Factorizable R-Matrices for Small Quantum Groups
Representations of small quantum groups uq(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...
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| description | Representations of small quantum groups uq(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.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 076, 25 pages
Factorizable R-Matrices for Small Quantum Groups
Simon LENTNER and Tobias OHRMANN
Fachbereich Mathematik, University of Hamburg,
Bundesstraße 55, 20146 Hamburg, Germany
E-mail: simon.lentner@uni-hamburg.de, tobias.ohrmann@uni-hamburg.de
Received January 16, 2017, in final form September 15, 2017; Published online September 25, 2017
https://doi.org/10.3842/SIGMA.2017.076
Abstract. Representations of small quantum groups uq(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.
Key words: factorizable; R-matrix; quantum group; modular tensor category; transparent
object
2010 Mathematics Subject Classification: 17B37; 20G42; 81R50; 18D10
1 Introduction
Hopf algebras with R-matrices, so called quasitriangular Hopf algebras, give rise to tensor cate-
gories 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 equiva-
lent to the fact that there are no transparent objects V , i.e., no objects where the double-braiding
c2 : 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 SL2(Z) are constructed from θ and c2, 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 uq(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 R0Θ̄, where the fixed element Θ̄ ∈ uq(g)− ⊗ uq(g)+ is constructed from
a dual basis of PBW generators, while R0 ∈ uq(g)0⊗ uq(g)0 is a free parameter subject to some
constraints. He gives one canonical solution for R0 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.
mailto:simon.lentner@uni-hamburg.de
mailto:tobias.ohrmann@uni-hamburg.de
https://doi.org/10.3842/SIGMA.2017.076
2 S. Lentner and T. Ohrmann
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 Car-
tan part of the quantum group by a choice of a lattice ΛR ⊆ Λ ⊆ ΛW between root- and
weight-lattice, respectively a choice of a subgroup of the fundamental group π1 := ΛW /ΛR,
corresponding to a choice of a Lie group between adjoint and simply-connected form. These ex-
tensions 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 R0 in this spirit (under
some additional assumptions). As it turns out, the solutions can be parametrized by subgroups
H1, H2 ⊂ π1 and group pairings between H1, H2, 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 Bn behaves differently for n odd or even, and that D2n with
non-cyclic fundamental group allows several more solutions with non-symmetric R0.
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 π1
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 finite-
dimensional (semi-)simple complex Lie algebra g and a primitive `-th root of unity q Lusztig has
introduced in [12] the small quantum group uq(g) which has a triangular decomposition u+
q u
0
qu
−
q
where the (exponentiated) Cartan algebra u0
q is the groupring of the root lattice ΛR modulo some
suitable sublattice and u±q are generated by simple root vectors Eαi , Fαi fulfilling q-deformed
Serre relations. In [13, Section 32] he gives an ansatz for an R-matrix in the form R0Θ̄, where Θ̄
consists of dual PBW basis’ and R0 ∈ u0
q ⊗ u0
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 quan-
tum group uq(g,Λ,Λ
′) with any choice of lattice between root- and weight-lattice ΛR ⊆ Λ ⊆ ΛW
and any possible choice of quotient by a subgroup Λ′ ⊆ Λ in the Cartan part u0 = C[Λ/Λ′].
Later, we prove that Λ′ is in fact unique if we want a quasitriagular structure (Corollary 3.6).
The R0-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 uq(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]:
Theorem 3.3. The R0-matrix is necessarily of the form
f(µ, ν) =
1
d
q−(µ,ν)g(µ̄, ν̄)δµ̄∈H1δν̄∈H2 ,
Factorizable R-Matrices for Small Quantum Groups 3
where H1, H2 are subgroups of Λ/ΛR ⊆ π1 with |H1| = |H2| =: d (not necessarily isomorphic!)
and g : H1 ×H2 → 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
pairing f̂ := |Λ/Λ′| · f between the preimages Gi := Λi/Λ
′ of the groups Hi is non-degenerate
(which is no surprise). In particular we show that this condition fixes Λ′ uniquely. In later
applications we often encounter f̂ as a natural identification of G1 and the dual Ĝ2, e.g., when
studying representation theory.
To find all solutions f with this property we develop a machinery to push f̂ into the funda-
mental group π1, which encapsulates all the `-dependence: In Definition 3.8 we give an abstract
characterization of a centralizer transfer map
A` : Λ/ΛR
∼−→ Cent`Λ(ΛR)/Cent`ΛR(Λ)
(without proving that it always exists). In a generic case this is just multiplication by `, but it se-
verely 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
that f̂ 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 Zn 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 = ΛW
we have a generic construction, the cases An with their large fundamental group Zn+1 is treated
by hand, as isD2n with non-cyclic fundamental group, which has the only cases allowing Λ1 6= Λ2.
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 matrixR21R for
an arbitrary choice of R-matrix in terms the R0-part. This gives a purely lattice theoretic prob-
lem 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 SymG
(
f̂
)
of f̂ . 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 H1 = H2 and f , g are
symmetric. Other cases appear only in some of the non-cyclic Z2×Z2-extension for type g = D2n
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 SymG
(
f̂
)
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/ΛR as well as for g = Bn, Λ = ΛR, ` ≡ 2 mod 4 including A1.
In Section 5.4 we prove the following result:
Lemma 5.19. The transparent objects in the category of representations of the Hopf algebra
uq(g,Λ) with R-matrix given by Lusztig’s ansatz are 1-dimensional objects Cχ and are the f -
transformed of the radical of SymG
(
f̂
)
:
χ(µ) = f(µ, ξ), ξ ∈ Rad
(
SymG
(
f̂
))
.
In the following we summarize our results by a table containing all quasitriangular quantum
groups uq(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 = ΛR and the following new factorizable cases:
4 S. Lentner and T. Ohrmann
(` odd, E6, Λ = ΛW ) and (` ≡ 2 mod 4, g = Bn, Λ = ΛR) (including A1) and (` odd,
g = D2n, Λ1 6= Λ2). All other cases can be modularized as discussed in Question 7.3.
The columns of the following table are labeled by
1) the finite-dimensional simple complex Lie algebra g,
2) the natural number `, determining the root of unity q = exp
(
2πi
`
)
,
3) the number of possible R-matrices for the Lusztig ansatz,
4) the subgroups Hi ⊆ H = Λ/ΛR introduced in Theorem 3.3,
5) the subgroups Hi in terms of generators given by multiples of fundamental dominant
weights λi ∈ ΛW ,
6) the group pairing g : H1 ×H2 → C× determined by its values on generators,
7) the group of transparent objects T ⊆ Λ/Λ′ introduced in Lemma 5.19.
g ` # Hi ∼= Hi (i=1,2) g T ⊆ Λ/Λ′
all
` odd 1
Z1 〈0〉 g = 1
0
` ≡ 0 mod 4 1 Zn2
∞
Zn−1
2 , 2 - x
Zd 〈d̂λn〉 g(d̂λn, d̂λn) = exp
(
2πik
d
)
An≥1 Zn2 , 2 |x
π1 = Zn+1
d |n+ 1 d̂ = n+1
d
gcd
(
d, k`−d̂n
gcd(`,d̂)
)
= 1
x = d`
gcd(`,d̂)
` ≡ 2 mod 4 1 Z1 〈0〉 g = 1 0
` ≡ 2 mod 4 2 Z2 〈λn〉 g(λn, λn) = ±1 Z2
Bn≥2 ` ≡ 0 mod 4 2 Z2 〈λn〉 g(λn, λn) = ±1 Zn2π1 = Z2
` odd 1 Z2 〈λn〉 g(λn, λn) = (−1)n+1 Z2
` ≡ 2 mod 4 1 Z1 〈0〉 g = 1 Zn−2
2
` ≡ 2 mod 4 1
Z2 〈λn〉
g(λn, λn) = 1 Zn−1
2
Cn≥3 ` ≡ 0 mod 4 2 g(λn, λn) = ±1 Zn2π1 = Z2
` odd 1 g(λn, λn) = −1 Z2
` ≡ 2 mod 4 1 Z1 〈0〉 g = 1 Z2(n−1)
2
` ≡ 2 mod 4 1
Z2
H1
∼= 〈λ2n−1〉
g(λ2n−1, λ2n) = (−1)n
Z2n
2
` ≡ 0 mod 4 2δ2 |n g(λ2n−1, λ2n) = ±1, n even
H2
∼= 〈λ2n〉` odd 1 g(λ2n−1, λ2n) = −1 0
D2n≥4
π1 = Z2 × Z2
` ≡ 2 mod 4 1
Z2 〈λ2n〉
g(λ2n, λ2n) = (−1)n+1 Z2n−1
2
` ≡ 0 mod 4 2δ2-n g(λ2n, λ2n) = ±1, n odd Z2n
2
` odd 1 g(λ2n, λ2n) = −1 Z2
` even 16
Z2 × Z2 〈λ2n, λ2n+1〉
g(λ2(n−1)+i, λ2(n−1)+j) = ±1 Z2n
2
` odd 6
det(K) = K12 +K21 = 0 mod 2 Z2
det(K) = K12 +K21 = 1 mod 2 Z2
2
Factorizable R-Matrices for Small Quantum Groups 5
` ≡ 2 mod 4 1 Z1 〈0〉 g = 1 Z2n
2
` ≡ 2 mod 4 1
Z2 〈2λ2n+1〉
g(2λ2n+1, 2λ2n+1) = 1
Z2n+1
2
` ≡ 0 mod 4 2 g(2λ2n+1, 2λ2n+1) = ±1
D2n+1≥5 ` odd 1 g(2λ2n+1, 2λ2n+1) = −1 Z2
π1 = Z4
` even 4
Z4 〈λ2n+1〉
g(λ2n+1, λ2n+1) = c, c4 = 1 Z2n+1
2
` odd 2 g(λ2n+1, λ2n+1) = ±1 Z2
` ≡ 2 mod 4 1 Z1 〈0〉 g = 1 Z6
2
` ≡ 0 mod 3 3
Z3 〈λn〉
g(λn, λn) = c, c3 = 1 Z6
2, 2 | `
E6
` ≡ 1 mod 3 2 g(λn, λn) = 1, exp
(
2πi2
3
)
π1 = Z3
0, 2 - `
` ≡ 2 mod 3 2 g(λn, λn) = 1, exp
(
2πi
3
)
` ≡ 2 mod 4 1 Z1 〈0〉 g = 1 Z6
2
` even 2
Z2 〈λn〉
g(λn, λn) = ±1 Zn2E7
π1 = Z2 ` odd 1 g(λn, λn) = 1 Z2
E8 ` ≡ 2 mod 4 1 Z1 〈0〉 g = 1 Z8
2
F4 ` ≡ 2 mod 4 1 Z1 〈0〉 g = 1 Z2
2
G2 ` ≡ 2 mod 4 1 Z1 〈0〉 g = 1 Z2
2
Table 1: Solutions for R0-matrices.
In the case D2n, Λ = ΛW , g is uniquely defined by a (2 × 2)-matrix K ∈ gl(2,F2), s.t.
g(λ2(n−1)+i, λ2(n−1)+j) = exp
(2πiKg
ij
2
)
for i, j ∈ {1, 2}.
2 Preliminaries
2.1 Lie theory
Throughout this article, g denotes a finite-dimensional simple complex Lie algebra. We fix
a choice of simple roots ∆ = {αi | i ∈ I}, so that the Cartan matrix C is given by Cij = 2
(αi,αj)
(αi,αi)
,
where ( , ) denotes the normalized Killing form. For a root α, we define dα := (α,α)
2 and set
di = dαi . By ΛR := Z[∆] and Λ∨R := Z[∆∨] we denote the (co)root lattice of g.
By ΛW , we denote the weight lattice spanned by fundamental dominant weights λi, which are
defined by the equation (λi, αj) = δi,jdi. Finally, we define the co-weight lattice Λ∨W as the Z-
span of the elements λ∨i := λi
di
. The quotient π1 := ΛW /ΛR is called the fundamental group of g.
One can easily see that the Killing form restricts to a perfect pairing ( , ) : Λ∨W ×ΛR → Z and
that we get a string of inclusions ΛR ⊆ Λ∨R ⊆ ΛW ⊆ Λ∨W .
2.2 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 R-
matrix of Uq(g). Namely, for a specific element Θ̄ ∈ U≥0
q ⊗U≤0
q from a dual basis and a suitable
(not further specified) element in the coradical R0 ∈ U0
q ⊗ U0
q we are looking for R-matrices of
6 S. Lentner and T. Ohrmann
the form
R = R0Θ̄.
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 uq(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 Uq(g,Λ,Λ
′), we have
R0 ∈ C[Λ/Λ′]⊗ C[Λ/Λ′].
This ansatz has been worked out by Müller in his dissertation [14, 15, 16] for small quantum
groups uq(g) which we will use in the following, leading to a system of quadratic equation on R0
that are equivalent to R being an R-matrix:
Theorem 2.1 (cf. [16, Theorem 8.2]). Let u := uq(g).
(a) There is a unique family of elements Θβ ∈ u−β ⊗ u
+
β , β ∈ ΛR, such that Θ0 = 1 ⊗ 1 and
Θ =
∑
β Θβ ∈ u⊗ u satisfies ∆(x)Θ = Θ∆̄(x) for all x ∈ u.
(b) Let B be a vector space-basis of u−, such that Bβ = B ∩ u−β is a basis of u−β for all β.
Here, u−β refers to the natural ΛR-grading of u−. Let {b∗ | b ∈ Bβ} be the basis of u+
β dual
to Bβ under the non-degenerate bilinear form ( · , · ) : u− ⊗ u+ → C. We have
Θβ = (−1)trβqβ
∑
b∈Bβ
b− ⊗ b∗+ ∈ u−β ⊗ u
+
β .
Theorem 2.2 (cf. [16, Theorem 8.11]). Let Λ′ ⊂ {µ ∈ Λ |Kµ central in uq(g,Λ)} a subgroup
of Λ, and G1, G2 subgroups of G := Λ/Λ′, containing ΛR/Λ
′. In the following, µ, µ1, µ2 ∈ G1
and ν, ν1, ν2 ∈ G2.
The element R = R0Θ̄ with an arbitrary R0 =
∑
µ,ν
f(µ, ν)Kµ⊗Kν is a R-matrix for uq(g,Λ,Λ
′),
if and only if for all α ∈ ΛR and µ, ν the following holds:
f(µ+ α, ν) = q−(ν,α)f(µ, ν), f(µ, ν + α) = q−(µ,α)f(µ, ν), (2.1)∑
ν1+ν2=ν
f(µ1, ν1)f(µ2, ν2) = δµ1,µ2f(µ1, ν),
∑
µ1+µ2=µ
f(µ1, ν1)f(µ2, ν2) = δν1,ν2f(µ, ν1),∑
µ
f(µ, ν) = δν,0,
∑
ν
f(µ, ν) = δµ,0.
3 Conditions for the existence of R-matrices
3.1 A first set of conditions on Λ/Λ′
The target of our efforts is a Hopf algebra called small quantum group uq(g,Λ,Λ
′) with Cartan
part u0
q = C[Λ/Λ′]. It is defined, e.g., in [11] and depends on lattices Λ, Λ′ defined below. For
Λ = ΛR 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 = R0Θ̄-matrix for the quantum group
uq(g,Λ,Λ
′) can be obtained from an R0-matrix of the form
R0 =
∑
µ,ν∈Λ
f(µ, ν)Kµ ⊗Kν ∈ C[Λ/Λ′]⊗ C[Λ/Λ′].
In the following we collect necessary and sufficient conditions for R = R0Θ̄ to be an R-matrix.
Factorizable R-Matrices for Small Quantum Groups 7
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
ΛR ⊆ Λ ⊆ ΛW .
These choices have a nice geometric interpretation as quantum groups associated to different
Lie groups associated to the Lie algebra g.
Another interesting choice is ΛR ⊆ Λ ⊆ Λ∨W
∼= Λ∗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 Λ1,Λ2 ⊆ Λ∨W we
define the sublattice
CentΛ1(Λ2) := { ν ∈ Λ1 | (ν, µ) ∈ ` · Z ∀µ ∈ Λ2}.
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 uq(g,Λ,Λ
′) is well-defined is Λ′ ⊆ CentΛR(ΛR).
Theorem 3.3 (cf. [11, Theorem 3.4]). The R0-matrix is necessarily of the form
f(µ, ν) =
1
d|ΛR/Λ′|
· q−(µ,ν)g(µ̄, ν̄)δµ̄∈H1δν̄∈H2 ,
where H1, H2 are subgroups of H := Λ/ΛR ⊆ π1 with equal cardinality |H1| = |H2| =: d (not
necessarily isomorphic!) and g : H1 ×H2 → C× is a pairing of groups.
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 π1 cyclic in [10] and for Z2×Z2
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
RadL(g) := {λ ∈ G | g(λ, η) = 1 ∀ η ∈ H}.
Similarly, the right radical is defined as
RadR(g) := {η ∈ H | g(λ, η) = 1 ∀λ ∈ G}.
The pairing g is called non-degenerate if RadL(g) = 0. If in addition RadR(g) = 0, g is called
perfect.
For an R0-matrix of this form, a sufficient condition is that they fulfill the so-called diamond-
equations (see [11, Definition 2.7]) for each element 0 6= ζ ∈ (Cent(ΛR) ∩ Λ)/Λ′.
However, we will now go into a different, more systematic direction that makes use of the
following observation:
Lemma 3.5. An R0-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 R0Θ̄ iff the restriction to the support
f̂ := d|ΛR/Λ′| · f : G1 ×G2 → C×
is a perfect group pairing, where Gi := Λi/Λ
′ ⊆ Λ/Λ′ =: G.
8 S. Lentner and T. Ohrmann
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
f(µ+ α, ν) = q−(ν,α)f(µ, ν), f(µ, ν + α) = q−(µ,α)f(µ, ν).
For the other equations the sums get only contributions in the support Λ1/Λ
′×Λ2/Λ
′. The quan-
tities f(µ, ν) ·d|ΛR/Λ′| for fixed ν (or µ) are characters on the respective support, and by the as-
sumed non-degeneracy all ν 6= 0 give rise to different nontrivial characters. Then the second and
third relations follows from orthogonality of characters. Note that since d|ΛR/Λ′| = |G1| = |G2|
(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 |G1| = |G2|. �
Corollary 3.6. A first consequence of the perfectness of f̂ (i.e., a necessary condition for quasi-
triangularity) is
CentΛR(Λ1) = CentΛR(Λ2) = Λ′.
This fixes Λ′ uniquely. Moreover in cases Λ1 6= Λ2, which can only happen for g = D2n, where
π1 is noncyclic, we get an additional constraint relating Λ1, Λ2.
In our case, the only possibility for Λ1 6= Λ2, s.t. G1
∼= G2 is g = D2n. In this case, we have
CentΛR(ΛW ) = CentΛR(ΛR) 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 for f̂ to a non-degeneracy condition for g on H1, H2 ⊂ π1
that can be checked explicitly.
3.2 A natural form on the fundamental group
We now define for each triple (Λ,Λ1,Λ2) and each `th root of unity q a natural pairing a` on
the subgroups Hi := Λi/ΛR of the fundamental group π1 := ΛW /ΛR. 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 ΛR) to Hi by a suitable isomorphism A`.
This isomorphism A` will encapsulate the crucial dependence on the common divisors of `, |H|
and the root lengths di; 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 iso-
morphism A` to the Hi and thus define forms ag` on H. The main result of this section is in
Theorem 3.13 that the non-degeneracy condition in Lemma 3.5 for R0(f) depending on Hi, g is
equivalent to ag` being non-degenerate.
Definition 3.7. Let Λ ⊆ Λ∨W be a sublattice, s.t. ΛR ⊆ Λ. By Λ̂ ⊂ Λ∨W we denote the unique
sublattice, s.t. the symmetric bilinear form ( · , · ) : Λ∨W ×Λ∨W → Q induces a commuting diagram
ΛR Λ̂ Λ∨W
Λ∨W
∗ Λ∗ Λ∗R,
∼= ∼= ∼=
where Λ∗ := HomZ(Λ,Z). In particular, we have Λ̂R = Λ∨W and Λ̂∨W = ΛR.
Factorizable R-Matrices for Small Quantum Groups 9
Definition 3.8. A centralizer transfer map is an group endomorphism A` ∈ EndZ(Λ), s.t.
1) A`(Λ)
!
= Λ ∩ ` · Λ̂R = Cent`Λ(ΛR),
2) A`(ΛR)
!
= ΛR ∩ ` · Λ̂ = Cent`ΛR(Λ).
Such a A` induces a group isomorphism
Λ/ΛR
∼−→ Cent`Λ(ΛR)/Cent`ΛR(Λ).
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:
Example 3.10. For Λ = Λ∨W we have A` = ` · id.
For Λ = ΛR 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.
Lemma 3.11. Assume gcd(`, |Λ∨W /Λ|) = 1, then A` = ` · id. In particular this is the case if `
is prime to all root lengths and all divisors of the Cartan matrix.
Moreover if ` = `1`2 with gcd(`1, |Λ∨W /Λ|) = 1, then A` = `1 ·A`2.
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 |Λ̂/ΛR| = |Λ∨W /Λ|.
For the second claim we simply consider the inclusion chains
A`(Λ) ⊂ Λ ∩ `2 · Λ̂R ⊂ Λ ∩ ` · Λ̂R,
A`(ΛR) ⊂ Λ ∩ `2 · Λ̂ ⊂ ΛR ∩ ` · Λ̂,
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:
Theorem 3.12. Let ΛR ⊆ Λ1, Λ2 ⊆ ΛW be intermediate lattices, s.t. the condition in Corol-
lary 3.6 is fulfilled, i.e., CentΛR(Λ1) = CentΛR(Λ2) = Λ′. Assume we have a centralizer transfer
map A`.
1. The following form is well defined on the quotients:
a`g : Λ1/ΛR × Λ2/ΛR −→ C×,
(λ̄, µ̄) 7−→ q−(λ,A`(µ)) · g(λ,A`(µ)).
2. Let
CentgΛ1
(Λ2) :=
{
λ ∈ Λ1 | q(λ,µ) = g(λ, µ) ∀µ ∈ Λ2
}
.
Then the inclusion CentgΛ1
(Λ2) ↪→ Λ1 induces an isomorphism
CentgΛ1
(Λ2)/Λ′ ∼= Rad
(
a`g
)
.
10 S. Lentner and T. Ohrmann
Corollary 3.13. The quasitriangularity conditions for a choice R0 are by Lemma 3.5 equivalent
to the non-degeneracy of the group pairing on Λ1/Λ
′ × Λ2/Λ
′:
f̂(λ, µ) = q−(λ,µ)g(λ, µ).
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
ΛR/Λ
′ Λ1/Λ
′ Λ1/ΛR
(Λ2/CentΛ2(ΛR))∧ (Λ2/Λ
′)∧ (CentΛ2(ΛR)/Λ′)∧ ,
q−(·,·) f̂ f̂ ′
where G∧ denotes the dual group of a group G.
Now, by the five lemma we know that f̂ is an isomorphism if and only if the induced map f̂ ′
is an isomorphism. Post-composing this map with the dualized centralizer transfer matrix
A∧` : (CentΛ2(ΛR)/Λ′)∧ ∼= (Λ2/ΛR)∧ gives a`g. �
4 Explicit calculation for every g
In the following, we want to compute the endomorphism A` ∈ EndZ(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.
4.1 Technical tools
We choose the basis of simple roots αi for ΛR and the dual basis of fundamental coweights λ∨i
for the dual lattice Λ∨W with (αi, λ
∨
j ) = δi,j .
For any choice Λ ⊂ ΛW ⊂ Λ∨W , let AΛ be a basis matrix, i.e., any Z-linear isomorphism
Λ∨W → Λ sending the basis λ∨i of Λ∨W to some basis µi of Λ. It is unique up to pre-composition
of a unimodular matrix U ∈ SLn(Z).
The dual basis AΛ̂ of Λ̂ is defined by(
AΛ̂
(
λ∨i
)
, AΛ
(
λ∨j
))
= δij .
Explicitly, AΛ̂ is given by AΛ̂ =
(
A−1
Λ AR
)T
, where (AR)ij = (αi, αj). Now, let AΛ = PΛSΛQΛ
be the unique Smith decomposition of AΛ, which means: PΛ, QΛ are unimodular and SΛ is
diagonal with diagonal entries (SΛ)ii =: dΛ
i , such that dΛ
i | dΛ
j for i < j.
Example 4.1. For the root lattice the dΛR
i are the divisors of scalar product matrix (αi, αj).
Their product is∏
i
dΛR
i =
∣∣Λ∨W /ΛR∣∣ =
(∏
i
di
)
· |π1|, di =
(αi, αi)
2
.
For the coweight lattice all d
Λ∨W
i = 1. For the weight lattice we recover the familiar dΛW
i = di.
Factorizable R-Matrices for Small Quantum Groups 11
Without loss of generality, we will assume the basis matrices AΛ to be symmetric, i.e.,
QΛ = P TΛ . We then have the following lemma:
Lemma 4.2. Let ΛR ⊆ Λ ⊆ Λ∨W be a lattice. We define lattices
ACent :=
(
P TΛ
)−1
D`P
−1
Λ , D` := Diag
(
`
gcd
(
`, dΛ
i
)) .
Then,
CentΛR(Λ) = ARACentΛ
∨
W , CentΛ(ΛR) = AΛACentΛ
∨
W .
Proof. We compute explicitly,
CentΛR(Λ) = ΛR ∩ ` · Λ̂ = ARΛ∨W ∩
(
A−1
Λ AR
)T
`Λ∨W
=
(
A−1
Λ AR
)T (((
A−1
Λ AR
)T )−1
ARΛ∨W ∩ `Λ∨W
)
= ARA
−1
Λ
(
AΛ ∩ `Λ∨W
)
= AR
(
PΛSΛP
T
Λ
)−1(
PΛSΛP
T
Λ Λ∨W ∩ `Λ∨W
)
= AR
(
P TΛ
)−1
S−1
Λ
(
SΛΛ∨W ∩ `Λ∨W
)
= AR
(
P TΛ
)−1
S−1
Λ Diag(lcm(SΛii , `))Λ
∨
W = AR
(
P TΛ
)−1
D`Λ
∨
W = ARACentΛ
∨
W .
On the other hand,
CentΛ(ΛR) = Λ ∪ `Λ̂R = Λ ∪ `Λ∨W = AΛΛ∨W ∪ `Λ∨W = PΛSΛP
T
Λ Λ∨W ∪ `Λ∨W
= PΛ
(
SΛΛ∨W ∪ `Λ∨W
)
= PΛSΛD`Λ
∨
W = AΛ
(
P TΛ
)−1
D`Λ
∨
W = AΛACentΛ
∨
W .
In particular, this means AΛ̂ CentΛ(ΛR) = CentΛR(Λ). �
4.2 Case Λ = ΛW
In order to exhaust all cases that appear in our setting, we continue with Λ = ΛW :
Lemma 4.3. In the case Λ = ΛW , the centralizer transfer matrix A` is of the following form:
A` =
{
AΛWACentQ
T
CP
−1
C A−1
ΛW
, gcd(`, |π1|) 6= 1,
` · id, else.
Here, C = PCSCQC denotes the Smith decomposition of the Cartan matrix of g.
Proof. As we noted in Example 4.1, we have AΛW = Diag(di), for di being the ith root length.
Since di ∈ {1, p} for some prime number p, up to a permutation AΛW is already in Smith normal
form: this means that PΛW is a permutation matrix of the form (PΛW )ij = δj,σ(i) for some
σ ∈ Sn, s.t. dσ(1) ≤ · · · ≤ dσ(n). It follows that ACent = Diag
(
`
gcd(`,di)
)
.
Using the definition Cij =
(αi,αj)
di
, in the case gcd(`, |π1|) 6= 1 we obtain
ACentC
T = CACent.
Thus,
A`AR = AΛWACentQ
T
CP
−1
C A−1
ΛW
AR = ARC
−1ACentQ
T
CP
−1
C C
= ARACent
(
CT
)−1
QTCP
−1
C C = ARACent.
By the previous lemma, this proves the first condition for A`. The second condition follows
immediately from the previous lemma.
The case gcd(`, |π1|) = 1 follows from Lemma 3.11 and the fact that |π1| = |Λ∨W /Λ∨R|. �
12 S. Lentner and T. Ohrmann
4.3 Case An
In the following example, we treat the case g = An with fundamental group ΛW /ΛR = Zn+1 for
general intermediate lattices ΛR ⊆ Λ ⊆ ΛW .
Example 4.4. In order to compute the centralizer transfer map A`, we first compute the Smith
decomposition of AR:
AR =
2 −1 0 . . . 0
−1 2 −1 0 0
0 −1 2
. . .
. . .
...
0 0
. . .
. . . −1 0
...
. . . −1 2 −1
0 . . . 0 −1 2
=
−1 0 0 . . . 0
2 −1 0 0
0 2 −1
. . .
...
0 0
. . .
. . . 0
...
. . . 2 −1 0
0 . . . 0 2 1
1 0 0 . . . 0
0 1 0 0
0 0 1
. . .
...
...
. . .
. . .
. . . 1 0
0 . . . 0 n+ 1
−2 1 0 . . . 0
−3 0 1
. . . 0
−4 0 0
. . .
...
...
...
. . . 1 0
−n 0 1
1 0 . . . 0 0
.
A sublattice ΛR ( Λ ( ΛW is uniquely determined by a divisor d |n + 1, so that Λ/ΛR ∼= Zd
and is generated by the multiple d̂λn, where d̂ := n+1
d . Then
dΛ
i =
{
1, i < n,
d, i = n.
Since An is simply laced with cyclic fundamental group, the formula AΛ = PRSΛP
T
R gives us
symmetric basis matrices of sublattices ΛR ⊆ Λ ⊆ ΛW . We also substitute the above basis matrix
of the root lattice AR by AR(QR)−1P TR . It is then easy to see that the definition A` := PRD`P
T
R
gives a centralizer transfer matrix. We calculate it explicitly
(A`)ij =
(
PRD`P
−1
R
)
ij
=
δij , i < n,
(n+ 1− j)
(
`
gcd(`, d)
− 1
)
, i = n and j < n,
`
gcd(`, d)
, i = j = n.
Now a form g is uniquely determined by a dth root of unity g(χ, χ) = exp
(
2πi·k
d
)
= ζkd with
some k. Then we calculate the form a`g on the generator
a`g(χ, χ) = q−(χ,A`(χ))g(χ,A`(χ)) = q
− (n+1)2·`
d2 gcd(`,d̂)
(λ∨n ,λ
∨
n) · g(χ, χ)
`
gcd(`,d̂)
= exp
(
2πi · (k`− d̂n)
d · gcd(`, d̂)
)
.
For example the trivial g (i.e., k = 0) gives an R-matrix for all lattices Λ (defined by d̂d = n+1)
iff d̂
gcd(`,d̂)
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.
Factorizable R-Matrices for Small Quantum Groups 13
4.4 Case Dn
Finally, we consider the root lattice Dn. Since we have π1(D2n≥4) ∼= Z2×Z2 and π1(D2n+1≥5) ∼=
Z4, it is appropriate to split this investigation in two steps. We start with D2n≥4. In order to
compute the respective Smith decompositions, we used the software Wolfram Mathematica.
Example 4.5. In the case D2n≥4, we have three different possibilities for the lattices ΛR ⊆
Λ1,Λ2 ⊆ ΛW :
1. Λ1 6= Λ2, H1
∼= H2
∼= Z2: In this case, the subgroups Λi/ΛR ⊆ ΛR are spanned by
the fundamental weights λ2(n−1)+i. As in the case An, we define the centralizer transfer map
A` := PRD`P
−1
R on H2. This is possible since the symmetric basis matrix AΛ2 = PRSΛ2P
T
R
of Λ2 is already in Smith normal form. Using the software Wolfram Mathematica in order to
compute PR, we obtain A`(λ2n) = `
gcd(2,`) . Combining this with (λ2n−1, λ2n) = n−1
2 , we get
a`g(λ2n−1, λ2n) = exp
(
2πi · (kl − 2(n− 1))
2 · gcd(2, `)
)
for g(λ2n−1, λ2n) = exp
(
2πik
2
)
.
2. Λ1 = Λ2, Hi
∼= Z2: Without restrictions and in order to use the same definition for A`
as above, we choose Λi, s.t. the group Λi/ΛR is spanned by λ2n. Combining the above result
A`(λ2n) = `
gcd(2,`) with (λ2n, λ2n) = n
2 , we obtain
a`g(λ2n, λ2n) = exp
(
2πi · (kl − 2n)
2 · gcd(2, `)
)
for g(λ2n, λ2n) = exp
(
2πik
2
)
.
3. Λ1 = Λ2 = ΛW , H ∼= Z2×Z2: A group pairing g : (Z2×Z2)× (Z2×Z2)→ C× is uniquely
defined by a matrix K ∈ gl(2,F2), so that
g(λ2(n−1)+i, λ2(n−1)+j) = exp
(
2πiKij
2
)
.
Since Dn is simply-laced, we have A` = ` · id. Using (λ2(n−1)+i, λ2(n−1)+j) mod 2 = δi+jodd, we
obtain
ag` (λ2(n−1)+i, λ2(n−1)+j) = exp
(
2πi ·Kij`
2
)
(−1)i+j .
The last step is the case D2n+1≥5:
Example 4.6. Since it it is simply-laced and its fundamental group is cyclic, the case D2n+1≥5
can be treated very similar to An. We distinguish two cases:
1. Λ1 = Λ2, Hi = 〈2λ2n+1〉 ∼= Z2. As in the case An, we define the centralizer transfer map
A` := PRD`P
−1
R on H2. Using (λ2n+1, λ2n+1) = 2n+1
4 , we obtain
a`g(2λ2n+1, 2λ2n+1) = exp
(
2πi · (k`− 2(2n+ 1))
2 · gcd(2, `)
)
.
for g(2λ2n+1, 2λ2n+1) = exp
(
2πik
2
)
.
2. Λ1 = Λ2 = ΛW , H = 〈λ2n+1〉 ∼= Z4. By an analogous argument as above, we obtain
a`g(λ2n+1, λ2n+1) = exp
(
2πi · (k`− (2n+ 1))
4
)
for g(λ2n+1, λ2n+1) = exp
(
2πik
4
)
.
14 S. Lentner and T. Ohrmann
4.5 Table of all quasitriangular quantum groups
In the following table, we list all simple Lie algebras and check for which non-trivial choices
of Λ, Λi, ` and g the element R0Θ̄ is an R-matrix. As before, we define Hi := Λi/ΛR and
H := Λ/ΛR. In the cyclic case, if xi are generators of the Hi, then the pairing is uniquely
defined by an element 1 ≤ k ≤ |Hi|, s.t. g(x1, x2) = exp
(
2πik
|Hi|
)
. In the case D2n, Λ = ΛW , g is
uniquely defined by a 2× 2-matrix K ∈ gl(2,F2), s.t. g(λ2(n−1)+i, λ2(n−1)+j) = exp
(2πiKg
ij
2
)
for
i, j ∈ {1, 2}.
The columns of the following table are labeled by
1) the finite-dimensional simple complex Lie algebra g,
2) the natural number `, determining the root of unity q = exp
(
2πi
`
)
,
3) the number of possible R-matrices for the Lusztig ansatz,
4) the subgroups Hi ⊆ H = Λ/ΛR introduced in Theorem 3.3,
5) the subgroups Hi in terms of generators given by multiples of fundamental dominant
weights λi ∈ ΛW ,
6) the group pairing g : H1 ×H2 → C× determined by its values on generators,
7) the group pairing a`g ⊆ Λ/Λ′ introduced in Theorem 3.12 determined by its values on
generators.
g ` # Hi ∼= Hi (i=1,2) g a`g
all 1 Z1 〈0〉 g = 1 1
∞
Zd 〈d̂λn〉 g(d̂λn, d̂λn) = exp
(
2πik
d
)
exp
( 2πi·(k`−d̂n)
d·gcd(`,d̂)
)An≥1
π1 = Zn+1 d |n+ 1 d̂ = n+1
d
gcd
(
d, k`−d̂n
gcd(`,d̂)
)
= 1
` even 2
Z2 〈λn〉
g(λn, λn) = ±1 −1
Bn≥2
π1 = Z2
` odd 1 g(λn, λn) = (−1)n+1 exp
( 2πi·(k`−n)
2
)
` ≡ 2 mod 4 1
Z2 〈λn〉
g(λn, λn) = 1 exp
( 2πi·(k `
2
+1)
2
)
Cn≥3 ` ≡ 0 mod 4 2 g(λn, λn) = ±1 −1
π1 = Z2
` odd 1 g(λn, λn) = −1 exp
( 2πi·(k`−2n)
2
)
` ≡ 2 mod 4 1
Z2
H1
∼= 〈λ2n−1〉
g(λ2n−1, λ2n) = (−1)n
exp
( 2πi·(k `
2
−n+1))
2
)
` ≡ 0 mod 4 2δ2 |n g(λ2n−1, λ2n) = ±1, n even
H2
∼= 〈λ2n〉` odd 1 g(λ2n−1, λ2n) = −1 exp
( 2πi·(k`−2(n−1))
2
)
D2n≥4
π1 = Z2 × Z2 ` ≡ 2 mod 4 1
Z2 〈λ2n〉
g(λ2n, λ2n) = (−1)n+1
exp
( 2πi(k `
2
−n)
2
)
` ≡ 0 mod 4 2δ2-n g(λ2n, λ2n) = ±1, n odd
` odd 1 g(λ2n, λ2n) = −1 exp
( 2πi(k`−2n)
2
)
` even 16
Z2 × Z2 〈λ2n, λ2n+1〉
g(λ2(n−1)+i, λ2(n−1)+j) = ±1
exp
( 2πi·Kij`
2
)
(−1)i+j
` odd 6 det(K) = K12 +K12 mod 2
Factorizable R-Matrices for Small Quantum Groups 15
` ≡ 2 mod 4 1
Z2 〈2λ2n+1〉
g(2λ2n+1, 2λ2n+1) = 1
exp
( 2πi·(k `
2
−2n−1)
2
)
` ≡ 0 mod 4 2 g(2λ2n+1, 2λ2n+1) = ±1
D2n+1≥5 ` odd 1 g(2λ2n+1, 2λ2n+1) = −1 exp
( 2πi·(k`−2(2n+1))
2
)
π1 = Z4
` even 4
Z4 〈λ2n+1〉
g(λ2n+1, λ2n+1) = c, c4 = 1
exp
( 2πi·(k`−(2n+1))
4
)
` odd 2 g(λ2n+1, λ2n+1) = ±1
` ≡ 0 mod 3 3
Z3 〈λn〉
g(λn, λn) = c, c3 = 1
exp
( 2πi·(k`−1)
3
)E6 ` ≡ 1 mod 3 2 g(λn, λn) = 1, exp
(
2πi2
3
)
π1 = Z3
` ≡ 2 mod 3 2 g(λn, λn) = 1, exp
(
2πi
3
)
` even 2
Z2 〈λn〉
g(λn, λn) = ±1
exp
( 2πi·(k`−1)
2
)E7
π1 = Z2 ` odd 1 g(λn, λn) = 1
Table 2: Solutions for R0-matrices.
The Lie algebras E8, F4 and G2 have trivial fundamental groups and thus have no non-
trivial solution. We want to emphasize once more that the choice Λi = ΛR always leads to
a quasitriangular quantum group.
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
Λ′Lusz = 2 CentΛR(2ΛW ).
This coincide with our choice Λ′ = CentΛR(Λ1 + Λ2), if and only if
2 gcd
(
`, dΛ
i
)
= gcd
(
`, 2dWi
)
, (4.1)
where the dΛ
i denote the invariant factors of Λ∨W /Λ and the dWi denote the invariant factors of
Λ∨W /ΛW (i.e., ordered root lengths).
In particular, for ` odd these choices never coincide. For Λ = ΛW , Λ′ = Λ′Lusz holds if and
only if 2di | `. 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′ = CentΛR(Λ1 + Λ2) = CentΛR(Λ). We have
2 CentΛR(2ΛW ) = 2
(
ΛR ∩ 2̂ΛW
)
= AR2
(
Λ∨W ∩A−1
W
`
2
Λ∨W
)
= AR Diag
(
2`
gcd(`, 2dWi )
)
Λ∨W .
By Lemma 4.2, this coincides with Λ′ if and only if equation (4.1) holds. �
5 Factorizability of quantum group R-matrices
We first recall the definition of factorizable braided tensor categories and factorizable Hopf
algebras, respectively.
16 S. Lentner and T. Ohrmann
Definition 5.1 ([3]). A braided tensor category C is factorizable if the canonical braided tensor
functor G : C � Cop → Z(C) is an equivalence of categories.
In [17], Schneider gave a different characterization of factorizable Hopf algebras in terms of
its Drinfeld double, leading to the following theorem:
Definition 5.2. A finite-dimensional quasitriangular Hopf algebra (H,R) is called factorizable
if its monodromy matrix M := R21 ·R ∈ H ⊗H is non-degenerate, i.e., the following linear map
is bijective
H∗ → H, φ 7→ (id⊗φ)(M).
Equivalently, this means we can write M =
∑
iR
i
1 ⊗Ri2 for two basis’ Ri1, R
j
2 ∈ H.
Theorem 5.3. Let (H,R) be a finite-dimensional quasitriangular Hopf algebra. Then the cate-
gory of finite-dimensional H-modules H − modfd is factorizable if and only if (H,R) is a fac-
torizable Hopf algebra.
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.
5.1 Monodromy matrix in terms of R0
In order to obtain conditions for the factorizability of the quasitriangular small quantum groups
(uq(g,Λ,Λ
′), R0(f)Θ̄) as in Theorem 2.2 in terms of g, q, Λ and f , we start by calculating the
monodromy matrix M := R21 ·R ∈ uq(g,Λ,Λ′)⊗ uq(g,Λ,Λ′) in general as far as possible:
Lemma 5.4. For R = R0(f)Θ̄ 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′:
mµ,ν :=
∑
µ′,ν′∈Λ/Λ′
f(µ− µ′, ν − ν ′)f(ν ′, µ′).
Proof. We first plug in the expressions for R0 from Theorem 3.3 and Θ̄ from Theorem 2.2 and
simplify:
M := R21 ·R = (R0)21 · Θ̄21 ·R0 · Θ̄
=
∑
µ1,ν1∈Λ
f(µ1, ν1)Kν1 ⊗Kµ1
∑
β1∈Λ+
R
(−1)trβ1qβ1
∑
b1∈Bβ2
b∗+1 ⊗ b
−
1
×
∑
µ2,ν2∈Λ
f(µ2, ν2)Kµ2 ⊗Kν2
∑
β2∈Λ+
R
(−1)trβ2qβ2
∑
b2∈Bβ2
b−2 ⊗ b
∗+
2
=
∑
β1,β2∈Λ+
R
(−1)trβ1+β2qβ1qβ2
∑
µ1,µ2,ν1,ν2∈Λ
f(µ1, ν1)f(µ2, ν2)qβ1(ν2−µ2)Kν1+µ2 ⊗Kµ1+ν2
×
∑
b1∈Bβ1
,b2∈Bβ2
b∗+1 b−2 ⊗ b
−
1 b
∗+
2
,
Factorizable R-Matrices for Small Quantum Groups 17
where Λ+
R = N0[∆]. The last equation holds since b−1 ∈ u−β1
and hence fulfills Kν2b
−
1 =
q−β1ν2b−1 Kν2 and similarly for b∗+1 . We have two triangular decompositions
uq = u0
qu
−
q u
+
q , uq = u0
qu
+
q u
−
q ,
and the Λ+
R-gradation on u±q induces a gradation
uq ⊗ uq ∼=
⊕
β1,β2
(
u0 ⊗ u0
)(
u+
q β1
u−q β2
⊗ u−q β1
u+
q β2
)
.
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 Mβ1,β2 ∈ (uq ⊗ uq)(β1,β2) for every β1, β2 ∈ Λ+
R as follows
Mβ1,β2 :=
∑
µ1,µ2,ν1,ν2∈Λ
f(µ1, ν1)f(µ2, ν2)qβ1(ν2−µ2)Kν1+µ2 ⊗Kµ1+ν2
×
∑
b1∈Bβ1
,b2∈Bβ2
b∗+1 b−2 ⊗ b
−
1 b
∗+
2
.
Since the second sum in Mβ1,β2 runs over a basis in u+
q β1
u−q β2
⊗u−q β1
u+
q β2
, the invertibility of M
is equivalent to the invertibility for all β1 ∈ Λ+
R the following element:
Mβ1
0 :=
∑
µ1,µ2,ν1,ν2∈Λ/Λ′
qβ1(ν2−µ2)f(µ1, ν1)f(µ2, ν2)Kν1+µ2 ⊗Kµ1+ν2
=
∑
µ,ν∈Λ/Λ′
Kν ⊗Kµ ·
∑
µ′,ν′∈Λ/Λ′
qβ1(µ′−ν′)f(µ− µ′, ν − ν ′)f(ν ′, µ′)
.
Since Kν ⊗Kµ is a vector space basis of u0
q ⊗ u0
q = C[Λ/Λ′]⊗C[Λ/Λ′], this in turn is equivalent
to the invertibility of the following family of matrices mβ1 for all β1 ∈ Λ+
R with rows/columns
indexed by elements in µ, ν ∈ Λ/Λ′:
mβ1
µ,ν :=
∑
µ′,ν′∈Λ/Λ′
f(µ− µ′, ν − ν ′)f(ν ′, µ′)qβ1(µ′−ν′).
We now use the fact that R was indeed an R-matrix. By property (2.1) in Theorem 2.2 we have
mβ1
µ,ν =
∑
µ′,ν′∈Λ/Λ′
f(µ− µ′, ν − ν ′)f(ν ′ + β1, µ
′)q−β1ν′ .
Since the invertibility of a matrix mµ,ν is equivalent to the invertibility of any matrix mµ,ν+β1 ,
we may substitute ν ′ 7→ ν ′ + β1, ν 7→ ν + β1, pull the constant factor q−β
2
1 in front (which also
does not affect invertibility) and hence eliminate the first β1 from the condition. Hence the
invertibility of R is equivalent to the invertibility of the following family of matrices m̃β1 for all
β1 ∈ Λ+
R:
m̃β1
µ,ν :=
∑
µ′,ν′∈Λ/Λ′
f(µ− µ′, ν − ν ′)f(ν ′, µ′)q−β1ν′ .
18 S. Lentner and T. Ohrmann
We may now use the same procedure to eliminate the second β1, 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′:
mµ,ν :=
∑
µ′,ν′∈Λ/Λ′
f(µ− µ′, ν − ν ′)f(ν ′, µ′).
This was the assertion we wanted to prove. �
Definition 5.5. Let g : G1 ×G2 → C× be a group pairing. It induces a symmetric form on the
product G1 ×G2 we denote by Sym(g):
Sym(g) : (G1 ×G2)×2 −→ C×,
((µ1, µ2), (ν1, ν2)) 7−→ g(µ1, ν2)g(ν1, µ2).
Lemma 5.6. If g : G1 × G2 → 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 G1 × G2 to Ĝ2 × Ĝ1. The
symmetric form Sym(g) is given by the composition of this isomorphism with the canonical
isomorphism Ĝ2 × Ĝ1
∼= Ĝ1 ×G2. This proves the claim. �
Consider for a finite abelian group G and subgroups G1, G2 ≤ G the canonical exact sequence
0→ G1 ∩G2 → G1 ×G2 → G1 +G2 → 0. (5.1)
For µ ∈ G1 +G2, we denote its fiber by
(G1 ×G2)µ := {(µ1, µ2) ∈ G1 ×G2 |µ1 + µ2 = µ}.
Moreover, we define
Rad :=
{
(µ1, µ2) ∈ G1 ×G2 | Sym
(
f̂
)
((µ1, µ2), x) = 1 ∀x ∈ (G1 ×G2)0
}
,
Radµ := Rad∩(G1 ×G2)µ,
Rad⊥0 := {µ1 + µ2 ∈ G | (µ1, µ2) ∈ Rad}.
Lemma 5.7. We have two split exact sequences:
0→ Rad0 → Rad→ Rad⊥0 → 0,
0→ Rad⊥0 → G→ Rad0 → 0.
Proof. The first sequence is exact by definition of the three groups. Moreover, we know
Rad = ker
(
ι̂ ◦ Sym
(
f̂
)) ∼= ker(ι̂) = im(π̂) ∼= Ĝ ∼= G,
where ι̂, π̂ denote the duals of the inclusion and projection in (5.1). In Example 5.11 we will
see that in the case G1 = G2 = G, f̂ symmetric, Rad0 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. If f̂ is asymmetric, we will see in Section 5.3 that Rad0 is isomorphic to Zk2 for some
k ≥ 2, thus
Rad⊥0 −→ Rad, x 7−→
∑
x̃∈Radx
x̃
is a section of the first exact sequence. Here we used that the sum over all elements in Zk2
vanishes. Again, it follows that both diagrams split. Finally, if G1 6= G2 (i.e., in the case D2n),
then f̂ = q−(·,·) on G1 ∩ G2. By the same argument as in Example 5.11, Rad0 is the 2-torsion
subgroup of G1∩G2. But we have G ∼= G1∩G2×π1 in this case, hence both sequences split. �
Factorizable R-Matrices for Small Quantum Groups 19
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:
SymG
(
f̂
)
: G×G −→ C×,
(µ, ν) 7−→ Sym
(
f̂
)
(β ◦ α(µ), β ◦ α(ν)).
Moreover, we have Rad
(
SymG
(
f̂
)) ∼= Rad0.
Theorem 5.9. We have shown in Theorem 2.2 and Lemma 3.5 that the assumption that R =
R0(f)Θ̄ is an R-matrix is equivalent to the existence of subgroups G1, G2 ⊂ Λ/Λ′ of same order
some d|ΛR/Λ′| and f restricting up to a scalar to a non-degenerate pairing f̂ : G1 × G2 → C×
and f vanishes otherwise.
In this notation the matrix m as defined in the previous lemma can be rewritten as
mµ,ν =
1
d2|ΛR/Λ′|2
∑
µ̃∈(G1×G2)µ
ν̃∈(G1×G2)ν
Sym
(
f̂
)
(µ̃, ν̃).
It is invertible if and only if Rad0 = 0. In this case,
mµ,ν =
|G1 ∩G2|
d2|ΛR/Λ′|2
SymG
(
f̂
)
.
We first note that Rad0 = 0 implies Rad⊥0 = G and thus G = G1 + G2. Together with
Corollary 3.6 this implies
Corollary 5.10.
Λ′ = CentΛR(Λ).
Before we proof the theorem, we first give a simple example:
Example 5.11. Let G1 = G2 = G (correspondingly Λ1 = Λ2 = Λ) and assume f̂ is symmetric
non-degenerate, then the radical measures 2-torsion:
Rad
(
SymG
(
f̂
)) ∼= Rad0 = {µ ∈ G | 2µ = 0}.
Again, this is the only case appearing for cyclic fundamental groups. Hence in all cases except
g = D2n 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 = G1 + G2, otherwise the matrix has zero-columns and rows, differently formulated: the
fibers (G1 ×G2)µ in the short exact sequence must be non-empty for all µ ∈ G. If on the other
hand, Rad0 = 0, then Rad⊥0 = G and thus G1 + G2 = G must also hold, thus we assume this
from now on. By the short exact sequence the fiber (G1 ×G2)0
∼= G1 ∩G2, other fibers are of
the explicit form µ̃+ (G1 ×G2)0 for some choice of representative µ̃. Therefore,
mµ,ν =
1
d2|ΛR/Λ′|2
∑
µ̃∈(G1×G2)µ
ν̃∈(G1×G2)ν
Sym
(
f̂
)
(µ̃, ν̃)
=
1
d2|ΛR/Λ′|2
∑
ν̃∈(G1×G2)ν
Sym
(
f̂
)
(µ̃, ν̃)
∑
η̃∈(G1×G2)0
Sym
(
f̂
)
(η̃, ν̃)
20 S. Lentner and T. Ohrmann
=
|G1 ∩G2|
d2|ΛR/Λ′|2
∑
ν̃∈(G1×G2)ν
Sym
(
f̂
)
(µ̃, ν̃) · δSym(f)(ν̃, )|G1∩G2
=1 = (∗).
Fix as above a representative ν̃ of the fiber of ν, i.e., ν̃ ∈ (G1×G2)ν such that Sym(f)(ν̃, )|G1∩G2
= 1 holds. Two elements fulfilling this property differ by an element in the subgroup Rad0 ≤
G1 ∩G2, thus
(∗) =
|G1 ∩G2|
d2|ΛR/Λ′|2
Sym
(
f̂
)
(µ̃, ν̃)
∑
ξ̃∈Rad0
Sym
(
f̂
)
(ξ̃, ν̃) · δSym(f)(ν̃, )|G1∩G2
=1
=
|G1 ∩G2||Rad0 |
d2|ΛR/Λ′|2
Sym
(
f̂
)
(µ̃, ν̃) · δSym(f̂)(ν̃, )|G1∩G2
=1 δSym(f̂)(µ̃, )|Rad0
=1.
Since m is symmetric, we have
mµ,ν =
|G1 ∩G2||Rad0 |
d2|ΛR/Λ′|2
Sym
(
f̂
)
(µ̃, ν̃) · δSym(f̂)(ν̃, )|G1∩G2
=1δSym(f̂)(µ̃, )|G1∩G2
=1
=
|G1 ∩G2||Rad0 |
d2|ΛR/Λ′|2
SymG
(
f̂
)
(µ, ν)δRadµ 6=∅δRadν 6=∅
and this is invertible if an only if Rad0
∼= Rad
(
SymG
(
f̂
))
= 0. �
5.2 Factorizability for symmetric R0(f)
For R0 =
∑
µ,ν f(µ, ν)Kµ ⊗ Kν being the Cartan part of an R-matrix, assume that f̂ = |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:
Lemma 5.12. Let ΛR ⊆ Λ ⊆ ΛW be an arbitrary intermediate lattice for a certain irreducible
root system. Then the order of the group G = Λ/CentΛR(Λ) is odd if and only if both of the
following conditions are satisfied:
1) |Λ/ΛR| is odd,
2) ` is either odd or (` ≡ 2 mod 4, g = Bn, Λ = ΛR) including A1.
Proof. We saw that in all our cases, there exists an isomorphism
Λ/ΛR ∼= CentΛ(ΛR)/CentΛR(Λ).
Moreover, from Lemma 4.2 we know that |Λ/CentΛ(ΛR)| = det(D`), where D` was the diagonal
matrix Diag
(
`
gcd(`,dΛ
i )
)
) with dΛ
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,
|G| = |Λ/CentΛR(Λ)| = |Λ/CentΛ(ΛR)||CentΛ(ΛR)/CentΛR(Λ)|
= |Λ/CentΛ(ΛR)||Λ/ΛR| = det(D`)|Λ/ΛR| =
n∏
i=1
`
gcd(`, dΛ
i )
|Λ/ΛR|.
Clearly, this term is odd if ` and |Λ/ΛR| are odd. In the case (` ≡ 2 mod 4, g = Bn, Λ = ΛR),
the Smith normal form SR of the basis matrix AR 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 multipli-
city m` of the prime 2 in
n∏
i=1
`
gcd(`,dΛ
i )
is at most the multiplicity mπ1 of the prime 2 in |π1|. We
check this condition for rank n > 1:
Factorizable R-Matrices for Small Quantum Groups 21
• For g simply-laced (or triply-laced g = G2) we have all di = 1, hence n |m` (equality for
` = 2 mod 4). The cases Dn with mπ1 = 2 have rank n ≥ 4, all others except An have
mπ1 = 0, 1, so the necessary condition m` ≤ mπ1 is never fulfilled. The cases An have
2mπ1 |(n+ 1) ≤ (m` + 1)
!
≤ (mπ1 + 1) which can only be true in rank n = 1 treated above.
• For g doubly-laced of rank n > 1, we always have always mπ1 = 0, 1 but m` can be
considerably smaller than above, namely for ` = 2 mod 4 equal to the number of short
simple roots dαi = 1 (otherwise m` again increases by n for every factor 2 in `), hence the
necessary condition m` ≤ mπ1 can be fulfilled only for Bn (which would also include A1
above for n = 1). More precisely, since m` = mπ1 and the decomposition for Λ/Λ′ has an
additional factor |Λ/ΛR|, it can only be odd for Λ = ΛR.
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 = ΛR. In the previous section we have seen that f̂ = q−(·,·) gives always
an R-matrix in this case. By the proof of the previous lemma, we have
Rad0
∼=
n∏
i=1
Z
gcd
(
2, `
gcd(`,dR
i
)
),
where the dRi denote the invariant factors of Λ∨W /ΛR.
5.3 Factorizability for D2n, R0 antisymmetric
The split case g = D2n, G = G1 × G2 is clearly factorizable, so the only remaining case for
which we have to check factorizabilty is g = D2n, Λ = ΛW for f̂ being not symmetric. We
know that in this case, the corresponding form g on Λ/ΛR is uniquely defined by a 2× 2-matrix
K ∈ gl(2,F2), s.t. g(λ2(n−1)+i, λ2(n−1)+j) = exp
(2πiKij
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 D2n, Λ = ΛW .
Lemma 5.14. For g simply-laced and Λ = ΛW , let f̂ = q−(·,·)g : G×G→ C× be a non-degenerate
form as in Theorem 3.3 and Lemma 3.5, s.t. the form g : π1 × π1 → C× is asymmetric. Then,
Rad0
∼=
n⊕
i=1
Zgcd(2,`dRi ),
where the dRi denote the invariant factors of π1. In particular, Rad0 = 0 holds if and only if
gcd(2, `|π1|) = 1.
Proof. We recall the definition of Rad0
(
SymG
(
f̂
))
in this case:
Rad0(SymG(f̂)) =
{
µ ∈ G | f(ν, µ)−1 = f(µ, ν) ∀ ν ∈ G
}
=
{
µ ∈ G | q(ν,µ)g(ν, µ)−1 = q−(µ,ν)g(µ, ν) ∀ ν ∈ G
}
=
{
µ ∈ G | q(ν,µ) = q−(µ,ν) ∀ ν ∈ G
}
=
{
µ ∈ G | q(2µ,ν) = 1 ∀ ν ∈ G
}
=
{
µ ∈ G | 2µ ∈ Cent2ΛW (ΛW )/2 CentΛR(ΛW )
}
= (∗).
For g is simply-laced, we have ΛW = Λ∨W , thus
(∗) ∼= Cent2ΛW (ΛW )/2 CentΛR(ΛW ) = (2ΛW ∩ `ARΛW )/2`ARΛW
= PR Diag
(
lcm
(
2, `dRi
))
ΛW /PR2`SRΛW = ΛW /Diag
(
gcd
(
2, `dRi
))
ΛW .
This proves the claim. �
22 S. Lentner and T. Ohrmann
5.4 Transparent objects in non-factorizable cases
In this section, we determine the transparent objects in the representation category of uq(g,Λ)
with our R-matrix given by R0Θ̄ and R0 = 1
|Λ/Λ′|
∑
µ,ν∈Λ/Λ′
f̂ with f̂ a group pairing Λ1/Λ
′ ×
Λ2/Λ
′ → C×.
Definition 5.15. Let C be a braided monoidal category with braiding c. An object V ∈ C is
called transparent if the double braiding cW,V ◦ cV,W is the identity on V ⊗W for all W ∈ C.
The following theorem by Shimizu gives a very important characterization of factorizable
categories:
Theorem 5.16 ([18, Theorem 1.1]). A braided finite tensor category is factorizable if and only
if the transparent objects are direct sums of finitely many copies of the unit object.
Corollary 5.17. In particular, for a Hopf algebra H the representation category H −modfd is
factorizable if and only if the transparent objects are multiples of the trivial representation and
vice versa.
Since in our cases Λ1 6= Λ2 can only appear in D2n, and we know those are factorizable, we
shall in the following restrict ourselves to the case Λ1 = Λ2 = 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 uq(g)-module V , with a highest-weight vector v and Kµv = χ(Kµ)v, is
a transparent object, then necessarily the 1-dimensional Λ/Λ′-module Cχ is a transparent object
over the Hopf algebra C[Λ/Λ′] with R-matrix R0. If V is 1-dimensional, then V is transparent
if and only if Cχ is.
Proof. Let V be transparent. For every ψ : Λ/Λ′ → C× we have another finite-dimensional
module W := uq(g) ⊗uq(g)+ Cψ with highest weight vector w = 1 ⊗ 1ψ which we can test this
assumption against
c2 : V ⊗W →W ⊗ V → V ⊗W.
We calculate the effect of c2 on the highest-weight vectors v ⊗ w:
c2(v ⊗ w) = τW⊗VR0Θ̄τV⊗WR0Θ̄(v ⊗ w).
Because v, w were assumed highest-weight vectors, the Θ̄ act trivially. Hence follows that Cχ, Cψ
have a trivial double braiding over the Hopf algebra C[Λ/Λ′] with R-matrix R0. Because we
could achieve this result for any ψ this means that Cχ is transparent as asserted.
Now, let V = Cχ be 1-dimensional over uq(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 = 1χ
is a highest weight vector, and one time because it is also a lowest weight vector. But if the
double-braiding of v = 1χ with any element w is trivial, then V = Cχ is already transparent
over uq(g). �
Lemma 5.19. Cχ is a transparent object over the Hopf algebra C[Λ/Λ′] with R-matrix R0 iff it
is an f -transformed of the radical of SymG
(
f̂
)
, i.e.,
χ(µ) = f(µ, ξ), ξ ∈ Rad0 .
Factorizable R-Matrices for Small Quantum Groups 23
Proof. Since f is nondegenerate, we can assume χ(µ) = f(µ, ξ) and wish to prove Cχ is
transparent iff ξ ∈ Rad0. We test transparency against any module Cψ and also write ψ(µ) =
f(λ, µ) (note the order of the argument). We evaluate the double-braiding on 1χ ⊗ 1ψ and get
the following scalar factor, which needs to be = 1 for all ψ in order to make Cχ transparent:
1
|G|2
∑
µ,ν
χ(µ)ψ(ν)
∑
µ1+µ2=µ
ν1+ν2=µ
Sym
(
f̂
)
((µ1, µ2), (ν1, ν2))
=
1
|G|2
∑
µ,ν
f(µ, ξ)f(λ, ν)
∑
µ1+µ2=µ
ν1+ν2=µ
f(µ1, ν1)f(ν2, µ2)
=
1
|G|2
∑
µ,ν
f(µ, ξ)f(λ, ν)
∑
ν1,µ1
f(µ1, ν1)f(ν, µ)f−1(ν1, µ)f−1(ν, µ1)f(ν1, µ1)
=
1
|G|
∑
ν
f(λ, ν)
∑
ν1,µ1
f(µ1, ν1)δξ=−ν+ν1f
−1(ν, µ1)f(ν1, µ1)
=
1
|G|
∑
ν
f(λ, ν)
∑
µ1
f(µ1, ξ + ν)f(ξ, µ1) = f−1(λ, ξ)f−1(ξ, λ) = SymG
(
f̂
)
(λ, ξ).
This scalar factor of the double braiding is equal +1 for all λ (and hence all Cψ) iff ξ ∈ Rad0 as
asserted. �
The previous two lemmas combined imply that any irreducible transparent uq(g)-module
has necessarily the characters χ(µ) = f(µ, ξ), ξ ∈ Rad0 as highest-weights, and conversely if
such a character χ gives rise to 1-dimensional uq(g)-modules (i.e., χ|2ΛR = 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 Rad0
(
SymG
(
f̂
))
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 D2n) then the transparent objects are
all 1-dimensional Cχ where the characters χ are the f -transformed of the elements in the radical
of the bimultiplicative form Sym
(
f̂
)
|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 D2n) the fact that Rad0 is the
2-torsion of Λ/Λ′ and f -transformation is a group isomorphism shows:
The group T of transparent objects consists of Cχ where χ|2Λ = 1, i.e., the two-torsion of the
character group.
The remaining case in D2n with f nonsymmetric and has been done by hand in Lemma 5.14.
6 Quantum groups with a ribbon structure
In [16, Theorem 8.23], the existence of ribbon structures for uq(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 uq(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
ν0
u is a ribbon element
in uq(g,Λ).
24 S. Lentner and T. Ohrmann
Proof. We consider the natural N0[αi | i ∈ I]-grading on the Borel parts u± := uq(g,Λ)± [13].
Since u± is finite-dimensional, there exists a maximal ν0 ∈ N0[αi | i ∈ I], s.t. the homogeneous
component u±ν0
is non-trivial. More explicitly ν0 is of the form
ν0 =
∑
α∈Φ+
(`α − 1)α,
where `α := `
gcd(`,2dα) .
Using the formulas u =
(∑
f(µ, ν)Kµ+ν
)−1
ϑ and S(u) =
(∑
f(µ, ν)Kµ+ν
)−1
S(ϑ), where
ϑ =
∑
Θ̄(2)S−1
(
Θ̄(2)
)
, Müller proves the formula K2
−ν0
= u−1S(u). Using the fact that u
commutes with all grouplike elements, this implies v2 = uS(u). In order to show that v is central,
we first show that K−1
ν0+2ρ is a central element. By the K,E-relations, this is equivalent to
ν0 + 2ρ ∈ CentΛ(ΛR),
where ρ = 1
2
∑
α∈Φ+
α is the Weyl vector.
We calculate directly that this is always the case:
(ν0 + 2ρ, β) = q
∑
α∈Φ+
(`α−1+1)(α,β)
= q
`
∑
α∈Φ+
1
gcd(`,2dα)
·2dα(α∨,β)
= 1.
Since K2ρux = xK2ρu holds for all x ∈ uq(g,Λ) (see [16, Lemmas 8.22 and 8.19]), we have
vx = K−1
ν0
ux = K−1
ν0+2ρK2ρux = K−1
ν0+2ρxK2ρu = xK−1
ν0+2ρK2ρu = xv,
hence v is central. �
7 Open questions
Question 7.1. It was surprising to us that the case D2n = so4n(C) has so many more solutions
that the other cases, in particular with non-symmetric R0, due to the non-cyclic fundamental
group. Do these additional modular tensor categories appear elsewhere? Does the non-symmetry
have interesting implications on the category?
Question 7.2. Our procedure would be similarly possible for any diagonal Nichols algebra. The
Lusztig ansatz can in these cases be found in [1].
Question 7.3. In each case where uq(g,Λ), R is not factorizable, we can modularize (see [2])
the corresponding representation category and get a modular tensor category, which should be
representations over some “quasi-quantum group” uq(g, Λ̃, ω), R which is a quasi-Hopf algebra
where the group ring C[Λ̃] is deformed by a 3-group-cocycle ω. Can we describe this quasi-Hopf
algebra in a closed form? Moreover, is every factorizable quasi-quantum group the modularization
of a quasi-triangular quantum group from our list?
More technically:
Question 7.4. The centralizer transfer map A` in Definition 3.8 (and correspondingly the
form a`) had a very general characterization, but we could only prove existence by a construc-
tion using the classification of simple Lie algebras (and distinguishing three cases). We strongly
suspect that these maps exist under rather general assumptions.
Also the result Theorem 3.3 from our previous article [11] has only been proven there for
cyclic groups (and by hand for Z2 × Z2) although we strongly suspect it holds for every abelian
group.
Factorizable R-Matrices for Small Quantum Groups 25
Acknowledgements
Both authors thank Christoph Schweigert for helpful discussions and support. They also thank
the referees, who gave a relevant contribution to improve the article with their comments. The
first author was supported by the DAAD P.R.I.M.E program funded by the German BMBF
and the EU Marie Curie Actions as well as the Graduiertenkolleg RTG 1670 at the University
of Hamburg. The second author was supported by the Collaborative Research Center SFB 676
at the University of Hamburg.
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1 Introduction
2 Preliminaries
2.1 Lie theory
2.2 Lusztig's ansatz for R-matrices
3 Conditions for the existence of R-matrices
3.1 A first set of conditions on /'
3.2 A natural form on the fundamental group
4 Explicit calculation for every g
4.1 Technical tools
4.2 Case =W
4.3 Case An
4.4 Case Dn
4.5 Table of all quasitriangular quantum groups
5 Factorizability of quantum group R-matrices
5.1 Monodromy matrix in terms of R0
5.2 Factorizability for symmetric R0(f)
5.3 Factorizability for D2n, R0 antisymmetric
5.4 Transparent objects in non-factorizable cases
6 Quantum groups with a ribbon structure
7 Open questions
References
|
| id | nasplib_isofts_kiev_ua-123456789-148764 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T16:50:27Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Lentner, S. Ohrmann, T. 2019-02-18T18:47:57Z 2019-02-18T18:47:57Z 2017 Factorizable R-Matrices for Small Quantum Groups / S. Lentner, T. Ohrmann // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 19 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B37; 20G42; 81R50; 18D10 DOI:10.3842/SIGMA.2017.076 https://nasplib.isofts.kiev.ua/handle/123456789/148764 Representations of small quantum groups uq(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. Both authors thank Christoph Schweigert for helpful discussions and support. They also thank
 the referees, who gave a relevant contribution to improve the article with their comments. The
 first author was supported by the DAAD P.R.I.M.E program funded by the German BMBF
 and the EU Marie Curie Actions as well as the Graduiertenkolleg RTG 1670 at the University
 of Hamburg. The second author was supported by the Collaborative Research Center SFB 676
 at the University of Hamburg. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Factorizable R-Matrices for Small Quantum Groups Article published earlier |
| spellingShingle | Factorizable R-Matrices for Small Quantum Groups Lentner, S. Ohrmann, T. |
| title | Factorizable R-Matrices for Small Quantum Groups |
| title_full | Factorizable R-Matrices for Small Quantum Groups |
| title_fullStr | Factorizable R-Matrices for Small Quantum Groups |
| title_full_unstemmed | Factorizable R-Matrices for Small Quantum Groups |
| title_short | Factorizable R-Matrices for Small Quantum Groups |
| title_sort | factorizable r-matrices for small quantum groups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148764 |
| work_keys_str_mv | AT lentners factorizablermatricesforsmallquantumgroups AT ohrmannt factorizablermatricesforsmallquantumgroups |