl}(2)sl(2) at roots of unity

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On invariants of graphs related to quantum $\mathfrak{sl}(2)$ at roots of unity