Factorizable $R$-Matrices for Small Quantum Groups

Representations of small quantum groups $u_q({\mathfrak{g}})$ at a root of unity and their extensions provide interesting tensor categories, that appear in different areas of algebra and mathematical physics. There is an ansatz by Lusztig to endow these categories with the structure of a braided tensor category. In this article we determine all solutions to this ansatz that lead to a non-degenerate braiding. Particularly interesting are cases where the order of $q$ has common divisors with root lengths. In this way we produce familiar and unfamiliar series of (non-semisimple) modular tensor categories. In the degenerate cases we determine the group of so-called transparent objects for further use.


Introduction
Hopf algebras with R-matrices, so called quasitriangular Hopf algebras, give rise to tensor categories with a braiding c : V ⊗ W ∼ −→ W ⊗ V . Of particular interest are braided tensor categories where the braiding fulfills a certain non-degeneracy condition, see Definition 5.1, which is equivalent to the fact that there are no transparent objects V , i.e., no objects where the double-braiding c 2 : V ⊗ W ∼ −→ V ⊗ W is the identity for all W . A C-linear tensor category with a nondegenerate braiding, as well as finiteness conditions and another natural transformation θ : V ∼ −→ V (twist), is called a modular tensor category. Note that we do not require the category to be semisimple.
Modular tensor categories have many interesting applications: They give rise to topological invariants and mapping class group actions [7,19]. For example, the standard generators T , S of the mapping class group of the torus SL 2 (Z) are constructed from θ and c 2 , respectively. A different source for modular tensor categories in mathematical physics are vertex algebras. There are only few example classes of modular tensor categories, in particular non-semisimple ones.
The aim of the present article is to provide modular tensor categories from small quantum groups u q (g) at a primitive -th root of unity q for a finite-dimensional simple complex Lie algebra g. Lusztig [12] has constructed these finite-dimensional Hopf algebras and provided an ansatz for an R-matrix R 0Θ , where the fixed elementΘ ∈ u q (g) − ⊗ u q (g) + is constructed from a dual basis of PBW generators, while R 0 ∈ u q (g) 0 ⊗ u q (g) 0 is a free parameter subject to some constraints. He gives one canonical solution for R 0 whenever has no common divisors with root lengths, otherwise there are cases where no R-matrix exists [8] and the quantum group becomes more interesting [9], involving, e.g., the dual root system. Of particular interest in conformal field theory [4,5,6] is the most extreme case where all root lengths (α, α) divide . In particular our article addresses the question, which modular tensor category appear in these cases. We find indeed, e.g., in Lemma 4.7 that these extremal cases give especially nice R-matrices; although in general they are not factorizable and will require modularization (see for example [2]) to match the CFT side.
But even if there are no common divisors with the root length, the resulting braided tensor categories may not fulfill the non-degeneracy condition and hence provides no modular tensor category.
Both obstacles (existence and non-degeneracy) can be be resolved by extending the Cartan part of the quantum group by a choice of a lattice Λ R ⊆ Λ ⊆ Λ W between root-and weight-lattice, respectively a choice of a subgroup of the fundamental group π 1 := Λ W /Λ R , corresponding to a choice of a Lie group between adjoint and simply-connected form. These extensions are already present in [12] as the choice of two lattices X, Y with pairing X × Y → C × (root datum). In this way the number of possible R matrices increases and the purpose of our paper is to study them all.
In a previous article [11] we have already constructed some solutions R 0 in this spirit (under some additional assumptions). As it turns out, the solutions can be parametrized by subgroups H 1 , H 2 ⊂ π 1 and group pairings between H 1 , H 2 , and the set of solutions depends on the common divisors of not only with root lengths, but also divisors of the Cartan matrix. Some cases admit no braided structure, while others have multiple in-equivalent solutions. An interesting occurrence was for example that B n behaves differently for n odd or even, and that D 2n with non-cyclic fundamental group allows several more solutions with non-symmetric R 0 .
In the present article we conclude this effort: First we introduce more systematical techniques that allow us to compute a list of all quasitriangular structures (without additional assumptions, so we find more solutions). Then our new techniques allow us to determine, which of these choices fulfill the non-degeneracy condition. We also determine which cases have a ribbon structure. A main role in the first part is played by a natural pairing a on the fundamental group π 1 which depends only on the common divisors of with the fundamental group and encapsulates the essential -dependence. Then the non-degeneracy of the braiding turns out to depend only on the 2-torsion of the abelian group in question.
Our result produces a list of modular tensor categories for representations of quantum groups. Moreover we use our methods to explicitly describe the group of transparent objects if the category is not modular, which is for example a prerequisite for modularization.
We now discuss our methods and results in more detail: In Section 2 we briefly recall the Lie theory and Hopf algebra preliminaries: For every finitedimensional (semi-)simple complex Lie algebra g and a primitive -th root of unity q Lusztig has introduced in [12] the small quantum group u q (g) which has a triangular decomposition u + q u 0 q u − q where the (exponentiated) Cartan algebra u 0 q is the groupring of the root lattice Λ R modulo some suitable sublattice and u ± q are generated by simple root vectors E α i , F α i fulfilling q-deformed Serre relations. In [13, Section 32] he gives an ansatz for an R-matrix in the form R 0Θ , whereΘ consists of dual PBW basis' and R 0 ∈ u 0 q ⊗ u 0 q is an arbitrary element in the Cartan part that has to fulfill certain relations.
Our goal is to study the existence and non-degeneracy of R-matrices of this form for the quantum group u q (g, Λ, Λ ) with any choice of lattice between root-and weight-lattice Λ R ⊆ Λ ⊆ Λ W and any possible choice of quotient by a subgroup Λ ⊆ Λ in the Cartan part u 0 = C[Λ/Λ ]. Later, we prove that Λ is in fact unique if we want a quasitriagular structure (Corollary 3.6).
The R 0 -matrix has the following interpretation: It is an R-matrix for the groupring C[Λ/Λ ] and it appears as the braiding between highest-weight vectors in our u q (g)-modules. Thus the previous theorem clarifies which choices for an R-matrix for the group ring lift to the quantum group.
In Section 3 we address the question of constructing quasi-triangular R-matrices. First we briefly recall the following general combinatorial result in [11]: where H 1 , H 2 are subgroups of Λ/Λ R ⊆ π 1 with |H 1 | = |H 2 | =: d (not necessarily isomorphic!) and g : H 1 × H 2 → C × is a pairing of groups.
Then we proceed differently than in the previous article: Using the previous result, we prove in Lemma 3.5 that the quasitriangularity of R is equivalent to the assertion that the group pairingf := |Λ/Λ | · f between the preimages G i := Λ i /Λ of the groups H i is non-degenerate (which is no surprise). In particular we show that this condition fixes Λ uniquely. In later applications we often encounterf as a natural identification of G 1 and the dualĜ 2 , e.g., when studying representation theory.
To find all solutions f with this property we develop a machinery to pushf into the fundamental group π 1 , which encapsulates all the -dependence: In Definition 3.8 we give an abstract characterization of a centralizer transfer map (without proving that it always exists). In a generic case this is just multiplication by , but it severely depends on common divisors of with root length and divisors of the Cartan matrix. With this matrix we can transfer q −(µ,ν) to a natural form a g on the fundamental group. We prove thatf is non-degenerate iff a g (µ, ν) = q −(µ,A (ν)) · g(µ, A (ν)) is non-degenerate. This explains why the set of solutions, say for fundamental group Z n always looks like the subset of invertible elements Z × n but it is shifted (namely by A ) depending on and the root system in question. In Section 4 the remaining computational work is done for quasitriagularity: We calculate a list containing a g for all simple g, depending on common divisors of with root length and divisors. We thus write down all solutions for f and hence R-matrices. The calculation starts with the Smith normal form for the Cartan matrix in question and uses three cases: For Λ = Λ W we have a generic construction, the cases A n with their large fundamental group Z n+1 is treated by hand, as is D 2n with non-cyclic fundamental group, which has the only cases allowing Λ 1 = Λ 2 .
In Section 5 we address our main issue of factorizability with our new tools: In Section 5.1 we introduce factorizability. Then we calculate the monodromy matrix R 21 R for an arbitrary choice of R-matrix in terms the R 0 -part. This gives a purely lattice theoretic problem equivalent to the factorizability of such an R-matrix. Then we prove in the main Theorem 5.9 that factorizability is equivalent to the non-degeneracy of a symmetrization Sym G f off . As will turn out later, the radical of this form is isomorphic to the group of transparent objects. In Section 5.2 we restrict ourselves to the symmetric case where H 1 = H 2 and f , g are symmetric. Other cases appear only in some of the non-cyclic Z 2 ×Z 2 -extension for type g = D 2n and are dealt with in Section 5.3 and give surprising new solutions.
The main result for the symmetric case is that the radical of the form Sym G f is in this case simply the 2-torsion of Λ/Λ (Example 5.11) and that this is non-degenerate precisely for odd and odd Λ/Λ R as well as for g = B n , Λ = Λ R , ≡ 2 mod 4 including A 1 . In Section 5.4 we prove the following result: Lemma 5. 19. The transparent objects in the category of representations of the Hopf algebra u q (g, Λ) with R-matrix given by Lusztig's ansatz are 1-dimensional objects C χ and are the ftransformed of the radical of Sym G f : In the following we summarize our results by a table containing all quasitriangular quantum groups u q (g, Λ) with their group of transparent objects. In Section 6 we show that all our quasitriangular quantum groups admit a ribbon element. The factorizable solutions and thus modular tensor categories are odd, Λ = Λ R and the following new factorizable cases: ( odd, E 6 , Λ = Λ W ) and ( ≡ 2 mod 4, g = B n , Λ = Λ R ) (including A 1 ) and ( odd, g = D 2n , Λ 1 = Λ 2 ). All other cases can be modularized as discussed in Question 7.3.

Lie theory
Throughout this article, g denotes a finite-dimensional simple complex Lie algebra. We fix a choice of simple roots ∆ = {α i | i ∈ I}, so that the Cartan matrix C is given by we denote the (co)root lattice of g. By Λ W , we denote the weight lattice spanned by fundamental dominant weights λ i , which are defined by the equation (λ i , α j ) = δ i,j d i . Finally, we define the co-weight lattice Λ ∨ W as the Zspan of the elements λ ∨ i := λ i d i . The quotient π 1 := Λ W /Λ R is called the fundamental group of g. One can easily see that the Killing form restricts to a perfect pairing ( , ) : Λ ∨ W × Λ R → Z and that we get a string of inclusions

Lusztig's ansatz for R-matrices
The starting point for our work [11] was Lusztig's ansatz in [13, Section 32.1] for a universal Rmatrix of U q (g). Namely, for a specific elementΘ ∈ U ≥0 q ⊗ U ≤0 q from a dual basis and a suitable (not further specified) element in the coradical R 0 ∈ U 0 q ⊗ U 0 q we are looking for R-matrices of the form We remark that there is no claim that all possible R-matrices are of this form. However they are an interesting source of examples, motivated by the interpretation of u q (g) as a quotient of a Drinfeld double and thus well-behaved with respect to the triangular decomposition. This ansatz has been successfully generalized to general diagonal Nichols algebras in [1]. In our more general setting U q (g, Λ, Λ ), we have This ansatz has been worked out by Müller in his dissertation [14,15,16] for small quantum groups u q (g) which we will use in the following, leading to a system of quadratic equation on R 0 that are equivalent to R being an R-matrix: } a subgroup of Λ, and G 1 , G 2 subgroups of G := Λ/Λ , containing Λ R /Λ . In the following, µ, µ 1 , µ 2 ∈ G 1 and ν, ν 1 , if and only if for all α ∈ Λ R and µ, ν the following holds: 3 Conditions for the existence of R-matrices

A f irst set of conditions on Λ/Λ
The target of our efforts is a Hopf algebra called small quantum group u q (g, Λ, Λ ) with Cartan part u 0 It is defined, e.g., in [11] and depends on lattices Λ, Λ defined below. For Λ = Λ R the root lattice and this is the usual small quantum group; the choice of Λ differs in literature.
In the previous section we have discussed an R = R 0Θ -matrix for the quantum group u q (g, Λ, Λ ) can be obtained from an R 0 -matrix of the form In the following we collect necessary and sufficient conditions for R = R 0Θ to be an R-matrix. Definition 3.1. We fix once-and-for-all a finite-dimensional simple complex Lie algebra g and a lattice Λ between root-and weight-lattice These choices have a nice geometric interpretation as quantum groups associated to different Lie groups associated to the Lie algebra g.
R , which would below pose no additional complications and may produce further interesting factorizable R-matrices.
Definition 3.2. We fix once-and-for-all a primitive -th root of unity q. For Λ 1 , Informally, this is the centralizer with respect to the braiding q −(ν,µ) .
Contrary to [11] we do not fix Λ but we prove later Corollary 3.6 that there is a necessary choice for Λ . In this way, we get more solutions than in [11]. The only condition necessary to ensure that the Hopf algebra The necessity of this form (in particular that the support of f is indeed a subgroup!) amounts to a combinatorial problem of its own interest, which we solved for π 1 cyclic in [10] and for Z 2 ×Z 2 by hand; a closed proof for all abelian groups would be interesting.
Definition 3.4. Let g : G × H → C × be a finite group pairing, then the left radical is defined as Similarly, the right radical is defined as The pairing g is called non-degenerate if Rad L (g) = 0. If in addition Rad R (g) = 0, g is called perfect.
For an R 0 -matrix of this form, a sufficient condition is that they fulfill the so-called diamondequations (see [11,Definition 2.7 However, we will now go into a different, more systematic direction that makes use of the following observation: Lemma 3.5. An R 0 -matrix of the form given in Theorem 3.3 is a solution to the equations in Theorem 2.2, and hence produces an R-matrix R 0Θ iff the restriction to the support Proof . We first show that a solution with restriction to the support a nondegenerate pairing solves the equation.
The first equations are obviously fulfilled for the form assumed For the other equations the sums get only contributions in the support Λ 1 /Λ ×Λ 2 /Λ . The quantities f (µ, ν) · d|Λ R /Λ | for fixed ν (or µ) are characters on the respective support, and by the assumed non-degeneracy all ν = 0 give rise to different nontrivial characters. Then the second and third relations follows from orthogonality of characters. Note that since d|Λ R /Λ | = |G 1 | = |G 2 | (equality of the latter was an assumption!) we were able to chose the right normalization. For the other direction assume a solution of the given form to the equations. Then already the third equation shows that no f (−, ν) may be the trivial character and hence the form on the support is nondegenerate and hence perfect by Corollary 3.6. A first consequence of the perfectness off (i.e., a necessary condition for quasitriangularity) is This fixes Λ uniquely. Moreover in cases Λ 1 = Λ 2 , which can only happen for g = D 2n , where π 1 is noncyclic, we get an additional constraint relating Λ 1 , Λ 2 .
In our case, the only possibility for and thus the above condition is always fulfilled.
Our main goal for the new approach on quasitriangularity as well as the later modularity is to reduce this non-degeneracy condition forf to a non-degeneracy condition for g on H 1 , H 2 ⊂ π 1 that can be checked explicitly.

A natural form on the fundamental group
We now define for each triple (Λ, Λ 1 , Λ 2 ) and each th root of unity q a natural pairing a on the subgroups H i := Λ i /Λ R of the fundamental group π 1 := Λ W /Λ R . The simplest example is a = e −2πi(µ,ν) . In general it is a transportation of the natural form q −(µ,ν) (which does not factorize over Λ R ) to H i by a suitable isomorphism A .
This isomorphism A will encapsulate the crucial dependence on the common divisors of , |H| and the root lengths d i ; moreover, for different H these forms are not simply restrictions of one another.
Then, we can moreover transport any given pairing g together with q −(µ,ν) along the isomorphism A to the H i and thus define forms a g on H. The main result of this section is in Theorem 3.13 that the non-degeneracy condition in Lemma 3.5 for R 0 (f ) depending on H i , g is equivalent to a g being non-degenerate.
Such a A induces a group isomorphism Of course A is not unique.
Question 3.9. Are there abstract arguments for the existence of these isomorphism and for its explicit form?
We will calculate explicit expressions for A depending on the cases in the next section. At this point we give the generic answers: For Λ = Λ R the two conditions are equivalent, so existence is trivial (resp. obviously the two trivial groups are isomorphic) and we may simply take for A any base change between left and right side. The expression may however be nontrivial.
In particular this is the case if is prime to all root lengths and all divisors of the Cartan matrix. Moreover This means we only have to calculate A for all divisors of |Λ ∨ W /Λ|, which is a subset of all divisors of root lengths times divisors of the Cartan matrix.
Proof . For the first condition we need to show for any λ ∈ Λ ∨ W that λ ∈ Λ already implies λ ∈ Λ. But if by assumption the order of the quotient group Λ ∨ W /Λ is prime to , then · is an isomorphism on this abelian group, hence follows the assertion. For the second condition applies the same argument noting that |Λ/Λ R | = |Λ ∨ W /Λ|. For the second claim we simply consider the inclusion chains where a first isomorphism is given by A 2 and again 1 · is a second isomorphism because it is prime to the index.
Our main result of this chapter is the following: 1. The following form is well defined on the quotients: A (µ)).

Let
Corollary 3.13. The quasitriangularity conditions for a choice R 0 are by Lemma 3.5 equivalent to the non-degeneracy of the group pairing on Λ 1 /Λ × Λ 2 /Λ : By the previous theorem this condition is now equivalent to the nondegeneracy of a g .
This condition on the fundamental group, which is a finite abelian group and mostly cyclic, can be checked explicitly once a g has been calculated.
Proof of Theorem 3.12. The first part of the theorem is a direct consequence of the definition of the centralizer transfer matrix A . For the second part, we first notice that by assumption we have a commutative diagram of finite abelian groups where G ∧ denotes the dual group of a group G. Now, by the five lemma we know thatf is an isomorphism if and only if the induced mapf is an isomorphism. Post-composing this map with the dualized centralizer transfer matrix

Explicit calculation for every g
In the following, we want to compute the endomorphism A ∈ End Z (Λ) and the pairing a on the fundamental group explicitly in terms of the Cartan matrices and the common divisors of with root lengths and divisors of the Cartan matrix. We will finally give a list for all g.

Technical tools
We choose the basis of simple roots α i for Λ R and the dual basis of fundamental coweights λ ∨ i for the dual lattice Λ ∨ W with (α i , λ ∨ j ) = δ i,j . For any choice Λ ⊂ Λ W ⊂ Λ ∨ W , let A Λ be a basis matrix, i.e., any Z-linear isomorphism Λ ∨ W → Λ sending the basis λ ∨ i of Λ ∨ W to some basis µ i of Λ. It is unique up to pre-composition of a unimodular matrix U ∈ SL n (Z).
The dual basis AΛ ofΛ is defined by Explicitly, AΛ is given by be the unique Smith decomposition of A Λ , which means: P Λ , Q Λ are unimodular and S Λ is diagonal with diagonal entries (S Λ ) ii =: are the divisors of scalar product matrix (α i , α j ). Their product is For the coweight lattice all d Without loss of generality, we will assume the basis matrices A Λ to be symmetric, i.e., Q Λ = P T Λ . We then have the following lemma: W be a lattice. We define lattices Then, Proof . We compute explicitly, On the other hand, In particular, this means AΛ Cent Λ (Λ R ) = Cent Λ R (Λ).

Case Λ = Λ W
In order to exhaust all cases that appear in our setting, we continue with Λ = Λ W : In the case Λ = Λ W , the centralizer transfer matrix A is of the following form: else.
Here, C = P C S C Q C denotes the Smith decomposition of the Cartan matrix of g.
Proof . As we noted in Example 4.1, we have A Λ W = Diag(d i ), for d i being the ith root length. Since d i ∈ {1, p} for some prime number p, up to a permutation A Λ W is already in Smith normal form: this means that P Λ W is a permutation matrix of the form (P Λ W ) ij = δ j,σ(i) for some σ ∈ S n , s.t. d σ(1) ≤ · · · ≤ d σ(n) . It follows that A Cent = Diag gcd( ,d i ) .
Using the definition C ij = Thus, By the previous lemma, this proves the first condition for A . The second condition follows immediately from the previous lemma. The case gcd( , |π 1 |) = 1 follows from Lemma 3.11 and the fact that

Case A n
In the following example, we treat the case g = A n with fundamental group Λ W /Λ R = Z n+1 for general intermediate lattices Example 4.4. In order to compute the centralizer transfer map A , we first compute the Smith decomposition of A R : A sublattice Λ R Λ Λ W is uniquely determined by a divisor d | n + 1, so that Λ/Λ R ∼ = Z d and is generated by the multipledλ n , whered := n+1 d . Then Since A n is simply laced with cyclic fundamental group, the formula A Λ = P R S Λ P T R gives us symmetric basis matrices of sublattices Λ R ⊆ Λ ⊆ Λ W . We also substitute the above basis matrix of the root lattice A R by A R (Q R ) −1 P T R . It is then easy to see that the definition A := P R D P T R gives a centralizer transfer matrix. We calculate it explicitly Now a form g is uniquely determined by a dth root of unity g(χ, χ) = exp 2πi·k d = ζ k d with some k. Then we calculate the form a g on the generator d · gcd( ,d) .
For example the trivial g (i.e., k = 0) gives an R-matrix for all lattices Λ (defined bydd = n + 1) iffd is coprime to d. For coprime to the divisor n+1 this amounts to all lattices associated to decompositions of n + 1 into two coprime factors.

Case D n
Finally, we consider the root lattice D n . Since we have π 1 (D 2n≥4 ) ∼ = Z 2 × Z 2 and π 1 (D 2n+1≥5 ) ∼ = Z 4 , it is appropriate to split this investigation in two steps. We start with D 2n≥4 . In order to compute the respective Smith decompositions, we used the software Wolfram Mathematica.
The following lemma connects our results with Lusztig's original result: Lemma 4.7. In Lusztig's definition of a quantum group he uses the quotient This coincide with our choice Λ = Cent Λ R (Λ 1 + Λ 2 ), if and only if where the d Λ i denote the invariant factors of Λ ∨ W /Λ and the d W i denote the invariant factors of Λ ∨ W /Λ W (i.e., ordered root lengths).
In particular, for odd these choices never coincide. For Λ = Λ W , Λ = Λ Lusz holds if and only if 2d i | . This is the most extreme case of divisibility and it is precisely the case appearing in logarithmic conformal field theories.
Proof . We first note that in our cases, Λ = Cent By Lemma 4.2, this coincides with Λ if and only if equation (4.1) holds.

Factorizability of quantum group R-matrices
We first recall the definition of factorizable braided tensor categories and factorizable Hopf algebras, respectively.
Equivalently, this means we can write Shimizu [18] has recently proven a number of equivalent characterizations of factorizability for arbitrary (in particular non-semisimple) braided tensor categories. Besides the two previous characterizations (equivalence to Drinfeld center and nondegeneracy of the monodromy matrix), factorizability is equivalent to the fact that the so-called transparent objects are all trivial, see Theorem 5.16 below, which will become visible during our analysis later.

Monodromy matrix in terms of R 0
In order to obtain conditions for the factorizability of the quasitriangular small quantum groups (u q (g, Λ, Λ ), R 0 (f )Θ) as in Theorem 2.2 in terms of g, q, Λ and f , we start by calculating the monodromy matrix M := R 21 · R ∈ u q (g, Λ, Λ ) ⊗ u q (g, Λ, Λ ) in general as far as possible: Lemma 5.4. For R = R 0 (f )Θ as in Theorem 2.2, the factorizability of R is equivalent to the invertibility of the following complex-valued matrix m with entries indexed by elements in µ, ν ∈ Λ/Λ : Proof . We first plug in the expressions for R 0 from Theorem 3.3 andΘ from Theorem 2.2 and simplify: . The last equation holds since b − 1 ∈ u − β 1 and hence fulfills K ν 2 b − 1 = q −β 1 ν 2 b − 1 K ν 2 and similarly for b * + 1 . We have two triangular decompositions and the Λ + R -gradation on u ± q induces a gradation The factorizability of R is equivalent to the invertibility of M interpreted as a matrix indexed by the PBW basis. The grading implies a block matrix form of M , so the invertibility M is equivalent to the invertibility of M β 1 ,β 2 ∈ (u q ⊗ u q ) (β 1 ,β 2 ) for every β 1 , β 2 ∈ Λ + R as follows Since the second sum in M β 1 ,β 2 runs over a basis in u + , the invertibility of M is equivalent to the invertibility for all β 1 ∈ Λ + R the following element: Since K ν ⊗ K µ is a vector space basis of u 0 q ⊗ u 0 q = C[Λ/Λ ] ⊗ C[Λ/Λ ], this in turn is equivalent to the invertibility of the following family of matrices m β 1 for all β 1 ∈ Λ + R with rows/columns indexed by elements in µ, ν ∈ Λ/Λ : We now use the fact that R was indeed an R-matrix. By property (2.1) in Theorem 2.2 we have Since the invertibility of a matrix m µ,ν is equivalent to the invertibility of any matrix m µ,ν+β 1 , we may substitute ν → ν + β 1 , ν → ν + β 1 , pull the constant factor q −β 2 1 in front (which also does not affect invertibility) and hence eliminate the first β 1 from the condition. Hence the invertibility of R is equivalent to the invertibility of the following family of matricesm β 1 for all β 1 ∈ Λ + R : We may now use the same procedure to eliminate the second β 1 , hence the invertibility of R is equivalent to the invertibility of the following matrix with rows/columns induced by elements in µ, ν ∈ Λ/Λ : This was the assertion we wanted to prove.
Definition 5.5. Let g : G 1 × G 2 → C × be a group pairing. It induces a symmetric form on the product G 1 × G 2 we denote by Sym(g): Lemma 5.6. If g : G 1 × G 2 → C × is a perfect pairing of abelian groups, then the symmetric form Sym(g) is perfect.
Proof . By assumption, g × g defines an isomorphism between G 1 × G 2 to G 2 × G 1 . The symmetric form Sym(g) is given by the composition of this isomorphism with the canonical isomorphism G 2 × G 1 ∼ = G 1 × G 2 . This proves the claim.
Consider for a finite abelian group G and subgroups G 1 , G 2 ≤ G the canonical exact sequence For µ ∈ G 1 + G 2 , we denote its fiber by Moreover, we define Rad ⊥ 0 := {µ 1 + µ 2 ∈ G | (µ 1 , µ 2 ) ∈ Rad}. Lemma 5.7. We have two split exact sequences: The first sequence is exact by definition of the three groups. Moreover, we know whereι,π denote the duals of the inclusion and projection in (5.1). In Example 5.11 we will see that in the case G 1 = G 2 = G,f symmetric, Rad 0 is the 2-torsion subgroup of G, and the second map in the second exact sequence is just the projection, hence both diagrams split in this case. Iff is asymmetric, we will see in Section 5.3 that Rad 0 is isomorphic to Z k 2 for some k ≥ 2, thus is a section of the first exact sequence. Here we used that the sum over all elements in Z k 2 vanishes. Again, it follows that both diagrams split. Finally, if G 1 = G 2 (i.e., in the case D 2n ), thenf = q −(·,·) on G 1 ∩ G 2 . By the same argument as in Example 5.11, Rad 0 is the 2-torsion subgroup of G 1 ∩ G 2 . But we have G ∼ = G 1 ∩ G 2 × π 1 in this case, hence both sequences split.
Corollary 5.8. Using the projection α : G → Rad ⊥ 0 and the inclusion β : Rad ⊥ 0 → Rad from the above lemma, we can define a symmetric form on G: Moreover, we have Rad Sym G f ∼ = Rad 0 .
Theorem 5.9. We have shown in Theorem 2.2 and Lemma 3.5 that the assumption that R = R 0 (f )Θ is an R-matrix is equivalent to the existence of subgroups G 1 , G 2 ⊂ Λ/Λ of same order some d|Λ R /Λ | and f restricting up to a scalar to a non-degenerate pairingf : G 1 × G 2 → C × and f vanishes otherwise.
In this notation the matrix m as defined in the previous lemma can be rewritten as It is invertible if and only if Rad 0 = 0. In this case, We first note that Rad 0 = 0 implies Rad ⊥ 0 = G and thus G = G 1 + G 2 . Together with Corollary 3.6 this implies Corollary 5.10.
Before we proof the theorem, we first give a simple example: Example 5.11. Let G 1 = G 2 = G (correspondingly Λ 1 = Λ 2 = Λ) and assumef is symmetric non-degenerate, then the radical measures 2-torsion: Again, this is the only case appearing for cyclic fundamental groups. Hence in all cases except g = D 2n factorizability is equivalent to |Λ/Λ | being odd.
Proof of Theorem 5.9. The first part of the theorem follows by applying Lemma 3.5 to the matrix m as given in the previous lemma. Now, assume that m is invertible. We must have G = G 1 + G 2 , otherwise the matrix has zero-columns and rows, differently formulated: the fibers (G 1 × G 2 ) µ in the short exact sequence must be non-empty for all µ ∈ G. If on the other hand, Rad 0 = 0, then Rad ⊥ 0 = G and thus G 1 + G 2 = G must also hold, thus we assume this from now on. By the short exact sequence the fiber (G 1 × G 2 ) 0 ∼ = G 1 ∩ G 2 , other fibers are of the explicit formμ + (G 1 × G 2 ) 0 for some choice of representativeμ. Therefore, Fix as above a representativeν of the fiber of ν, i.e.,ν ∈ (G 1 ×G 2 ) ν such that Sym(f )(ν, )| G 1 ∩G 2 = 1 holds. Two elements fulfilling this property differ by an element in the subgroup Rad 0 ≤ G 1 ∩ G 2 , thus Since m is symmetric, we have and this is invertible if an only if Rad 0 ∼ = Rad Sym G f = 0.

Factorizability for symmetric R 0 (f )
For R 0 = µ,ν f (µ, ν)K µ ⊗ K ν being the Cartan part of an R-matrix, assume thatf = |G|f on G is symmetric. We have shown in Example 5.11 that factorizability is equivalent to |G| being odd. We now want to give a necessary and sufficient condition for this: 2) is either odd or ( ≡ 2 mod 4, g = B n , Λ = Λ R ) including A 1 .
Proof . We saw that in all our cases, there exists an isomorphism Moreover, from Lemma 4.2 we know that |Λ/ Cent Λ (Λ R )| = det(D ), where D was the diagonal matrix Diag gcd( ,d Λ i ) ) with d Λ i being the invariant factors of the lattice Λ (i.e., the diagonal entries of the Smith normal form of a basis matrix of Λ). Thus, Clearly, this term is odd if and |Λ/Λ R | are odd. In the case ( ≡ 2 mod 4, g = B n , Λ = Λ R ), the Smith normal form S R of the basis matrix A R is given by 2 · id. Thus, |G| is odd in this case. On the other hand, let |G| be odd: We first consider the case even. A necessary condition for |Λ/Λ | odd is that the multiplicity m of the prime 2 in is at most the multiplicity m π 1 of the prime 2 in |π 1 |. We check this condition for rank n > 1: • For g simply-laced (or triply-laced g = G 2 ) we have all d i = 1, hence n | m (equality for = 2 mod 4). The cases D n with m π 1 = 2 have rank n ≥ 4, all others except A n have m π 1 = 0, 1, so the necessary condition m ≤ m π 1 is never fulfilled. The cases A n have 2 mπ 1 |(n + 1) ≤ (m + 1) ! ≤ (m π 1 + 1) which can only be true in rank n = 1 treated above.
• For g doubly-laced of rank n > 1, we always have always m π 1 = 0, 1 but m can be considerably smaller than above, namely for = 2 mod 4 equal to the number of short simple roots d α i = 1 (otherwise m again increases by n for every factor 2 in ), hence the necessary condition m ≤ m π 1 can be fulfilled only for B n (which would also include A 1 above for n = 1). More precisely, since m = m π 1 and the decomposition for Λ/Λ has an additional factor |Λ/Λ R |, it can only be odd for Λ = Λ R .
On the other hand, if is odd, then the whole product term is odd. But since |G| was assumed to be odd, also |Λ/Λ | must be odd.
Corollary 5.13. Let Λ = Λ R . In the previous section we have seen thatf = q −(·,·) gives always an R-matrix in this case. By the proof of the previous lemma, we have where the d R i denote the invariant factors of Λ ∨ W /Λ R .

Factorizability for D 2n , R 0 antisymmetric
The split case g = D 2n , G = G 1 × G 2 is clearly factorizable, so the only remaining case for which we have to check factorizabilty is g = D 2n , Λ = Λ W forf being not symmetric. We know that in this case, the corresponding form g on Λ/Λ R is uniquely defined by a 2 × 2-matrix K ∈ gl(2, F 2 ), s.t. g(λ 2(n−1)+i , λ 2(n−1)+j ) = exp 2πiK ij 2 for i, j ∈ {1, 2}. From this we see that if g is not symmetric, it must be antisymmetric, i.e., g(µ, ν) = g(ν, µ) −1 . Thus, the following lemma applies in this case, and hence there are no factorizable R-matrices for D 2n , Λ = Λ W .
Proof . We recall the definition of Rad 0 Sym G f in this case: For g is simply-laced, we have Λ W = Λ ∨ W , thus This proves the claim.

Transparent objects in non-factorizable cases
In this section, we determine the transparent objects in the representation category of u q (g, Λ) with our R-matrix given by R 0Θ and R 0 = 1 |Λ/Λ | µ,ν∈Λ/Λ f withf a group pairing Λ 1 /Λ × Definition 5.15. Let C be a braided monoidal category with braiding c. An object V ∈ C is called transparent if the double braiding c W,V • c V,W is the identity on V ⊗ W for all W ∈ C.
The following theorem by Shimizu gives a very important characterization of factorizable categories:  Since in our cases Λ 1 = Λ 2 can only appear in D 2n , and we know those are factorizable, we shall in the following restrict ourselves to the case Λ 1 = Λ 2 = Λ. The proof below works also in the more general case, but requires more notation. As usual we first reduce the Hopf algebra question to the group ring and then solve the group theoretical problem.
Lemma 5.18. If a u q (g)-module V , with a highest-weight vector v and K µ v = χ(K µ )v, is a transparent object, then necessarily the 1-dimensional Λ/Λ -module C χ is a transparent object over the Hopf algebra C[Λ/Λ ] with R-matrix R 0 . If V is 1-dimensional, then V is transparent if and only if C χ is.
Proof . Let V be transparent. For every ψ : Λ/Λ → C × we have another finite-dimensional module W := u q (g) ⊗ uq(g) + C ψ with highest weight vector w = 1 ⊗ 1 ψ which we can test this assumption against We calculate the effect of c 2 on the highest-weight vectors v ⊗ w: Because v, w were assumed highest-weight vectors, theΘ act trivially. Hence follows that C χ , C ψ have a trivial double braiding over the Hopf algebra C[Λ/Λ ] with R-matrix R 0 . Because we could achieve this result for any ψ this means that C χ is transparent as asserted. Now, let V = C χ be 1-dimensional over u q (g) and transparent over C[Λ/Λ ], and let w be any element in any module W , then again the two Θ act trivially, one time because v = 1 χ is a highest weight vector, and one time because it is also a lowest weight vector. But if the double-braiding of v = 1 χ with any element w is trivial, then V = C χ is already transparent over u q (g).
The previous two lemmas combined imply that any irreducible transparent u q (g)-module has necessarily the characters χ(µ) = f (µ, ξ), ξ ∈ Rad 0 as highest-weights, and conversely if such a character χ gives rise to 1-dimensional u q (g)-modules (i.e., χ| 2Λ R = 1), then these are guaranteed transparent objects. Hence the final step is to give more closed expressions for the f -transformed characters χ of the radical depending on the case and check the 1-dimensionality condition.
In all cases where f is symmetric we have seen in Example 5.11 that Rad 0 Sym G f is the 2-torsion subgroup of Λ/Λ , so in these cases χ gives rise to a 1-dimensional object.
Corollary 5.20. If f is symmetric (true for all cases except D 2n ) then the transparent objects are all 1-dimensional C χ where the characters χ are the f -transformed of the elements in the radical of the bimultiplicative form Sym f | G on G = Λ/Λ . In particular the group of transparent objects is isomorphic to this radical as an abelian group.
Corollary 5.21. In the case of symmetric f (all cases except D 2n ) the fact that Rad 0 is the 2-torsion of Λ/Λ and f -transformation is a group isomorphism shows: The group T of transparent objects consists of C χ where χ| 2Λ = 1, i.e., the two-torsion of the character group.
The remaining case in D 2n with f nonsymmetric and has been done by hand in Lemma 5.14.

Quantum groups with a ribbon structure
In [16,Theorem 8.23], the existence of ribbon structures for u q (g, Λ) is proven. In this section we construct a ribbon structure for all cases. In the proof, we use several auxiliary results from [16]. Theorem 6.1. Let u q (g, Λ) be quasitriangular Hopf algebra, with an R-matrix satisfying the conditions in Theorem 2.2 and let u := S(R (2) )R (1) . Then v := K −1 ν 0 u is a ribbon element in u q (g, Λ).