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Topological invariants from non-restricted quantum groups

We introduce the notion of a relative spherical category. We prove that such a category gives rise to the generalized Kashaev and Turaev-Viro-type 3-manifold invariants defined in arXiv:1008.3103 and arXiv:0910.1624, respectively. In this case we show that these invariants are equal and extend to what we call a relative Homotopy Quantum Field Theory which is a branch of the Topological Quantum Field Theory founded by E. Witten and M. Atiyah. Our main examples of relative spherical categories are the categories of finite dimensional weight modules over non-restricted quantum groups considered by C. De Concini, V. Kac, C. Procesi, N. Reshetikhin and M. Rosso. These categories are not semi-simple and have an infinite number of non-isomorphic irreducible modules all having vanishing quantum dimensions. We also show that these categories have associated ribbon categories which gives rise to re-normalized link invariants. In the case of sl(2) these link invariants are the Alexander-type multivariable invariants defined by Y. Akutsu, T. Deguchi, and T. Ohtsuki.


Introduction
A principal feature of quantum topology is its interplay between tensor categories and lowdimensional topology. The fundamental example of such interplay is the modular category formed from finite dimensional representation of the restricted quantum group U q (sl 2 ) at a root of unity and the corresponding Reshetikhin-Turaev and Turaev-Viro 3-manifold invariants. Loosely speaking, this modular category is the category of all representations over U q (sl 2 ) quotient the ideal of representation with zero quantum dimension. This category is semi-simple with a finite number of isomorphism classes of irreducible representations all having non-vanishing quantum dimensions. Generalizations to other semi-simple categories were done by several authors including H. Andersen, V. Turaev, H. Wenzl, J. Barrett, B. Westbury, A. Ocneanu, S. Gelfand, D. Kazhdan, and others.
Less progress has been made in categories which are not semi-simple and have objects with vanishing quantum dimensions. This is true even though the Volume Conjecture naturally arises in such a setting (see [23]). In the case of sl 2 , 3-manifold invariants arising from representations with zero quantum dimensions have been studied by two approaches: 1) R. Kashaev's geometric and physical construction in [20,21], which was extended by S. Baseilhac and R. Benedetti [2] and 2) the topological and algebraic Turaev-Viro-type invariants of N. Geer, B. Patureau-Mirand, V. Turaev [17]. In [12], the notion of aΨ-system is used to generalize Kashaev's construction to a categorical setting. This paper has five main results: First, we show that the non-restricted quantum groups associated to simple Lie algebras considered by C. De Concini and V. Kac, in [8], admit Turaev-Viro-type invariants arising from representations with zero quantum dimensions. Second, we prove that such quantum groups also lead toΨ-systems and so from [12] admit generalized Kashaev invariants. Third, we show that in this situation the generalized Kashaev and TV-type invariants are equal. Fourth, we extend the above mentioned TV-type invariants to a kind of Homotopy Quantum Field Theory. Finally, we prove that certain representations (with zero quantum dimensions) over nonrestricted quantum groups give rise to re-normalized Reshetikhin-Turaev link invariants. All of these results are proved in a general categorical language which also contains as examples the usual modular categories formed from representations of restricted quantum groups. This paper has two major components. The aim of first component is to introduce and study "relative spherical categories." These categories are generalizations of the usual modular categories associated to restricted quantum groups (see Theorem 4). However, in general they are not necessarily semi-simple and can have an infinite number of non-isomorphic simple objects all having vanishing quantum dimensions. In Subsection 3.4, we will show that a relative spherical category C leads to a Turaev-Viro-type invariant T V of triples (a closed oriented 3-manifold M , a link in M , a conjugacy class of homomorphisms π 1 (M ) → G) where G is a group which is part of the data of the category. Then in Subsection 3.6 we will also show that C gives rise to aΨ-system. Thus, the construction of [12] gives a Kashaev type invariant K and in Subsection 3.6 we prove that T V = K. The final piece of the first component of this paper is to show that T V can be extended to what we call a "relative Homotopy Quantum Field Theory" (relative HQFT), see Section 4. This relative HQFT is similar in nature to Turaev's [29] Homotopy Quantum Field Theory which is a branch of Topological Quantum Field Theory founded by E. Witten and M. Atiyah. However, we have cobordism with graphs inside. Our relative HQFT is also similar to the Quantum Hyperbolic Field Theory coming from the Borel subalgebra of quantum sl 2 defined in [3]. One can hope to make a precise relationship with these different field theories.
The aim of the second component of this paper is to construct invariants of links and 3-manifolds from representations over non-restricted quantum groups which are not semi-simple and have vanishing quantum dimensions. In the series of papers [8]- [11], C. De Concini, V. Kac, C. Procesi, N. Reshetikhin and M. Rosso establish a deep body of work pertaining to quantum groups and their representation theory, at odd ordered roots of unity. In Section 5 we recall some of their results and prove that the representations studied in these papers give rise to three topological invariants: T V , K and a Reshetikhin-Turaev or Akutsu-Deguchi-Ohtsuki type link invariant (see Theorem 7). The representations considered here have zero quantum dimensions and so the usual associated topological invariants are trivial. To overcome this difficultly we show that these representations are part of an ambidextrous pair with modified dimension d. This modified dimension is used to renormalize the usual link invariants and the quantum 6j-symbols. Let us be more precise. For an odd ordered root of unity, let U and U H be the quantum groups associated to a fixed simple Lie algebra, described in Subsection 5.1. We prove that the finite dimensional weight modules over U H contain a ribbon category which admits an ambidextrous pair. Combining this with a direct calculation of the open Hopf link we prove that the results of [16] can be applied and lead to link invariants (see Theorem 7 and Proposition 44). Then incorporating all the results of Section 5 we show that the finite dimensional weight modules over U form a relative spherical category and so the results described in the previous paragraph can be applied. In particular, this shows that the category of finite dimensional weight modules over U has several Ψ-systems (giving a positive answer to Conjecture 42 of [12] for U) and proves that the corresponding generalized Kashaev and Turaev-Viro-type 3-manifold invariants are equal.
The work of N. Geer was partially supported by the NSF grants DMS-0706725 and DMS-1007197. B. Patureau thanks Utah State University and the members of its mathematics department for their generous hospitality.

G-spherical categories andΨ-systems
In this section we give the basic categorical definitions used in this paper.
1.1. Linear tensor categories. A tensor category C is a category equipped with a covariant bifunctor ⊗ : C × C → C called the tensor product, an associativity constraint, a unit object I, and left and right unit constraints such that the Triangle and Pentagon Axioms hold. Let k be an integral domain. A tensor category C is said to be k-linear if its hom-sets are k-modules, the composition and tensor product of morphisms are k-bilinear, and End C (I) is a free k-module of rank one. Then we identify k = End C (I), via the ring isomorphism k → End C (I), k → k Id I and call k the ground ring of C. Let k × be the set of invertible elements of k. An object V of C is simple if End C (V ) = k Id V . An object which is a direct sum of simple objects is called semi-simple. For any simple object V and f ∈ End(V ), there is a unique x ∈ k such that f = x Id V . This x is denoted f .
If C is an k-linear category and {V i } i∈I is a set of simple objects numerated by a set I, for i, j, k ∈ I we consider the following k-modules . A pivotal category is a tensor category with duality morphisms b V : I → V ⊗V * , d V : V * ⊗V → I, b V : I → V * ⊗ V and d V : V ⊗ V * → I which satisfy compatibility conditions (see for example [17]).
Let C be a pivotal category. A morphism f : V 1 ⊗ · · · ⊗ V n → W 1 ⊗ · · · ⊗ W m in C can be represented by a box and arrows: Boxes as above are called coupons. By a C-colored ribbon graph in an oriented surface Σ, we mean a graph embedded in Σ whose edges are colored by objects of C and whose vertices lying in Int Σ = Σ \ ∂Σ are thickened to coupons colored by morphisms of C. Let Gr C be the category of C-colored ribbon graphs in R × [0, 1] and G : Gr C → C be the Reshetikhin-Turaev k-linear functor preserving the duality morphism of C (see [17]).
A ribbon category is a pivotal category with a braiding c V,W : V ⊗ W → W ⊗ V and twist θ V : V → V which are compatible with the pivotal structure (see [28,Chapter 1]). If C is a ribbon category, the Reshetikhin-Turaev functor G extend as G : Rib C → C where Rib C is the category of C-colored ribbon graphs in R 2 × [0, 1].

Ambidextrous pairs and traces.
We recall the process of re-normalizing colored ribbon graphs first used in [14,15] then generalized using the notion of ambidextrous entities in [13,16,17,18].
Let C be a k-linear pivotal (resp., ribbon) category and let T ⊂ S 2 (resp., T ⊂ S 3 ) be a closed C-colored ribbon graph. Let e be an edge of T colored with a simple object V of C. Cutting T at a point of e, we obtain a C-colored ribbon graph T V in R × [0, 1] (resp., in R 2 × [0, 1]) where G(T V ) ∈ End(V ) = k Id V . We call T V a cutting presentation of T and let T V ∈ k denote the isotopy invariant of T V defined from the equality G(T V ) = T V Id V .
Let A be a class of simple objects of C and d : Let T be a closed C-colored ribbon graph which admit two cutting presentations T V and T V with both V and V in A. Then we consider if the following equality holds: First, suppose that C is a k-linear ribbon category and let L A be the class of C-colored ribbon graphs in S 3 with at least one edge colored by an element of A. Suppose that Equation (1) holds for all T ∈ L A . Then we say that (A, d) is an ambidextrous pair or ambi for short.
On the other hand, suppose that C is a k-linear pivotal category. In [17] a smaller class of graphs is considered: A ribbon graph is trivalent if all its coupons are adjacent to 3 half-edges. Let T A be the class of C-colored connected trivalent ribbon graphs in S 2 such that the colors of all edges belong to A. Suppose that Equation (1) holds for all T ∈ T A . Then we say that (A, d) is a trivalent-ambidextrous pair or t-ambi for short.
If (A, d) is ambi (resp., t-ambi) we define a function G : L A → k (resp., G : where T V is any cutting presentation of T with V ∈ A. The above definitions imply that G is well defined. With Equation (2) in mind, we call d a modified quantum dimension. Let Proj be the full subcategory of C consisting of projective objects. A trace on Proj is a family of k-linear functions t = {t V : End C (V ) → k} V ∈Proj which is suitably compatible with the tensor product and composition of morphisms (see [13,18]). Let A Proj be the set of simple projective objects of C.
ambi and the corresponding isotopy invariant G is determined by the trace: where T V is any cutting presentation of T (see [13,18]).
1.3. G-spherical categories. We now fix a group G.
Definition 1. A pivotal category is G-graded if for each g ∈ G we have a non-empty full subcategory if V ∈ C g , V ∈ C g and Hom C (V, V ) = 0, then g = g .
Let X ⊂ G be a subset with the following properties: (1) X is symmetric: G can not be covered by a finite number of translated copies of X , in other words, for any g 1 , . . . , g n ∈ G, we have n i=1 (g i X ) = G. Let us write G = G \ X . Definition 2. Let C be a G-graded k-linear pivotal category. Let A be the class of all simple objects in g∈G C g . We say that C is (X , d)-relative G-spherical if (1) for each g ∈ G , C g is semi-simple (i.e. all objects of C g are semi-simple) with finitely many isomorphism classes of simple objects, for any isomorphic objects V, V ∈ A and for any g 1 , g 2 , g 1 g 2 ∈ G \ X and V ∈ G g 1 g 2 we have where irr(C g i ) denotes a representing set of the isomorphism classes of simple objects of D g i . If C is a category with the above data, for brevity we say C is a relative G-spherical category.
The map b always exists when k is a field of characteristic 0 and C is a category whose objects are finite dimensional k-vector spaces. In particular, in [17] it is shown that, for any g ∈ G , the map is well defined and satisfies all the properties above. A representative set for A is a family {V i } i∈I of simple objects of A numerated by elements of a set I such that any element of A is isomorphic to a unique element of {V i } i∈I . Let I → I, i → i * be the involution determined by V i * V * i . For each g ∈ G let I g = {i ∈ I : V i ∈ C g } then I = g∈G I g . For i ∈ I g , we call i = g the degree of i.
By basic data in C we mean a representative set {V i } i∈I for A and a family of isomorphisms Lemma 3. If no object of A is isomorphic to its dual, then C contains a basic data. In particular, basic data exists if X contains the set {g ∈ G : g = g −1 }.
Proof. Choose a representative for each isomorphism class of simple objects of g∈G C g . Let these representatives {V i } i∈I be numerated by a set I. Then I is a representative set for A. The hypothesis of the lemma imply that i * = i for all i ∈ I. Hence, for any unordered pair (i, i * ), we can take an arbitrary isomorphism V i → (V i * ) * for w i and choose w i * so that it satisfies Equation (4). For the second statement, let g ∈ G and let V be a simple object of C g . Since g −1 = g, the G-grading of C implies that V and V * are not isomorphic. Thus, we can apply the above argument.
If C is a (X , d)-relative G-spherical category with basic data {V i , w i } i∈I then the functions d : A → k and b : A → k can and will be considered as functions from I to k given by d(i) = d(V i ) and b(i) = b(V i ), respectively.
The following theorem shows that relative G-spherical categories are generalizations of the usual modular categories associated to restricted quantum groups. Theorem 4. LetŨ q (g) be the restricted quantum group associated to a simple Lie algebra g of type A, B, C, D, at a primitive root of unity q of even order (see [28,XI.6.3]). Let C be the modular category formed from finite dimensional representations ofŨ q (g) modulo negligible morphisms (see [28,XI]). Then C is a (∅, qdim)-relative G-spherical category with basic data where qdim is the usual quantum dimension of C and G = {1} is the trivial group of one element. Here the map b can be chosen to be the map defined in (3) or the map defined by (3) where the dimension dim k is replaced with the usual quantum dimension qdim.
Proof. First, C is a C-linear pivotal category (see [28,XI]). We assign C 1 = C and then by definition C is G-graded. Also, C is semi-simple with finitely many isomorphism classes of simple objects and so Property 1 of Definition 2 holds. Since qdim is non zero for each simple object of C then Equation (1) holds by definition of G and so (A, qdim) is a t-ambi pair. Finally, it is shown in [31] that basic data exists.
Remark 5. In Subsection 3.4 we show that a relative G-spherical category gives rise to a modified Turaev-Viro invariant. Let C be the modular category of Theorem 4. The original Turaev-Viro invariant is not equal to the modified invariant corresponding to C where b is defined as in Equation (3). However, the original Turaev-Viro invariant is equal to the modified invariant arising from C when b is taken to be the map given by (3) where the dimension dim k is replaced with the usual quantum dimension qdim. In this case, the modified invariant does not depend on the graph in the 3-manifold.
1.4.Ψ-systems. A Ψ-system in a k-linear category C consists of (1) a distinguished set of simple objects {V i } i∈I such that Hom(V i , V j ) = 0 for all i = j; (2) an involution I → I, i → i * ; (3) two families of morphisms {b i : (4) for any i, j ∈ I such that H ij k = 0 for some k ∈ I, the identity morphism Consider the k-module H =Ĥ ⊕Ȟ, wherê Let π k ij : H → H k ij , π ij k : H → H ij k ,π : H →Ĥ,π : H →Ȟ be the obvious projections. The k-module H has a symmetric bilinear pairing , given by for any x, y ∈ H. A transpose of f ∈ End(H) is a map f * ∈ End(H) such that f x, y = x, f * y for all x, y ∈ H. Note that if a transpose f * of f exists, then it is unique and (f * ) * = f (see [12]). An operator f ∈ End(H) such that f * = f is called symmetric. An operator f ∈ End(H) such that f (H ij k ) ⊂ H ij k and f (H k ij ) ⊂ H k ij for all i, j, k ∈ I is called grading-preserving. We define linear maps A, B : H → H by In [12], it is shown that both A and B are involutive and have transposes. Also, from [12] we have that the operators L = A * A, R = B * B, C = (AB) 3 ∈ End(H) are symmetric, grading-preserving and invertible. Definition 6 ( [12]). AΨ-system in C is a Ψ-system in C together with a choice of invertible, symmetric, grading-preserving operators C T R where L 1 2 is defined to be the invertible operator L 1 2 = BAR − 1 2 AB ∈ End(H).

Modified Reshetikhin-Turaev link invariants from projective ambidextrous objects
Let C be a pivotal k-linear category with duality morphisms b V : For every morphism f : V → W in C let f * : W * → V * be the dual (or transposed) morphism given by The axioms of a pivotal category imply that for each object V of C there is a canonical functorial isomorphism V → V * * , see [18]. To simplify notation we will use this isomorphism to identify V * * with V . Therefore, for any object V of C, we have [18] it is shown that when C is ribbon this definition is equivalent to the definition of ambidextrous given in [16].
Theorem 7. Let C be a ribbon category and let A Proj be the set of simple projective objects of C. Suppose that there is a projective ambidextrous object in C. Then there is an unique (up to a scalar of k × ) map d : A Proj → k × such that (A Proj , d) is an ambidextrous pair. Hence the map G : L A Proj → k given in Subsection 1.2 is a well defined isotopy invariant.
Proof. From Theorem 3.3.2 of [13] we have the existence and uniqueness of a trace on the subcategory of projective objects Proj. As in Subsection 1.2 this trace defines a function d : A Proj → k × . The properties of the trace imply that d(V ) = d(V * ) for all V ∈ A Proj and that (A Proj , d) is an ambi pair. The uniqueness of the trace implies the uniqueness of d.
Let J be a projective ambidextrous object in a ribbon category C. Following [16], for any simple objects V, W of C, define S (V, W ) = V W then for any V ∈ A Proj with S (J, V ) invertible, we have that is the value of G on the Hopf link colored by J and V . There is a large class of examples where the category C is not semi-simple and the usual Reshetikhin-Turaev ribbon functor G restricted to the domain L A Proj of G is zero. For example, let g be a simple Lie algebra. In Section 5, for each rth root of unity e 2 iπ r , we define an algebra U H which we call the unrolled Drinfeld-Jimbo quantum group associated to g. Then in Subsection 5.3, we introduce a ribbon category D θ of nilpotent U H -weight modules which satisfy the hypothesis of Theorem 7. In particular, we use S to give an explicit formula for the modified quantum dimension d in this case (see Theorem 45). The corresponding invariant G restricted to framed links colored with projective simple modules can be considered as a generalized colored Alexander invariant which in the case of sl(2) is the hierarchy of invariants defined in [1] (see Theorem 35 of [16]).
In [7] we will extend the link invariants of Theorem 7 to a modified Reshetikhin-Turaev-type 3-manifold invariant. More precisely, in [7] we will give a notion of a relative G-modular category and show that such categories have Kirby colors which give rise to 3-manifold invariants.

Topological invariants from relative G-spherical categories
In this section we will introduce a state sum (15) of 6j-symbol associated to a relative spherical category. This sum is a topological invariant called modified Turaev-Viro invariant (cf. Theorem 15). Then in Subsection 3.6 we will show that a relative spherical category inducesΨ-structures (cf. Theorem 19) and that the associated Kashaev-type state sums (18) are equal to the modified Turaev-Viro invariant (cf. Theorem 20).
3.1. Modified 6j-symbols. In [17] the authors show that certain pivotal categories give rise to modified 6j-symbols. In this subsection, we show the techniques of [17] can be applied to the situation of relative spherical categories.
Let C be a (X , d)-relative G-spherical category with basic data {V i , w i } i∈I . For any i, j, k ∈ I, recall the notation H ijk = Hom(I, V i ⊗ V j ⊗ V k ). The k-modules H ijk , H jki , H kij are canonically isomorphic. Indeed, let σ(i, j, k) be the isomorphism Using the functor G : Gr C → C, one easily proves that Identifying the modules H ijk , H jki , H kij along these isomorphisms we obtain a symmetrized multiplicity module H(i, j, k) depending only on the cyclically ordered triple (i, j, k). Remark that the G-grading of C implies that for all i, j, k ∈ I such that i j k = 1 ∈ G, then H(i, j, k) = {0}.
By a labeling of a graph we mean a function assigning to every edge of the graph an element of I. By a trivalent graph we mean a (finite oriented) graph whose vertices all have valency 3. Let Γ be a labeled trivalent graph in S 2 . Using the standard orientation of S 2 (induced by the right-handed orientation of the unit ball in R 3 ), we cyclically order the set X v of 3 half-edges adjacent to any given vertex v of Γ. The labels of the edges determine a function f v : X v → I as follows: if a halfedge e adjacent to v is oriented towards v, then f v (e) = i is the label of the edge of Γ containing e; if a half-edge e adjacent to v is oriented away from v, where v runs over all vertices of Γ.
Consider now a labeled trivalent graph Γ ⊂ S 2 endowed with a family of vectors where v runs over all vertices of Γ. We thicken Γ into a C-colored ribbon graph on S 2 as follows. First, we insert inside each edge e of Γ a coupon with one edge outgoing from the bottom along e and with one edge outgoing from the top along e in the direction opposite to the one on e. If e is labeled with i ∈ I, then these two new (smaller) edges are labeled with V i and V i * , respectively, and the coupon is labeled with w i : V i → V * i * as in Figure 1.

Figure 1
Next, we thicken each vertex v of Γ to a coupon so that the three half-edges adjacent to v yield three arrows adjacent to the top side of the coupon and oriented towards it. If i, j, k ∈ I are the labels of these arrows (enumerated from the left to the right), then we color this coupon with the image of h v under the natural isomorphism H v (Γ) → H ijk . Denote the resulting C-colored ribbon graph by Ω Γ,h . Then G(Γ, h) = G(Ω Γ,h ) is an isotopy invariant of the pair (Γ, h) independent of the way in which the vertices of Γ are thickened to coupons.
The invariant G defined in Section 1.3 can be be extended to a bigger class of C-colored ribbon graphs in S 2 . We say that a coupon of a ribbon graph is straight if both its bottom and top sides are incident to exactly one arrow. We can remove a straight coupon and unite the incident arrows into a (longer) edge, see Figure 2. We call this operation straightening. A quasi-trivalent ribbon graph is a ribbon graph in S 2 such that straightening it at all straight coupons we obtain a trivalent ribbon graph. −→ Figure 2. Straightening a quasi-trivalent ribbon graph Lemma 8. Let T I be the class of connected quasi-trivalent ribbon graphs in S 2 such that the colors of all edges belong to the set {V i } i∈I and all straight coupons are colored with isomorphisms in C. Then Formula (2) determines a well-defined function G : Proof. This is proved in Lemma 2 of [17].
We can combine the invariant G with the thickening of trivalent graphs to obtain invariants of trivalent graphs in S 2 . Suppose that Γ ⊂ S 2 is a labeled connected trivalent graph. We define where v runs over all vertices of Γ. The C-colored ribbon graph Ω Γ,h constructed above belongs to the class T I defined in Lemma 8. Set By the properties of G , the vector G (Γ) ∈ H(Γ) is an isotopy invariant of Γ. Both H(Γ) and G (Γ) are preserved under the reversion transformation inverting the orientation of an edge of Γ and replacing the label of this edge, i, with i * . This can be easily deduced from Formula (4).
We define the modified 6j-symbol of the tuple (i, j, k, l, m, n) to be i j k l m n = G (Γ) ∈ H(Γ) .
It follows from the definitions that the modified 6j-symbols have the symmetries of an oriented tetrahedron. In particular, These equalities hold because the labeled trivalent graphs in S 2 defining these 6j-symbols are related by isotopies and reversion transformations described above. Also, it follows from Equation (7) that where x ∈ H(i, j, k) = H ijk and y ∈ H(k * , j * , i * ) = H k * j * i * and Θ = Θ i,j,k is the theta graph with vertices u, v and three edges oriented from v to u and labeled with i, j, k, see Figure 3b. Clearly, so that we can use x, y as the colors of u, v, respectively. It follows from the definitions that the pairing (, ) ijk is invariant under cyclic permutations of i, j, k and (x, Given indices i, j, k ∈ I 3 and a tensor product of several k-modules such that among the factors there is a matched pair H(i, j, k), H(k * , j * , i * ), we may contract any element of this tensor product using the pairing (10). This operation is called the contraction along H(i, j, k) and denoted by * ijk . For example, an element The rest of this section contains properties of the modified 3j and 6j-symbols defined above.
Lemma 9. For any elements i, j, k of I, the pairing (10) is non-degenerate.
Proof. Let us assume that H(i, j, k) = 0 otherwise the lemma is trivial. Then by Equation (7), Then the lemma follows from Lemma 3 of [17]. To show that the hypothesis of the lemma in [17] are satisfied it is enough to show that the pair (i, j) has a quality called good. A pair of indices (l, m) ∈ I 2 is called good if V l ⊗ V m splits as a direct sum of some V n 's (possibly with multiplicities) such that n ∈ I and d(V n ) is invertible in k. Hence, by definition of a relative spherical category the pair (i, j) is good and the proof follows Lemma 3 of [17].
Theorem 10 (The Biedenharn-Elliott identity). Let j 0 , j 1 , . . ., j 8 be elements of I. Assume that Here both sides lie in the tensor product of the six k-modules Proof. The proof follows from Theorem 7 of [17]. We now explain why the hypothesis of this theorem are satisfied. First, the definitions above imply that the set J of [17] is equal to I j 2 j 3 . Second, as in the proof of Lemma 9 certain pairs of elements of I must be good. As above, if l, m, n ∈ I with H(l, m, n) = 0 then the pair (l, m) is good. Moreover, if the pair (l, m) is not good then for any n ∈ I the space H(l, m, n) is zero. Thus, from (9) it follows that both sides of Equation (12) are zero unless all required pairs are good and so the hypothesis of [17] are satisfied.
Theorem 11 (The orthonormality relation). Let i, j, k, l, m, p be elements of I. Assume j l / ∈ X , then determined by the duality pairing.
Proof. The proof follows from Theorem 8 of [17]. Again here, as in the proof of Theorem 10 the definition of a relative G-spherical category implies the hypothesis of [17].
The following lemma is an algebraic analog of the H-bubble move.
Lemma 12. Let g 1 , g 2 , g 3 , g 4 , g 5 , g 6 ∈ G \ X with g 3 = g 1 g 2 , g 6 = g 2 g 4 and g 5 = g 1 g 6 . If i ∈ I g 1 , Proof. The proof follows from a direct manipulation of the orthonormality relation using the properties of the map b. In particular, the proof of [17,Lemma 25] can easily be adapted to the case when G is not commutative.

3.2.
Topological preliminary. Let M be an oriented compact smooth 3-manifold. Here a graph Y is a finite 1-dimensional CW-complex disjoint union a finite number of circles without vertices. We denote by Y 0 the set of vertices of Y and ∂Y the set of univalent vertices of Y . By a graph in M , we mean an embedding of a graph in M such that Y ∩ ∂M ⊂ ∂Y and Y meets all connected components of M . We say that Y is rooted in Y ∩ ∂M . By a triangulation T of M , we mean a smooth ∆-complex structure on M (as in [19]). Loosely speaking, a ∆-complex structure is a quotient space of a collection of disjoint simplices obtained by identifying certain of their faces. In particular, the interior of the simplices of T are embedded in M but their faces are not necessarily distincts and two different simplices might meet on several faces. We say that T is quasi-regular if any simplex of T is embedded in M . This is equivalent to requiring that the two endpoints of any edge of T are distinct vertices of T .
Let Y be a graph in M . Let T be a quasi-regular triangulation of M and T ∂ be a triangulation of ∂M . Let Y be a set of unoriented edges of T such that the union of these edges is the graph Y in M . We say the pair (T , Y) is an H-triangulation of (M, Y ) relative to T ∂ if the following four conditions hold: (1) the vertices of Y are also vertices of T , (2) all the vertices of T are contained in Y (i.e. all the vertices are incident to an edge of Y), (3) any edge of T with both endpoints in Y 0 is contained in ∂M and (4) T restricts to T ∂ .
Theorem 13. Any triplet (a compact orientable 3-manifold M , a quasi-regular triangulation T ∂ of ∂M , a graph Y in M rooted in T ∂ ) admits an H-triangulation of (M, Y ) relative to T ∂ .
The proof of Theorem 13 will be given in Subsection 4.3.

Fundamental groupoid and space of representations in G.
A groupoid is a small category in which all morphisms are isomorphisms. Let us consider two examples of groupoids. First, every group G can be seen as a groupoid with only one object whose set of endomorphisms is the group G. Second, let W be a non empty locally path connected topological space. The fundamental groupoid π 1 (W ) of W is the small category whose objects are points of W and whose morphisms are homotopy classes of paths in W , where composition is concatenation of paths. Let Z be a subset of W . If Z is nonempty, let π 1 (W, Z) be the full subcategory of π 1 (W ) whose objects are points of Z. When Z is a single point w of W then π 1 (W, Z) = π 1 (W, {w}) is the usual fundamental group. If W 1 , . . . , W n are the connected components of W and Z meets all of these component, When W is connected, let where w is any point of W and g ∈ G acts on ρ : Remark that for two different choices of the base point w, the corresponding spaces of representations of W are canonically isomorphic. We set We extend this definition for non connected spaces by Let E be a set of paths in π 1 (W, Z), we say that Let W be a triangulated manifold. A G-coloring of W is a map Φ from the set of oriented edges of W to G such that (1) Φ(−e) = Φ(e) −1 for any oriented edge e of W , where −e is e with opposite orientation and (2) if e 1 , e 2 , e 3 are ordered edges of a face of W endowed with orientation induced by the order, then Φ(e 1 ) Φ(e 2 ) Φ(e 3 ) = 1.
Let M be an oriented compact smooth 3-manifold and T ∂ a quasi-regular triangulation of its boundary. Let Y be a graph in M rooted in T ∂ . Let Y 0 be the set of vertices of Y and T ∂ 1 be the set of edges of T ∂ . An admissible representation of (M, In particular, this is true for a closed 3-manifold: If L is a link in a

Modified Turaev-Viro-invariants.
We start from the algebraic data described in the Subsection 3.1 and produce a topological invariant of a quadruple (M, T ∂ , Y, ρ), where M is a compact oriented 3-manifold, T ∂ is a quasi-regular triangulation of its boundary, Y ⊂ M is a graph in M rooted in T ∂ , and ρ is an admissible representation of (M, T ∂ , Y ).
A state of an admissible G-coloring Φ of a triangulated manifold W is a map ϕ assigning to every oriented edge e of W an element ϕ(e) of I Φ(e) such that ϕ(−e) = ϕ(e) * for all e. The set of all states of Φ is denoted St(Φ). The identities d(ϕ(e)) = d(ϕ(−e)) and b(ϕ(e)) = b(ϕ(−e)) allow us to use the notation d(ϕ(e)) and b(ϕ(e)) for non-oriented edges.
The representation ρ restricts to an admissible G-coloring Φ ∂ of the triangulated surface (∂M, T ∂ ). Given a state ϕ of the G-coloring Φ ∂ of T ∂ , the trivalent graph Γ ϕ dual to the 1-skeleton of T ∂ becomes a labeled graph in an oriented surface. We adopt the convention that an oriented edge e of Γ ϕ has the same label than the oriented edge of T ∂ crossing e from the left to the right. We set where be an H-triangulation of (M, Y ) relative to T ∂ and let Φ be an admissible G-coloring of T that restrict to ρ. We now define a certain partition function (state sum) as follows. For each tetrahedron t of T , we choose its vertices This 6j-symbol belongs to the tensor product of 4 multiplicity modules associated to the faces of t and does not depend on the choice of the numeration of the vertices of t compatible with the orientation of M . Note that any face of T \ T ∂ belongs to exactly two tetrahedra of T , and the associated multiplicity modules are dual to each other. The tensor product of the 6j-symbols |t| ϕ associated to all tetrahedra t of T can be contracted using this duality. We denote by cntr the tensor product of all these contractions. Let T 1 be the set of unoriented edges of T that are not in T ∂ and let T 3 the set of tetrahedra of T . Set Theorem 15. T V (T , Y, Φ) depends only on the isotopy class of Y in M , on the triangulation T ∂ of ∂M and on the admissible representation ρ of (M, T ∂ , Y ). We set Moreover, this invariant extends to a Relative Quantum Field Theory given in (31).
A proof of this theorem will be given in Section 4.3.
3.5. Kashaev type 3-manifold invariants. In [12], Geer, Kashaev and Turaev introduce the notion of aΨ-system in a tensor category (see Definition 6 above). It is shown that such a system, plus some algebraic data, leads to the existence of a Kashaev-type invariant of a link in a 3-manifold.
In this subsection we briefly recall the construction of this invariant given in [12]. Throughout this subsection, M is a closed connected orientable 3-manifold and L is a non-empty link inside M . Here we regard L as a graph in M with no vertices. Let (T , L) be an H-triangulation of (M, L). Let t be an tetrahedron of T . An integral charge on t is a 1 2 Z-valued map c defined on the edges of t such that c(e) = c(e ) for opposite edges e and e , and for any edges e 1 , e 2 , e 3 of t forming the boundary of a face of t we have c(e 1 ) + c(e 2 ) + c(e 3 ) = 1/2. Let E(T ) be the set of edges of T and let E(T ) be the set consisting of the edges of all the tetrahedra {t i } of T . Any edge e of T gives rise to n elements of E(T ) where n is the number of tetrahedra of T adjacent to e. Let T : E(T ) → E(T ) be the natural projection. An integral charge on an H-triangulation (T , L) of (M, L) is a map c : E(T ) → 1 2 Z such that the restriction of c to any tetrahedron t of T is an integral charge on t, and for each edge e of T we have e ∈ −1 T (e) c(e ) = c e , where c e = 0, if e belongs to L and c e = 1, otherwise. Each charge c on (T , L) determines a cohomology class [c] ∈ H 1 (M ; Z/2Z) as follows. Let s be a simple closed curve in M which lies in general position with respect to T and such that s never leaves a tetrahedron t of T through the same 2-face by which it entered. Thus, each time s passes through t, it determines a unique edge e belonging to both the entering and departing faces. The sum of the residues 2 c| t (e)(mod 2) ∈ Z/2Z over all passages of s through tetrahedra of T depends only on the homology class of s and is the value of [c] on s. For any non-empty link L ⊂ M and any ζ ∈ H 1 (M ; Z/2Z), each H-triangulation of (M, L) has a charge representing ζ. The theory of integral charges is based on the work of Neumann [24,25].
We now describe the algebraic data needed to define the Kashaev-type invariant. Let C be a k-linear category where k is a field. Recall the vector spaces H ij k , H i jk , H, etc., from Section 1. Fix a Ψ-system in C with distinguished simple objects {V i } i∈I . Fix a family {I g } g∈G of finite subsets of the set I numerated by elements of a group G and satisfying the following conditions: for any finite family {g r ∈ G} r , there is g ∈ G such that I ggr = ∅ for all r; (5) we are given a map b : I → k such that b(i) = b(i * ) for all i ∈ I, and for any g 1 , g 2 ∈ G, k ∈ I g 1 g 2 such that I g 1 = ∅ and . We define two linear forms T, T , : H ⊗4 → k by the following diagrammatic formulae: for any u, v, x, y ∈ H, As remarked in the previous subsection, the G-coloring Φ induces the conjugacy class of a representation [Φ] ∈ M(M, G). Fix a total ordering of the vertices of T . We recall the partition function defined, in [12], from the above data.
For any tetrahedron t of T , let v 1 , v 2 , v 3 , v 4 be its vertices in increasing order (induced from the total ordering). We say that t is right oriented if the tangent vectors Then for each right oriented tetrahedron t of T and state ϕ the restriction of This vector is denoted by |t| c ϕ and depends on ϕ, c and the ordering of the vertices of T . Similarly, if t a left oriented tetrahedron then one uses The multiplicity modules of |t| c ϕ are associated to the faces of t. Note that any face of T belongs to exactly two tetrahedra of T , and the associated multiplicity modules are dual to each other as in Lemma 1 of [12]. The tensor product of the 6j-symbols |t| c ϕ associated to all tetrahedra t of T can be contracted using this duality. We denote by cntr the tensor product of all these contractions. Set where t runs over all tetrahedra of T .
Theorem 16 ([12]). Suppose that there exists a scalar q ∈ k such that q is equal to the operator Then 3.6.Ψ-system in relative G-spherical categories. In this subsection we prove that a relative G-spherical category with basic data has a naturalΨ-system for any choice of a square root of the modified dimension d. Moreover, for this situation, we show that corresponding Kashaev-type and modified TV invariants are equal. Let C be a (X , d)-relative G-spherical category with basic data {V i , w i } i∈I where the ground ring k is a field. For every pair (i, i * ) ∈ I × I define the morphisms It is easy to see that Equation (4) Recall the definition of a Ψ-system given in Subsection 1.4.
Proof. By definition {V i , b i , d i } i∈I satisfies the first three properties in the definition of a Ψ-system. Let i, j ∈ I and suppose H ij k = 0 for some k. Then Equation (7) implies i j = k ∈ G and so V i ⊗ V j is an object of C k . Thus, since C k is semi-simple with a finite number of isomorphism classes of simple objects given by {V l : l ∈ I k } we have that Id V i ⊗V j has the desired property. Let δ A = A δ A and δ B = B δ B.

Recall the operators
Lemma 18. δ, δ A , δ B are symmetric commuting invertible operators satisfying are obviously commuting.
Notice that on H k ij ⊕ H j * i * k * , δ acts by the scalar d 1 2 (k) = d 1 2 (k * ). As S stabilizes these summands of H, S and δ commute which is the first equality of (19). Similarly, the other equality in (19) and Equations (20), (21) are easily deduced from the relation satisfied by A, B, S and the definitions of δ, δ A , δ B .
We will now show that the equalities in (22) are a consequence of the fact that (A, d) is a t-ambi pair. Consider the isomomorphisms Using these isomorphisms the pairings (, ) ijk : H(i, j, k) ⊗ k H(k * , j * , i * ) → k and , : H ⊗ H → k given in Equations (5) and (11), respectively are related by: Remark that the isomorphisms of (25) commute with A and B and so we have Thus, Ly = δ 2 (δ A ) −2 (y), R = δ 2 (δ B ) −2 (y) and this implies Equation (22).
The form T on H ⊗4 is 0 on most of its summands. In particular, it can only be non-zero if the morphisms are composable which mean that the summand has the form for some (i, j, k, l, m, n) ∈ I 6 . Equations (23), (24) follow from the fact that on these summand the operators δ, δ A , δ B act by easily identifiable scalars. For example, on the summand of (27) the operators δ B 1 and δ 2 act as δ B Proof. The Equations (6a),(6b) are trivial except for BR 1 2 B = R − 1 2 which is a consequence of the fact that conjugation by B exchange δ and δ B (see Equations (21)). Equations (6c),(6d) follow from Equations (23), (24). For example, T R are equal. Finally, Proof. First, using Lemma 18 and Theorem 19 it is easy to compute the operator q : H → H: Remark that as q is responsible for the ambiguity of q in K, we see that in our context, this ambiguity disappears. From duality and the equality d(m) After identifying the modules H ij k and H k * j * i * and with the H(i, j, k * ) (using the isomorphisms in Equation (25)) we have that Note the 6j-symbol in the last equality correspond to the same spherical graph with opposite orientation. Let (T , L) be an H-triangulation of (M, L). Let Φ be an admissible G-coloring of T such that [Φ] = ρ ∈ M(M, G). Let c be a integral charge on (T , L) such that [c] = ζ ∈ H 1 (M ; Z/2Z). For each state ϕ of Φ let us write T V ϕ (resp. K ϕ ) for the summand of T V (M, L, ρ) (resp. K(M, L, ρ, ζ)) corresponding to ϕ. In other words, we have Let ϕ be a state of Φ. We will show that T V ϕ = K ϕ . First, notice that the factors of T V ϕ and K ϕ corresponding to b are identical. Let t be a tetrahedron of T with ordered vertices v 1 , v 2 , v 3 , v 4 . We will now compute the vector |t| c ϕ which is used in the construction of K. Let a = c(v 1 v 2 ), b = c(v 2 v 3 ) and i, j, k, l, m, n be the indices given in Equation (16). By definition we have Using Lemma 18 and the fact that δ, δ A and δ B all act as scalars on a multiplicity space we have  (28) we have |t| c ϕ = S t 1 S t 2 |t| ϕ where |t| ϕ is the tensor used in the construction of T V . Now for each tetrahedron t of T we have a scalar S t 1 as described above. Each of these scalars is the product e d(ϕ(e)) c(e) where e runs over the 6 edges of t. Let e be an edge of T . The portion of t∈T 3 S t 1 corresponding to the edge e is where c e = 0, if e belongs to L and c e = 1, otherwise (for notation see Subsection 3.5). Thus, . For each t ∈ T 3 , the contribution of S t 2 can be considered as the operator f δ −1 f where f runs over the set of the 4 faces of t and δ f is δ applied to the multiplicity module H associated to the face f . This factor is exactly what is needed to relate the pairs (, ) ijk and , , see Equation (26). Combining all the statements above we see that T V ϕ = K ϕ and so the invariants are equal.

Relative Homotopy Quantum Field Theory
In this section we will extend the invariant T V of 3-manifolds with triangulated boundary to an invariant of 3-manifolds bounding surfaces with dots. We will also prove that main the topological theorems of the paper.

4.1.
A 2 + 1-cobordism category Cob ∆ of triangulated surfaces. The first step in extending T V is the introduction of an "oscillating path" in a triangulated surface.
An oscillating path in a triangulated surface Σ is a map η : {edges of Σ} → {−1, 0, 1} that satisfies the following conditions: Let P = η −1 ({−1, 1}) be the support of η and |P | be the 1dimensional simplicial complex which is the closure of P . Then we require that any connected component of |P | is homeomorphic to a segment and contains an even number of edges. We also require that η takes alternatively values +1 and −1 on the sequence of edges of any of these paths.
We can represent an oscillating path on a triangulation by coloring some of its edges with + and − (see Figure 4). We define the number of strands of (Σ, η) to be the cardinal of the set of vertices Σ 0 of Σ minus the number of edges in the support of η. This is the number of connected components of Σ 0 ∪ |P |.
Let Σ 1 and Σ 2 be two triangulated oriented surfaces with admissible G-colorings and oscillating paths. Let Σ * 2 be the surface Σ 2 with opposite orientation. An enriched cobordism from Σ 1 to Σ 2 is a quadruplet (M, f, Y, ρ) where: (1) M is an oriented compact 3-manifold, (3) Y ⊂ M is a graph in M rooted in the triangulation T ∂ of ∂M induced from Σ 1 Σ * 2 under f such that if e is an edge of Σ 1 (resp. Σ 2 ) whose value under the oscillating path is +1 (resp. −1) then there exits a connected component Y e of Y and a trivializing disk D 2 embedded in M with Int(D 2 ) ∩ Y = ∅ and ∂D 2 = e ∪ Y e (see Figure 5), (4) ρ is an admissible representation of (M, T ∂ , Y ) that restricts to both of the admissible G-colorings of Σ 2 and Σ 1 .
The triplet (M, Y, ρ) is regarded up to diffeomorphism that is trivial on the boundary (where we use f to identify ∂M and Σ 1 Σ * 2 ). One can glue an enriched cobordism (M, f, Y, ρ) from Σ 1 to Σ 2 with an enriched cobordism (M , f , Y , ρ ) from Σ 2 to Σ 3 . The result is an enriched cobordism (M , f , Y , ρ ) from Σ 1 to Σ 3 where Y is the union of the graphs Y and Y . The set of vertices of Y is where Y 0 and Y 0 are the vertices of Y and Y , respectively and Z = Y 0 ∩ Y 0 is the subset of these vertices that are on Σ 2 → M . Remark that Van Kampen's theorem for groupoids (see [5]) imply that π(M , Y 0 ∪ Y 0 ) is the pushout of the maps π(Σ 2 , Z) → π(M, Y 0 ) and π(Σ 2 , Z) → π(M , Y 0 ). Thus, there exists an unique representation ρ of (M , Y 0 ∪ Y 0 ) which restrict to both ρ and ρ . The representation ρ is the restriction of ρ to (M , Y 0 ). Loosely speaking, a path γ in π(M , Y 0 ) can be cut to a finite composition of smaller paths in π(M, Y 0 ) and π(M, Y 0 ), and ρ (γ) is the product of the images by ρ and ρ of these smaller paths.

Theorem 21.
There is a monoidal category Cob ∆ whose objects are quadruplets (Σ, T , η, Φ) formed by an oriented surface Σ with a quasi-regular triangulation T , an oscillating path η and an admissible G-coloring Φ of T , and whose morphisms are enriched cobordisms. The tensor product is given by disjoint union and the composition by gluing along the boundaries.
Proof. It is well known that cobordisms of surfaces form a monoidal category. It is easy to adapt the proof of this result to enriched cobordisms. The only non-trivial point is to prove that objects have an identity morphism. Let us construct the identity morphism M = (Σ × [0, 1], Y, ρ) of (Σ, T , η, Φ) as follows. The underlying 3-manifold is the cylinder Σ × [0, 1]. The graph Y is a tangle as in the schematic picture of Figure 6. Let Σ = (Σ, T , η, Φ) be an object of Cob ∆ . The dual object Σ * of Σ in Cob ∆ is defined as follows. Let Σ * be the surface Σ with opposite orientation. Let T * = T be the triangulation of Σ * and let Φ * = Φ be the G-coloring of T * . Finally, let η * = −η be the oscillating path of T * . We define Σ * = (Σ * , T * , η * , Φ * ).
Consider the operator d Σ on H( Σ) given by Define the non degenerate pairing

4.2.
A 2 + 1-cobordism category Cob of marked surfaces. In this subsection, we remove the requirement that surfaces are triangulated.
A marked surface is a closed oriented surface Σ with a finite set of points m ⊂ Σ such that m has a nonempty intersection with any connected component of Σ. The elements of m are called marks. We consider marked surfaces equipped with a representation in G that is an element of M(Σ, m, G).
If Σ 1 and Σ 2 are two marked surfaces with representations in G, a marked cobordism from Σ 1 to Σ 2 is a quadruplet (M, f, Y, ρ) where: (1) M is an oriented compact 3-manifold, (2) f : Σ 1 Σ * 2 → ∂M is a diffeomorphism, (3) Y ⊂ M is a graph rooted in the subset m ⊂ ∂M formed by the image under f of the marks of Σ 1 and Σ 2 , (4) ρ is an element of M(M, Y 0 , G) that restricts to the representations of Σ 1 and Σ 2 in G. The triplet (M, Y, ρ) is regarded up to diffeomorphism which is trivial with respect to the boundary (where we use f to identify Σ 1 Σ * 2 with ∂M ). Let Cob be the monoidal category whose objects are marked surfaces with representation in G and morphisms are marked cobordisms. There is an obvious forgetful semi-functor Cob ∆ → Cob sending the set of vertices of a triangulation to the set of marks. We modify this semi-functor to construct an equivalence of categories. The main point here is that the number of marks must be reduced.
If (Σ, T ) is a triangulated surface with an oscillating path η, we can associate a set of marks m containing one point in each connected component of the support |P | of η as follows: m is the set of vertices v of T such that for any edge e containing v, η(e) ≥ 0. Hence the number of marks of (Σ, T , η) is equal to the number of strands of (Σ, η). Restricting the representation in G of an object Σ of Cob ∆ , we get an object E( Σ) of Cob. In order to extends this map to a functor E : Cob ∆ → Cob we need to define some special cobordisms.
Let Σ = (Σ, T , η, Φ) be an object of Cob ∆ and let Σ = E( Σ). Consider the enriched cobordism Σ×[0, 1] that represent the identity of Σ (see Figure 6). Using the forgetful semi-functor Cob ∆ → Cob we can view this enriched cobordism as marked cobordism of (Σ, T 0 , Φ). Cutting this marked cobordism along Σ × { 1 2 }, we can see it as the composition of two morphisms C Σ We define E on morphisms as follows. Let Σ be another object of Cob ∆ and let Σ = E( Σ ). For The functor E : Cob ∆ → Cob is an equivalence of monoidal categories.
Proof. We need to show that E is a full, faithful and essentially surjective. Consider the map Hom Cob ∆ ( Σ, Σ ) → Hom Cob (E( Σ), E( Σ )) induced by E. This map is invertible with inverse Therefore, E is full and faithful. The functor E is essentially surjective because any marked surface has a quasi-regular triangulation T with oscillating path containing those vertices of T which are not in m.
In fact, if two objects of Cob ∆ have the same underlying marked surface Σ, then they are canonically isomorphic (using the cylinders of Lemma 22) and thus we can define Q(Σ) as the result of the identification of these spaces along these isomorphisms. This construction naturally extends to a functor Q : Cob → Vect. The elementary moves (see [17]) are the so-called H-bubble moves (Figure 7a), the H-Pachner 2 ↔ 3 moves (Figure 7b) and the H-lune moves (Figure 7c). It is understood that a negative move is only allowed if it is the inverse of a positive move (hence these moves do not change the graph). The H-lune move may be expanded as a composition of H-bubble moves and H-Pachner moves (see [2], Section 2.1), but it will be convenient for us to use the H-lune moves directly.
Proof of Theorem 13. As in the statement of the theorem, let T ∂ be a quasi-regular triangulation of ∂M and let Y be a graph in M rooted in T ∂ .
In the proof, we shall use the language of skeletons of 3-manifolds dual to the language of triangulations (see, for instance [28,2]). A skeleton of M is a 2-dimensional polyhedron P in M such that M \ P is a disjoint union of open (half) 3-balls and locally P looks like a plane, or a union of 3 half-planes with common boundary line in R 3 , or a cone over the 1-skeleton of a tetrahedron. A typical skeleton of M is constructed from a triangulation T of M by taking the union P T of the 2-cells dual to its edges. This construction establishes a bijective correspondence T ↔ P T between the quasi-regular triangulations T of M and "quasi-regular" skeletons P of M such that every 2-face of P is a disk adjacent to two distinct components of M − P . If (T , Y) is an H-triangulation of (M, Y ), to specify the graph Y in dual skeleton we provide some faces of P T with dots such that each component of M − P T is adjacent to at least two (distinct) dotted faces. These dots correspond to the intersections of Y with the 2-faces. Here the vertices Y 0 of Y become a set of distinguished components of M \ P T . The elementary moves on the H-triangulations may be translated to this dual language and give the well-known Matveev-Piergallini moves on skeletons adjusted to the setting of Hamiltonian graphs, see [2].
Take a collar spine S of (M , Γ) as in [30,Theorems 6.4 for an open tubular neighborhood V (Y ) of Y and Γ is a graph in the boundary of M which is the disjoint union of the trivalent graph dual to the triangulation of ∂M union one meridian circle for each edge (and circle component) of Y . Then we fill the meridian circle of S with a doted disk and get a skeleton S of M in the complement of which a (half) 3-ball has at least one region in its boundary with a dot. Taking the dual of S we obtain an H-triangulation (T , Y) of (M, Y ) relative to T ∂ which may not be quasi-regular. If T is not quasi-regular then S contains a region which has the same (half) 3-cell on both sides or a region with both sides in Y 0 . We call such a region bad.
The skeleton S can be modified by the Matveev-Piergallini moves dual to the elementary moves to obtain a skeleton which is dual to a quasi-regular H-triangulation. In the case when M is closed this was done by Baseilhac and Benedetti [2] in the proof of Proposition 4.20. In particular, for a 3-cell with a bad region they do a dual H-bubble move on S which introduces a new 3-cell which is partially bounded by a two cell which they call a capping disk. Then they use dual H-Pachner moves to slide a portion of the capping disk over the bad region, thus dividing a (half) 3-cell with   [2] can be applied in the case when M has a boundary and Y is a graph.
Proposition 23. Let M be a compact 3-manifold, T ∂ a quasi-regular triangulation of ∂M and Y a graph in M rooted in T ∂ . Any two H-triangulations of (M, Y ) relative to T ∂ can be related by a finite sequence of H-bubble moves and H-Pachner moves in the class of H-triangulations of (M, Y ) relative to T ∂ .
Proof. Let (T 1 , Y 1 ) and (T 2 , Y 2 ) be H-triangulation of (M, Y ) relative to T ∂ . By doing positive H-bubble moves if necessary we can assume that each edge of Y is realized by the same number of edges in both Y 1 and Y 2 . Let P T i be the skeleton dual to T i where each edge of Y i corresponds to a dotted face as in the proof of Theorem 13. Replacing a neighborhood of each of these dots with a circle we obtain a collar spine S i of (M \ V (Y ), Γ i ) where Γ i is the graph in the boundary of M \ V (Y ) formed by these circles. The assumption on the edges of Y 1 and Y 2 implies that Γ 1 and Γ 2 are isotopic. Then Theorem 6.4B of [30] implies that two spines S 1 and (32) Proof. The proof is exactly the same as the one of [17,Lemma 18]. That is we can translate the H-Pachner, H-lune, and H-bubble move into algebraic identities: the Biedenharn-Elliott identity, the orthonormality relation and Equation (13), respectively.
Let (T , Y) be a H-triangulation of (M, Y ). A G-gauge of T is a map from the set of vertices T 0 of T to G. The G-gauges of T form a multiplicative group which acts on the set of G-colorings of T as follows. If δ is a G-gauge of T and Φ is a G-coloring of T , then the G-coloring δ Φ is given by is the initial (resp. terminal) vertex of an oriented edge e. Let T i 0 be the set of vertices of T that are not vertices of Proof of Theorem 14. Take any G-coloring Φ of T which restricts to ρ. We say that a vertex v of T \ Y 0 is bad for Φ if there is an oriented edge e in T outgoing from v such that Φ(e) ∈ X . Since (T , Y) is an H-triangulation any edge not in ∂M has at least one endpoint not in Y 0 . Thus, it is clear that Φ is admissible if and only if Φ has no bad vertices. We show how to modify Φ to reduce the number of bad vertices. Let v be a bad vertex for Φ and let E v be the set of all oriented edges of T outgoing from v. Pick any Let δ v,g to be the G-valued gauge defined by Then δ v,g Φ takes values in G \ X on all edges of T incident to v and takes the same values as Φ on all edges of T not incident to v. Here we use the fact that the edges of T are not loops which follows from the quasi-regularity of T . The transformation Φ → δ v,g Φ decreases the number of bad vertices. Repeating this argument, we find a G-coloring without bad vertices which restricts to ρ.
Remark that for Φ ∈ M(M, T 0 , G), and for g ∈ G T i 0 , the G-colorings Φ and δ g Φ restrict to the same representation in M (M, Y 0 , G).
Proof. We assume in the proof, without lose of generality, that M is connected. When M is closed and Y 0 is empty, the elements of M(M, G) bijectively correspond to the G-colorings of T considered up to gauge transformations (see for example [27]). Therefore, assume Y 0 is not empty. We first distinguish a set of edge Y. Let start with Y = Y. Suppose that some vertex of T is not connected to Y 0 by a path in Y. We say that such a vertex is isolated. Then there exists an edge of T whose ends are an isolated vertex on one side and a vertex not isolated in the other side. We then add this edge to Y. We repeat this process until there is no more isolated vertex. One can check that he had added one edge of T for each circle component of Y. A path of length n is a sequence of n oriented edges e 1 , . . . , e n where the target of e i is equal to the source of e i+1 . Choose an orientation of the edges of Y . This orientation can be extended to a compatible orientation of the edges of Y such that for every vertex v of T there exists an unique shortest path γ v in Y from some vertex of Y 0 to v. To be more precise, recall that every oriented edge of Y is realized by a path in Y. So there is an unique compatible orientation of the edges of Y. As for the added edges of Y \ Y, we choose the orientation toward the originally isolated vertex. The union of paths γ v where v is a vertex of T is a disjoint union of trees rooted in Y 0 . We say a vertex v of T is good if Φ 1 (e) = Φ 2 (e) for every edge e of γ v . In particular, every vertex of Y 0 is good.
Suppose all vertices of T are good. Then Φ 1 = Φ 2 because for any oriented edge e of T from v 1 to v 2 , we have . Thus, it remains to show that we can modify Φ 1 by a gauge transformation to increase the number of good vertices. Let e be an edge of Y from v 1 to v 2 where v 1 is good and v 2 is not good. Then γ v 2 = γ v 1 e and so δ v 2 ,g Φ 1 (e) = Φ 2 (e) where g = Φ 2 (e) −1 Φ 1 (e). If v is a good vertex the path γ v can not pass through v 2 as Φ 1 (e) = Φ 2 (e). Therefore, the set of good vertices strictly increases when one replace Φ 1 with δ v 2 ,g Φ 1 .
Proof. A similar claim is proved in Lemma 27 of [17] when G is a commutative group. Since v 0 is in T i 0 the vertex v 0 is the endpoint for exactly two edges e 1 , e 2 of Y . As shown in the proof of Lemma 27 of [17] there is a sequence of elementary moves through quasi-regular triangulation with admissible colorings from (T , Y, Φ) to (T , Y, δ c Φ). This sequence starts with a positive H-bubble move at e 1 (where we introduce the color c(v 0 )) and ends with a negative H-bubble move at e 2 .
Lemma 27. If Φ and Φ are two admissible G-colorings of T that restrict to the same representation Proof. Since Φ and Φ represent the same element of M(M, Y 0 , G) by Lemma 25 there exists pairs (v i , g i ) ∈ {vertices of T } × G such that Φ = δ vn,gn δ v n−1 ,g n−1 · · · δ v 1 ,g 1 Φ.
where δ v i ,g i is the G-valued gauge, defined as in (33), which takes non-trivial values at a single vertex v i , for all i = 1, . . . , n. We prove the desired equality by induction on n. If n = 0 then Φ = Φ and the equality is clear. Otherwise, let E 1 be the set of (oriented) edges of T beginning at v 1 . Pick any Then δ v 1 ,g Φ and δ vn,gn δ v n−1 ,g n−1 · · · δ v 2 ,g 2 δ v 1 ,g Φ = δ v 1 ,gg −1 1 Φ are admissible colorings. Lemma 26 and the induction assumption imply that Proof. For concreteness, assume that T is obtained from T by a negative move. The admissible G-coloring Φ of T restricts to an admissible G-coloring Φ of T which restrict to the same element Proof of Theorem 15. From Proposition 23 we know that any two H-triangulation of (M, Y ) are related by a finite sequence of elementary moves. Then the result follows from the Theorem 28 by induction on the number of moves.

Quantum groups at root of unity
In this section we first recall some of the deep results established by De Concini, Kac, Procesi, Reshetikhin and Rosso in the series of papers [8]- [11]. We then prove that the modules studied in these papers give rise to the topological invariants described in Sections 2, 3 and 4.
5.1. The quantum groups U and U H . Let g be a simple finite-dimensional complex Lie algebra of rank n and dimension 2N + n with a root system. Fix a set of simple roots {α 1 , . . . , α n } and let ∆ + be the corresponding set of positive roots. Also, let A = (a ij ) 1≤i,j≤n be the Cartan matrix corresponding to these simple roots. There exists a diagonal matrix D = diag(d 1 , . . . , d n ) such that DA is symmetric and positive definite. Let h be the Cartan subalgebra of g generated by the vectors H 1 , . . . , H n where H j is determined by α i (H j ) = a ji . Let L R be the root lattice which is the Z-lattice generated by the simple roots {α i }. Let , be the form on L R given by α i , α j = d i a ij . Let L W be the weight lattice which is the Z-lattice generated by the elements of h * which are dual to the elements H i , i = 1 · · · n. Let ρ = 1 2 α∈∆ + α ∈ L W . Let r be an odd integer such that r ≥ 3 (and r / ∈ 3Z if g = G 2 ). Let q = e 2iπ/r and for i = 1, . . . , n, let q i = q d i . For x ∈ C and k, l ∈ N we use the notation: Remark for x ∈ C, {x} = 0 if and only if x ∈ r 2 Z.
Next we consider two quantum groups associated to g. The Drinfeld-Jimbo quantum group U is defined as the algebra with generators K β , X i , X −i for β ∈ L W , i = 1, . . . , n and relations where σ = ±1. The algebra U is a Hopf algebra with coproduct ∆, counit and antipode S defined by The unrolled quantum group U H is the algebra generated by K β , X i , X −i , H i for β ∈ L W , i = 1, . . . , n with Relations (34),(35),(36) plus the relations where σ = ±1. The algebra U H is a Hopf algebra with coproduct ∆, counit and antipode S defined as above on K β , X i , X −i and defined on the elements H i for i = 1, . . . , n by is the symmetric algebra of h ⊗ Z C. The obvious map U → U H is an injective morphism of Hopf algebra. Using this morphism we can identify U with a Hopf subalgebra of U H .

5.2.
The PBW basis. Let α 1 , . . . , α n be any ordering of the simple positive roots and let s i 1 s i 2 · · · s i N be a reduced decomposition of the longest element of the Weyl group. Then β 1 = α i 1 , β 2 = s i 1 α i 2 , . . . , β N = s i 1 s i 2 · · · s i N −1 α i N is a total ordering of the set of positive roots ∆ + . We call β * = (β 1 , . . . , β N ) a convex order of ∆ + . For each i = 1, . . . , N let X ±β i be the positive (resp. negative) root vector of U (see for example Section 8.1 and 9.1 of [6]). We say that (X ±β ) β∈β * is a convex set of root vectors which depends of the order of α 1 , . . . , α n and s i 1 , s i 2 , . . . , s i N .
If y 1 , . . . , y m is a set of elements in an algebra then let Alg y i : i = 1, . . . , m and Span C y i : i = 1, . . . , m be the algebra generated and the complex vector space spanned by these elements, respectively. We will use the following notation (for U, U H , U h see Section 5.1): ..,n is the set of fundamental weights .
We will need the following weak version of the PBW theorem, for a proof see for example [10,6].
Note that in Theorem 29 one can reverse the order of the products for the monomials of the basis of U ± and still obtain a basis. From Proposition 5.6 of [9] we have that Z 0 is a Hopf subalgebra of U contained in the center U. Moreover, the algebra Z 0 is also in the center of U H .
Let G be the simply connected group associated to g and let T = exp h be the maximal torus of G. Let U + and U − be the unipotent radicals of the opposite Borel subgroups B + and B − , respectively. Recall from [9] that, as a Hopf algebra, Z 0 is the coordinate ring of the dual group G which is the subgroup of B + × B − given by the kernel of the composition B + × B − → T × T → T where T × T → T is the multiplication and B + × B − → T × T is the quotient modulo the unipotent radical (also see [11,Section 6.3]). As a variety, G can be identified with Thus, G can be identified with the group of ring homomorphisms g : Z 0 → C where the group structure is given by gh = (g⊗h)•∆. Moreover, T can be identified with the subgroup of G consisting of elements g which are zero on any element of Z 0 ∩ (U ++ ∪ U −− ).
The authors of [11] define a subset of G called the unramified locus. Let X be the complement of the unramified locus in G. The set G \ X is a Zariski dense open subset of G.
We will use the group G throughout the rest of the paper. In particular, we will show G gives a grading on a certain category of U-modules where graded pieces corresponding to the unramified locus are semi-simple. With this in mind we recall the following theorem. Theorem 30. ([8,9,10] and [11,Prop. 5.5]). For any g ∈ G the following are equivalent: (1) U ⊗ g:Z 0 →C C is a semi-simple algebra.
(2) g / ∈ X . (3) There are (at least) r n non isomorphic irreducible U-modules of dimension r N on which any element of z ∈ Z 0 acts by the scalar g(z).
In Corollary 34 we will give a partial description of X .

Weight modules.
A multiplicative weight is a C-algebra homomorphism κ : U 0 → C. Given a U-module V and a multiplicative weight κ, let E κ (V ) be the weight space consisting of elements v ∈ V such that any x ∈ U 0 acts on v as the scalar κ(x). A U-module V is called a weight module if V splits as a direct sum of its weight spaces and if all elements of Z 0 act diagonally on it. On the other hand, an additive weight is a C-algebra homomorphism λ : U h → C. Given a U Hmodule V and an additive weight λ, let E λ (V ) be the weight space consisting of elements v ∈ V such that any x ∈ U h acts on v as the scalar λ(x). A U H -module V is called a weight module if V splits as a direct sum of its weight spaces, all elements of Z 0 act diagonally on it, and for any β = i b i α i ∈ L W , the element K β acts on V as the scalar i q b i H i i . An additive weight λ induces a multiplicative weight κ = q λ defined by the rule κ( If γ is a multiplicative or additive weight we call a vector v ∈ E γ (V ) a weight vector with (multiplicative or additive) weight γ. The vector space U H with the adjoint action is a weight module over U H . If x ∈ E λ (U H ) we say x has weight λ.
Let D and D H be the categories of finite dimensional weight modules over U and U H , respectively. Let D T be the full subcategory of D whose objects are the modules on which X r ±β acts by 0, for all β ∈ β * . For g ∈ G, let D g be the full subcategory of D whose objects are the modules for which any x ∈ Z 0 acts by the scalar g(x). Then D = g∈G D g is G-graded. Also, if t ∈ T ⊂ G we have that D t ⊂ D T and D T = t∈T D t .
Let V be an object of D H . The first equality in Equation (37) implies that the action of X j , for j = 1, . . . , n, translates the weight spaces of V . Similarly, any x ∈ U ++ ∪U −− translate non trivially the weights. Combining this with the fact that V is finite dimensional one sees that the operator x : V → V is nilpotent. In particular for any β ∈ β * , the operator X r ±β : V → V is nilpotent and thus the zero operator since X r ±β ∈ Z 0 . By forgetting the action of U h the module V becomes a weight module over U. Moreover, since X r ±β : V → V is the zero operator, V is an object in D T . Thus, we have a forgetful functor F : D H → D T (38) and we say that objects of D T and D H are nilpotent modules.
For any nilpotent module V there exists a vector v ± such that X ±i v ± = 0 for all i = 1, . . . , n. Such vectors v + and v − are called highest weight vectors and lowest weight vectors, respectively. If V is simple then v ± is unique up to a scalar.
Proof. The first two items follows from the fact that Z 0 is a Hopf subalgebra of U. The third is a consequence of Theorem 30.

Typical modules.
Let I H be the ideal of U H generated by the central elements X r ±β , β ∈ β * . Let I = I H ∩ U the corresponding ideal in U. The ideals I and I H do not depend on the choice of β * because they are generated by (Z 0 ∩ U ++ ) ∪ (Z 0 ∩ U −− ). The set of monomials of the PBW basis that contain at least one factor X k ±β , with k ≥ r, is a basis of I. Hence U ±< and I ∩ U ± are complementary vector subspaces of U ± (note U ±< is defined in Subsection 5.2). Any nilpotent module is annihilated by I.
Let κ ∈ Hom alg (U 0 , C) be a multiplicative weight and consider the one dimensional U 0 U + -module v 0 whose action is determined by xv 0 = κ(x)v 0 for x ∈ U 0 and yv 0 = (y)v 0 for y ∈ U + . This module induces the Verma module V κ which is an infinite dimensional weight module over U with highest weight vector v 0 of multiplicative weight κ. The module V κ contains an unique maximal submodule not containing v 0 . The quotient of this submodule is an irreducible U-module V κ with a highest weight vector of weight κ. Similarly, an additive weight λ ∈ Hom alg (U h , C) defines an one dimensional module over U h U 0 U + whose action is given by λ on U h , q λ on U 0 and on U + . The same process gives the Verma module V λ with irreducible quotient V λ , both having highest weight vectors of weight λ.
Proposition 32. Let λ ∈ Hom alg (U h , C) be an additive weight and κ = q λ ∈ Hom alg (U 0 , C) its induced multiplicative weight. Then V λ ∈ Ob(D H ) and V κ ∈ Ob(D |κ| ) are the unique (up to isomorphism) irreducible nilpotent module with highest weights λ and κ, respectively. Furthermore, F(V λ ) is isomorphic to V κ , where F is the forgetful functor defined in (38).
Proof. Let V be the module V λ or V κ with height weight vector v 0 . Since the ideal I annihilates V we have that V = U − v 0 = U −< v 0 . Therefore, the set {X k 1 −β 1 · · · X k N −β N v 0 : 0 ≤ k i < r} contains a finite basis of weight vectors. The uniqueness of V comes from the universal property of its Verma module V : V can be mapped to any module W with a highest weight vector of the same weight as v 0 . If W is irreducible then the kernel of this map is a maximal proper submodule and so W is isomorphic to V .
Similarly, the map F( V λ ) If the kernel of this map was nonzero then it would be a nilpotent module with a highest weight vector w. But w would also be a highest weight vector of V λ generating a proper submodule of V λ . This is not possible because V λ is irreducible.
Proposition 32 implies that any irreducible nilpotent module in D is isomorphic to F(V λ ) for some λ. However, it is not clear that every nilpotent U-module V is isomorphic to F(W ) for some U H weight module W .
Proposition 33. [8,Th. 3.2] Let λ ∈ Hom alg (U h , C) be an additive weight and κ = q λ ∈ Hom alg (U 0 , C) its induced multiplicative weight. Let v + and v − be the highest and lowest weight vectors of V κ , respectively. Then the following are equivalent.
Proof. In [8, Section 3.2] De Concini and Kac define the so called diagonal module V κ = V κ /I V κ with highest weight vector of weight κ. They prove that the third condition is equivalent to the module V κ being irreducible. Thus, the proposition follows from the following facts: 1) when V κ is irreducible it is isomorphic to V κ , 2) the maps in the proposition are surjective, and 3) If λ and κ = q λ are weights such that one of the equivalent conditions of Proposition 33 is satisfied then we say that V λ and V κ are typical modules. The following corollary combined with Proposition 32 implies that any irreducible U-module V ∈ D g with g ∈ T \ X is typical.
Proof. There exist (non unique) additive weights λ i , for i = 1, . . . , r n , with induced multiplicative weights κ i such that |κ i | = g, for all i = 1, . . . , r n . Then for all β ∈ ∆ + and i = 1, . . . , r n we have g(K r β ) = κ i (K r β ) = q r λ i ,β . Combining this with the hypothesis g(K r β ) = ±1 we have q 2 λ i +ρ,β +m β,β = 1 for all β ∈ ∆ + , m ∈ {0, . . . r − 1} and i ∈ {1, . . . , r n }. Thus, Proposition 33 implies that for each i = 1, . . . , r n the module V κ i has dimension r N and the result follows from Theorem 30. 5.5. The pivotal structure. It is well known that 2ρ, for all x in U H . Equation (39) is a modification (by the central element K r 2ρ ) of a similar well known equation. We impose this modification to make the modified dimension d "spherical" and the twist invariant by the antipode (see Subsections 5.7 and 5.8).
It is a general fact that a category of modules over a Hopf algebra in which the square of the antipode is equal to the conjugation by a group-like element is a pivotal (or sovereign) category (see [4,Proposition 2.9]). Hence, from Equation (39) it follows that D and D H are both pivotal k-linear categories with ground ring C. In particular, for any object V in D, the dual object and the duality morphisms are defined as follows: V * = Hom C (V, C) and , where {v j } is a basis of V and {v * j } is the dual basis of V * . 5.6. Ambidextrous modules. Recall the definition of an ambidextrous object given in Section 2. In this subsection we show that typical modules over U H and U are ambidextrous. With this in mind we give a general theorem which allows one to prove certain objects are ambidextrous.
We will now assume that C is a category of H-modules for some Hopf C-algebra H. If V is any object of C we have the bilinear pairing , on V ⊗V * = (V ⊗V * ) * given by w, w = d V ⊗V * (w⊗w ).
Then for any f ∈ End(V ⊗ V * ) we have f (w), w = w, f * (w ) .
Theorem 35. Let V be an irreducible H-module. Assume there exists w 0 , w 0 ∈ V ⊗ V * and x, y ∈ H such that w 0 , w 0 = 1 and x.w 0 , y.w 0 are nonzero H-invariant vectors (i.e. any h ∈ H acts on x.w 0 and y.w 0 by the scalar ε(h)). Then the module V is ambidextrous if either of the following two conditions hold: (a) Cw 0 and Cw 0 are invariant lines under the action of End C (V ⊗ V * ), (b) ker x ⊂ ker ·, w 0 and ker y ⊂ ker w 0 , · , here x, y are operators on V ⊗ V * .
If Condition (a) holds we can assume v = v =0. Then α = f (w 0 ), w 0 = w 0 , f * (w 0 ) = α . On the other hand, if Condition (b) is satisfied, v, w 0 = w 0 , v = 0 and so α = f (w 0 ), w 0 = w 0 , f * (w 0 ) = α . Thus, in both case we have f * Lemma 36. Consider the two following elements of U: If γ * is any convex order of ∆ + and {X α } α∈γ * is the corresponding convex set of root vectors then there exists a, b ∈ C * such that Proof. This is a consequence of (PBW) Theorem 29. Indeed, the weight space of U + /I with weight 2(r − 1)ρ is one dimensional, generated by α∈γ * X r−1 α for any convex order γ * of ∆ + . This implies the equivalence for X − , the equivalence for X + is obtained analogously with U − /I. Theorem 37. Any typical module V over U H or U is ambidextrous.
Proof. Let v + , v + be (the unique up to a scalar) highest weight vectors of V, V * , respectively.
choose scalars of the vectors). We want to apply Theorem 35 where w 0 = v + ⊗ v + , w 0 = v − ⊗ v − and x, y are any elements of U proportional to α∈γ * X r−1 −α , α∈γ * X r−1 α , respectively, for any convex Then each x ∈ (U H ) ⊗k defines a linear map ψ(x) : V → V and we have a finite dimensional representation ψ : For any x ∈ (U H ) ⊗k , let ψ(q x ) be the limit as m goes to infinity of the absolutely convergent series m j=0 1 j! ψ 2iπx r j . In this situation, we say that q x is an operator on D H . If q x , q y are two operators such that ψ(q x ) = ψ(q y ) for all finite dimensional representations ψ then we say the operators are equal and write q x = q y . One can multiply operator and take their coproduct and antipode using the rules ∆q x = q ∆x and S(q x ) = q S(x) . We will also call the linear map ψ(x) : V → V an operator and denote it by x.
Let H be the operator on D H defined as Lemma 39. For any x, y ∈ U H with weights α, β ∈ L W , respectively, we have the equalities of operators on D H : H(x ⊗ y) = q α,β (xK β ⊗ yK α )H. Proof. The proof is a direct computation using the following three facts: 1) for any weight vectors v λ and w µ of weight λ and µ, respectively.
LetŘ be the truncated quasi R-matrix of U H given by Theorem 40. The operator R = HŘ leads to a braiding The element R < does not have a pole at q when q is a primitive root of unity of order r. Thus, the relations of Equation (46) hold when q is a root of unity. Finally, Lemma 39 implies that the operators HH h and H satisfy the same commutator relations on U h (g) and U H , respectively. Thus, the relations of Equation (45) hold.
In particular, since u has weight 0, the operators u H and u commute.
Let ν and θ be the operators ν = u H u and θ = νK r−1 2ρ . Then Proposition 41. Let V λ be a typical U H -module then : V λ → V λ is the identity.
Theorem 42. Let D θ be the full subcategory of D H formed by modules on which acts as the identity. Then D θ is a C-linear ribbon category with braiding c V,W : Proof. Recall that I H is the ideal of U H generated by the central elements X r ±β , β ∈ β * . Remark that is central and group-like in the quotient Hopf algebra U H /I H . Hence D θ is a tensor subcategory of D H . The braiding of D H given Theorem 40 restricts to a braiding on D θ . Finally, since acts as the identity on any object in D θ we have that operators θ and S(θ) are equal. It follows that θ V is a twist on D θ .
Remark that from Proposition 41 the category D θ contains all the typical modules of D H .

Conjecture 43.
= 1 and D H = D θ is a ribbon category.
5.8. The modified dimension. As irreducible projective modules of D and D H are ambidextrous, there is an unique non trivial trace on the ideal of projective objects (cf [18]). On simple objects, the trace is given by the modified dimension which we calculate for the special case of nilpotent modules. Then we show that the modified dimension is the same for an object and its dual, thus leading to a relative G-spherical structure in D.
Proof. Let {v i } be a basis of V λ such that v i is a weight vector with additive weight λ i . Let w µ be a highest weight vector of V µ . From Equation (44) we have We now give two facts. Let v be any weight vector of V λ of weight η. Fact 1: R(w µ ⊗ v) = q <µ,η> (w µ ⊗ v). This fact follows from the definition ofŘ, Equation (48) and the property that E α w µ = 0 for α ∈ ∆ + . Fact 2: All the pure tensors of the element ( where w is a weight vector of V µ and v is a weight vector of V λ whose weight is of strictly higher order than that of the weight of v. Fact 2 is true because E n α v (for α ∈ ∆ + and n ∈ N >0 ) is zero or a weight vector whose weight is of strictly higher order than the weight of v. We will now compute S (λ, µ) directly. Let V be a typical module and recall that the duality morphisms d V : V ⊗ V * → C is defined as where v is a weight vector of V λ of additive weight η. Consider the element S ∈ End C (V µ ) given by • (Id Vµ ⊗b V λ ). To simplify notation set S = (X 1 )(X 2 )(X 3 )(X 4 ) where X i is the corresponding morphism in the above formula. The morphism S is determined by its value on the highest weight w µ . By definition S(w µ ) = S (λ, µ)w µ , so it suffices to compute S(w µ ).
where z k = v * i (for some i), v k is a weight vector of V λ whose weight is of strictly higher weight than that of the weight of z * k and w k is a weight vector of V µ . Moreover, the second equality of the above equation follows from Fact 1, the third from (48) and Fact 2, and finally the fourth from Equation (49) and the fact that z k (v k ) = 0. The key observation in this proof is that Facts 1 and 2 imply that in the above computation the only contribution of the action of the operator R comes from H.
We now can apply the results of Section 2 to D θ . From Theorem 37 we know that any typical U H -module J is ambidextrous. Therefore, we can apply Theorem 7 to D θ and obtain an ambi pair (A Proj , d). As discussed in Section 2 the functions d and d J are proportional. Now, let λ ∈ Hom alg (U h , C) be an additive weight. Combining the formula in Proposition 44 with the formula α∈∆ − (q 2 λ+(1−r)ρ,α − 1) = q λ+(1−r)ρ,−2ρ α∈∆ + { λ + (1 − r)ρ, α } we can see that the function d can be multiplied by a scalar in C × so that it is given by the formula for V λ with |λ| / ∈ X . Thus we have proved: Theorem 45. The category D θ is a ribbon category with a projective ambidextrous object and so gives rise to an ambi pair (A Proj , d) and isotopy invariant G where d satisfies Equation (51).
It is clear from Formula (51) that d(V λ ) depends only on the multiplicative weight κ = q λ ∈ Hom alg (U 0 , C) induced by λ. Thus we can define, d(V κ ) = d(V λ ) whenever |κ| / ∈ X . Let A be the set of irreducible U-modules V such that V ∈ D g for some g ∈ G \ X . Theorem 31 implies that each module in A is projective. Let Proj be the full subcategory of D consisting of projective U-modules. Recall the notion of a trace given in Subsection 1.2.
Proof. Let U ∈ D g be a typical projective U-module with g / ∈ X . Consider the linear map t : End C (U ) → C given by f → d(U ) f where, as above, f is defined by f = f Id U . From Theorem 37 we have that U is ambidextrous. Then from [18] we have that t determines a unique trace t = {t V } on Proj such that t = t V .
On the other hand, F defines a trace t = {t V } on the projective modules of D H given by t V (f ) = F (L f ) where L f is ribbon graph which is the closure of coupon of f . Then by definition, for any additive weight λ with |λ| / ∈ X , we have By construction of t, for each κ = q λ with |λ| / ∈ X , we have t Vκ = t F (V λ ) where F : D H → D is the forgetful functor defined above. Moreover, t F (V λ ) (F(f )) = t V λ (f ) for all f ∈ End(V λ ). Thus, t Vκ (Id Vκ ) = t V λ (Id V λ ) = d(V λ ) = d(V κ ).
Lemma 46 implies the assignment d on nilpotent U-modules can be extended to a function d : A → C given by d(V ) = t V (Id V ). Since V ∈ A is projective it follows from [18] whenever W is a direct summand of U ⊗ V (see [18]).
Lemma 47. For any V ∈ A, we have s(V ) = 1 and so d(V ) = d(V * ).
Proof. First, we will show that for an irreducible V ∈ D g the slope s(V ) depends only of the degree g ∈ G. In [11] it is shown that for any g, g , gg ∈ G\X and any irreducible modules V ∈ D g , V ∈ D g we have where irr(D gg ) denote a representing set of the isomorphism classes of irreducible modules of D gg . This implies that if W 1 , W 2 ∈ D g are irreducible modules then there exists projective irreducible modules U, V such that both W 1 and W 2 are direct summands of U ⊗ V . Then by Equation (52) we have s(W 1 ) = s(W 2 ) = s(U )s(V ). Thus, the slope factor through a map on G \ X also denoted by s which satisfies s(gg ) = s(g)s(g ) and s(g −1 ) = 1/s(g). Next we show that s extend to a character on the whole group G: If x ∈ X , choose g ∈ G \ (X ∪ X x −1 ) then we can define s(x) as the ratio s(gx)/s(g). This is well defined: if h ∈ G \ (X ∪ X x −1 ) then there exists k ∈ G \ (X ∪ X x −1 ∪ X g ∪ X h) for which k, kx, kg −1 , kh −1 / ∈ X and thus s(kx)/s(gx) = s(kg −1 ) = s(k)/s(g), s(kx)/s(hx) = s(kh −1 ) = s(k)/s(h) which imply that s(gx)/s(g) = s(kx)/s(k) = s(hx)/s(h).
But now Equation (51) implies that T ⊂ ker s. As the normal subgroup of the borel B + ⊂ G generated by its Cartan subgroup T is B + itself, this implies that B + ⊂ ker s. Similarly B − ⊂ ker s thus ker s ⊃ B + B − = G.
Proof. From Lemmas 46 and 47 there exists a trace t = {t V } V ∈Proj such that d(V ) = d(V * ) for all V ∈ Proj. Then from [18] we have that (A Proj , d) is t-ambi.
Let us increase X by adding all g ∈ G with g 2 = 1. Then G \ X is still a Zariski open dense subset of G and by Lemma 3 there exists basic data for D.
Theorem 49. The category D of finite dimensional weight modules over U is a (X , d)-relative G-spherical category which admit basic data.
Proof. The category D is a C-linear pivotal category where the pivotal structure is given in Subsection 5.5. The G-grading of D is defined in Subsection 5.3. Item (1) of Definition 2 follows from Theorem 31. Lemma 48 implies Item (2). Finally, since the objects of D are finite dimensional vector spaces it follows from [17] that the map b : A → C defined in Equation (3) is well defined and satisfies all the properties of Item (3).