Quantum invariants of framed links from ternary self-distributive cohomology
aa r X i v : . [ m a t h . G T ] F e b QUANTUM INVARIANTS OF FRAMED LINKS FROMTERNARY SELF-DISTRIBUTIVE COHOMOLOGY
EMANUELE ZAPPALA
Abstract.
The ribbon cocycle invariant is defined by means of a par-tition function using ternary cohomology of self-distributive structures(TSD) and colorings of ribbon diagrams of a framed link, followingthe same paradigm introduced by Carter, Jelsovsky, Kamada, Lang-for and Saito in Transactions of the American Mathematical Society2003;355(10):3947-89, for the quandle cocycle invariant. In this articlewe show that the ribbon cocycle invariant is a quantum invariant. Wedo so by constructing a ribbon category from a TSD set whose twistingand braiding morphisms entail a given TSD 2-cocycle. Then we showthat the quantum invariant naturally associated to this braided categorycoincides with the cocycle invariant. We generalize this construction tosymmetric monoidal categories and provide classes of examples obtainedfrom Hopf monoids and Lie algebras. We further introduce examplesfrom Hopf-Frobenius algebras, objects studied in quantum computing.
Contents
1. Introduction 21.1. Main results 51.2. Organization of the article 52. Preliminaries 62.1. Racks, quandles and cocycle invariants 62.2. Framed links and their diagrams 82.3. Ternary racks and their self-distributive cohomology 92.4. Braided monoidal categories and ribbon categories 113. Ribbon cocycle invariants 124. Ribbon categories from self-distributive ternary operations 155. The Ribbon cocycle invariant is a quantum invariant 226. Examples and Computations 246.1. Examples from compositions of binary quandles 246.2. Examples from heap structures 266.3. Computations of quantum invariants 287. Generalized construction 297.1. TSD objects in symmetric monoidal cateogries 307.2. Examples of TSD objects 327.3. Categorical 2-cocycle condition 337.4. Examples of categorical 2-cocycles 34
Introduction
Self-distributivity of binary operations is well known to be an algebraicformulation of the Reidemeister move III in knot theory. Sets with self-distributive operations (i.e. shelves) satisfying extra conditions encodingReidemeister moves I and II have been used, starting in the 1980’s, to con-struct invariants of knots and links. For instance, Joyce and Matveev inde-pendently defined what is now known as the fundamental quandle of a knot[15, 20], whose construction is given as a presentation where the generatorscorrespond to the arcs of a knot diagram, while the relations formally corre-spond to the conjugation operation in a group. Shelves satisfying the alge-braic Reidemeister move II condition are called racks, while those satisfyingalso the algebraic counterpart of Reidemeister move I are called quandles.More recently, the notion of (co)homology of quandle has been introduced,and a state-sum invariant of links that utilizes quandle cohomology has beenconstructed in [4]. The resulting “cocycle invariant” is obtined as a sumover all the colorings of a knot diagram, the states, of all the products ofBoltzmann weights, determined by quandle 2-cocycles. Although computingcocycle invariants introduces a new problem, that of obtaining nontrivialquandle second cohomology classes, it is in general easier to compare twococycle invariants rather than comparing the fundamental quandle of twoknots.Moreover, it is known that quandles induce solutions to the set-theoreticYang-Baxter equation and therefore, upon linearizing the corresponding set-theoretic map, they produce Yang-Baxter operators [7]. In fact, given aquandle and a 2-cocycle α , one can construct a Yetter-Drinfel’d module (i.e.a particular instance of a ribbon category) [13] and, consequently, one canobtain quantum link invariants associated to the ribbon category followinga standard procedure as in [26]. It naturally arises the question of whetherthe two types of invariant are somehow related. A positive answer has beengiven in [13], where it is shown that the invariants coincide in a suitablesense.Ternary self-distributive structures are generalizations of binary shelves tothe setting of ternary operations. A suitable diagrammatic interpretation ofcrossing of ribbons in terms of ternary operations translates the fundamentalmoves for the isotopy equivalence of framed links into a ternary analogue ofrack. A corresponding state-sum invariant that uses cohomology of ternaryracks and colorings of ribbon diagrams associated to framed links is thenconstructed [27] following the same reasoning as in the binary case. This UANTUM INVARIANTS FROM TSD COHOMOLOGY 3 invariant, called ribbon cocycle invarinat , has been studied for a fundamentalclass of ternary racks, called group heaps, and it has been seen to detectnontrivial framing of links [24].On the other hand, group heaps can be generalized to certain structures,named quantum heaps , that naturally arise from involutory Hopf algebras,i.e. having antipode that squares to the identity map. A correspondingconstruction for Hopf monoids in symmetric monoidal categories exists [9],providing a large class of examples for ternary self-distributive objects insymmetric monoidal categories, in the sense of [8] Section 8. It is there-fore possible to develop an analogue of the ternary set-theoretic theory insymmetric monoidal categories.The scope of this article is that of using ternary self-distributive (TSD)structures and their ternary cocycles to construct ribbon categories whosetwisting morphisms are nontrivial, and study the corresponding link invari-ants. The starting point of this study follows the paradigm that has beenused in [13] to prove that the cocycle invariants are indeed quantum invari-ants. We prove, in fact, that set-theoretic TSD structures and a choice ofa ternary 2-cocycle are linearized to obtain a braiding in a suitably con-structed symmetric monoidal category. The construction is similar to thatof the braid category [12, 16], but braiding and twisting are induced by theTSD structure following the doubling functorial procedure in [8], and usingTSD cocycles to twist the morphisms obtained. Analogously to the fact thatthe ribbon cocycle invariant detects nontrivial framings [24], we obtain thatthe twisting defined in this category is nontrivial, as opposed to the case ofYetter-Drinfel’d modules associated to (binary) quandle operations.On the one hand, there is no strict need of defining a ribbon categoryout of the data of a TSD and a ternary 2-cocycle, in the sense that we canobtain a representation of the framed braid group in a similar fashion as in[26], from which a corresponding quantum invariant would naturally arise.On the other hand, though, this construction easily generalises to multipleobjects where “self-distributive” ternary actions are defined. These producea more general family of ribbon categories where the twists are obtained byTSD operations as in the previous case, while the braidings are obtainedfrom ternary actions. Moreover, the braiding and twisting morphisms canbe deformed by cohomological classes that twist the weights and entail theoperations and mutual actions of the underlying structure. Among the ex-amples that we present in this paper, we find mutually distributive structuresand their labeled cohomology, whose algebraic properties were studied in [8],and G -families of quandles and their cohomology theory, extensively studiedin connection with knotted handlebody invariants [11, 21].The approach mentioned above, in addition, is particularly suitable to begeneralized to the case of TSD objects in symmetric monoidal categories.As observed above, in fact, the notion of heap has a counterpart obtainedfrom involutory Hopf monoids in symmetric monoidal categories, therefore EMANUELE ZAPPALA providing a fertile ground for a general theory that associates a ribbon cat-egory to a symmetric monoidal category along with a TSD object in it.Using the TSD morphism we obtain, in fact, a Yang-Baxter operator in thetensor product of the TSD object we start with, and use this to define thebraiding of the ribbon category. The twist is obtained via the same procue-dure by interpreting twists as self-intersections of a ribbon. In other words,Reidemeister move I does not hold when we consider framed links, but it isreplaced by a twisting which can be defined using a variation of the braiding.We have mentioned that we utilize TSD cohomology classes to deformthe braiding and twisting in the case of linearized TSD operations. Whenworking in a symmetric monoidal category, we can introduce a categoricalversion of the 2-cocycle condition. The setting, here, generalizes the set-theoretic one in two fundamental ways. Recall the set-theoretic 2-cocyclecondition, which reads ψ ( x, y, z ) − ψ ([ x, u, v ] , [ y, u, v ] , [ z, u, v ]) − ψ ( x, u, v ) + ψ ([ x, y, z ] , u, v ) = 0for all x, y, z, u, v ∈ X , where ( X, [ − , − , − ]) is a TSD set. Firstly, observethat certain elements appear in more than one term, and therefore are re-peated. This is no particular concern when dealing with set-theoretic struc-tures, but in a general symmetric monoidal category, it is required thateach instance of a repreated element is replaced by an instance of a comul-tiplication morphism. In fact, the definition of TSD object, see for instaceSection 8 in [8] for n -ary case, implements this perspective already, and itis somehow natural to expect that it carries on to the 2-cocycle condition.Secondly, in the set-theoretic case coefficients of cohomology are taken in agroup, and linearization naturally requires the coefficients to be representedin the ground field. In an arbitrary symmetric monoidal category, we inter-pret this situation as an equality holding in the unit object of the category.The object of coefficients naturally acts on the TSD object allowing the“cocycle” to perturb the Yang-Baxter operator associated to the TSD mor-phism. If one thinks of the group algebra associated to a group as being aHopf algebra where the comultiplication is simply the splitting of an elementin two identical copies, then the categorical interpretation of the 2-cocyclecondition seems to be on the same footing as the 2-cocycle condition in theground field of the linearization of a set-theoretic operation.In the general situation, one further assumption is necessary, in order toapply the same construction as in the category of vector spaces. Namely,one needs to assume that the category is I -linear, where I is the unit object.Then the 2-cocycles are assumed to take values in the ground object I and,moreover, they are supposed to satisfy a convolution inversion formula, inorder to allow the definition of inverses. This is naturally satisfied in thelinearized case, since comultiplication is simply diagonal, and coefficients ina group are automatically invertible.Naturally, as in the set-theoretic case one can obtain ribbon categoriesfrom multiple TSD sets having suitable ternary actions and families of UANTUM INVARIANTS FROM TSD COHOMOLOGY 5 ternary 2-cocycles, we can generalize the previous construction in a sym-metric monoidal category where multiple TSD objects along with certainternary morphisms are defined. An interesting class of examples arises fromternary augmented racks, where the axioms of augmentation can be easilytranslated from the case of vector spaces and Hopf algebras to that of Hopfmonoids in a symmetric monoidal category.1.1.
Main results.
We proceed now to concisely summarize the main re-sults of the present article.The first result (Theorem 4.3) is that starting from a TSD set (
X, T ) anda given ternary 2-cocycle α ∈ Z ( X, A ) with coefficients in an abelian group A , we construct a ribbon category R ∗ α ( X ) whose braidings are constructedout of a Yang-Baxter operator arising from ( X, T ) and deformed by the co-cycle α . Moreover, it is shown that the ribbon category is well-defined, up toequivalence of braided categories, with respect to the cohomology class of α ,in the sense that if β represents the same cohomology class, then there ex-ists a braided functor that gives an equivalence of categories between R ∗ α ( X )and R ∗ β ( X ). Moreover, a similar construction is shown to hold when startingwith a family of TSD sets { X i } i ∈ I along with maps T ij : X i × X j × X j −→ X i satisfying a generalized version of TSD condition (Theorem 4.12). In this sit-uation, we define the notion of ternary 2-cocycles for the family { X i } i ∈ I anduse them to deform the Yang-Baxter operator associated to it. We thereforeconstruct a braiding and a twisting in order to obtain a ribbon categorywhose families of objects and morphisms are larger than those of R ∗ α ( X ). InTheorem 5.1 we show that the quantum invariants associated to the ribboncategory R ∗ α ( X ) coincide with the (state-sum) ribbon cocycle invariant in asuitable sense, i.e. when we take a representation of the coefficient abeliangroup of cohomology in the ground field. This construction is generalizedto the setting of TSD objects in symmetric monoidal categories, of whichTSD sets in the category of sets are a particular instance. It is shown thatin this setup, from a TSD object we obtain a Yang-Baxter operator whichis deformed by means of what is hereby called a categorical 2-cocycle. Thenew Yang-Baxter operator is used then to construct the braiding in what isshown to be a ribbon category (Theorem 7.16).1.2. Organization of the article.
This article is structured as follows.We review some preliminary material in Section 2, where we recall binaryand ternary self-distributive structures, the cocycle invariant and some ba-sic notions regarding symmetric monoidal categories. In Section 3 we give adetailed account of the construction of the ribbon invariant of framed linksas well as a proof of its being well-posed. In Section 4 we show that startingfrom the data of a TSD set and a ternary 2-cocycle, there exists a ribboncategory determined up to equivalence of categories with respect to coho-mology class of the 2-cocycle. Moreover, it is shown that a similar constructexists starting from a family of TSD structures with some extra compati-bility conditions and an analogue of the notion of ternary 2-cocycle. The
EMANUELE ZAPPALA corresponding ribbon cateogry has a wider class of objects and morpshimswith respect to the previous one. We then proceed to show, in Section 5,that the (state-sum) ribbon cocycle invariant coincides with the quantuminvariant associated to the ribbon category arising in Section 4. Section 6presents various examples to elucidate the construction in practice. Section 7is devoted to generalizing the theory developed in the previous sections inthe context of symmetric monoidal cateories and TSD objects. The notionof categorical 2-cocycle condition is introduced in order to deform the braid-ings obtained from TSD objects, in a fashion that follows the paradigm ofSection 4. Quantum invariants associated to this class of ribbon categoriesare discussed in Section 8. Finally, further examples arising from ternaryracks are given in the Appendix.
Acknowledgements.
This research has been funded by the Estonian Re-search Council under the grant: MOBJD679. The author is grateful to M.Elhamdadi and M. Saito for useful conversations.2.
Preliminaries
In this section we provide preliminary material that is used throughoutthe article.2.1.
Racks, quandles and cocycle invariants.
Racks are (non-associative) magmas satisfying the self-distributive property given by ( x ∗ y ) ∗ z = ( x ∗ z ) ∗ ( y ∗ z ) for all x, y, z , such that the right multiplication maps arebijections. Self-distributivity is an “algebraization” of the topological notionof Reidemeister move III, while the requirement that right multiplications bebujective corresponds to imposing Reidemeister move II. Idempotent racksare called quandles, where idempotence corresponds to the remaining Rei-demeister move I. It is well known that knot and link isotopy classes in R ,or S , can be characterized combinatorially via their diagrams, i.e. projec-tions on the plane satisfying certain regularity properties, and Reidemeistermoves I, II and III. Consequently, quandles have been used in [4] to con-struct state-sum invariants of links, named cocycle invariants. Fundamentalroles in the definition and validity of the cocycle invariant are played by thenotion of quandle coloring of a knot/link diagram, and a cohomology theoryassociated to racks and quandles. In fact, loosely speaking, the invariant isdefined by considering all possible colorings of a fixed given diagram of alink L , and multiplying the weights of each crossing of the diagram, each ofwhich defined by applying a pre-determined 2-cocycle to the colors meetingat the crossing. When applying any of the Reidemeister moves to pass fromone diagram of L to the other, i.e. when performing an isotopy on L , thecolors of the diagrams correspond bijectively by virtue of the axioms defin-ing a quandle, and the weights remain unchanged because of the definitionof quandle cohomology. UANTUM INVARIANTS FROM TSD COHOMOLOGY 7 x x ∗ yy xy x ∗ y Figure 1.
Coloring condition for positive crossings (left)and negative crossings (right).We proceed to briefly review the notion of quandle coloring of a linkdiagram, and the definition of cohomology associated to a quandle Q . Areference for both definitions is the article [4], where the cohomology utilizesabelian coefficients, while the case with non-abelian coefficients is treatedin [5]. Let L be an oriented link, let D indicate an oriented diagram of L , and let Q be a quandle, with operation denoted by the symbol ∗ . Acoloring of D by Q is a map C : R −→ Q , where R denotes the set of arcs ofthe diagram D , satisfying the conditions given in Figure 1, for positive andnegative crossings.Let Q be a quandle and define chain groups C n ( Q ) to be the free abeliangroup generated by the elements of Q n for each n . Then, we define the n th -differential ∂ n on generators according to the assignment ∂ n ( x , . . . , x n )= n X i =2 ( − n [( x , . . . , x i − , ˆ x i , x i +1 , . . . , x n ) − ( x ∗ x i , . . . , x i − ∗ x i , ˆ x i , x i +1 , . . . , x n )]where we have used ˆ to indicate omission of an element. Observe thatthe first term in the sum is the “usual” simplicial term, while the secondterm contains the information associated to the operation ∗ , determiningthe quandle structure. One proves directly that the maps ∂ n satisfy the pre-simplicial conditions and it follows automatically that ∂ n − ◦ ∂ n = 0, fromwhich we have a well defined chain complex and an associated homologytheory called rack homology . Quandle homology is obtained by quotientingout the sub-complex C q n ( Q ) generated as a free abelian group by n -tuplesof Q n where x i = x i +1 for some i . In fact, it is the indempotency conditionthat induces well defined maps ∂ n when restricting on the subgroups C q n ( Q ).Taking A to be an abelian group and dualizing the rack and quandle chaincomplexes, one obtains associated cohomology theories which we denoteby H n ( Q ; A ) and H n q ( Q ; A ), respectively. A representative φ of a secondcohomology class [ φ ] ∈ H ( Q ; A ) satisfies the 2-cocycle condition, whichtakes the form φ ( x, y ) − φ ( x ∗ z, y ∗ z ) − φ ( x, z ) + φ ( x ∗ y, z ) = 0 , EMANUELE ZAPPALA
Figure 2.
Crossing of a blackboard framing of a framed link.for all x, y, z ∈ Q . The 2-cocycle condition is related via a diagrammaticinterpretation to Reidemeister move III, as depicted in Figure 1 in [4], whilethe 3-cocycle condition, which we do not explicitly consider herein, is relatedto the tetrahedron move and shadow colorings.Fix now a coloring C by a quandle Q , defining at each crossing τ i ofFigure 1 a Boltzmann sum, B ( τ i , C ), as ψ ( x, y ), for positive crossing (leftpanel), and ψ ( x, y ) − for negative crossing (right panel), where ψ ∈ Z ( Q, A )is a quandle 2-cocycle, one defines the state-sum (or partition function) X C Y i B ( τ i , C )for any given diagram of a knot, and where the sum runs over all the colorings C of the fixed diagram, and the product runs over all the crossings. Thisstate-sum, called cocycle invariant , is shown to be an invariant of knots in[4], where it has been firstly introduced. When dealing with a link, oneproceeds analogously for each component and defines an invariant that is avector with as many entries as the components of the given link.2.2. Framed links and their diagrams.
Framed links are embeddingsof finitely many copies of S × D , i.e. solid tori, in the three dimensionalspace R , or its compactification S . Alternatively, framed links can bedefined as links along with a choice of a section of their normal bundle.Diagrammatically, a frmed link L is represented by a link diagram whosearcs are thickened to be ribbons. This thickening is obtained in a standardway by doubling each arc so to obtain a second copy of the link diagram,parallel to the first one. The one lets the second copy mimick the over/underpassing information of the first diagram. Such a thickened diagram is called blackboard framing . A crossing of a diagram whose arcs have been thickenedinto a ribbon is represented in Figure 2. From the figure is clear that thecoloring paradigm corresponding to that of quandles changes. We can thinkof each crossing as two arcs, each of them underpassing two arcs. Thecoloring rule is suitably defined by means of ternary racks. This concept,introduced to the author by M. Saito, is formalized in Section 3, where it isalso given a construction of the ribbon cocycle invariant. UANTUM INVARIANTS FROM TSD COHOMOLOGY 9 ∼ = +1 Figure 3.
Self-crossing of a ribbon introduces twists. − ∼ = Figure 4.
Twists with opposite signs annihilate each other.Reidemeister moves (R moves for short) of type II and III translate di-rectly into analogously defined moves where each arc is thickened to a rib-bon, while R move I does not hold in the context of framed links, since itintroduces a twist, i.e. a change in the framing. This is depicted in Fig-ure 3. Throughout this article we will depict positive, resp. negative, twistsby a rectangle inserted in a ribbon with a positive, resp. negative, integerindicating the number of twists and their orientations. Isotopy equivalenceof framed links is characterized by moves RII, RIII and the cancellation oftwists depicted in Figure 4, where each twist is thought of as a self-crossingas in Figure 3 (with negative twists obtained by kinks in the opposite direc-tion).2.3.
Ternary racks and their self-distributive cohomology.
Ternaryracks are generalizations of racks to sets with ternary operations. Specif-ically, a set X endowed with a ternary operation T : X × X × X −→ X satisfying the conditions T ( T ( x, y, z ) , u, v ) = T ( T ( x, u, v ) , T ( y, u, v ) , T ( z, u, v )) , is said to be a ternary self-distributive (TSD) set. A TSD set such that themap X −→ X defined by T ( • , x, y ) is a bijection for all x, y ∈ X is saidto be a ternary rack. A notable example of TSD structure is the heap of agroup, defined as ( x, y, z ) xy − z . Heap operations have been consideredand studied in [24], in relation with their ribbon cocycle invariants (seeSection 3 below). TSD operations naturally arise also by composing binaryself-distributive operations. For instance, if ( Q, ∗ ) is a rack, or quandle, thenthe operation T ( x, y, z ) := ( x ∗ y ) ∗ z can be seen to endow Q with a TSDstructure [8].We recall the notion of TSD (co)homology of ternary racks [14] and, morespecifically, the TSD 2-cocycle condition, since the ribbon cocycle invariantutilizes 2-cocycles as weights, in a fashion similar to the original cocycleinvariant introduced in [4]. Let ( X, T ) be a TSD set, and define C n ( X ) tobe the free abelian group generated by (2 n + 1)-tuples of elements of X .Define maps ∂ n : C n ( X ) −→ C n − ( X ) by ∂ n ( x , . . . , x n +1 ) n X i =1 ( − i [( x , . . . , x i − , \ x i , x i +1 , x i +2 , . . . , x n +1 ) − ( T ( x , x i , x i +1 ) , . . . , T ( x i − , x i , x i +1 ) , \ x i x i +1 , x i +2 , . . . , x n +1 )] , and extended by Z -linearity. A (long) direct computation shows that themaps ∂ n are obtained as the alternating sum ∂ n = P ni =1 ( − i ∂ in , wherethe maps ∂ in satisfy the usual pre-simplicial module axioms and, conse-quently, ( C n ( X ) , ∂ n ) defines a chain complex whose associated homology,written H n ( X ), is called TSD homology. Given an abelian group A weobtain, upon dualizing the previous chain complex, TSD cochain groupsand associated cohomology. We indicate these groups with the symbols C n ( X ; A ) and H n ( X ; A ), respectively. The 2-cocycle condition, for a 2-cochain ψ : X −→ A , takes the form ψ ( x, y, z ) − ψ ( T ( x, u, v ) , T ( y, u, v ) , T ( z, u, v ))= ψ ( x, u, v ) − ψ ( T ( x, y, z ) , u, v )for all x, y, z, u, v ∈ X . As it will be seen in Section 3, the ternary 2-cocycle condition, along with an appropriate interpretation of colorings ofblackboard framings by ternary racks, is invariant under moves RII, RIII andcancellation move. It is therefore possible to define Boltzmann weights bymeans of ternary 2-cocycles and introduce a state-sum invariant of framedlinks. Given a 1-cochain f : X −→ A , the first cohomology differential δ maps it to the function( x, y, z ) δ f ( x, y, z ) := f ( x ) − f ( T ( x, y, z )) . UANTUM INVARIANTS FROM TSD COHOMOLOGY 11
Therefore two 2-cocycles ψ and φ are in the same second cohomology classif they differ by a term δ f as above, for some 1-cochain f . As it willbe proved in Section 3, changing the representative of a second cohomologyclass changes the ribbon cocycle invariant by a well understood term, so thatthe invariant is a well defined function, up to a known equivalence relation,of the cohomology group H ( X ; A ). This observation did not appear inthe original construction in [27], and has been proven when T is the heapoperation in [24].2.4. Braided monoidal categories and ribbon categories.
Recall thatgiven a monoidal category ( C , ⊗ ), a braiding in C is a natural family ofisomorphisms c X,Y : X ⊗ Y −→ Y ⊗ X such that the Hexagon Axiom issatisfied [16], Chapter XIII. Specifically, it is required that the diagram X ⊗ ( Y ⊗ Z ) ( Y ⊗ Z ) ⊗ X ( X ⊗ Y ) ⊗ Z Y ⊗ ( Z ⊗ X )( Y ⊗ X ) ⊗ Z Y ⊗ ( X ⊗ Z ) c X,Y ⊗ Z α Y,Z,X α X,Y,Z c X,Y ⊗ α Y,X,Z ⊗ c X,Z where α indicates the associativity constraint of the category C , and we haveomitted the subscript of the identity morphism, as no confusion arises. Asimilar diagram for the inverse of c X,Y is required to commute, but thiscan be obtained from the commutativity of the previous one. Therefore,it is not an independent axiom, see comment in [16] right above Defini-tion XIII.1.1. A monoidal category endowed with a braiding is said to be a braided monoidal category . Observe that for any object X ∈ C , the braiding c X,X is a solution to the braid (Yang-Baxter) equation. In fact, the diagram-matic interpretation coincides with the RIII move for knot/link diagrams.In this article we will consider our monoidal categories to be strict, so thatthe associativity constraints will not be written now on. This assumptionis not particularly restrictive, as any monoidal category can be seen to beequivalent to a struct monoidal category.Typical examples of braided monoidal categories arise from braided bial-gebras (see Chapter VIII in [16]), where the category of H -modules, of abraided bialgebra H , has a braided structure associated to the universal R -matrix of H . Another important class of examples arises from crossed G -sets, where the braiding is obtained using the crossed action. Linearizingthese structures produces bradings in some subcategory of vector spaces.More generally, one can use a quandle operation, which generalizes the ax-ioms of crossed G -set.A left dual of X in a braided monoidal category is an object X ∗ such thatthere exist morphisms coev : I −→ X ⊗ X ∗ and ev : X ∗ ⊗ X −→ I such thatthe equalities ( ⊗ ev ) ◦ ( coev ⊗ ) = and ( coev ⊗ ) ◦ ( ⊗ ev ) = hold. A similar definition for right duals can be made. A category such that left andright duals exist for all objects is said to be autonomous , and in this caseleft and right duals coincide. In what follows, we will refer to left dualitysimply as “duality”, if not otherwise specified.In a braided monoidal category, the notion of dual introduces a diagram-matic interpretation with different types of crossing orientations. The cor-responding RIII moves with new orientations are seen to be induced by theoriginal diagrammatic RIII, as in Figure 10 in [26]. See [16], Chapter XIV,for the diagrammatic interpretation of duality.Given a braided monoidal category with duals, a twist is a natural familyof isomorphisms θ X such that θ X ⊗ Y = ( θ X ⊗ θ Y ) ◦ c Y,X ◦ c X,Y . Moreover, θ is required to behave well with respect to dual objects, in the sense that θ X ∗ = ( θ X ) ∗ .In the previous example of braided monoidal categories arising from lin-earization of crossed G -sets, linear duals, and evaluation and coevaluationmaps in vector spaces give a ribbon cateogry structure along with trivialtwists. In fact, triviality of twists can be interpreted as a consequence of thefact that quandles are idempotent following Figure 3. A twist is introducedby means of a self-crossing, and the corresponding effect of applying a quan-dle operation is trivial, due to idempotence. As it will be seen below, usingternary operations and their diagrammatic interpretation gives rise to rib-bon categories, following a simialar paradigm, whose twisting morphisms arenontrivial. Consequently the corresponding invariants detect the framing ofknot/link. 3. Ribbon cocycle invariants
Following [4, 5], it is introduced in this section an invariant of framedlinks, using colorings of ribbon diagrams by ternary quandles, and ternaryquandle 2-cocycles. This invariant was originally introduced in [27], andstudied in [24] in the case of heap invariants. We give the details of theconstruction, as they are relevant for the rest of the article.Framed links are represented in the rest of the paper by their blackboardframing . Therefore the arcs of a projection on the plane are representedby ribbons bounded by two parallel arcs. Orientetions of the ribbons arespecified by orientations of the parallel arcs, which will be always assumedto be concordant. The framing of a ribbon, which is an integer number,is obtained by twisting the two arcs delimiting the ribbon. This is givenby consecutive self-intersections, and therefore a specified orientation of theribbon determines whether n consecutive twists are positive or negative. Adiagram whose edges are specified by two parallel arcs, therefore defininga ribbon, is called ribbon diagram . It follows from the definitions that theblackboard framing of a framed link is a ribbon diagram.Let X be a tarnary quandle and let D be a diagram of a framed link.Suppose for the moment that the link has a single component, in other words UANTUM INVARIANTS FROM TSD COHOMOLOGY 13 it is a diagram of a framed knot. To each ribbon arc in D , associate a color bya pair of elements ( x , x ) ∈ X × X , corresponding to each side of the ribbon.At a positive crossing τ of D , where the arcs colored by ( x τ , x τ ) and ( y τ , y τ )meet, let the overpassing ribbon mantain the same color, while change thecolor of the underpassing ribbon to ( T ( x , y , y ) , T ( x , y , y )). When acrossing τ is negative, we change the color of the underpassing ribbon to( z , z ), where z i is the unique element of X such that T ( y i , x , x ) = z i ,whose existence is guaranteed by the axioms of ternary quandle. We nowpose the following. Definition 3.1.
Let D be a ribbon diagram whose set of ribbon arcs isdenoted by R , and let X be a ternary quandle. Then, a coloring of D by X ,is a (set-theoretic) map C : R −→ X × X, that is consistent with the coloring rule above. The set of colorings of aframed link is defined to be the set of colorings of a ribbon diagram of thelink. Lemma 3.2.
Let X be a ternary quandle and D a ribbon diagram of aframed link. Then the set of colorings C of D by X is invariant underReidemeister moves II and III, and moreover it respects cancelling of kinks. As a consequence, the notion of coloring of a framed link is well posed,and the set of colorings of a framed link is an isotopy invariant.Suppose φ is a ternary quandle 2-cocycle of X , with coefficients in A .For a given crossing τ , define the Boltzmann weight at τ , depending on thecoloring C and the 2-cocycle φ by ( φ ( x τ , y τ , y τ ) ǫ ( τ ) , φ ( x τ , y τ , y τ ) ǫ ( τ ) ) ∈ A × A ,where the sign ǫ ( τ ) is that of the crossing. Now we can define the 2-cocycleinvariant of a link L with respect to a ternary quandle X and a ternary2-cocycle ψ ∈ Z ( X, A ) as follows. First we give the definition in the caseof framed knots, and then generalize it to framed links.
Definition 3.3.
Let D be a ribbon diagram of a framed knot K having k crossings τ , · · · , τ k , and let ψ ∈ Z ( X, A ) be a ternary 2-cocycle. Definethe cocycle invariant of K , by the 2-cocycle ψ asΘ D ( X, T, ψ ) = X C ( k Y i =1 ψ ( x τ i , y τ i , y τ i ) ǫ ( τ i ) , k Y i =1 ψ ( x τ i , y τ i , y τ i ) ǫ ( τ i ) ) , where ǫ ( τ i ) is the sign of the i th crossing, and the sum runs over all colorings C of the diagram D . We sometimes shorten notation by writing Θ ψ insteadof Θ D ( X, T, ψ ), and name Θ D ( X, T, ψ ) the ribbon cocycle invariant. Eachterm ψ ( x τ i j , y τ i , y τ i ) ǫ ( τ i ) , j = 1 ,
2, is also called the
Boltzmann weight at τ ,relative to the coloring C and the 2-cocycle ψ . It is denoted by the symbol B j ( C , τ ).It remains to prove that the 2-cocycle invariant does not depend on thechoice of ribbon diagram D used in Definition 3.3. Theorem 3.4.
The cocycle invariant does not depend on the equivalenceclass of the ribbon diagram D . Therefore, it is well defined and it is aninvariant of framed knots. Moreover, changing a -cocycle ψ to anotherrepresentative of the same second cohomology group [ ψ ] ∈ H ( X, A ) , changesthe invariant to an integer multiple of the unit ( e, e ) ∈ A × A .Proof. The Theorem is proved by showing that the state sum is invari-ant with respect to Reidemeister moves II and III, and with respect tokink cancellation. Compare with the moves T1-T5 and T6 f given in [12],page 166, and with rel -rel in [23], page 14. As seen in Lemma 3.2, ap-plying a Reidemeister move II or III, or applying the kink cancellationmove transforms one coloring of a diagram to another coloring. Assum-ing that the colors at the top strands, hen changing colorings between Rei-demeister move II, are ( x, y ) × ( z, w ), we see that the consecutive cross-ings τ and τ contribute with a cocycle weight of ( ψ ( x, z, w ) , ψ ( y, z, w ) and( ψ ( x, z, w ) − , ψ ( y, z, w ) − ) respectively. So the contributions cancel. Rei-demeister mover III coincides with with the ternary 2-cocycle condition,and the corresponding invariance is guaranteed. Cancellation of kinks issimilar to the case of Reidemeister move II, and the sign in Definition 3.3ensures that the state sum does not change. Suppose ψ is a coboundary,i.e. ψ = δf , for some f : X −→ A . Let C be a fixed coloring and sup-pose we order the crossings τ , . . . , τ n starting from an arbitrarily chosenarc, and numbering the crossings as we encounter them following the arcalong the knot. Consider τ i and τ i +1 , and assume without loss of generalitythat τ i is a positive crossing. Then the Boltzmann sum contribution onthe first entry of the state sum is given by f ( x ) f ( T ( x, z, w )) − , assumingthat the colorings on top of τ i are ( x, y ) × ( z, w ). Likewise, the contri-bution on the second entry is f ( y ) f ( T ( y, z, w )) − . Now, suppose τ i +1 ispositive, and let ( z ′ , w ′ ) overpass at τ i +1 . Then the Boltzmann sum is givenby f ( T ( x, z, w )) f ( T ( T ( x, z, w ) , z ′ , w ′ )) − , which shows that the two terms f ( T ( x, z, w )) and f ( T ( x, z, w )) − cancel. Similarly for the second entry ofthe Boltzmann weight. If τ i +1 is negative, and we let ( z ′ , w ′ ) be underpass-ing, we see that the terms f ( T ( x, z, w )) and f ( T ( y, z, w )) again appear inboth crossings with opposite signs, and cancel out again. Proceeding in thisfashion we see that when we have τ n and τ the remaining terms cancelout, since have assumed that C is a colorng, hence it is well defined. Eachcoloring in the state sum contributes with a trivial term, so the invariantof a coboundary simply counts the number of colorings. Consequently, it isan integer multiple of ( e, e ). This fact implies that if ψ and φ differ by acoboundary, their invariants differ by an integer multiple of ( e, e ). A moreconcise argument, based on an interpretation of the state-sum invariant interms of Kronecker pairing first described in [4], is given in Theorem 5.8 of[24] for the case when T is the heap of a group. The reader can verify thatit is applicable also for general TSD operations. (cid:3) UANTUM INVARIANTS FROM TSD COHOMOLOGY 15
Remark 3.5.
The invariant Θ D ( X, T, ψ ) is, by construction, an elementof the group ring Z [ A × A ]. Since there is an isomorphism Z [ A × A ] ∼ = Z [ A ] ⊗ Z [ A ], we can identify Θ D ( X, T, ψ ) with a sum of tensor products ofelements of A .We now generalize Definition 3.3 to framed links with multiple compo-nents. First, if a link L has t components L = K ∪ · · · ∪ K t , we label thecrossings τ of L with the number of the component the underpassing ribbonbelongs to. Therefore, for example, if at the crossing τ the underpassingribbon belongs to the component i , τ is denoted by τ i . For j = 1 , B ( i ) j ( C , τ i ) relative to the crossing τ i , in the i th component, as ψ ( x j , y , y ) ǫ ( τ i ) , where ( y , y ) is the coloring of the over-passing ribbon (not necessarily in the component K i ) and ( x , x ) is thecoloring of the underpassing ribbon (in the component K i by assumption).In the following definition, we denote a vector with multiple entries beingpairs with the notation ( a, b ) × · · · × ( a ′ , b ′ ). Definition 3.6.
Let the notation be as in the previous paragraph. Thenthe (vector) ribbon cocycle invariant of L , relative to the ternary quandle X and the ternary 2-cocycle ψ is defined as ~ Θ D ( X, T, ψ ) = X C × ti =1 ( k ( i ) Y s =1 B i ( C , τ i ( s )) , k ( i ) Y s =1 B i ( C , τ i ( s ))) , where k ( i ) is the number of crossings in the i th component, τ i ( s ) indicatesthe s th crossing in the i th component, and the sum indicates that in eachcomponent of the vector we are summing over all possible colorings C .An argument similar to that of Theorem 3.4, applied to each component,shows that the (vector) ribbon cocycle invariant does not depend on theisotopy class of the framed link L and it is therefore well defined. More-over, changing ψ to another 2-cocycle in the same second cohomology classchanges the invariant by an integer multiple of the vector ( e, e ) × · · · × ( e, e ),where e is the neutral element of the coefficient group used for cohomology.4. Ribbon categories from self-distributive ternary operations
In this section it is given a generalization of known constructions thatallow to define a Yetter-Drinfeld module from a set-theoretic quandle [13].More specifically, the aim of this section is to define a ribbon category given aternary self-distributive object in the symmetric monoidal category of vectorspaces. As opposed to the case of a quandle, the ribbon category that isobtained in this procedure admits nontrivial twisting morphisms. Theseare defined by means of a self-crossing, similar to the Reidemeister move I,which in this case is not the trivial map. In this section, along with sections 5and 6 we focus on the case of ternary self-distributive objects obtained vialinearization of set-theoretic structures. We use ternary cohomology in the usual sense [8]. It will be shown in Section 7 that this construction canbe generalized to the case of ternary self-distributive objects in symmetricmonoidal categories endowed with duals, using a generalized version of 2-cocycle condition with coefficients in a group object of the given category.Let Q := { x , . . . , x n } be a finite ternary quandle with operation T : Q × Q × Q −→ Q . Then linearizing the operation over a ground field k givesa ternary self-distributive object in the category of k -vector spaces, ( X, T ),where X := k h x , . . . , x n i and the linearized operation is indicated with thesame symbol T .Let H ( Q, A ) indicate the second ternary self-distributive cohomologygroup of Q , with coefficients in the multiplicative (abelian) group A . Fix anontrivial 2-cocycle α : Q × Q × Q −→ A . In the examples treated belowwe have the case when A ⊂ k × or, more generally, when it is given a groupcharacter of A in the group of units of k . The group A therefore acts on thevector space X and its tensor products via scalar multiplication of k . We willuse the symbol · to indicate this action, for clarity, and we use juxtapositionto denote the multiplication in A .Using this data, we define the brading c α : X ⊗ ⊗ X ⊗ −→ X ⊗ ⊗ X ⊗ by the assignment( x ⊗ y ) ⊗ ( z ⊗ w ) α ( x, z, w ) α ( y, z, w ) · ( z ⊗ w ) ⊗ ( T ( x, z, w ) ⊗ T ( y, z, w )) , having used a comma to separate the entries of T instead of the symbol ⊗ to shorten notation. As it is proved below, the morphism c α satisfiesthe braid equation if and only if α is a 2-cocycle, i.e. [ α ] ∈ H ( Q, A ) (cf.with Proposition 3.3 in [13]). The twisting morpshism θ α : X ⊗ −→ X ⊗ isdefined by extending the assignement x ⊗ y α ( x, x, y ) α ( y, x, y ) · T ( x, x, y ) ⊗ T ( y, x, y ) . This definition is motivated by introducing a complete twist in a ribbonby self crossing, Figure 3.We introduce now the ribbon category whose objects are all even tensorpowers of the vector space X and generated by braiding and twisting givenabove. See[16] Chapter XII, Section XII.1, for the general definition ofpresentation of a category. We give a very explicit definition below, todescribe the morphisms in detail. Definition 4.1.
Let (
X, T ) be a ternary self-distributive object in V k , asabove, and [ α ] ∈ H ( Q, A ). Define the category R α ( X ) as follows. Theobjects are even powers X ⊗ n of X in the category V k . The tensor prod-uct of two objects Y and Z , written Y ⊠ Z , is defined to be the tensorproduct ⊗ in V k . The trivial power X is set to be k by definition. Themorphisms of this category are defined as follows. The set Hom( X ⊗ , X ⊗ )consists of the identity map and twists ( θ α ) ◦ m , where m ∈ Z and ◦ indicatescomposition. The set Hom( Y, Z ) is empty if Y = Z . The morphism setHom( X ⊗ , X ⊗ ) is the free monoid generated under composition by twofoldtensor products of f, g ∈ Hom( X ⊗ , X ⊗ ), and the braiding c α := c α , . UANTUM INVARIANTS FROM TSD COHOMOLOGY 17
The set Hom( X ⊗ n , X ⊗ n ) is defined inductively as the free monoid gen-erated under composition by tensor products f ∈ Hom( X ⊗ m , X ⊗ m )and g ∈ Hom( X ⊗ m , X ⊗ m ) with m + m = n . The braidings c αn,m : X ⊗ n ⊠ X ⊗ m −→ X ⊗ m ⊠ X ⊗ n are the morphisms X n + m ) −→ X n + m ) corresponding to block permutation switching X ⊗ n and X ⊗ m , obtainedby subsequent applications of c α , .Endow the category R α ( X ) with duals by setting ( X ⊗ n ) ∗ := ( X ∗ ) ⊗ n ,where X ∗ is the linear dual of X . The evaluation map ev is determinedby x ⊗ y ⊠ f ⊗ g f ( y ) g ( x ), and the coevaluation map coev by 1 x j ⊗ x i ⊠ x i ⊗ x j , where the Einstein summation convention is used. Let R ∗ α ( X ) indicate the category R α ( X ) with the addition of duals. Remark 4.2.
Direct inspection shows that the twisting morphism θ α de-fined above coincides with the self-intersection morphism obtained by meansof ev and coev maps as composition: ( ev ⊗ ⊗ ) ◦ ( ⊗ ⊗ c α ) ◦ ( coev ⊗ ⊗ ),cf. Figure 3. Theorem 4.3.
With the same notation as above, the category R ∗ α ( X ) withbraiding induced by c α and twisting morphisms induced by θ α is a ribboncategory. Moreover, if [ α ] = [ β ] , then the two categories R ∗ α ( X ) and R ∗ β ( X ) are equivalent.Proof. The fact that R α ( X ) is a tensor category is a consequence of the factthat V k is a tensor category, and the definitions. To verify naturality of thebraiding, it is enough to verify the commutativity of the square X ⊗ n ⊠ X ⊗ m X ⊗ m ⊠ X ⊗ n X ⊗ n ⊠ X ⊗ m X ⊗ m ⊠ X ⊗ nc α n, m f ⊠ g g ⊠ fc α n, m for all morphisms f and g . This follows from the fact that the morphismsare defined as block braidings induced by c α . So the naturality is a directconsequence of the definition of morphisms, and the fact that c α satisfiesthe braid equation (proved below) in the same way it is proved for the braidcategory. The fact that c α is a family of isomorphisms is a consequence ofthe invertibility of T axiom, holding for ternary racks. To verify that c α isindeed a braiding, it has to be shown that it satisfies the braid equation. Tothis objective, observe first that it is enough to prove that c α , satisfies thebraid equation( c α , ⊠ ) ◦ ( ⊠ c α , ) ◦ ( c α , ⊠ ) = ( ⊠ c α , ) ◦ ( c α , ⊠ ) ◦ ( ⊠ c α , ) , since the general case would follow from iterations of this specific case. Usingthe shorthand notation T y,zx = T ( x, y, z ), on a general basis vector x ⊗ y ⊠ ∼ = Figure 5.
A twist can be slid over a crossing. z ⊗ w ⊠ u ⊗ v , the left hand side equals( c α , ⊠ ) ◦ ( ⊠ c α , ) ◦ ( c α , ⊠ ) x ⊗ y ⊠ z ⊗ w ⊠ u ⊗ v = α ( x, z, w ) α ( y, z, w ) α ( T z,wx , u, v ) α ( T z,wy , u, v ) α ( z, u, v ) α ( w, u, v ) · u ⊗ v ⊠ T u,vz ⊗ T u,vw ⊠ T u,vT z,wx ⊗ T u,vT z,wy , while the right hand side equals( ⊠ c α , ) ◦ ( c α , ⊠ ) ◦ ( ⊠ c α , ) x ⊗ y ⊠ z ⊗ w ⊠ u ⊗ v = α ( z, u, v ) α ( w, u, v ) α ( x, u, v ) α ( y, u, v ) α ( T uvx , T uvz , T uvw ) α ( T uvy , T uvz , T uvw ) · u ⊗ v ⊠ T u,vz ⊠ T u,vw ⊠ T T u,vz ,T u,vw T u,vx ⊗ T T u,vz ,T u,vw T u,vy . The two terms are seen to coincide by applying the 2-cocycle condition to x, z, w, u, v and y, z, w, u, v separately, and the definition of self-distributivityof T . The fact that the duals turn R α ( X ) into a rigid category is standard.It is left to prove that the twisting morphisms are naturural with respect tothe braiding, i.e. that ( θ α ⊠ θ α ) ◦ c α = c α ◦ ( θ α ⊠ θ α ) , and that θ αY ⊠ Z = c αZ,Y ◦ c αY,Z ◦ ( θ αY ⊠ θ αZ ) . The latter follows immediately from the definition of θ α as extension of θ α .To prove naturality, observe that it is enough to show that( θ α ⊠ ) ◦ c α , = c α , ◦ ( ⊠ θ α )and ( ⊠ θ α ) ◦ c α , = c α , ◦ ( θ α ⊠ ) , since the general naturality with respect of the braiding is obtained dia-grammatically by sliding twists below and above a crossing, as in Figure 4.The case of a twist being slid below a crossing is similarly depicted. Onsimple tensors x ⊗ y ⊠ z ⊗ w , the left hand side of the first equality becomes( θ α ⊠ ) ◦ c α , x ⊗ y ⊠ z ⊗ w = α ( x, x, y ) α ( y, x, y ) α ( T z,wx , z, w ) α ( T z,wy , z, w ) · z ⊗ w ⊠ T z,wT z,wx ⊗ T z,wT z,wy , UANTUM INVARIANTS FROM TSD COHOMOLOGY 19 while the right hand side is c α , ◦ ( ⊠ θ α ) x ⊗ y ⊠ z ⊗ w = α ( x, z, w ) α ( y, z, w ) α ( T z,wx , T z,wx , T z,wy ) α ( T z,wy , T z,wx , T z,wy ) · T z,wz ⊗ T z,ww ⊠ T T z,wx ,T z,wy T z,wx ⊗ T T z,wx ,T z,wy T z,wy . These are seen to coincide upon applying the 2-cocycle condition twice to x, x, y, z, w and y, x, y, z, w , and using self-distributivity of T . Similarly, forthe second equality to be verified one has on the left hand side( ⊠ θ α ) ◦ c α , x ⊗ y ⊠ z ⊗ w = α ( z, z, w ) α ( w, z, w ) α ( x, T z,wz , T z,ww ) α ( y, T z,wz , T z,ww ) · T z,wz ⊗ T z,ww ⊠ T T z,wz ,T z,ww x ⊠ T T z,wz ,T z,ww x . To verify that they are the same, apply the 2-cocycle condition to T z,wx , z, w, z, w and observe that T ( x, z, w )= T ( T ( T ( x, z, w ) , z, w ) , z, w )= T ( T ( T ( x, z, w ) , z, w ) T ( z, z, w ) , T ( w, z, w ))= T ( x, T ( z, z, w ) , T ( w, z, w )) , and similarly T ( y, z, w ) = T ( y, T ( z, z, w ) , T ( w, z, w )). This sequence ofequalities is motivated diagrammatically by sliding a ribbon beneath a selfcrossing.Suppose now that [ α ] = [ β ], i.e. there exists f : X −→ A such that α ( x, y, z ) = β ( x, y, z ) f ( x ) f ( T ( x, y, z )) − for all x, y, z . Set ˜ f : X ⊗ −→ X ⊗ as ˜ f ( x ⊗ y ) := f ( x ) f ( y ) · x ⊗ y , and extended by linearity. The map ˜ f has aninverse given by ˜ f − ( x ⊗ y ) := f ( x ) − f ( y ) − · x ⊗ y . The definition of ˜ f andits inverse clearly extends to objects of R ∗ α ( X ) and R ∗ β ( X ). Define a functor F : R ∗ β ( X ) −→ R ∗ α ( X ) as follows. On objects F is defined to be the identity.On morphisms τ ∈ Hom( X ⊗ n , X ⊗ n ) define F ( τ ) = ( ˜ f − ) ⊗ n ◦ τ ◦ ( ˜ f ⊗ n ).This assignment is functorial, and moreover maps θ β to θ α and c β , to c α , ,since ( ˜ f − θ β ˜ f )( x ⊗ y )= ( f ( x ) f ( y ) β ( x, x, y ) β ( y, x, y ) f ( T ( x, x, y )) − f ( T ( y, x, y )) − ) · x ⊗ y = α ( x, x, y ) α ( y, x, y ) · x ⊗ y = θ α ( x ⊗ y ) ( ˜ f − c β , ˜ f )( x ⊗ y ⊠ z ⊗ w )= ( f ( x ) f ( y ) β ( x, z, w ) β ( y, z, w ) f ( T (( x, z, w )) − f ( T (( y, z, w )) − ) · z ⊗ w ⊠ T (( x, z, w ) ⊗ T (( y, z, w )= α ( x, z, w ) α ( y, z, w ) · z ⊗ w ⊠ T (( x, z, w ) ⊗ T (( y, z, w )= θ α , ( x ⊗ y ⊠ z ⊗ w ) . The definition of F clearly respects tensor products and due to the inductivedefinition of twists and braiding in the categories R ∗ α ( X ) and R ∗ β ( X ), it fol-lows that F is a braided tensor functor. Since F is essentially surjective onobjects (it is the identity), and fully faithful on morphisms (due to invert-ibility of ˜ f ), F is an equivalence of braided tensor categories that respectsthe twisting structure. This completes the proof. (cid:3) Remark 4.4.
Observe that the twisting morphisms θ α are in general non-trivial, even for a trivial α .The main purpose of the construction of Theorem 4.3, is to show thatthe natural invariants associated to the ribbon category coincide with thecocycle invariants defined in Section 3. It is in fact possible to bypass thisconstruction and prove that the twisting morphsism θ α and the braidingmorphism c α , induce a representation of the infinite framed braid group(the inductive limit of the framed braid groups F B n of [18]) similarly to[26], using the linear map Φ b of Section 5 (given below). It is convenient,though, to define a ribbon category from a ternary self-distributive structureand a ternary 2-cocycle since this is suitable for a generalization to multiplecompatible self-distributive structures. Definition 4.5. A compatible system of ternary self-distributive structures is a finite family { ( Q i , T i ) } i ∈ I of ternary self-distributive sets along withactions T ij : Q i × Q j × Q j −→ Q i , of Q j × Q j on Q i for all i, j = 1 , . . . n , satisfying the compatibility condition T ik ◦ ( T ij × × ) = T ij ◦ ( T ik × T jk × T jk ) ◦ (cid:1) ◦ ( × × ∆ × ∆ ) , where ∆ = (∆ × ) ◦ ∆ and (cid:1) is the permutation map corresponding toternary self-distributivity. Such a system is denoted { Q i , T ij } , where it isimplicitly assumed that T ii := T i .Linearizing such a system of compatible structues gives a multi-objectanalogue of ternary self-distributive object in the category of vector spaces.We will sometime call the linearized object a compatible system of self-distributive structures, since no confusion will arise. Remark 4.6.
There are infinitely many examples of the structure given inDefinition 4.5 arising from mutually distributive structures as in [8]. In factlet (
X, T , T ) be a mutually self-distributive set (with X finite). Linearizing T and T over a field k and defining T = T = T and T = T = T over the vector space, it is seen by direct inspection that the structure { k h X i , T ij } i,j =0 , is a compatible system of ternary self-distributive struc-tures.In Section 6 it will be seen that compositions of G -families of quandlesprovide other natural examples of these structures. Furthermore, in theAppendix, more examples from augmented Hopf modules will be introduced.In particular it will be seen that there are compatible systems with multiplebase spaces.To the notion of compatible system of ternary self-distributive structures,there corresponds the notion of compatible system of ternary 2-cocycles asfollows. Definition 4.7.
Let { ( Q i , T ij ) } be a compatible system of ternary self-distributive structures. A compatible system of -cocycles with coefficientsin A (abelian group with multiplicative notation) is a family of maps α ij : X i × X j × X j −→ A such that α ij ( x, y, z ) · α ik ( T ij ( x, y, z ) , u, v )= α ik ( x, u, v ) · α ij ( T ik ( x, u, v ) , T jk ( y, u, v ) , T jk ( z, u, v )) , for all x ∈ X i , y, z ∈ X j and u, v ∈ X k and all i, j, k . Such a family ofmaps is denoted by the symbol { α ij } , where parentheses can be omitted toshorten notation. Definition 4.8.
A compatible system of 2-cocycles { α ij } is said to be trivial,or cobounded, if there exists a family of maps f i : X i −→ A such that δf i ( x × y × y ) := f i ( x ) f i ( T ij ( x × y × y )) − = α ij ( x × y × y )for all i, j and all x ∈ X i , y , y ∈ X j .Two systems of 2-cocycles, { α ij } and { β ij } are said to be equivalent ifthe system { α ij β − i,j } is trivial. Remark 4.9.
Observe that when i = j = k it follows that each α ii is aternary 2-cocycle for the ternary self-distributive structure T i , and more-over the triviality condition gives that α ii represents the trivial class insecond ternary self-distributive cohomology group. It follows that if α ii isnot cobounded in the ternary self-distributive cochain, then a compatiblesystem of 2-cocycles is nontrivial in the sense of Definition 4.8. Remark 4.10.
Definitions 4.7 and 4.8 are clearly reminiscent of a coho-mology theory. It is natural to ask whether such a theory derives from adeformation theory of compatible systems. The answer is no, in that infin-itesimal deformations of compatible systems require more conditions to besatisfied, than just the 2-cocycle condition of Definition 4.7.
Remark 4.11.
In the case of Remark 4.6, when a system of compatiblestructures is defined on the same base space, a compatible system of 2-cocycles is the same as a 2-cocycle in the labeled cohomology of [8].
Let { ( Q i , T ij ) } be a compatible system of ternary self-distributive struc-tures and let { α ij } be a compatible system of 2-cocycles. Let X i = k h Q i i and denote the linearized maps T ij by the same symbols. To these data it ispossible to associate twisting morphisms and braidings as follows. For each X i there is a twisting θ α ij : X i ⊗ X i −→ X i ⊗ X i defined by extending theassignment x ⊗ y α ii ( x, x, y ) α ii ( y, x, y ) · T i ( x, x, y ) ⊗ T i ( y, x, y ) . For each pair of objects X i and X j there is a braiding c α ij : X ⊗ i ⊗ X ⊗ j −→ X ⊗ j ⊗ X ⊗ i induced by( x ⊗ y ) ⊗ ( z ⊗ w ) α ij ( x, z, w ) α ij ( y, z, w ) · ( z ⊗ w ) ⊗ ( T ij ( x, z, w ) ⊗ T ij ( y, z, w )) . The construction of Definition 4.1 is adapted to this case and the corre-sponding category is denoted by R ∗{ α ij } ( { X i } ). Theorem 4.12.
The category R ∗{ α ij } ( { X i } ) with braiding and twisting morp-shisms defined above is a ribbon category. Moreover, if two systems { α ij } and { β i,j } are equivalent, then R ∗{ α ij } ( { X i } ) and R ∗{ β ij } ( { X i } ) are equivalent.Proof. The proof is substantially the same as that of Theorem 4.3, wherethe 2-cocycle condition of α is replaced by compatibility condition of thefamily { α ij } and self-distributivity of T is replaced by compatibility of thesystem { T ij } . When { α ij } and { β ij } are equivalent, one can construct maps˜ f i as in Theorem 4.3 and show that these induce an equivalence of ribboncategories. (cid:3) Observe that R ∗{ α ij } ( { X i } ) is a ribbon category with “distinguished” ob-jects X i ⊗ X i , and other objects given by tensor products obtained from thedistinguished ones.5. The Ribbon cocycle invariant is a quantum invariant
The ribbon category R ∗ α ( X ) allows to define an invariant of framed linksfrom any ternary self-distributive operation and a fixed ternary 2-cocycle,following standard procedures [26]. Let ( X, T ) be a ternary self-distributiveobject arising from a set-theoretic ternary quandle Q as above, and [ α ] ∈ H ( Q, A ) be fixed. A framed link is represented by the closure of an element b ∈ F B n of the framed braid group on n ribbons [18] where, since twistingof the ribbon and crossings commute, it is assumed that the twists are ontop of the braid. Using the same notation of [18], b = t r · · · t r n n · τ, where r i are integers indicating the number of twists of the i th ribbon and τ is an element of the braid group B n .Then, the quantum invariant associated to ( X, T ) and α , is obtained byconsidering the object X ⊠ · · · ⊠ X ( n -fold product) in R ∗ α ( X ) and takingthe trace of the morphism Φ b : X ⊠ · · · ⊠ X −→ X ⊠ · · · ⊠ X corresponding UANTUM INVARIANTS FROM TSD COHOMOLOGY 23 to b as follows. Each generator t r i i corresponds to ( θ α ) r i , and each crossing σ ± i in the product defining τ corresponds to ( c α , ) ± . This invariant isdenoted by the symbol Ψ D ( X, T, α ) , where D is the ribbon diagram representing b .The following theorem establishes that two procedures described in Sec-tion 3 and this section coincide. Theorem 5.1.
Let L be a framed link, with presentation given by a framedbraid b as above, ( X, T ) be a ternary self-distributive structure that is lin-earized over k , and α a -cocycle of ( X, T ) with coefficitents in A . Sup-pose that χ : A −→ k × is a group character. Fix a diagram D of L .Then the ribbon cocycle invariant χ Θ D ( X, T, α ) and the quantum invari-ant Ψ D ( X, T, χ ◦ α ) coincide.Proof. The proof is very similar to that of Theorem 3.5 in [13]. Observe thatin order to compute Ψ D ( X, T, α ), one has to consider all combinations ofbasis vectors x ǫ ( i ) i ⊗ · · · ⊗ x ǫ ( i n ) i n apply the endomorphism Φ b and thereforeapply it to all the possible combinations x ǫ ( j ) j ⊗ · · · ⊗ x ǫ ( j n ) j n . It followsthat the only nontrivial contributions to Ψ D ( X, T, α ) are obtained when x ǫ ( j ) j ⊗ · · · ⊗ x ǫ ( j n ) j n = x − ǫ ( i ) i ⊗ · · · ⊗ x − ǫ ( i n ) i n , by definition of ev and coev mapsin R ∗ α ( X ). At each crossing, whether corresponding to a twisting t r i or oneof the factors of the braid τ , Φ b contributes with a scalar given by either α ( z , z , z ) α ( z , z , z ) , or α ( z , w , w ) α ( w , w , w ) , where it has been assumed that θ α is applied to the vector z ⊗ z and c α isapplied to the vector z ⊗ z ⊠ w ⊗ w . The vector output is given by either T ( z , z , z ) ⊗ T ( z , z , z ) , or w ⊗ w ⊠ T ( z , w , w ) ⊗ T ( w , w , w ) . It therefore follows that the only nontrivial contribution to Ψ D ( X, T, α )corresponds to the colorings of D by X and each contribution equals one ofthe summands that define Θ D ( X, T, α ). The proof is complete. (cid:3)
The quantum invariant Ψ D ( X, T, α ) does not only provide a differenntinterpretation of the ribbon cocycle invariant Θ D ( X, T, α ), but it is alsosuitable for a generalization to framed braids (not necessarily closed).
Theorem 5.2.
Let F be a diagram of a framed braid b ∈ F B n . Then Φ b defined as above is an invariant of F . Examples and Computations
Examples from compositions of binary quandles.
In this sub-section it is showed that the notions of G -family of quandles and G -familyof 2-cocycles [11, 21] provide interesting examples of compatible systems ofTSD structures. An explicit example using Nosaka cocycles is also shownin detail.Recall first, that a G -family of quandles is a set X along with a familyof quandle operations ∗ g indexed by a group G (i.e. g ∈ G for all g ), andsatisfying the compatibility conditions( x ∗ g y ) ∗ h y = x ∗ gh y, ( x ∗ g y ) ∗ h z = ( x ∗ h z ) ∗ h − gh ( y ∗ h z ) . Given a G -family of quandles one can construct a compatible system ofternary self-distributive operations as follows. Linearize the base set X toobtain V = k h X i . Then define maps T gh : V ⊗ V ⊗ V −→ V by theassignment x ⊗ y ⊗ z ( x ∗ h y ) ∗ h − z . A direct computation shows thatthe system so defined is indeed compatible. Observe that T gg being self-distributive is an instance of the fact that composing mutually distributivebinary operations produces a ternary self-distributive operation, as in [8]. Definition 6.1.
The compatible system constructed above from the G -family of quandles ( X, ∗ g ) g ∈ G is called the compatible system associated toa G -family .Since the main construction of Section 4 associate a ribbon category toa compatible system of distributive structures by means of a system of 2-cocycles, it is fundamental to obtain such objects for compatible systemassociated to G -families. As shown in the next result, it is possible to doso using the notion of G -family 2-cocycles, see [11, 21] for the definition of G -family (co)homology. In what follows it is assumed that the X -set Y appearing in G -family cohomology is a singleton endowed with the trivial G -family action. The set Y is therefore omitted without further notice, butthis should not cause any difficulties. Proposition 6.2.
Let ( X, ∗ g ) g ∈ G and ( V, T gh ) g,h ∈ G be a G -family of quan-dles and the associated compatible system of self-distributive structures, re-spectively. Let θ be a G -family -cocycle. Then there is an associated com-patible system of -cocycles θ gh defined as follows θ gh ( x × y × z ) := θ (( x, e ) × ( y, g )) + θ (( x ∗ g y, e ) × ( z, h )) . Remark 6.3.
A couple of observations are due. Firstly, notice that since X acts trivially on Y , the two terms corresponding to deleting the first entryof θ , according to the definition of G -family 2-cocycle cancel each other, sothat it is reasonable to arbitrarily choose on element of G to label all thefirst entries x , where the obvious choice falls upon the neutral element e of G . Secondly, there is a parallel between labels assignment in the definition UANTUM INVARIANTS FROM TSD COHOMOLOGY 25 of chain maps in “labeled cohomology” of Theorem 5.3 in [8], particularlyclear from Remark 5.7 in the same article, and the group element enriching y or z to a pair ( y, g ) or ( z, h ), respectively. Proof of 6.2.
Using Remark 6.3, the proof is almost immediate. In fact, thecondition that θ gh has to satisfy is (in additive notation) θ fg ( x × y × z ) + θ fh ( T ij ( x × y × z ) × u × v )= θ fh ( x × u × v ) + θ fg ( T fh ( x × u × v ) , T gh ( y × u × v ) × T gh ( z × u × v )) . Using the definition of θ gh , the G -family 2-cocycle condition becomes equiv-alent to labeled cohomology 2-cocycle condition, since terms obtained bydeleting x cancel. Now the same proof as in Remark 5.7 of [8] can be ap-plied, mutatis mutandis, to complete. (cid:3) Let now G = SL(2; Z ) and X = Z × Z , with operations {∗ g } g ∈ G definedas follows [11], x ∗ g y := gx + ( − g ) y for all x, y ∈ X and g ∈ G , where G acts on X by matrix multiplication on column vectors, and is theidentity of G . A direct computation shows that these operations define a G -family structure on ( X, G ) (Proposition 2.3 in [11]). From this data, Nosakahas constructed [21] a G -family 2-cocycle, that has been employed in [11]to compute cocycle invariants of certain handlebody knots and distinguishthem from their mirror images. As pointed out above, it is not restrictiveto omit the singleton set Y := { y } in the original construction. Define α : ( X × G ) × −→ Z by( x, g ) × ( y, h ) λ ( g )det( x − y, (1 − h ) − y ) , where λ is the abelianization function, defined by λ ( A ) = ( a + d )( b − c )(1 − bc ),for a matrix A := (cid:18) a bc d (cid:19) . Then α is a G -family 2-cocycle and it followsthat { α gh } defined as in Proposition 6.2 is a compatible system of 2-cocycles. Example 6.4.
Applying Theorem 4.12 to the compatible system of 2-cocycles { θ gh } g,h ∈ G associated to Nosaka’s G -family 2-cocycle α via Propo-sition 6.2 one obtains a ribbon category. The braiding at level 2 is givenexplicitly by the maps c α gh , ( x ⊗ y ⊠ z ⊗ w ) = α (( x, e ) , ( z, g ) α (( x ∗ g z, e ) , ( w, h )) α (( y, e ) , ( z, g ) α (( x ∗ g z, e ) , ( w, h )) · z ⊗ w ⊠ ( x + ( h − − ) z + ( − h ) w ) ⊗ ( y + ( h − − ) z + ( − h ) w ) , while twisting morphisms are given by θ α gh ( x ⊗ y ) = α (( x, e ) , ( x, g ) α (( x, e ) , ( y, h )) α (( y, e ) , ( x, g ) α (( y ∗ g x, e ) , ( w, h )) · ( h − x + ( − h − ) y ) ⊗ (( h − − ) x − h − y ) . Examples from heap structures.
Recall that given a group G , theheap operation G × G × G −→ G defined by x × y × z xy − z , defines aternary quandle structure on G . Linearizing this assignment over a field k produces a ternary quandle object in the category of vector spaces, where thediagonal map is induced by x x ⊗ x ⊗ x . This definition in fact coincideswith the quantum heap of the Hopf algebra structure on the group ring k [ G ],as it can be seen directly. Example 6.5.
Let Z be the cyclic group of order 2 and let C [ Z ] bethe structure defined above, obtained by linearizing the heap operation of Z . The elements of Z are identified with the symbols e x , with x ∈ Z ,generating the two dimensional vector space. A direct computation showsthat H ( Z , Z ) = Z ⊕ Z with generators corresponding to the equiva-lence classes of characteristic functions χ (0 , , and χ (0 , , . Fix the cocycle(0 , ∈ H ( Z , Z ), i.e. the map φ ( x, y, z ) = 1 if ( x, y, z ) = (0 , ,
1) and φ ( x, y, z ) = 0 otherwise. Identifying Z with ( − i ∈ C × one gets a nontriv-ial cohomology class in H ( Z , C × ), still denoted by φ . The correspondingtwisting morphisms and braiding morphism are as follows θ φ ( e x ⊗ e y ) = ( − φ ( x,x,y )+ φ ( y,x,y ) · e x + x + y ⊗ e y + x + y = e y ⊗ e x , that is, the twisting morphism is given by transposition, and c φ ( e x ⊗ e y ⊠ e z ⊗ e w ) = ( − φ ( x,z,w )+ φ ( y,z,w ) · e z ⊗ e w ⊠ e x + z + w ⊗ e y + z + w = ( e z ⊗ e w ⊠ e x ⊗ e y z = we z ⊗ e w ⊠ e x +1 ⊗ e y +1 otherwise Example 6.6.
Let X be the abelian group Z n with group heap structure,and set A = Z taken with multiplicative notation, with generator g . Suppose ρ is a given group character mapping A in the group of units of k . In [24],Lemma 3.7, it is shown that the 2-cochain φ i : X × −→ Z defined by theformula φ i := X x ∈ Z n [ n − X j =0 χ ( x,j,j + i ) ] , where χ ( x,y,z ) is the characteristic function at the triple ( x, y, z ) ∈ X × is anontrivial 2-cocycle for any choice of i = 1 , . . . , n −
1. It is in fact therebyproved that [ φ i ] = [ φ k ] in the second cohomology group, whenever i = k in Z n . The ribbon category corresponding to φ , for some choice of n ∈ N and0 = i ∈ Z n is determined by braiding and twisting morphisms obtained asfollows. For all e x , e y , e z , e w ∈ k [ X ], the linearization of X over k coincidingwith the group algebra of Z n , c φ i , maps simple tensors according to the UANTUM INVARIANTS FROM TSD COHOMOLOGY 27 assignment c φ i , ( x ⊗ y ⊠ z ⊗ w ) = ρ ( g φ i ( x,z,w ) ) ρ ( g φ i ( y,z,w ) ) e z ⊗ e w ⊠ e x − z + w ⊗ e y − z + w = ρ ( g φ ( x,z,w ) ) e z ⊗ e w ⊠ e x − z + w ⊗ e y − z + w = ( ρ ( g ) e z ⊗ e w ⊠ e x + i ⊗ e y + i if w − z = ie z ⊗ e w ⊠ e x + k ⊗ e y + k if w − z = k = i The twisting morphism θ φ i maps simple tensors as θ φ i ( e x ⊗ e y ) = ρ ( g φ i ( x,x,y ) ) ρ ( g φ i ( y,x,y ) ) e y ⊗ e y − x = ρ ( g φ i ( x,x,y ) ) e y ⊗ e y − x = ( ρ ( g ) e y ⊗ e y + i if y − x = ie y ⊗ e y + k if y − x = k = i We note that the twisting morphism θ φ i is determined, up to scalar mul-tiplication, by the Takasaki quandle operation x ∗ y = 2 y − x associatedwith the abelian group Z n . This is in fact a general feature of the twistingmorphism of an abelian heap. It is also easy to see that for non-abelianheaps one obtains the core of a quandle, instead of the Takasaki structure.Generalization of the preceding braiding and twisting structure to the caseof linear combinations of φ i ’s is easily obtained from the previous equations. Example 6.7.
Let X = D be the dihedral group on 6 elements, with pre-sentation h s, r | s = r = 1 , srs = r − i . Once again consider the groupheap strucutre on X and linearize it to obtain a quantum heap on the groupring C [ X ]. Denote characteristic functions χ ( x,y,z ) , those cochains with co-efficients in Z defined by χ ( x,y,z ) ( x ′ , y ′ , z ′ ) = 1 if ( x ′ , y ′ , z ′ ) = ( x, y, z ), and0 otherwise. Define the 2-cochain ψ = X x [ χ ( x, ,r ) + χ ( x,r,r ) + χ ( x,r , ] + X x [ χ ( x,s,sr ) + χ ( x,sr,sr ) + χ ( x,sr ,s ) ] . A direct computation shows that ψ satisfies the 2-cocycle condition and itis therefore a 2-cocycle. moreover ψ is nontrivial [24], Example 5.13 andProposition 5.14, so that [ ψ ] = 0. Mapping Z to the 3 rd roots of unity G we obtain ψ ∈ Z ( Z , G ), where G acts on C [ Z ], and therefore on C [ Z ] ⊗ C [ Z ], by scalar multiplication. Twisting morphisms are obtainedas θ ψ ( x ⊗ y ) = e πi ( ψ ( x,x,y )+ ψ ( y,x,y )) · y ⊗ yx − y, where the scalar e πi ( ψ ( x,x,y )+ ψ ( y,x,y )) is nontrivial if and only if x − y = r ,in which case we obtain e πi ( ψ ( x,x,y )+ ψ ( y,x,y )) = e πi . Observe that yx − y isthe core quandle operation. In fact, θ ψ corresponds (up to the multiplyingscalar) to the R matrix obtained by linearizing the set-theoretic solution ofYang-Baxter equation corresponding to the core of group D . The braiding c ψ , is given explicitly by c ψ , ( x ⊗ y ⊠ z ⊗ w ) = e πi ( ψ ( x,z,w )+ ψ ( y,z,w )) · z ⊗ w ⊠ xz − w ⊗ yz − w, where the scalar multiple is nontrivial if and only if z − w = r , in which casewe have e πi ( ψ ( x,z,w )+ ψ ( y,z,w )) = e πi , similarly to the case of twisting.6.3. Computations of quantum invariants.
In this subsection we il-lustrate the theory by computing some invariants of a few basic framedlinks corresponding to linearized ternary self-distributive structures and 2-cocycles given above. We compare with the invariants computed in [24]. Infact it will be apparent how the two types of invariants encode the sameinformation, as proved in Theorem 5.1.
Example 6.8.
Let us consider the unknot with frame n ∈ N . Let H = Z m for some arbitrary m , considered with abelian heap structure, i.e. T ( x, y, z ) = x − y + z . As already argued in Example 6.6, for each i = 1 , . . . , m − Z given by φ i := P x ∈ Z m [ P m − j =0 χ ( x,j,j + i ) ]. Let us fix an arbitrary i and let ζ ∈ C denote aprimitive m th -root of unity. We assume Z to be generated (in multiplica-tive notation) by g , and define the character ˜ χ : Z −→ C × by reducing theexponents of g modulo m and identifying it with the corresponding powerof ζ . The map that is associated to the n -framed unknot is θ n , where weset θ := θ φ i for simplicity. We denote the basis vectors of the vector space X = k h H i generated by H by the symbols e x , for x ∈ H . Then θ n is easilyseen to be given by the map e x ⊗ e y q ( x, y ) · e ny − ( n − x ⊗ e ( n +1) y − nx ,where the unit q ( x, y ) ∈ C × is determined below. For a vector to contributeto the trace of θ n we need e x = e ny − ( n − x and e y = e ( n +1) y − nx , which gives n ( y − x ) = 0. If ( n, m ) = 1 then there are | X | vectors that satisfy this con-dition. We have that tr q ( θ n ) = | X | , since i = 0. Observe that the conditionof e x ⊗ e y contributing to the trace of θ n can be rephrased as x, y giving acoloring of the diagram of the n -framed unknot. So, when choosing m co-prime with n the invariant is simply counting colorings. If ( n, m ) = d = 1,then we have d elements α divisible by m/d . Each of these solutions gives | H | = m vectors that contribute nontrivially to the trace of θ n . Moreover,whenever i = α for one of the previous solutions, we have a contribution of ζ n . We have tr q ( θ n ) = ( dm ( i, m/d ) = m/d ( d − m + ζ n m ( i, m/d ) = m/d The cocycle invariant Ψ, by direct computation, is seen to be equal to me ⊗ e when m and n are coprime, andΨ = ( md · e ⊗ e ( i, m/d ) = m/dm ( d − · e ⊗ e + mg n ⊗ g n ( i, m/d ) = m/d UANTUM INVARIANTS FROM TSD COHOMOLOGY 29 when ( m, n ) = d . We see that applying χ to Ψ we obtain tr q ( θ n ), as required.Moreover, we can choose m and i such that tr q ( θ n ) = dm + ζ n m and ζ n = 1,so the invariant detects twisting. Example 6.9.
Let us now consider the torus link T (2 , n ) on two strings,with even number of crossings. We compute the quantum invariant cor-responding to the cocycle φ i ∈ Z ( Z m , Z ), for some i = 1 , . . . , n −
1. Set X = C h Z m i . The framed braid whose closure gives T (2 , n ) corresponds tothe endomorphism c n : X ⊗ ⊠ X ⊗ −→ X ⊗ ⊠ X ⊗ , obtained by composingthe braiding of two ribbons 2 n times. We use the symbol ⊠ to distinguishpairs corresponding to the two edges of a ribbon, following previous conven-tions. We choose, as before, an integer m and a primitive m th -root of unitiy ζ and we use again the map ˜ χ that sends a generator, say g , of Z in mul-tiplicative notation to ζ . On basis vectors we have, by direct computation, c n ( e x ⊗ e y ⊠ e z ⊗ e w ) = q ( x, y, z, w ) · e x + n ( w − z ) ⊗ e y + n ( w − z ) ⊠ e z + n ( y − x ) ⊗ e w + n ( y − x ) = q ( x, y, z, w ) · e x ⊗ e y ⊠ e z ⊗ e w , where q ( x, y, z, w ) ∈ C × is deter-mined as follows. We shorten notation by omitting the variables, thereforewriting just q . When y − x = w − z = i we have q = ζ n , when just oneof w − z or y − x equals i , q = ζ n , while q = 1 whenever both w − z and y − x are different from i . Then c n is diagonal and each vector contributesto the trace so, to complete the computation, one needs just to count howmany vectors contribute with either of the weights q given above. We have tr q ( c n ) = n ζ n +2 n ( n − ζ n + n + n . Observe that the computation paral-lels perfectly that of Example 5.12 in [24], where the fact that each initial arccoloring defines a full coloring of T (2 , n ) translates into the statement that c n is diagonal, and where in the cocycle invariant the weights contribute tothe entries of the tensor product of the group algebra Z n [ Z ] depending onthe components of the link.7. Generalized construction
In this section, we develop a generalized version of the theory describedabove, where we consider symmetric monoidal categories. To this objective,we first need to provide a framework for ternary self-distributivity in sym-metric monoidal categories and introduce a categorical version of 2-cocyclecondition. The first construction has been introduced by the author, alongwith M. Elhamdadi and M. Saito, in [8], while the second construction willbe introduced herein.Our main interest in generalizing the construction from the linearized casedescribed above to more general objects in symmetric monoidal categorieslies in the following discussion. When computing the quantum invariantassociated to a TSD set we see that a vector contributes to the invariant,i.e. to the (quantum) trace of the linear map X ⊗ n −→ X ⊗ n associatedto a framed braid diagram with n doubled-strings, if the coloring conditionon the edges of the diagram is satisfied. This is due to the fact that thecomultiplication of X is simply given by producing two copies of an element of X . For a general TSD object, say in the category of vector spaces, thecomultiplication is usually not the same as x x ⊗ x , so that there canbe contributions to the quantum trace that do not correspond to coloringsof the framed diagram by X . We expect that this phenomenon producesstronger invariants than the ribbon cocycle invariant. In fact, while a trivialcocycle does not produce new non-trivial invariants in the linearized case (itsimply counts the colorings of a diagram, which is a known invariant), witha general comultiplication we can obtain nontrivial invariants correspondingto the YB operator associated to the TSD object even when this is notdeformed by a nontrivial cocycle.Let C be a symmetric monoidal category, with tensor functor denoted by ⊗ and let X be a fixed object in C . Then, associating the switching morphism τ : X ⊗ −→ X ⊗ to the transposition (12) ∈ S we obtain a “representation”of the infinite symmetric group S ∞ as follows. Let σ ∈ S n for some n ∈ N ,we decompose the permutation in a product of transpositions σ = σ · · · σ k for some k . Then we define the corresponding automorphism of X ⊗ n tobe τ ◦ · · · ◦ τ k , where τ i : X ⊗ n −→ X ⊗ n is the automorphism given by i − ⊗ τ X,X ⊗ n − i +1 . Then it is verified that this assignment does notdepend on the choice of decomposition of a permutation into transpositions,and it is therefore well defined. We obtain a correspondence (cid:1) : S ∞ −→∪ n Hom( X ⊗ n , X ⊗ n ), and (cid:1) σ is the automorphism of X ⊗ n that (cid:1) associatesto the permutation on n elements σ . In the rest of the section we will saythat (cid:1) σ is the morphism corresponding to the permutation σ .7.1. TSD objects in symmetric monoidal cateogries.
Let ( C , ⊗ , τ )be a symmetric monoidal category and let X be a ternary self-distributive(TSD) object in C , i.e. a comonoid object with a morphism T : X ⊗ X ⊗ X −→ X satisfying categorical self-distributivity and commuting withthe diagonal morphism ∆ := ∆ ◦ (∆ ⊗ ). Specifically, categorical self-distributivity means that X is endowed with a morphism ∆ : X −→ X ⊗ X which makes the same diagrams of coassociativity commute, a morphism ǫ : X −→ k which satisfies the same diagrams of counit in a coalgebra, andthe morphism T makes the following diagram commute (case n = 3 in [8]) X ⊗ X ⊗ X ⊗ X ⊗ n X ⊗ X (cid:1) t ⊗ ⊗ ∆ ⊗ T ⊗ ⊗ T ⊗ T ⊗ T TT
UANTUM INVARIANTS FROM TSD COHOMOLOGY 31 where ∆ := (∆ ⊗ ) ◦ ∆ = ( ⊗ ∆) ◦ ∆, and (cid:1) t is the shuffle correspondingto ternary self-distributivity t = (2 , , , , . In the rest of the article we will denote ∆ n := (∆ ⊗ n − ) ◦ · · · ◦ (∆ ⊗ ) ◦ ∆ : X −→ X ⊗ n , to indicate the n -fold diagonal of the comonoid object X . Remark 7.1.
In what follows we will assume our TSD objects to be cocom-mutative (as comonoids), as the main proof of the section will make use ofthis assumption. We point out that the preliminary definitions and resultsmake sense without this further assumption, unless otherwise specified.The final objective of this section is to generalize the construction of rib-bon category from a TSD operation to the case of TSD objects in symmetricmonoidal categories. When we linearize a TSD operation T over a field (orring), as we have seen in Section 4, the rack axiom of T which states that T ( x, y, z ) = d has a solution for x for every d , once we fix two elements y, z ,automatically provides a way of defining an inverse to the braiding mor-phisms. In the setting of symmetric monoidal categories, the definition ofTSD object given above does not provide any condition guaranteeing thepossibility of introducing an inverse to the braiding morphism. The nextdefinition provides an answer to this issue. Definition 7.2.
Let X be a TSD object in a symmetric monoidal category C . Then a ternary rack in C is a comonoid object X in C together with apair of morphisms T, T − : X ⊗ −→ X satisfying the TSD condition givenabove, along with the equation T − ◦ [ T ⊗ ⊗ ] ◦ ⊗ ∆ ⊗ ∆ = ⊗ ǫ ⊗ ǫ, where equality is meant as an equality of morphisms X ⊗ −→ X in C . Wealso require an analogous equation to be satisfied, where the roles of T and T − are exchanged. Example 7.3.
The fundamental example of TSD object in the symmetricmonoidal category of vector spaces is that of an involutory Hopf algebrawith quantum heap operation. This is a “categorical” version of the notionof heap of a group whose cocycle invariants have been introduced and studiedin [24]. The quantum heap operation is given by extending the operation x ⊗ y ⊗ z xS ( y ) z by linearity. A similar construction holds replacinga Hopf algebra by a Hopf monoid in a symmetric category C . This meansthat H is a bimonoid, i.e. an object that is both monoid and comonoid,and it is endowed with a morphism s : H −→ H that satisfies the samecommuative diagrams for the antipode as in the usual definition of Hopfalgebra. It has been proved in [9] that an involutory Hopf monoid gives riseto a TSD object by a generalization of the quantum heap construction (seeTheorem 7.12 therein). Both vector space and symmetric monoidal categoryversions of the proofs utilize the fact that T ◦ [ T ⊗ ⊗ ] ◦ [ ⊗ ˆ τ ] ◦ ⊗ ∆ ⊗ ∆ = ⊗ ǫ ⊗ ǫ, where ˆ τ is the morphism X ⊗ −→ X ⊗ corresponding to the permutation (cid:0) (cid:1) . See for instance the first part of the proof of Proposition 7.10, andLemma 7.11 in [9]. If X is the heap object corresponding to an involutoryHopf monoid in C , i.e. a TSD object, we have a ternary rack object in C by taking T − to be T ◦ [ ⊗ τ ], where τ denotes the switching morphismof C . Hopf algebras (or monoids) naturally give rise to TSD objects as wellas ternary racks. More generally, one can replace the notion of involutoryHopf monoid by the more general one of (categorical) heap, of which Hopfmonoids provide a fundamental example. See Definition 7.1 in [9].7.2. Examples of TSD objects.
There is a very natural situation in whichTSD objects arise in a symmetric monoidal category, as described in thefollowing example, which generalizes Example 7.3
Example 7.4.
We describe explicitly the quantum heap TSD morphism insymmetric monoidal categories mentioned above. Let X be an involutoryHopf monoid in C . Then it has been shown in [9] that X can be endowedwith a morphism T : X ⊗ −→ X that turns it into a TSD object in C asfollows. Set T := µ ◦ ( µ ⊗ ) ◦ ( ⊗ s ⊗ ), where µ is the multiplicationmorphism and s is the (involutive) antipode in X . Observe that when X is aHopf algebra in the category of vector spaces, we have that ( X, T ) coincideswith the quantum heap of Example 7.3. In fact, if in addition X is the groupalgebra of some group G , then T is the linearization of set-theoretic heapstructure of G , as in Subsection 6.2, so that this is a natural generalizationof objects already encountered. These structures will also be referred to as quantum heaps , as in the vector space case. Example 7.5.
Let X denote a Hopf monoid in C . Then, we can define anoperation akin to conjugation in a group, by means of the antipode S of X .We define q = µ ◦ ( ⊗ µ ) ◦ ( S ⊗ ⊗ ) (cid:1) ◦ ( ⊗ ∆), where (cid:1) corresponds tothe transposition (12) ∈ S . In the category of vector spaces or modules, i.e.when dealing with a Hopf algebra in the usual sense, this operation takesthe form x ⊗ y S ( y (1) ) xy (2) , where juxtaposition denotes multiplicationin X . We refer to this operation as quantum conjugation . Iterating theoperation q , i.e. defining T = q ◦ ( q ⊗ ) we obtain a ternary operation,called double quantum conjugation . In [8] Section 8, it is seen that q isbinary self-distributive, and it is also proven that composing binary self-distributive operations yields a TSD operation. Example 7.6.
Let C denote the category of vector spaces over a field , andlet L denote a Lie algebra. Set X = k ⊕ L and, denoting its elements bypairs ( a, x ), we can define a coproduct on X by the assingment( a, x ) ( a, x ) ⊗ (1 ,
0) + (1 , ⊗ (0 , x ) , and counit ǫ ( a, x ) = a . It is easy to see that this structure defines a coalgebrain C . Then, we apply Example 8.8 in [8] defining T by iteration of Lie bracket UANTUM INVARIANTS FROM TSD COHOMOLOGY 33 structure on basis vectors ( a, x ) by the assignment( a, x ) ⊗ ( b, y ) ⊗ ( c, z ) ( abc, bcx + c [ x, y ] + b [ x, z ] + [[ x, y ] , z ]) . Then T turns X into a TSD object in the category of vector spaces (proofin [8], Theorem 8.6 or Appendix A).In fact, it can be proved that the TSD structures of Example 7.6 areinvertible, and therefore are ternary racks. We will not consider these objectsin detail, leaving the proof of the previous claim to subsequent work, as theyare not cocommutative.7.3. Categorical 2-cocycle condition.
Let us now consider an I -linearsymmetric monoidal category C , where I denotes the unit object of C . Sup-pose X is a unitary comonoid object in C . This means that X is endowedwith morphisms ∆ : X −→ X ⊗ X , ǫ : X −→ I and η : I −→ X thatmake commute the diagrams defining a unitary coalgebra in the categoryof vector spaces. In this situation, we say that a morphism α : X −→ I is convolution invertible if there exists a morphism α − making the followingdiagram commute X ⊗ X ⊗ ⊗ X ⊗ I I ⊗ I (cid:1) ∆ ⊗ ǫ ⊗ α ⊗ α − where (cid:1) reorders the outputs of ∆ ⊗ as in the comultiplication of a tensorcoalgebra structure. Definition 7.7.
Let X be a TSD object in a symmetric monoidal category.Then a convolution invertible morphism α is called categorical 2-cocycle withcoefficients in I if the diagram X ⊗ X ⊗ I ⊗ X ⊗ I ⊗ I (cid:1) ◦ (∆ ) (cid:1) ◦ (∆ ∆ ) αα ◦ ( T ) αα ◦ ( T ) commutes, where (cid:1) = (2 , ,
5) and (cid:1) = (2 , , , , α ◦ ( η ⊗ ⊗ ) = α ◦ ( ⊗ η ⊗ ) = α ◦ ( ⊗ ⊗ η ) = ǫ ⊗ ǫ .We now illustrate the prevoius definitions in the category of vector spaces,using Sweedler notation to denote comultiplication. We will give examplesof the above structures later in the section, while we just assume in thefollowing example that such objects exist in the category of vector spaces. Example 7.8.
Suppose that C is the category of vector spaces over someground field k , and X is as above. We want to show how the commutativity of categorical 2-cocycle diagram translates in C . Using Sweedler notation,on a basis vector x ⊗ y ⊗ z ⊗ u ⊗ v we get α ( x (1) ⊗ y (1) ⊗ z (1) ) · α ( T ( x (2) ⊗ y (2) ⊗ z (2) ) ⊗ u ⊗ v )= α ( x (1) ⊗ u (1) ⊗ v (1) ) · α ( T ( x (2) ⊗ u (2) ⊗ v (2) ) ⊗ T ( y ⊗ u (3) ⊗ v (3) ) ⊗ T ( z ⊗ u (4) ⊗ v (4) ))where · indicates multiplication in k .The fact that α is convolution invertible in the “coefficients” object I playsa fundamental role in constructing inverses in the general construction, asit will be seen below.Lastly, we define an equivalence relation between categorical 2-cocycles. Definition 7.9.
Let α and β denote two categorical 2-cocycles. We saythat α and β are equivalent if there exists a convolution invertible morphism f : X −→ I such that the following diagram X ⊗ X ⊗ X X ⊗ ⊗ X ⊗ X ⊗ ⊗ X I (cid:1) ◦ ∆ ⊗ (234) ◦ (∆ ⊗ ⊗ ) α ⊗ ( f ◦ T ) β ⊗ f commutes.We show the previous definition in the category of vector spaces. Example 7.10.
Let (
X, T ) be a TSD object in the category of vector spaces.Then two categorical 2-cocycles α and β are equivalent, by definition, if itholds α ( x (1) ⊗ y (1) ⊗ z (1) ) f ( T ( x (2) ⊗ y (2) ⊗ z (2) )) = β ( x (1) ⊗ y ⊗ z ) f ( x (2) ),for some f , for all x, y, z ∈ X .7.4. Examples of categorical 2-cocycles.
We still need to provide exam-ples of categorical 2-cocycles, as per Definition 7.7. We begin by observingthat the setting of linearized operations and set-theoretic ternary 2-cocyclesof Section 4 provides first examples of such morphisms.
Example 7.11.
Let Q be a ternary (set-theoretic) quandle and let G be a(multiplicative) group. Suppose that α is a 2-cocycle, i.e. α ∈ Z ( Q, G ). Asin Section 4 we let X denote the linear space generated by the elements of Q and define T to be the linearized operation defined from the set-theoreticone of Q . Let χ : G −→ k × denote a group character. We define a linearmap ˜ α : X ⊗ X ⊗ X −→ k by the assignment on simple vectors x ⊗ y ⊗ z χα ( x, y, z ). Let us verify that ˜ α is indeed a categorical 2-cocycle. Since thediagonal in X is induced by the set-theoretic diagonal x x × x , applyingExample 7.8 it is enough to verify˜ α ( x ⊗ y ⊗ z ) · ˜ α ( T ( x ⊗ y ⊗ z ) ⊗ u ⊗ v )= ˜ α ( x ⊗ u ⊗ v ) · ˜ α ( T ( x ⊗ u ⊗ v ) ⊗ T ( y ⊗ u ⊗ v ) ⊗ T ( z ⊗ u ⊗ v ))which holds true using the definition of ˜ α and T as linearizations of set-theoretic structures. UANTUM INVARIANTS FROM TSD COHOMOLOGY 35
We consider now, temporarily, the binary version of the previous construc-tions. Let us set C to be the category of vector spaces over a ground field k .Recall, in this case, that a unital coalgebra ( X, ∆) of C along with a mor-phism of coalgebras q : X ⊗ X −→ X is said to be (binary) self-distributiveif the equality q ( q ( x ⊗ y ) ⊗ z ) = q ( q ( x ⊗ z (1) ) ⊗ q ( y ⊗ z (2) )) , holds for all x, y, z ∈ X . The (binary) categorical 2-cocycle condition for aconvolution invertible morphism σ : X ⊗ X −→ k is readily introduced asfollows α ( x (1) ⊗ y (1) ) α ( q ( x (2) ⊗ y (2) ) ⊗ z ) = α ( x (1) ⊗ z (1) ) α ( q ( x (2) ⊗ z (2) ) ⊗ q ( y ⊗ z (3) )) . As previously observed, in [8] Section 8, it has been proven that compos-ing binary self-distributive operations yields TSD operations. In the set-theoretic case, moreover, it is shown that binary 2-cocycles can be used toconstruct ternary 2-cocycles. We want to use a vector space version of theset-theoretic result to obtain examples of (ternary) categorical 2-cocycles.We consider the cocommutative case for simplicity.
Lemma 7.12.
In the setting above, suppose that α is a binary -cocycle.Then, defining ψ ( x ⊗ y ⊗ z ) := α ( x (1) ⊗ y (1) ) α ( q ( x (2) ⊗ y (2) ) ⊗ z ) , it followsthat ψ is a ternary -cocycle for the doubled TSD operation T = q ◦ ( q ⊗ ) .Proof. The proof of this fact follows the same lines of Remark 5.7 in [8],where the cocycles introduced in Remark 5.6 are assumed to coincide, andthe binary operations are taken both to be q . The geometric intepretationof the various applications of the 2-cocycle conditions are depicted as in Fig-ure 3 in [8], where extra care is to be taken to utilize the correct superscriptcorresponding to Sweedler notation for comultiplication. To prove that ψ isconvolution invertible, one needs the fact that q is a morphism of coalgebra,which is assumed by hypothesis. (cid:3) Example 7.13.
Let X be an involutory Hopf algebra over a ground field k , and let α denote a 2-cocycle with coefficients in k . Recall, from theintroduction of [1], or [17] Section 10.2.3, that this means that σ : X ⊗ X −→ k is convolution invertible, satisfies the equation σ ( x (1) ⊗ y (1) ) σ ( x (2) y (2) ⊗ z ) = σ ( x ⊗ y (1) z (1) ) σ ( y (2) ⊗ z (2) ) , for all x, y, z ∈ X , and it is normalized σ (1 ⊗ x ) = σ ( x ⊗
1) = ǫ ( x ). Let usconsider now the (double) quantum conjugation operation of Example 7.5.Then, assuming that the underlying coalgebra structure is cocommutativefor simplicity, one can verify that the morphism α : X ⊗ X −→ k defined as α ( x ⊗ y ) := σ ( x (1) ⊗ y (1) ) σ − ( y (2) ⊗ S ( y (3) ) x (2) y (4) ) satisfies the (binary) self-distributive cocycle condition. Applying Lemma 7.12 we obtain a (ternary)categorical 2-cocycle. In fact, the correspondence σ α is a quantumversion of the map given in [4], Theorem 7.1. This was originally motivatedby diagrammatic computations. Cocycles of Hopf algebras are used to twist the product structure and theantipode, to obtain a new Hopf algebra [1, 17]. It is known, and easily veri-fied, that when the underlying coalgebra structure is cocommutative, Hopf2-cocycles generate the same product of the Hopf algebra one starts with.Example 7.13 considers, in fact, a subclass of a family of 2-cocycles thatare referred to as invariant cocycles , or lazy cocycles [10]. These are definedas those 2-cocycles whose corresponding twisted Hopf algebra structure isunchanged. Invariant cocycles are defined up to an equivalence relation, andtheir equivalence classes constitute a group, as for the cohomology of groups,or algebras etc. It seems a relevant question to study equivalence classes ofcategorical 2-cocycles obtained from invariant cocycles. Moreover, as seenin the next section, categorical 2-cocycles are used to twist the Yang-Baxteroperator associated to a TSD object, and the ribbon category constructedbelow is well defined within an equivalence class. It might be of interestto relate the equivalence classes of invariant 2-cocycles to those of ribboncategories associated to TSD cocycles. We point out that we do not knowwhether it is possible to construct categorical TSD 2-cocycles from Hopf2-cocycles that are not invariant.7.5.
Construction of ribbon categories.
We define a general version ofribbon categories from TSD objects in symmetric monoidal categories. Thisconstruction gives the linearized definition in Section 4 when the symmetricmonoidal category is that of vector spaces over a field, the TSD object isdefined by linearizing a set-theoretic ternary rack/quandle, and the coeffi-cients are taken to be a subgroup of the units of the ground field, via a groupcharacter.Let C be an I -linear symmetric monoidal category, and let X be a ternaryself-distributive object in C . Let α be a categorical 2-cocycle in the sense ofDefinition 7.7. Then we define a braiding morphism c α , following the caseof linearized set-theoretic operations as follows c α , = ( ⊗ ⊗ ([ α ⊗ α ] ⊗ T ⊗ T )) (cid:1) c (∆ ⊗ ∆ ⊗ ) , where (cid:1) c is the morphism corresponding to the permutation σ = (cid:0) (cid:1) . Similarly we define a twisting morphism as follows θ α = ([ α ⊗ α ] ⊗ T ⊗ T ) (cid:1) θ (∆ ⊗ ) , where (cid:1) θ is the morphism corrsponding to the permutation σ = (cid:0) (cid:1) . Example 7.14.
We want to illustrate braiding and twisting morphisms,as given above, in the category of vector spaces. Let α be a categorical2-cocycle as in Example 7.8. We now explicitly give the form of switching UANTUM INVARIANTS FROM TSD COHOMOLOGY 37 and twisting morphisms on simple tensors. We have for c α , c α , ( x ⊗ y ⊗ ⊗ z ⊗ w ) = z (1) ⊗ w (1) ⊗ [ α ( x (1) ⊗ z (2) ⊗ w (2) ) · α ( y (1) ⊗ z (3) ⊗ w (3) )] · T ( x (2) ⊗ z (4) ⊗ w (4) ) ⊗ T ( y (2) ⊗ z (5) ⊗ w (5) ) , and for twisting morphism θ α θ α ( x ⊗ y ) = [ α ( x (1) ⊗ x (2) ⊗ y (2) ) · α ( y (1) ⊗ x (3) ⊗ y (3) )] · T ( x (4) ⊗ x (5) ⊗ y (5) ) ⊗ T ( y (4) ⊗ x (6) ⊗ y (6) ) . We suppose now, for simplicity, that the comultiplication of X is cocom-mutative. Although, in principle this condition can be weakened to someclasses of object whose comultiplication satisfies some special symmetriesin the next results, the proofs are significantly easier when dealing with acocommutative object. For instance, the comultiplication of Example 7.6is not cocommutative. It is possible to show nontheless that the followingconstructions hold. We consider this class of objects in a subsequent article. Definition 7.15.
Let C be an I -linear symmetric monoidal category withduals, ( X, T ) a TSD object in C , and α a categorical 2-cocycle of X withcoefficients in I . Let c α , and θ α be braiding a twisting morphisms, respec-tively, as previously defined in this section. Then we define R ∗ α ( X ) to be themonoidal category freely generated by X ⊗ X where morphisms are definedas in Definition 4.1 from c α , and θ α . Theorem 7.16. R ∗ α ( X ) is a ribbon category. Moreover, if α and β areequivalent, then R ∗ α ( X ) ∼ = R ∗ β ( X ) as ribbon categories. Before proceeding with the actual proof, we give some key steps of theproof in the setting of the category of vector spaces. This special caseis rather illuminating when considering the actual proof, and functions asprototype for the general case. We want to show that the braid equationholds, by computing the LHS and RHS on a simple tensor x ⊗ y ⊠ z ⊗ w ⊠ u ⊗ v ∈ X ⊗ ⊠ X ⊗ ⊠ X ⊗ . We obtain, using Sweedler’s notation, and applying one of the maps whose composition is the LHS of the braid equation x ⊗ y ⊠ z ⊗ w ⊠ u ⊗ v z (1) ⊗ w (1) ⊠ [ α ( x (1) ⊗ z (2) ⊗ w (2) ) α ( y (1) ⊗ z (3) ⊗ w (3) )] · T ( x (2) ⊗ z (4) ⊗ w (4) ) ⊗ T ( y (2) ⊗ z (5) ⊗ w (5) ) ⊠ u ⊗ v z (1) ⊗ w (1) ⊠ u (1) ⊗ v (1) ⊠ [ α ( T ( x (21) ⊗ z (41) ) ⊗ w (41) ⊗ u (2) ⊗ v (2) ) α ( T ( y (21) ⊗ z (51) ) ⊗ w (51) ⊗ u (3) ⊗ v (3) ) α ( x (1) ⊗ z (2) ⊗ w (2) ) α ( y (1) ⊗ z (3) ⊗ w (3) )] · T ( T ( x (22) ⊗ z (42) ⊗ w (42) ⊗ u (4) ⊗ v (4) ) ⊗ T ( T ( y (22) ⊗ z (52) ⊗ w (52) ⊗ u (5) ⊗ v (5) ) u (11) ⊗ v (11) ⊠ [ α ( z (11) ⊗ u (12) ⊗ v (12) ) α ( w (11) ⊗ u (13) ⊗ v (13) )] · T ( z (12) ⊗ u (14) ⊗ v (14) ) ⊗ T ( w (12) ⊗ u (15) ⊗ v (15) ) ⊠ [ α ( T ( x (21) ⊗ z (41) ) ⊗ w (41) ⊗ u (2) ⊗ v (2) ) α ( T ( y (21) ⊗ z (51) ) ⊗ w (51) ⊗ u (3) ⊗ v (3) ) α ( x (1) ⊗ z (2) ⊗ w (2) ) α ( y (1) ⊗ z (3) ⊗ w (3) )] · T ( T ( x (2) ⊗ z (42) ⊗ w (42) ⊗ u (4) ⊗ v (4) ) ⊗ T ( T ( y (22) ⊗ z (52) ⊗ w (52) ⊗ u (5) ⊗ v (5) ) . Let us now compute the RHS of the braid equation when evaluated on x ⊗ y ⊗ z ⊗ w ⊗ u ⊗ v . We do not write each step, but rather provide thefinal result( ⊠ c α , ) ◦ ( c α , ⊠ ) ◦ ( ⊠ c α , )( x ⊗ y ⊠ z ⊗ w ⊠ u ⊗ v )= u (11) ⊗ v (11) ⊠ [ α ( z (1) ⊗ u (2) ⊗ v (2) ) α ( w (1) ⊗ u (3) ⊗ v (3) )] · T ( z (21) ⊗ u (41) ⊗ v (41) ) ⊗ T ( w (21) ⊗ u (51) ⊗ v (51) ) ⊠ [ α ( T ( x (21) ⊗ u (141) ⊗ v (141) ) ⊗ T ( z (22) ⊗ u (42) ⊗ v (42) ) ⊗ T ( w (22) ⊗ u (52) ⊗ v (52) ))[ α ( T ( y (21) ⊗ u (151) ⊗ v (151) ) ⊗ T ( z (23) ⊗ u (43) ⊗ v (43) ) ⊗ T ( w (23) ⊗ u (53) ⊗ v (53) )) α ( x (1) ⊗ u (12) ⊗ v (12) ) α ( y (1) ⊗ u (13) ⊗ v (13) )] · T ( T ( x (12) ⊗ u (142) ⊗ v (142) ) ⊗ T ( z (24) ⊗ u (44) ⊗ v (44) ) ⊗ T ( w (24) ⊗ u (54) ⊗ v (54) )) ⊗ T ( T ( y (12) ⊗ u (152) ⊗ v (152) ) ⊗ T ( z (15) ⊗ u (45) ⊗ v (45) ) ⊗ T ( w (25) ⊗ u (55) ⊗ v (55) ))) . Now, comparing the two expressions we see that we cannot apply directlyternary self-distributivity of T and the categorical 2-cocycle condition to α , as the terms corresponding to comultiplications in Sweedler notation areshuffled differently. We can rearrenge them by means of the cocommutativityof X and then conclude that they coincide applying categorical 2-cocyclecondition and TSD property of T . UANTUM INVARIANTS FROM TSD COHOMOLOGY 39
A similar direct reasoning is applied to show that braiding morphism andtwisting commute. We will not add the details of this computation.The proof of Theorem 7.16 will consist of a similar reasoning, but replacingequations by commutative diagrams. We compare the expressions by writingthem as compositions of “all comultiplications”, “shuffle”, “ T ’s and α ’s” andfinally multiplying the α ’s by identifying two copies of the unit object I with I itself. Then using cocommutativity and naturality of shuffle morphismsin a symmetric monoidal category, one draws the conclusion that the twoterms are equal. Proof of Theorem 7.16.
As in the proof of Theorem 4.3, the crucial stepis to show that c α , satisfies the braid equation, and that θ α satisfies thecompatibility relation with respect to c α , . We proceed to prove the firstassertion. We need to show that the diagram X ⊗ ⊠ X ⊗ ⊠ X ⊗ X ⊗ ⊠ X ⊗ ⊠ X ⊗ X ⊗ ⊠ X ⊗ ⊠ X ⊗ X ⊗ ⊠ X ⊗ ⊠ X ⊗ X ⊗ ⊠ X ⊗ ⊠ X ⊗ X ⊗ ⊠ X ⊗ ⊠ X ⊗ c α , ⊠ ⊗ ⊗ ⊠ c α , ⊗ ⊠ c α , c α , ⊠ ⊗ c α , ⊠ ⊗ ⊗ ⊠ c α , is commutative. We will refer to the left perimeter, from top to bottom,of the preceding hexagon as the “left-hand side” of the braid diagram, orsimply LHS, and similarly for the right perimeter we will say the “right-hand side” or RHS of the braid diagram. In what follows we will keep usingthe same shortened notation in which a tensor product of maps of such as ⊗ ⊠ T ⊗ will be denoted by juxtaposition, T , where powers of tensorproducts of type ⊗ ⊗ in the exponents,although we keep the ⊠ symbol for clarity.The proof will consists of rewriting the LHS and the RHS of the braiddiagram in a convenient way, by using the axioms of the symmetric monoidacategory C , and the left Frobenius module axiom. Then we will argue thatthe two expressions coincide by means of the categorical 2-cocycle condition,and the rack axioms of T . We start with the LHS.Using the definition of c α , , the LHS of the braid diagram fits in thecommutative diagram of Figure 6. Now we rewrite the top-right perimeterof the previous diagram as in Figure 7, where we have used naturality ofthe shuffle morphisms in the category C , coassociativity of the morphism∆ and, in the bottom-left triangle, the fact that I commutes with objects X ⊠ X ⊠ X X ⊠ X ⊠ X ∼ = X ⊠ X ⊠ X X ⊠ X ⊠ X X ⊠ X ⊗ X ⊠ X X ⊠ X ⊠ X X ⊠ X ⊠ X X ⊠ X ⊠ X X ⊠ X ⊠ X c α , ) ◦ ( c α , ) ◦ ( c α , ) c , ∆ ∆
12 (cid:1) c α T µ c α , c α , Figure 6. of C . Similar considerations show that the diagram of Figure 8 commutes,where the horizontal maps are obtained from (appropriate) products of thecomultiplication morphism, and similarly for the hypotenuse of the centraltriangle. The morphism X ⊠ X ⊠ X −→ X ⊠ X ⊠ X is α T (( µ ◦ µ ) ◦ ( T ) ) T ◦ ( T ) . Pasting together the three diagrams along thecorresponding edges shows that the morphism ( c α , ⊠ ) ◦ ( ⊠ c α , ) ◦ ( c α , ⊠ ) can be rewritten by first aplying the comultiplication morphisms ∆ alltogether to produce the right number of copies of X , shuffle them in theright position and then apply all the morphisms T and α at the end. This isin fact better understood by thinking of the same proof for Hopf algebras. A(very tedious) direct inspection, simplified by diagrammatic reasoning, canbe applied to determine step by step the shuffles used in the commutativediagrams. Thus proceeding we see that we have obtained for the LHS of thebraid diagram ( c α , ⊠ ) ◦ ( ⊠ c α , ) ◦ ( c α , ⊠ ) = ( ⊗ ⊠ ( ◦ µ ⊗ ⊗ ) ⊠ ( µ ⊗ )) ◦ ( ⊠ α ⊗ T ⊗ ⊠ ( α ◦ ( T ⊗ ⊗ )) ⊗ ⊗ α ⊗ ⊗ ( T ◦ ( T ⊗ ⊗ ) ⊗ )) ◦ (cid:1) ◦ (∆ ⊗ ⊠ ∆ ⊗ ⊠ ∆ ⊗ ), where the shuffle morphism (cid:1) corresponds tothe permutation that rearrenges the terms in the last expression of thecomputation of ( c α , ⊠ ) ◦ ( ⊠ c α , ) ◦ ( c α , ⊠ ). A similar procedure shows thatthe RHS of the braid diagram is written as ( ⊠ c α , ) ◦ ( c α , ⊠ ) ◦ ( ⊠ c α , ) = UANTUM INVARIANTS FROM TSD COHOMOLOGY 41 X ⊠ X ⊠ X X ⊠ X ⊠ X X ⊠ X ⊠ X X ⊠ X ⊠ X X ⊠ X ⊠ X X ⊠ X ⊠ X X ⊠ X ⊗ X ⊠ X X ⊠ X ⊠ X X ⊠ I ⊗ X ⊠ X X ⊠ I ⊗ X ⊠ X X ⊠ X ⊠ X ⊗ I ⊗ X X ⊠ I ⊗ X ⊠ X X ⊠ X ⊠ X X ⊠ X ⊠ X X ⊠ X ⊠ X X ⊠ X ⊠ X X ⊠ X ⊠ X ∆ ∆ ∆ ∆ (∆ ) (∆ ) (cid:1)1 ∆ (cid:1) c (∆ ) (∆ ) ∆ ∆ α T α T (cid:1)1 ( T ) α ( T ) ∆ ∆ µ ( µ ) (cid:1) 1 µ ( ∆ ∆ ) (cid:1) c ( ( µ ) ) ◦ ( T ) ∆ ∆ c α , ( c α , ) ◦ ( c α , ) c α , Figure 7. ( ⊗ ⊠ ( µ ◦ α ⊗ ) ◦ T ⊗ ) ⊠ ( µ ◦ ( µ ◦ α ⊗ ⊗ µ ◦ α ⊗ ) ⊗ ( T ◦ T ⊗ ) ⊗ ) (cid:1) ◦ (∆ ⊗ ⊠ ∆ ⊗ ⊠ ∆ ⊗ ), where the shuffle morphism can be seen to coincide with themorphism corresponding to the permutation giving rise to the reordering ofelements given in the RHS of the computation preceding the present proof.To show that twist morphism θ α satisfies the compatibility condition withbraiding morphism c α , , we can proceed similarly. First we rewrite the twoperimeters of the diagram X ⊗ ⊠ X ⊗ X ⊗ ⊠ X ⊗ X ⊗ ⊠ X ⊗ X ⊗ ⊠ X ⊗ θ α ⊠ c α , c α , ⊠ θ α as composition of comultiplications in X and the appropriate shuffle, thenapplying the morphisms T , α and multiplication µ . Then we can comparethe two expressions by using TSD property of T and categorical 2-cocycle X ⊠ X ⊠ X X ⊠ X ⊠ X X ⊠ X ⊠ X X ⊠ X ⊠ X X ⊠ X ⊠ X X ⊠ X ⊠ X ⊗ I ⊗ X X ⊠ X ⊠ X ⊗ I ⊗ X X ⊠ X ⊠ X X ⊠ X ⊠ I ⊗ X X ⊠ X ⊠ I ⊗ X X ⊠ X ⊠ X (cid:1) 1 (∆ ) (∆ ) (cid:1)1 ( T ) α ( T ) ( µ ◦ µ ) α T µ ∆ ∆ c α , Figure 8. condition of α . Similar considerations are also applied to the diagram prov-ing that ( ⊠ θ α ) ◦ c α , = c α , ◦ ( θ α ⊠ ).We need to prove now that the morphism c α , has an inverse in the cat-egory C . Upon defining the morphism ˆ c α , := ( µ ⊗ ⊗ ⊠ ⊗ ) ◦ ( ˜ α ⊗ ⊠ ⊗ ) ◦ [( T ⊗ ) ⊗ ⊗ ( T ⊗ ⊠ ⊗ )] ◦ (cid:1) ◦ (∆ ⊗ ⊠ ∆ ⊗ ) where we have indi-cated ˜ α : X ⊗ −→ I the convolution inverse of α , and the shuffle morphismcorresponds to the permutation σ = (cid:0) (cid:1) , we see that ˆ c α , is a left and right inverse of c α , , which gives the requiredinvertibility. It is fundamental here, that α is convolution invertible.Once the fact that c α , satisfies the braid equation, the compatibilitycondition with θ α and invertibility, the rest of the proof proceeds as that ofTheorem 4.3, as the category R ∗ ( X ) is defined inductively as in Section 4.To show that, given equivalent 2-cocycles α and β , there exists an equiv-alence of ribbon categories R ∗ α ( X ) ∼ = R ∗ β ( X ) one proceeds as in the coun-terpart of the same proof of Theorem 4.3, where we use that a morphism UANTUM INVARIANTS FROM TSD COHOMOLOGY 43 f : X −→ I giving the equivalence between α and β is convolution invert-ible. (cid:3) Since over an algebraically closed fields of characteristic zero k , a co-commutative finite dimensional Hopf algebra X is isomorphic to the groupalgebra k [ G ] for some group G , in a sense, the construction of Section 4 isgeneral enough for TSD structures obtained via heaps. The ribbon cate-gory of Theorem 7.16 gives new results when applied to finite dimensionalHopf algebras over a ring/field that is not an algebraically closed field ofcharacteristic zero. Remark 7.17.
Observe that Theorem 7.16 generalizes the results of Sec-tion 4 to the setting of cohomology coefficients in a non-commutative group G , since from a 2-cocycle with coefficients in G and a group character χ : G −→ k × , we obtain a ribbon category in the category of vector spacesover k , where braiding and twisting are generated by c α , and θ α .8. Invariants of framed links in symmetric monoidal categories
Lastly, we describe in this section how the construction of Section 7, whosetwisting morphisms are revisited here, gives rise to quantum invariants offramed links in symmetric monoidal categories. This discussion follows stan-dard arguments, already introduced in Section 5 for instance.Recall that twists in a braided category, along with duals, uniquely de-termine a pivotal structure, i.e. an isomorphism of objects X with theirdouble duals X ∗∗ . The family of twists defined above, with the correspond-ing choice of pivotal structure can be used to define invariants of framedlinks. It is important to note, though, that duality of an object X in asymmetric monoidal category can be regarded as a finiteness condition thatensures the existence of quantum trace.In fact, for an important class of TSD objects, namely that of involutoryHopf monoids in a symmetric monoidal category, there often exists anothernatural choice of twists, and therefore pivotal structure, that can be usedto construct framed link invariants. Before explicitly giving the definitionof link invariants from the category R ∗ α ( X ), we discuss how to obtain newtwists for involutory Hopf algebras, and more generally Hopf monoids, undersome extra conditions.First recall that a Frobenius algebra ( X, µ, η, ∆ , ǫ ) in a symmetric monoidalcategory C is a bialgebra such that the following Frobenius axiom holds( ⊗ µ ) ◦ (∆ ⊗ ) = ∆ ◦ µ = ( µ ⊗ ) ◦ ( ⊗ ∆) . Given a Frobenius algebra X , we can define an (associative) pairing and acoparining, indicated as ∪ : X ⊗ X −→ I and ∩ : I −→ X ⊗ X respectively,as ∪ = ǫ ◦ µ and ∩ = ∆ ◦ η . Paring and coparing in a Frobenius algebrasatisfy the axiom( ⊗ ∪ ) ◦ ( ∩ ⊗ ) = = ( ∪ ⊗ ) ◦ ( ⊗ ∩ ) , which determines that ∪ is an associative non-degenerate pairing. It infact turns out that a monoid endowed with an associative pairing and acopairing satisfying the compatibility condition given above is equivalent tothe previous definition of Frobenius algebra.Suppose we are given an involutory Hopf algebra which is finitely gener-ated over a ground PID k . We know that a Frobenius algebra arises fromsuch a Hopf algebra, from traditional results of Larson and Sweedler [19].The Frobenius structure then gives an identification of X with its dual X ∗ and, consequently, a pivotal structure where X ∼ = X ∗∗ is obtained throughthe Frobenius pairing. Let us indicate by ∪ and ∩ the Frobenius pairings.Let us now consider an I -linear symmetric monoidal category C , and let X denote a (cocommutative) Hopf monoid in C . We define a braiding c usingthe quantum heap operation of X , as in the previous section. Suppose X hasan integral and a cointegral. These are, by definition, a point λ : I −→ X and a copoint γ : X −→ I , respectively, such that µ X ◦ ( λ ⊗ ) = ǫλ (1) ( γ ⊗ ) ◦ ∆ X = ηγ. (2)Then we can define a Frobenius structure on X with Frobenius pairing ∪ := γµ ( ⊗ S ), and copairing ∩ := ∆ λ . Such a construction also holdsfor finitely generated projective Hopf algebras (with trivial coinvariants),[22]. More generally, for an involutory Hopf monoid with (co)integrals ina symmetric monoidal category, under relatively mild assumptions, such as λγ = and λSγ = as in [6], the same constructions as above are stillvalid. It was shown in [25] that under such circumstances, the quantumheap braiding of X commutes with Frobenius paring and copairing. Wesummarize this in the following result. Lemma 8.1.
Let X denote a (cocommutative) Hopf monoid such that thereexist an integral λ and a cointegral γ such that setting ∪ = γµ ( ⊗ S ) and ∩ = ∆ λ , we obtain a Frobenius algebra structure on X . Then, we have thefollowing equalities ∪ ⊗ ⊗ = ( ⊗ ⊗ ∪ ) ◦ c ( ∪ ⊗ ⊗ ) ◦ c = ⊗ ⊗ ∪ . Similar conditions hold for ∩ and c .Proof. The diagrammatic proof in [25], Lemma 4.6, can be seen to apply inthis case as well. (cid:3)
We can now define θ := ( ⋒ ⊗ ⊗ ) ◦ ( ⊗ ⊗ c ) ◦ ( ⋓ ⊗ ⊗ ), where, bydefinition, we set ⋒ = ∪ ◦ ( ⊗ ∪ ⊗ ), and a similar definition holds for ⋓ .Consequently, it is easy to see (diagrammatically) that, applying Lemma 8.1,the braiding c and θ commute, in the sense that c ◦ ( ⊗ θ ) = ( θ ⊗ ) ◦ c and c ◦ ( θ ⊗ ) = ( ⊗ θ ) ◦ c . In fact, although the proofs in [25] are for algebrasin the category of modules, the diagrammatic proofs directly generalize to UANTUM INVARIANTS FROM TSD COHOMOLOGY 45 the case of Hopf-Frobenius algebras in more general symmetric monoidalcategories, such as those of [3, 6]. Now, defining a category ˜ R ∗ ( X ) as in Sec-tion 4, but replacing the twists with the new θ defined above and where thedeforming 2-cocycle is taken to be trivial, we have a proof analogous to thatof Theorem 7.16 implying that ˜ R ∗ ( X ) is a ribbon category, since braidingand twist satisfy the same coherence properties of those of Theorem 7.16.Let now L be a framed link and let, in the same notation of Section 5, b ( L ) = t k · · · t k n n τ be a framed braid whose closure is L , where τ ∈ B n .Decompose τ into a product of generators σ i of B n , namely τ = σ q i · · · σ q r i r for some positive integers r, q , . . . q r and 1 ≤ i , . . . , i r , ≤ n − r .We construct an endomorphism Φ L of X ⊗ n assigning to each power of σ j appearing in the decomposition of τ into product of generators of B n , themorpshism ⊗ ⊗ · · · ⊗ c α , ⊗ · · · ⊗ ⊗ , where c α , is applied to the copies of X ⊗ at positions i and i + 1. From the discussion in the previous paragraphit follows that Φ L so defined is an invariant of the link L . Analogously, thequantum trace of Φ L , Ψ L := tr q (Φ L ) is an invariant of L . Remark 8.2.
Observe that such a construction of quantum invariants alsoapplies to a Hopf-Frobenius structure where the twist is defined by means ofpairing and copairing morphisms, upon exchanging c α , and θ α with c and θ given above.We conclude by briefly turning our attention to the categorical version ofcompatible TSD systems encountered in Section 4. In fact, due to a currentlack of concrete examples, we have not explicitly mentioned such structuresin Section 7. A possible source of examples, namely that of ternary aug-mented racks, is discussed in the appendix below. The definitions of com-patible systems translate the definitions given in Section 4 in the setting ofsymmetric monoidal categories and allow to construct a braided monoidalcategory with certain base objects { X i } i ∈ I generating the object family, andmorpshims obtained inductively by combining all the possible compositionsand tensoring of braidings X i ⊗ X i ⊗ X j ⊗ X j . In order to obtain a quantuminvariant of framed links we proceed as in the basic case given above, butreplacing Φ L by a superposition of endomorphisms (hence we need to addthe additional requirement of abelian monoidal category, or with k -linearstructure), where the sum runs over all possible colorings X i ⊗ X i for eachdouble string of the framed braid chosen to represent the link L . Appendix A. Examples from ternary augmented racks
It has been observed in Section 4 that mutually distributive operationsprovide examples of compatible systems of self-distributive structures. Theassociated ribbon categories as in Theorem 4.12 have larger Hom sets thanribbon categories arising from a single ternary self-distributive structure,but consist of the same objects, i.e. a distinguished object X ⊗ with allthe tensor products arising from it. This class of examples still leaves open the question of whether there exist compatible systems with different basespaces, i.e. with X i = X j for some i and j . A possible answer comes fromternary augmented shelves [8]. We work in the category of vector spaces, butwe observe that our example is obtained via linearization of a set-theoreticstructure as given in Section 4. A related construction that is not obtainedfrom linearized operations can be performed, in the setting of symmetricmonoidal categories, by adapting the definitions in Section 4 in a spiritsimilar to those given in Section7. We believe that this case is of interest,but unfortunately we are not aware of an example of such a “non-linearized”structure at this time. We have therefore decided to include it hereby as anappendix.Suppose X and X are coaglebras and H is a Hopf algebra that acts onthem and, therefore, acts on X ⊗ i via comultiplication, for i = 1 ,
2. Suppose,further, that there exist morphisms of coalgebras p i : X ⊗ i −→ H satisfying p i ( z · ∆( h )) = S ( h (1) ) p i ( z ) h (2) , for every z ∈ X i and h ∈ H . Then the following result, extending TheoremB.6 in [8], shows that in this situation a compatible system arises naturally. Theorem A.1.
Let X , X and H be as above, with related morphisms p and p . Then, defining maps T ij : X i ⊗ X j ⊗ X j −→ X i by extending the assignment x ⊗ y ⊗ y x · p j ( y ⊗ y ) by linearity, itfollows that { X i , T ij } i,j =1 , is a compatible system of ternary self-distributivestructures. A similar result holds for a finite number of H modules X i , i = 1 , . . . , n with morphisms p i .Proof. The proof is analogous to that of Theorem B.6 in [8] and it is includedhere to show that having two morphisms p and p does not affect theprocedure. It is enough to prove compatibility condition on simple tensors x ⊗ y ⊗ y ⊗ z ⊗ z ∈ X i ⊗ X j ⊗ X j ⊗ X k ⊗ X k , where i, j, k = 1 ,
2, as follows T ij ( T ik ( x ⊗ z (1)1 ⊗ z (1)2 ) ⊗ T jk ( y ⊗ z (2)1 ⊗ z (2)2 ⊗ T jk ( y ⊗ z (3)1 ⊗ z (3)2 ))= ( x · p k ( z (1)1 ⊗ z (1)2 )) · p i ( y · p k ( z (2)1 ⊗ z (2)2 ) ⊗ y · p k ( z (3)1 ⊗ z (3)2 ))= x · [ p k ( z (1)1 ⊗ z (1)2 ) · p j (( y ⊗ y ) · ∆( p k ( z ⊗ z ) (2) )]= x · [ p k ( z ⊗ z ) (1) · S ( p k ( z ⊗ z ) (2) ) · p j ( y ⊗ y ) · p k ( z ⊗ z ) (3) ]= x · [ ǫ ( p k ( z ⊗ z ) (1) )1 · p j ( y ⊗ y ) · p k ( z ⊗ z ) ]= x · [ p j ( y ⊗ y ) · p k ( z ⊗ z )]= ( x · p j ( y ⊗ y )) · p k ( z ⊗ z )= T ik ( T ij ( x ⊗ y ⊗ y ) ⊗ z ⊗ z ) , where in the second equality has been used the fact that p k is a morphism ofcoalgebras, the third equality is obtained applying the defining property of p j and, finally, antipode axiom, counit axiom, associativity of the action of H UANTUM INVARIANTS FROM TSD COHOMOLOGY 47 and definition of maps T ij are used in the last four equalities. Coassociativityand associativity in H are used throughout, without explicit mention. (cid:3) Example A.2.
Let G n denote the cyclic group of order n in multiplicativenotation, therefore an element of G n is denoted by the symbol x k for some k determined by reduction modulo n . Let m and m be positive integers with( m , m ) = 1 and define G nm , G nm as above, with generators denoted bythe symbols y and y respectively. Then the group algebra H := k [ G n ]acts on X i := k [ G nm i ] via the map x k y m i ki and multiplication in G nm i .Define maps p i : X i ⊗ X i −→ H extending by linearity the assignment y k i ⊗ y k i x k − k . Then p i ( y k i ⊗ y k i · ∆( x k )) = p i ( y nk + k i ⊗ y nk + k i )= x m i k + k − m i k − k = x − k x k − k x k = S ( x k ) p i ( y k i ⊗ y k i ) x k . Moreover, since ( m , m ) = 1, it follows that X and X are not subrepre-sentations of each others, so that the compatible system arising from Theo-rem A.1 consists of two base objects that are independent as representationsof H . Example A.3.
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Institute of Mathematics and Statistics, University of Tartu, Narva mnt18, 51009 Tartu, Estonia
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