Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras
Let Σg,n be a compact oriented surface of genus g with n open disks removed. The algebra Lg,n(H) was introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche and is a combinatorial quantization of the moduli space of flat connections on Σg,n. Here we focus on the two building blocks L₀,₁(H) and L...
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| Cite this: | Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras / M. Faitg // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 39 назв. — англ. |
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| citation_txt | Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras / M. Faitg // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 39 назв. — англ. |
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| description | Let Σg,n be a compact oriented surface of genus g with n open disks removed. The algebra Lg,n(H) was introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche and is a combinatorial quantization of the moduli space of flat connections on Σg,n. Here we focus on the two building blocks L₀,₁(H) and L₁,₀(H) under the assumption that the gauge Hopf algebra H is finite-dimensional, factorizable, and ribbon, but not necessarily semisimple. We construct a projective representation of SL₂(Z), the mapping class group of the torus, based on L₁,₀(H), and we study it explicitly for H = Ūq(sl(2)). We also show that it is equivalent to the representation constructed by Lyubashenko and Majid.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 077, 39 pages
Modular Group Representations in Combinatorial
Quantization with Non-Semisimple Hopf Algebras
Matthieu FAITG
IMAG, Univ Montpellier, CNRS, Montpellier, France
E-mail: matthieu.faitg@gmail.com
Received February 02, 2019, in final form September 24, 2019; Published online October 03, 2019
https://doi.org/10.3842/SIGMA.2019.077
Abstract. Let Σg,n be a compact oriented surface of genus g with n open disks removed.
The algebra Lg,n(H) was introduced by Alekseev–Grosse–Schomerus and Buffenoir–Roche
and is a combinatorial quantization of the moduli space of flat connections on Σg,n. Here
we focus on the two building blocks L0,1(H) and L1,0(H) under the assumption that the
gauge Hopf algebra H is finite-dimensional, factorizable and ribbon, but not necessarily
semisimple. We construct a projective representation of SL2(Z), the mapping class group
of the torus, based on L1,0(H) and we study it explicitly for H = Uq(sl(2)). We also show
that it is equivalent to the representation constructed by Lyubashenko and Majid.
Key words: combinatorial quantization; factorizable Hopf algebra; modular group; restricted
quantum group
2010 Mathematics Subject Classification: 16T05; 81R05
1 Introduction
Let Σg,n be a compact oriented surface of genus g with n open disks removed and let G be
a connected, simply-connected Lie group. Fock and Rosly defined a Poisson structure on the
space of flat G-connections on Σg,n\D (where D is an open disk) in a combinatorial way, by
using a description of the surface as a ribbon graph, see [25, 26]. This construction is another
formulation of the Atiyah–Bott–Goldman Poisson structure on the character variety [8, 28]. The
algebra Lg,n is an associative non-commutative algebra which is a combinatorial quantization of
the algebra of functions on the space of flat G-connections, by deformation of the Fock–Rosly
Poisson structure.
These algebras were introduced and studied by Alekseev–Grosse–Schomerus [2, 3, 4, 5] and
Buffenoir–Roche [14, 15]. They replaced the Lie group G by the quantum group Uq(g), with
g = Lie(G), and defined non-commutative relations between functions on connections (also
called gauge fields) via matrix equations involving the R-matrix of Uq(g). More precisely, in the
Fock–Rosly graph description, a flat connection is described by the collection of its holonomies
along the edges of the graph. For instance, with the canonical graph shown in Fig. 1, a flat
G-connection on Σ1,0\D is represented by (xb, xa) ∈ G × G. The gauge group G acts on
connections by the adjoint action and dually on gauge fields (on the right). The algebra of
gauge fields is generated by matrix coefficients
V
Aij ,
V
Bi
j defined by
V
Aij(xb, xa) = ei(xa · ej) and
V
Bi
j(xb, xa) = ei(xb · ej), where V is a G-module with basis (ei) and dual basis (ei). Then the
algebra L1,0 is generated by the coefficients of the matrices
I
A,
I
B, where I is now a representation
of Uq(g), modulo certain matrix relations involving the R-matrix, see Definition 4.1 (also see
Remark 4.9 for an explicit formula). A connection is now xb ⊗ xa ∈ Uq(g)⊗2. The gauge
algebra Uq(g) acts on gauge fields on the right by the adjoint action. These relations give
a non-commutative Uq(g)-equivariant multiplication between gauge fields.
mailto:matthieu.faitg@gmail.com
https://doi.org/10.3842/SIGMA.2019.077
2 M. Faitg
xm
m b
a
xb xa
Figure 1. Surfaces Σ0,1\D and Σ1,0\D, with canonical curves and discrete connections.
These quantized algebras of functions Lg,n and their generalizations appear in various works
of mathematics and mathematical physics. Let us indicate papers which are related to this
work. In [16], they introduce multitangles (which are diagrams encoding transformations of the
graph and of the discrete connections) and use them to define a comultiplication on the discrete
connections (dual to the product in Lg,n), the holonomy of a path in the graph and the Wilson
loops. In [34], the axiomatic formulation of lattice gauge theory with Hopf algebras is given and it
generalizes the foundational works cited above. Their formalism does not require semi-simplicity
of the gauge algebra (except for some properties about the holonomy); however, contrarily to
the present work, they do not use matrix coefficients, they work with an arbitrary graph filling
the surface and the purpose of their paper is different (they do not discuss representations of
mapping class groups). In [10], the algebras Lg,n are recovered in a categorical setting based on
factorization homology; the case of the punctured torus is studied in [12], recovering a version
of L1,0. In [1], they study L1,0 with the super Hopf algebra gl(1|1) as well as the associated SL2(Z)
action and they conjecture that our result on the equivalence with the Lyubashenko–Majid
representation (Theorem 5.12) remains true with super Hopf algebras. [17] also discuss related
subjects.
The definition of Lg,n being purely algebraic, we can replace the gauge algebra Uq(g) by
any ribbon Hopf algebra H, thus obtaining an algebra Lg,n(H). In the foundational papers
on combinatorial quantization, it is always assumed that H is either Uq(g) with q generic or
a semisimple truncation of Uq(g) at a root of unity (defined in the context of weak quasi-Hopf
algebras). The moduli algebra, introduced in [3, 4, 5], is only defined when H is modular since
its definition uses the particular properties of the S-matrix in this case. Moreover, in [5], the
representation theory of Lg,n(H) is investigated and a projective representation of the mapping
class group of Σg,n on the moduli algebra is constructed, under the assumption that H is
a modular Hopf algebra (and in particular semisimple).
In this paper we assume that H is a finite-dimensional, factorizable, ribbon Hopf algebra
which is not necessarily semisimple, the guiding example being the restricted quantum group
H = U q(sl(2)). We consider L0,1(H), L1,0(H) and we generalize to this setting the projective
representation of the mapping class group of [5]; in this way we obtain a projective representation
of SL2(Z). The algebras L0,1(H) and L1,0(H) deserve particular interest because they are the
building blocks of the theory, thanks to the Alekseev isomorphism,
Lg,n(H) ∼= L1,0(H)⊗g ⊗ L0,1(H)⊗n.
This isomorphism was stated in [2] for H = Uq(g), q generic, but it can be generalized to our
assumptions on H (see [22, Proposition 3.5]).
After having recalled the definition of L0,1(H) and L1,0(H) and the H-action on them, we
prove that L0,1(H) ∼= H (Theorem 3.7) and that L1,0(H) is isomorphic to the Heisenberg
double of the dual Hopf algebra O(H) (Theorem 4.8). In particular, it implies that L1,0(H) is
(isomorphic to) a matrix algebra. We then define in Section 4.3 a representation of the algebra
Modular Group Representations in Combinatorial Quantization 3
of invariants, Linv1,0(H), on the space of symmetric linear forms SLF(H) (Theorem 4.10). We will
use this representation of the invariant elements to define a projective representation of SL2(Z)
on SLF(H).
The main ingredient of the construction of the mapping class group representation is the
natural action of the mapping class group MCG(Σ1,0\D) on L1,0(H), obtained by considering
the action of MCG(Σ1,0\D) on π1(Σ1,0\D) and by replacing the loops representing elements
in π1(Σ1,0\D) by the corresponding matrices in L1,0(H), up to some normalization [5]. It
turns out that MCG(Σ1,0\D) acts by automorphisms on L1,0(H), and thus, since it is a matrix
algebra, we get elements which implements these automorphisms by conjugation. Representing
these elements on SLF(H), we get a projective representation of SL2(Z); this is explained in
Section 5, with detailed proofs under our assumptions on H.
We show (Theorem 5.12) that this projective representation is equivalent to the Lyubashenko–
Majid representation [33]. This gives a natural and geometrical interpretation of the latter, which
was constructed by categorical methods.
Section 6 is devoted to the example of U q = U q(sl(2)) as gauge algebra. All the preliminary
facts about U q and the GTA basis, which is a suitable basis of SLF
(
U q
)
, are available in [21]. In
Theorem 6.5 we give the explicit formulas for the action of SL2(Z) on the GTA basis of SLF
(
U q
)
.
The multiplication formulas in this basis (see [21, Section 5], [27]), are the crucial tool to obtain
the result. The structure of SLF
(
U q
)
under the action of SL2(Z) is determined. Thanks to the
equivalence with the Lyubashenko–Majid representation and the work of [24], the representation
obtained in Section 6.3 is equivalent to the one studied in [24], which comes from logarithmic
conformal field theory. Finally, in Section 6.4, we formulate a conjecture about the structure
of SLF
(
U q
)
as a Linv1,0
(
U q
)
-module.
To sum up, the main results of this paper are:
– the construction of a projective representation of SL2(Z) (the mapping class group of the
torus) on the space of symmetric linear forms on H (Theorem 5.8),
– the equivalence of this representation to the one found by Lyubashenko and Majid in [33]
(Theorem 5.12),
– in the case of H = U q(sl(2)), an explicit realization of this representation and its structure
(Theorem 6.5 and Theorem 6.8).
In [22], we generalize Theorems 5.8 and 5.12 to higher genus. The projective representation
of the mapping class group is shown to be equivalent to that constructed by Lyubashenko using
the coend of a ribbon category [32]. The advantage of combinatorial quantization is that the
construction of the projective representation of the mapping class group is very explicit and
that the particular features of the algebra Lg,n(H) are helpful to compute the formulas of the
representation for a given H (see Section 6.3). In particular, for H = U q(sl(2)), one can hope
that the correspondence with logarithmic conformal field theory mentioned aboved still exists in
higher genus. Let us also mention that the subalgebra of invariant elements Linvg,n(H) is known to
be related to skein theory [16]; in fact, combinatorial quantization provides new representations
of skein algebras (work in progress).
Notations. If A is an algebra, V is a finite-dimensional A-module and x ∈ A, we denote
by
V
x ∈ EndC(V ) the representation of x on the module V . More generally, if X ∈ A⊗n and
if V1, . . . , Vn are A-modules, we denote by
V1...Vn
X the representation of X on V1 ⊗ · · · ⊗ Vn. As
in [18], we will use the abbreviation PIM for principal indecomposable module. Here we consider
only finite-dimensional representations.
Let M ∈ Matm(C)⊗ A = Matm(A), namely M is a matrix with coefficients in A. It can be
written as M =
∑
i,j E
i
j ⊗M i
j , where Eij is the matrix with 1 at the intersection of the i-th row
4 M. Faitg
and the j-th column and 0 elsewhere. More generally, every L ∈ Matm1(C)⊗· · ·⊗Matml(C)⊗A
can be written as
L =
∑
i1,j1,...,il,jl
Ei1j1 ⊗ · · · ⊗ E
il
jl
⊗ Li1...ilj1...jl
and the elements Li1...ilj1...jl
∈ A are called the coefficients of L. If f : A ! B is a morphism of
algebras, then we define f(L) ∈ Matm1(C)⊗ · · · ⊗Matml(C)⊗B by
f(L) =
∑
i1,j1,...,il,jl
Ei1j1 ⊗ · · · ⊗ E
il
jl
⊗ f
(
Li1...ilj1...jl
)
or equivalently f(L)i1...ilj1...jl
= f
(
Li1...ilj1...jl
)
.
Let M ∈ Matm(C)⊗A, N ∈ Matn(C)⊗A. We embed M , N in Matm(C)⊗Matn(C)⊗A by
M1 =
∑
i,j
Eij ⊗ In ⊗M i
j , N2 =
∑
i,j
Im ⊗ Eij ⊗N i
j ,
where Ik is the identity matrix of size k. In other words, M1 = M ⊗ In, N2 = Im ⊗ N ,
⊗ being the Kronecker product. The coefficients of M1 and N2 are respectively (M1)
ik
jl = M i
jδ
k
l ,
(N2)
ik
jl = δijN
k
l where δab is the Kronecker delta. Observe that M1N2 (resp. N2M1) contains
all the possible products of coefficients of M (resp. of N) by coefficients of N (resp. of M):
(M1N2)
ik
j` = M i
jN
k
` (resp. (N2M1)
ik
j` = Nk
`M
i
j). In particular, M1N2 = N2M1 if and only if the
coefficients of M commute with those of N . Now let Q ∈ Matm(C) ⊗Matn(C) ⊗ A; then we
denote Q12 = Q and
Q21 =
∑
i,j,k,l
Ekl ⊗ Eij ⊗Qikjl ∈ Matn(C)⊗Matm(C)⊗A.
so that (Q12)
ac
bd = Qacbd, (Q21)
ac
bd = Qcadb. Such notations are obviously generalized to bigger
tensors.
In order to simplify notations, we will use implicit summations. First, we use Einstein’s
notation for the computations involving indices: when an index variable appears twice, one time
in upper position and one time in lower position, it implicitly means summation over all the
values of the index. For instance if M ∈ Matm(C)⊗A and Q ∈ Matm(C)⊗Matn(C)⊗A, then
(M1Q12)
ac
bd = Ma
i Q
ic
bd; note that the repeated subscript 1 corresponds to matrix multiplication in
the first space. Second, we use Sweedler’s notation (see [30, Notation III.1.6]) without summation
sign for the coproducts, that is we write
∆(x) = x′ ⊗ x′′, (∆⊗ id) ◦∆(x) = (id⊗∆) ◦∆(x) = x′ ⊗ x′′ ⊗ x′′′, and so on.
Finally, we write the components of the R-matrix as R = ai ⊗ bi with implicit summation on i,
and we define R′ = bi ⊗ ai.
For q ∈ C\{−1, 0, 1}, we define the q-integer [n] (with n ∈ Z) by
[n] =
qn − q−n
q − q−1
.
We will denote q̂ = q − q−1 to shorten formulas. Observe that if q is a 2p-root of unity, then
[p] = 0 and [p− n] = [n].
As usual δs,t is the Kronecker symbol and In is the identity matrix of size n.
2 Some basic facts
We refer to [18, Chapters IV and VIII] for background material about representation theory.
Modular Group Representations in Combinatorial Quantization 5
2.1 Dual of a finite-dimensional algebra
Let A be a finite-dimensional C-algebra, V be a finite-dimensional A-module and
V
ρ : A !
EndC(V ) be the representation associated to V . In other words,
V
ρ is an element of EndC(V )⊗A∗
and if we choose a basis of V ,
V
ρ becomes an element, denoted
V
T , of Matn(C)⊗A∗ = Matn(A∗).
In other words
V
T is a matrix whose coefficients are linear forms on A. The linear form
V
T ij is
called a matrix coefficient of V . We will always consider the representation
V
x ∈ EndC(V ) of x
on V as the matrix
V
T (x) ∈ Matdim(V )(C):
V
xij =
V
T ij(x). Note that if a A-module V is a sub-
module or a quotient of a A-module W , then the matrix coefficients of W contain those of V
since
V
T is a submatrix of
W
T .
Since A∗ is finite-dimensional, it is generated as a vector space by the matrix coefficients of
the regular representation AA. Indeed, let {x1, . . . , xn} be a basis of A with x1 = 1 and let{
x1, . . . , xn
}
⊂ A∗ be the dual basis; it is readily seen that
AA
T i
1(xj) = δi,j and thus
AA
T i
1 = xi.
This implies that in the sequel one might restrict everywhere to the regular representation;
however, when studying examples it is very relevant to consider smaller representations in order
to greatly reduce the number of generators for the algebras that we will consider in the sequel
(like the fundamental representation X+(2) for A = U q(sl(2)), see Section 6).
Recall that the PIMs of A are the indecomposable projective A-modules. They are isomorphic
to the direct summands of the regular representation AA. In particular, the matrix coefficients
of AA are exactly those of the PIMs and it follows that the matrix coefficients of the PIMs
span A∗. Note however that the matrix coefficients of the PIMs (or equivalently of AA) do not
form a basis of A∗ in general. Indeed, even if we fix a family (Pα) representing each isomorphism
class of PIMs, it is possible for Pα and Pβ to have a composition factor S in common. In this
case, both
Pα
T and
Pβ
T contain
S
T as submatrix. This is what happens for H = U q(sl(2)), see, e.g.,
[21, Section 3]. In the semisimple case this phenomenon does not occur.
2.2 Braided Hopf algebras, factorizability, ribbon element
Let H be a braided Hopf algebra with universal R-matrix R = ai ⊗ bi (see, e.g., [30, Chap-
ter VIII]). Recall that
(∆⊗ id)(R) = R13R23, (id⊗∆)(R) = R13R12, (2.1)
(S ⊗ id)(R) =
(
id⊗ S−1
)
(R) = R−1, (S ⊗ S)(R) = R, (2.2)
R12R13R23 = R23R13R12 (2.3)
with R12 = ai ⊗ bi ⊗ 1, R13 = ai ⊗ 1⊗ bi, R23 = 1⊗ ai ⊗ bi ∈ H⊗3. The relation (2.3) is called
the (quantum) Yang–Baxter equation.
Consider
Ψ: H∗ ! H,
β 7! (β ⊗ id)(RR′),
where R′ = bi⊗ai. Ψ is called the Reshetikhin–Semenov-Tian-Shansky map [37], or the Drinfeld
map [19]. We say that H is factorizable if Ψ is an isomorphism of vector spaces. By the remarks
above, we can restrict β to be a matrix coefficient of some I.
Define R(+) = R, R(−) = (R′)−1, and let
I
L(±) =
( I
T ⊗ id
)(
R(±)) =
( I
a
(±)
i
)
b
(±)
i ∈ Matdim(I)(H) (2.4)
6 M. Faitg
with R(±) = a
(±)
i ⊗ b(±)i (note that
( I
a
(±)
i
)
b
(±)
i is the matrix obtained by multiplying each coef-
ficient (which is a scalar) of the matrix
( I
a
(±)
i
)
by the element bi ∈ H). Recall that R(−) is also
a universal R-matrix and in particular it satisfies the properties (2.1)–(2.3) above. We use the
letters I, J, . . . for modules over Hopf algebras. If H is factorizable, the coefficients of the matri-
ces
I
L(±) generate H as an algebra. These matrices satisfy nice relations which are consequences
of (2.1) and (2.3):
I⊗J
L
(ε)
12 =
I
L
(ε)
1
J
L
(ε)
2 ,
IJ
R
(ε)
12
I
L
(ε)
1
J
L
(σ)
2 =
J
L
(σ)
2
I
L
(ε)
1
IJ
R
(ε)
12 ∀ ε, σ ∈ {±},
IJ
R
(ε)
12
I
L
(σ)
1
J
L
(σ)
2 =
J
L
(σ)
2
I
L
(σ)
1
IJ
R
(ε)
12 ∀ ε, σ ∈ {±},
∆(
I
L(ε)a
b ) =
I
L(ε)i
b ⊗
I
L(ε)a
i . (2.5)
For instance, here is a proof of the first equality with ε = +:
I⊗J
L
(+)
12 =
(I⊗J
ai
)
12
bi =
( I
ai ⊗
J
aj
)
12
bibj =
( I
ai
)
1
( J
aj
)
2
bibj =
( I
ai
)
1
bi
( J
aj
)
2
bj =
I
L
(+)
1
J
L
(+)
2 ,
where we used (2.1); note that these are equalities between matrices in
Matdim(I)(C)⊗Matdim(J)(C)⊗H,
which imply equalities among the coefficients. If the representations I and J are fixed and
arbitrary, we will simply write these relations as
L
(ε)
12 = L
(ε)
1 L
(ε)
2 , R
(ε)
12L
(ε)
1 L
(σ)
2 = L
(σ)
2 L
(ε)
1 R
(ε)
12 , R
(ε)
12L
(σ)
1 L
(σ)
2 = L
(σ)
2 L
(σ)
1 R
(ε)
12 ,
the subscript 1 (resp. 2) corresponding implicitly to evaluation in the representation I (resp. J).
Recall that the Drinfeld element u and its inverse are
u = S(bi)ai = biS
−1(ai) and u−1 = S−2(bi)ai = S−1(bi)S(ai) = biS
2(ai). (2.6)
We assume that H contains a ribbon element v. It satisfies
v is central and invertible, ∆(v) = (R′R)−1v ⊗ v,
S(v) = v, ε(v) = 1, v2 = uS(u). (2.7)
The two last equalities can be deduced easily from the others. A ribbon element is in general
not unique. A ribbon Hopf algebra (H,R, v) is a braided Hopf algebra (H,R) together with
a ribbon element v.
We say that g ∈ H is a pivotal element if
∆(g) = g ⊗ g and ∀x ∈ H, S2(x) = gxg−1. (2.8)
Note that g is invertible because it is grouplike, and hence S(g) = g−1. A pivotal element is in
general not unique. But in a ribbon Hopf algebra (H,R, v) there is a canonical choice
g = uv−1. (2.9)
We will always take this canonical pivotal element g in the sequel.
Modular Group Representations in Combinatorial Quantization 7
2.3 Dual Hopf algebra O(H)
The canonical Hopf algebra structure on H∗ is defined by
(ψ · ϕ)(x) = (ψ ⊗ ϕ)(∆(x)), η(1) = ε,
∆(ψ)(x⊗ y) = ψ(xy), ε(ψ) = ψ(1), S(ψ) = ψ ◦ S
with ψ,ϕ ∈ H∗ and x, y ∈ H. When it is endowed with this structure, H∗ is called dual Hopf
algebra, and is denoted by O(H) in the sequel. In terms of matrix coefficients, we have
I⊗J
T12 =
I
T1
J
T2, η(1) =
C
T , ∆(
I
T ab ) =
I
T ai ⊗
I
T ib ,
ε(
I
T ) = Idim(I), S(
I
T ) =
I
T−1, (2.10)
where C is the trivial representation. For instance, here are proofs of the first and last equalities
(with h ∈ H):
(I⊗J
T12
)
(h) =
(I⊗J
h
)
12
=
( I
h′ ⊗
J
h′′
)
12
=
( I
h′
)
1
( J
h′′
)
2
=
I
T1(h
′)
J
T2(h
′′) =
( I
T1
J
T2
)
(h),(
S(
I
T )
I
T
)
(h) = S(
I
T )(h′)
I
T (h′′) =
I
T
(
S(h′)
) I
T (h′′) =
I
T
(
S(h′)h′′
)
= ε(h)
I
T (1) = ε(h)Idim(I).
By definition of the action on the dual I∗, it holds
S
( I
T
)
= t
I∗
T , (2.11)
where t is the transpose. Also recall the well-known exchange relation
IJ
R12
I
T1
J
T2 =
J
T2
I
T1
IJ
R12. (2.12)
As before, if the representations I and J are fixed and arbitrary, we will simply write T12 = T1T2
and R12T1T2 = T2T1R12.
3 The loop algebra L0,1(H)
We assume that H is a finite-dimensional factorizable Hopf algebra. The ribbon assumption is
not needed in this section.
3.1 Definition of L0,1(H) and H-module-algebra structure
Let T(H∗) be the tensor algebra of H∗, which by definition is linearly spanned by all the formal
products ϕ1 · · ·ϕn (with n ≥ 0 and ϕi ∈ H∗) modulo the obvious multilinear relations. There is
a canonical injection j : H∗ ! T(H∗) and we let
I
M ∈ Matdim(I)
(
T(H∗)
)
be the matrix defined
by
I
M = j
( I
T
)
, that is
I
Ma
b = j
( I
T ab
)
.
Definition 3.1. The loop algebra L0,1(H) is the quotient of T(H∗) by the following fusion
relations
I⊗J
M 12 =
I
M1
IJ
(R′)12
J
M2
IJ(
R′−1
)
12
for all finite-dimensional H-modules I, J .
8 M. Faitg
The right hand-side of the fusion relation in Definition 3.1 is the one of [13, Definition 1];
the one of [4, Definition 12] and [5, equation (3.11)] is different, due to different choices of the
action of H on L0,1(H) and to particular normalization of Clebsch–Gordan operators. Moreover
in the papers [4, 5, 13] the matrix
I⊗J
M did not appeared, instead it was always decomposed as
a sum of the matrices of the irreducible direct summands of I ⊗ J thanks to Clebsch–Gordan
operators, which is relevant in the semisimple case only; in [38], I and J are restricted to the
regular representation and the matrix
I⊗J
M is thus denoted by ∆a(M). In the semisimple setting,
the algebras resulting from each of these definitions are isomorphic.
Note that the fusion relation is a relation between matrices in Matdim(I)(C)⊗Matdim(J)(C)⊗
L0,1(H) (for all finite-dimensional I, J) which implies relations among elements of L0,1(H) (the
coefficients of these matrices). Explicitly, in terms of matrix coefficients it is written as
∀ I, J, a, b, c, d,
I⊗J
M ac
bd =
I
Ma
i
IJ
(R′)icjk
J
Mk
l
IJ
(R′−1)jlbd,
see the definition of the subscripts 1 and 2 in the Notations at p. 3. If the two representations I
and J are fixed and arbitrary, we will simply write
M12 = M1R21M2R
−1
21 (3.1)
the subscript 1 (resp. 2) corresponding implicitly to evaluation in the representation I (resp. J).
Finally, note that if f : I ! J is a morphism, it holds
f
I
M =
J
Mf, (3.2)
where we identify f with its matrix. Indeed, f
I
M = f j̄
( I
T
)
= j̄
(
f
I
T
)
= j̄
(J
Tf
)
= j̄
(J
T
)
f =
J
Mf
where j̄ is the linear map H∗ ! T(H∗)! L0,1(H).
Remark 3.2. One can check that
(I⊗J)⊗K
M =
I⊗(J⊗K)
M holds thanks to the Yang–Baxter equation.
We have an useful analogue of relation (2.12).
Proposition 3.3. The following exchange relations hold in L0,1(H):
IJ
R12
I
M1
IJ
(R′)12
J
M2 =
J
M2
IJ
R12
I
M1
IJ
(R′)12.
This relation is called the reflection equation. It can be written in a shortened way if the
representations I and J are fixed and arbitrary:
R12M1R21M2 = M2R12M1R21. (3.3)
Proof. Let PI,J : I ⊗J ! J ⊗ I be the flip map PIJ(v⊗w) = w⊗ v. We have the isomorphism
PIJ
IJ
R : I ⊗ J ! J ⊗ I, hence PIJ
IJ
R
I⊗J
M =
J⊗I
M PIJ
IJ
R thanks to (3.2). This gives
(PIJ)12
IJ
R12
I
M1
IJ
(R′)12
J
M2
IJ
(R′−1)12 =
J
M1
JI
(R′)12
I
M2
JI
(R′−1)12(PIJ)12
IJ
R12
= (PIJ)12
J
M2
JI
(R′)21
I
M1
JI
(R′−1)21
IJ
R12 = (PIJ)12
J
M2
IJ
R12
I
M1
as desired. �
Modular Group Representations in Combinatorial Quantization 9
Consider the following right action · of H on L0,1(H), which is the analogue of the right
action of the gauge group on the gauge fields
I
M · h =
I
h′
I
M
I
S(h′′) (3.4)
or more explicitly
I
Ma
b · h = (
I
h′)ai
I
M i
j
I
S(h′′)jb. As in [14], one can equivalently work with the
corresponding left coaction Ω: L0,1(H)! O(H)⊗ L0,1(H) defined by
Ω(
I
Ma
b ) =
I
T ai S
( I
T jb
)
⊗
I
M i
j ,
so that we recover · by evaluation: x · h = (〈?, h〉 ⊗ id) ◦ Ω(x). If we view O(H) and L0,1(H)
as subalgebras of O(H) ⊗ L0,1(H) in the canonical way, then Ω is simply written as Ω(
I
M) =
I
T
I
MS(
I
T ).
Proposition 3.4. The right action · is a H-module-algebra structure on L0,1(H). Equivalently,
Ω is a left O(H)-comodule-algebra structure on L0,1(H).
Proof. One must show for instance that Ω is a morphism of algebras, as in [14]. For the sake
of completeness we display the computation, where we use the shortened notation explained
before:
Ω(M)12 = T12M12S(T12) (definition)
= T1T2M1R21M2R
−1
21 S(T )2S(T )1 (equations (2.10) and (3.1))
= T1M1T2R21M2R
−1
21 S(T )2S(T )1 (commuting coefficients)
= T1M1T2R21M2S(T )1S(T )2R
−1
21 (equation (2.12))
= T1M1T2R21S(T )1M2S(T )2R
−1
21 (commuting coefficients)
= T1M1S(T )1R21T2M2S(T )2R
−1
21 (equation (2.12))
= Ω(M)1R21Ω(M)2R
−1
21 (definition). �
We say that an element x ∈ L0,1(H) is invariant if for all h ∈ H, x·h = ε(h)x (or equivalently,
Ω(x) = ε⊗ x). For instance, for every Φ ∈ EndH(I), the element tr(
I
gΦ
I
M) is invariant:
tr
(I
gΦ
I
M
)
· h = tr
(I
gΦ
I
h′
I
M
I
S(h′′)
)
= tr
(I
gΦ
I
S−1(h′′)
I
h′
I
M
)
= ε(h)tr
(I
gΦ
I
M
)
.
We denote by Linv0,1(H) the subalgebra of invariants elements of L0,1(H).
3.2 Isomorphism L0,1(H) ∼= H
Recall the right adjoint action of H on itself defined by a · h = S(h′)ah′′ with a, h ∈ H, whose
invariant elements are the central elements of H.
Proposition 3.5. If we endow H with the right adjoint action, the following map is a morphism
of (right) H-module-algebras
Ψ0,1 : L0,1(H) ! H,
I
M 7!
( I
T ⊗ id
)
(RR′) =
I
L(+)
I
L(−)−1.
Hence, Ψ0,1 brings invariant elements to central elements.
10 M. Faitg
We will call Ψ0,1 the Reshetikhin–Semenov-Tian-Shansky–Drinfeld morphism (RSD morphism
for short). The difference with the morphism Ψ of Section 2.2 is that the source spaces are
different.
Proof. Thanks to the relations of (2.5), we check that Ψ0,1 preserves the relation of Defini-
tion 3.1:
Ψ0,1(M)1R21Ψ0,1(M)2R
−1
21 = L
(+)
1 L
(−)−1
1 R21L
(+)
2 L
(−)−1
2 R−121
= L
(+)
1 L
(+)
2 R21L
(−)−1
1 L
(−)−1
2 R−121 = L
(+)
1 L
(+)
2 L
(−)−1
2 L
(−)−1
1
= L
(+)
12 L
(−)−1
12 = Ψ0,1(M)12.
For the H-linearity
Ψ0,1
( I
h′
I
M
I
S(h′′)
)
=
( I
T ⊗ id
)
(h′ ⊗ 1RR′S(h′′)⊗ 1)
=
( I
T ⊗ id
)
(h′ ⊗ 1RR′S(h)′′′ ⊗ S(h)′′h′′′′)
=
( I
T ⊗ id
)
(h′S(h)′′′ ⊗ S(h)′′RR′1⊗ h′′′′)
=
( I
T ⊗ id
)
(1⊗ S(h′)RR′1⊗ h′′) = S(h′)Ψ0,1(
I
M)h′′.
We used the basic properties of S and the fact that ∆opR = R∆, with ∆op(h) = h′′ ⊗ h′. �
Write T(H∗) =
⊕
n∈N Tn(H∗), where Tn(H∗) is the subspace generated by all the products
ψ1 · · ·ψn, with ψi ∈ H∗ for each i.
Lemma 3.6. Each element of T(H∗) is equivalent modulo the fusion relation of L0,1(H) to an
element of T1(H
∗). It follows that dim(L0,1(H)) ≤ dim(H∗).
Proof. It suffices to show that the product of two elements of T1(H
∗) is equivalent to a linear
combination of elements of T1(H
∗), and the result follows by induction. We can restrict to
matrix coefficients since they linearly span H∗. If we write R = ai ⊗ bi, then R′ = bi ⊗ ai and
the fusion relation is rewritten as
I⊗J
M12(
IJ
R′)12 =
I
M1(
IJ
R′)12
J
M2 =
I
M1
( I
bi
)
1
( J
ai
)
2
J
M2 =
( J
ai
)
2
I
M1
J
M2
( I
bi
)
1
.
Using S−1(aj)ai ⊗ bibj = 1⊗ 1, we get
I
M1
J
M2 =
J
S−1(ai)2
I⊗J
M12
(IJ
R′
)
12
( I
bi
)
1
(3.5)
and this give the result since
I
M1
J
M2 contains all the possible products between the coefficients
of
I
M and those of
J
M . �
Theorem 3.7. Recall that we assume that H is a finite-dimensional factorizable Hopf algebra.
Then the RSD morphism Ψ0,1 gives an isomorphism of H-module-algebras L0,1(H) ∼= H. It
follows that Linv0,1(H) ∼= Z(H).
Proof. Since H is factorizable, Ψ0,1 is surjective. Hence dim(L0,1(H)) ≥ dim(H). But by
Lemma 3.6, dim(L0,1(H)) ≤ dim(H∗) = dim(H). Thus dim(L0,1(H)) = dim(H). �
Modular Group Representations in Combinatorial Quantization 11
Let us point out obvious consequences. First, by comparing the dimensions, we see that the
canonical map H∗ ↪! T(H∗) � L0,1(H) is an isomorphism of vector spaces. Second, this shows
that the matrices
I
M are invertible since RR′ is invertible. More importantly, this theorem allows
us to identify L0,1(H) with H via
I
M =
I
L(+)
I
L(−)−1, where the matrices L(±) are defined in (2.4).
We will always work with this identification in the sequel.
Remark 3.8. Thanks to the isomorphism of vector spaces
I
T !
I
M and the relation (3.5), we
can see L0,1(H) as H∗ endowed with a product ∗ defined by
ϕ ∗ ψ = ϕ(?bjbi)ψ
(
S−1(ai)?aj
)
, xm 7! ϕ(x′mbjbi)ψ
(
S−1(ai)x
′′
maj
)
. (3.6)
This is the product of the gauge fields ϕ,ψ and its evaluation on the connection which assigns
xm ∈ H to the loop m, see Fig. 1. The action of the gauge algebra H on a gauge field ϕ is
ϕ · h = ϕ(h′?S(h′′)) and ∗ is H-equivariant.
We denote by SLF(H) the space of symmetric linear forms on H:
SLF(H) = {ψ ∈ H∗ | ∀x, y ∈ H, ψ(xy) = ψ(yx)}.
SLF(H) is obviously a subalgebra of O(H). Consider the following variant of the map Ψ of
Section 2.2, which will be useful in what follows
D : O(H) ! H,
ψ 7! (ψ ⊗ id) ((g ⊗ 1)RR′) ,
(3.7)
where g is the pivotal element (2.9). Since H is factorizable, D is an isomorphism of vector
spaces. A computation similar to that of the proof of Proposition 3.5 shows that D brings
symmetric linear forms to central elements. Moreover, it is not difficult to show that it induces
an isomorphism of algebras SLF(H) ∼= Z(H) = Linv0,1(H).
Let us fix a notation. Every ψ ∈ H∗ can be written as ψ =
∑
i,j,I λ
I
ij
I
Tij with λIij ∈ C. In
order to avoid the indices, define for each I a matrix ΛI ∈ Matdim(I)(C) by (ΛI)
i
j = λIji. Then ψ
can be expressed as ψ =
∑
I tr
(
ΛI
I
T
)
. We record these observations as a lemma.
Lemma 3.9. Every x ∈ L0,1(H) can be expressed as
x =
∑
I
tr
(
ΛI
I
g
I
M
)
such that D−1(x) =
∑
I tr
(
ΛI
I
T
)
. Moreover, if x ∈ Linv0,1(H), then D−1(x) ∈ SLF(H).
Remark 3.10. Let us stress that, due to non-semi-simplicity, this way of writing elements of
L0,1(H) and of SLF(H) is in general not unique, see the comments in Section 2.1.
4 The handle algebra L1,0(H)
We assume that H is a finite-dimensional factorizable ribbon Hopf algebra. Note however that
the ribbon assumption is not needed in Sections 4.1 and 4.2.
12 M. Faitg
4.1 Definition of L1,0(H) and H-module-algebra structure
Consider the free product L0,1(H) ∗ L0,1(H), and let j1 (resp. j2) be the canonical injection
in the first (resp. second) copy of L0,1(H). We define
I
B = j1(
I
M) and
I
A = j2(
I
M), that is
I
Ba
b = j1(
I
Ma
b ),
I
Aab = j2(
I
Ma
b ).
Definition 4.1. The handle algebra L1,0(H) is the quotient of L0,1(H)∗L0,1(H) by the following
exchange relations:
IJ
R12
I
B1
IJ
(R′)12
J
A2 =
J
A2
IJ
R12
I
B1
IJ
R−112
for all finite-dimensional H-modules I, J .
The exchange relation above is the same as in [13, Definition 1] except that A and B are
switched; the one of [4, Definition 12] and [5, equation (3.14)] is different, due to a different
choice of the action of H on L1,0(H). In the semisimple setting, the algebras resulting from
each of these definitions are isomorphic. This is a relation between matrices in Matdim(I)(C)⊗
Matdim(J)(C)⊗L1,0(H) (for all finite-dimensional I, J) which implies relations among elements
of L1,0(H), namely
∀ I, J, a, b, c, d,
IJ
Rac
ij
I
Bi
k(
IJ
R′)kjbl
J
Ald =
J
Aci
IJ
Rai
jk
I
Bj
l
(IJ
R−1
)lk
bd
.
Like the other relations before, we can write the L1,0(H)-exchange relation more simply as:
R12B1R21A2 = A2R12B1R
−1
12 . (4.1)
It immediately follows from (3.2) that if f : I ! J is a morphism, it holds
f
I
B =
J
Bf, f
I
A =
J
Af,
where we identify f with its matrix.
Similarly to L0,1(H), consider the following right action of H on L1,0(H), which is the ana-
logue of the action of the gauge group on the gauge fields:
I
B · h =
I
h′
I
B
I
S(h′′),
I
A · h =
I
h′
I
A
I
S(h′′). (4.2)
As above, it is equivalent to work with the corresponding left coaction Ω: L1,0(H) ! O(H) ⊗
L1,0(H) defined by
Ω(
I
B) =
I
T
I
BS(
I
T ), Ω(
I
A) =
I
T
I
AS(
I
T ).
Proposition 4.2. The right action · is a H-module-algebra structure on L1,0(H). Equivalently,
Ω is a left O(H)-comodule-algebra structure on L1,0(H).
Proof. One must show that Ω is an algebra morphism, as in [14]. This amounts to check that
Ω is compatible with the exchange relation, which is similar to the proof of Proposition 3.4. �
We denote by Linv1,0(H) the subalgebra of invariant elements of L1,0(H). For instance, the
elements
tr12
(
I⊗J
g12Φ
I
A1
IJ
(R′)12
J
B2
IJ
R12
)
(4.3)
with Φ ∈ EndH(I ⊗ J) and tr12 = tr⊗ tr, are invariant.
Modular Group Representations in Combinatorial Quantization 13
Finally, we describe a wide family of maps L0,1(H) ! L1,0(H). For w ∈ Linv0,1(H) = Z(H)
and m1, n1, . . . ,mk, nk ∈ Z, define
jwBm1An1 ...BmkAnk : L0,1(H) ! L1,0(H),
I
M 7!
I
w
I
Bm1
I
An1 · · ·
I
Bmk
I
Ank .
It is clear that these maps are morphisms of H-modules, but not of algebras in general. Hence
the restriction satisfies jwBm1An1 ...BmkAnk : Linv0,1(H) ! Linv1,0(H). This gives a particular type of
invariants in L1,0(H). We will more shortly write
xwBm1An1 ...BmkAnk = jwBm1An1 ...BmkAnk (x). (4.4)
We also use this notation for x ∈ H, thanks to the identification H = L0,1(H).
Remark 4.3. Recall from Remark 3.10 that the matrix coefficients do not form a basis of
L0,1(H). They just linearly span this space. However, the maps jwAm1Bn1 ...AmkBnk are well-
defined. Indeed, first observe that
jB : L0,1(H)
j1
↪−! L0,1(H) ∗ L0,1(H)
π
−!! L1,0(H),
jA : L0,1(H)
j2
↪−! L0,1(H) ∗ L0,1(H)
π
−!! L1,0(H)
are well-defined. Let us show for instance that the map jA−1B−1A is well-defined. Assume that
λab
I
T ba = 0. Applying the coproduct in O(H) twice and tensoring with idH , we get
λab
I
T bk ⊗ idH ⊗
I
T kl ⊗ idH ⊗
I
T la ⊗ idH = 0.
We evaluate this on (RR′)−1 ⊗ (RR′)−1 ⊗RR′:
λab
( I
M−1
)
b
k ⊗
( I
M−1
)
k
l ⊗
I
M l
a = 0.
Finally, we apply the map jA ⊗ jB ⊗ jA and multiplication in L1,0(H):
λab
( I
A−1
I
B−1
I
A
)
b
a = 0
as desired. A similar proof can be used to show that all the other maps defined by means of
matrix coefficients (like Ψ1,0 or α, β below etc..) are well-defined.
4.2 Isomorphism L1,0(H) ∼= H(O(H))
Let us begin by recalling the following definition (see for instance [35, Example 4.1.10]).
Definition 4.4. Let H be a Hopf algebra. The Heisenberg double of O(H), H(O(H)), is the
vector space O(H)⊗H endowed with the algebra structure defined by the following multiplica-
tion rules:
• The canonical injections H,O(H) ! O(H) ⊗ H are algebra morphisms. Thus we iden-
tify O(H) (resp. H) with O(H)⊗ 1 ⊂ H(O(H)) (resp. with 1⊗H ⊂ H(O(H))).
• Under this identification, we have the exchange relation
∀ψ ∈ O(H), ∀h ∈ H, hψ = ψ(?h′)h′′ = ψ′′(h′)ψ′h′′.
14 M. Faitg
There is a representation of H(O(H)) on O(H) defined by
h . ψ = ψ(?h) = ψ′′(h)ψ′, ϕ . ψ = ϕψ h ∈ H, ψ, ϕ ∈ O(H). (4.5)
This representation is faithful (see [35, Lemma 9.4.2]). Hence, if H is finite-dimensional, it
follows that H(O(H)) is a matrix algebra
H(O(H)) ∼= EndC(H∗). (4.6)
Under our assumptions on H, a natural set of generators for H(O(H)) consists of the matrix
coefficients of
I
T and of
I
L(±).
Lemma 4.5. With these generators, the exchange relation of H(O(H)) is
I
L
(±)
1
J
T2 =
J
T2
I
L
(±)
1
IJ
R
(±)
12
and the representation . is
I
L
(±)
1 .
J
T2 =
J
T2
IJ
R
(±)
12 ,
I
T1 .
J
T2 =
I⊗J
T12.
Proof. For the exchange relation, we apply the defining relation of H(O(H)) together with the
definition (2.4) of
I
L(±) and (2.1):
I
L
(+)
1
J
T2 =
( I
ai
)
1
bi
J
T2 =
( I
ai
)
1
J
T (?b′i)2b
′′
i =
( I
ai
)
1
J
T2
(J
b′i
)
2
b′′i =
( I
ai
I
aj
)
1
J
T2
( J
bj
)
2
bi
=
J
T2
( I
ai
)
1
bi
( I
aj
)
1
( J
bj
)
2
=
J
T2
I
L
(+)
1
IJ
R12.
We used that
J
T (?x) =
J
T
J
x (which just follows from the fact that
J
T is a morphism from H to
Matdim(J)(C)). The proof for
I
L(−) is exactly the same since R(−) is also a universal R-matrix.
For the second formula we just apply the definition of .:
I
L
(+)
1 .
J
T2 =
( I
ai
)
1
bi .
J
T2 =
( I
ai
)
1
J
T (?bi)2 =
( I
ai
)
1
J
T2
(J
bi
)
2
=
J
T2
( I
ai
)
1
(J
bi
)
2
=
J
T2
IJ
R12.
The proof for
I
L(−) is exactly the same. The last formula follows from (2.10). �
Thanks to [2], we know that there is a morphism from L1,0(H) to H(O(H)):
Proposition 4.6. The following map is a morphism of algebras:
Ψ1,0 : L1,0(H) ! H(O(H)),
I
B 7!
I
L(+)
I
T
I
L(−)−1,
I
A 7!
I
L(+)
I
L(−)−1.
Proof. One must check that the fusion and exchange relations are compatible with Ψ1,0. Ob-
serve that the restriction of Ψ1,0 to the first copy of L0,1(H) ⊂ L1,0(H) is just the RSD mor-
phism Ψ0,1, thus Ψ1,0 is compatible with the fusion relation over A. For the fusion relation
over B, we have
Ψ1,0(B)12 = L
(+)
12 T12L
(−)−1
12 (definition)
Modular Group Representations in Combinatorial Quantization 15
= L
(+)
1 L
(+)
2 T1T2L
(−)−1
2 L
(−)−1
1 (equations (2.5) and (2.10))
= L
(+)
1 T1L
(+)
2 R21T2L
(−)−1
2 L
(−)−1
1 (Lemma 4.5)
= L
(+)
1 T1L
(+)
2 R21T2R21L
(−)−1
1 L
(−)−1
2 R−121 (equation (2.5))
= L
(+)
1 T1L
(+)
2 R21L
(−)−1
1 T2L
(−)−1
2 R−121 (Lemma 4.5)
= L
(+)
1 T1L
(−)−1
1 R21L
(+)
2 T2L
(−)−1
2 R−121 (equation (2.5))
= Ψ1,0(B)1R21Ψ1,0(B)2R
−1
21 (definition).
The same kind of computation allows one to show that Ψ1,0 is compatible with the L1,0-exchange
relation. �
We will now show that Ψ1,0 is an isomorphism under our assumptions on H.
Lemma 4.7. Every element in L1,0(H) can be written as
∑
i(xi)B(yi)A with xi, yi ∈ L0,1(H).
It follows that dim(L1,0(H)) ≤ dim(L0,1(H))2 = dim(H)2.
Proof. This is the same proof as in Lemma 3.6. It suffices to show that an element like yAxB
can be expressed as
∑
i(xi)B(yi)A. The exchange relation can be rewritten as
I
A1
J
B2 =
I
S−1(ai)2
IJ
(R′)12
J
B2
IJ
R12
I
A1
IJ
(R′)12
J
(bi)1, (4.7)
and the result follows since
I
A1
J
B2 contains all the possible products between the coefficients of
I
A
and those of
J
B. �
Theorem 4.8. Recall that we assume that H is a finite-dimensional factorizable Hopf algebra.
Ψ1,0 gives an isomorphism of algebras L1,0(H) ∼= H(O(H)). It follows that L1,0(H) is a matrix
algebra: L1,0(H) ∼= Matdim(H)(C).
Proof. Observe that Ψ1,0 ◦ jA = iH ◦Ψ0,1 where iH : H ! H(O(H)) is the canonical inclusion.
Since Ψ0,1 is an isomorphism, there exist matrices
I
A(±) such that
Ψ1,0
( I
A(±)
)
=
I
L(±) ∈ Matdim(I)(H(O(H))).
Moreover, we have
Ψ1,0
( I
A(+)−1 IB
I
A(−)
)
=
I
T ∈ Matdim(I)(H(O(H))).
Thus Ψ1,0 is surjective, and hence dim(L1,0(H)) ≥ dim(H(O(H))) = dim(H)2. This to-
gether with Lemma 4.7 gives dim(L1,0(H)) = dim(H(O(H))). The last claim is a general
fact, see (4.6). �
Remark 4.9. Due to Theorem 4.8, there is an isomorphism of vector spaces L1,0(H)! H∗⊗H∗
given by
I
Bi
j
J
Akl 7!
I
T ij ⊗
J
T kl . This defines a product ∗ on H∗ ⊗H∗; due to (4.7), we get that it
satisfies
(ε⊗ ψ) ∗ (ϕ⊗ ε) = ϕ
(
S−1(ai)aj?bkal
)
⊗ ψ
(
bjak?blbi
)
.
Combining this with (3.6), we obtain the general formula
(ϕ1 ⊗ ψ1) ∗ (ϕ2 ⊗ ψ2) = ϕ1(?bmbn)ϕ2
(
S−1(ai)ajS
−1(an)?ambkal
)
16 M. Faitg
⊗ ψ1
(
bjak?bobpblbi
)
ψ2
(
S−1(ap)?ao
)
,
xb ⊗ xa 7! ϕ1(x
′
bbmbn)ϕ2
(
S−1(ai)ajS
−1(an)x′′bambkal
)
× ψ1
(
bjakx
′
abobpblbi
)
ψ2
(
S−1(ap)x
′′
aao
)
.
This is the product of the gauge fields ϕ1 ⊗ ψ1, ϕ2 ⊗ ψ2 and its evaluation on the connection
which assigns xb to the loop b and xa to the loop a, see Fig. 1. The action of the gauge algebra H
on a gauge field ϕ⊗ ψ is (ϕ⊗ ψ) · h = ϕ(h′?S(h′′))⊗ ψ(h′′′?S(h′′′′)) and ∗ is H-equivariant.
4.3 Representation of Linv
1,0(H) on SLF(H)
Recall from (4.5) that there is a faithful representation . of H(O(H)) on O(H). Using the
isomorphism Ψ1,0, we get a representation of L1,0(H) on O(H), still denoted .:
∀x ∈ L1,0(H), ∀ψ ∈ O(H), x . ψ = Ψ1,0(x) . ψ.
Thanks to Lemma 4.5, it is easy to get
I
A1 .
J
T2 =
J
T2
IJ
(RR′)12,
I
B1 .
J
T2 =
I
(ai)1
I⊗J
T12
I⊗J
(bi)12
IJ
(R′)12 =
I
(aiaj)1
I⊗J
T12
I
(bj)1
J
(bi)2
IJ
(R′)12, (4.8)
where as usual R = ai ⊗ bi and the last equality is obtained using (2.1).
In [2, Theorem 5] (which is stated in the case ofH = Uq(g), q generic), there is a representation
of Linv1,0(H) on a subspace of invariants in H∗. This can be generalized to our assumptions.
Theorem 4.10. The restriction of . to Linv1,0(H) leaves the subspace SLF(H) ⊂ H∗ stable
∀x ∈ Linv1,0(H), ∀ψ ∈ SLF(H), x . ψ ∈ SLF(H).
Hence, we have a representation of Linv1,0(H) on SLF(H). We denote it ρSLF.
Proof. For h ∈ H, define h̃ ∈ H(O(H)) by h̃ . ϕ = ϕ
(
S−1(h)?
)
for all ϕ ∈ H∗ (since the
representation . is faithful, this entirely defines h̃). It is easy to see that
∀ g ∈ H, ∀ψ ∈ O(H), g̃h̃ = g̃h, gh̃ = h̃g, h̃ψ = ψ
(
S−1(h′′)?
)
h̃′.
We define matrices
I
L̃(±) =
( I
a
(±)
i
)
b̃
(±)
i ∈ Matdim(I)
(
H(O(H))
)
with R(±) = a
(±)
i ⊗ b(±)i . By definition and (2.2), they satisfy
I
L̃
(±)
1 .
J
T 2 =
IJ
R
(±)−1
12
J
T 2. It is not
difficult to show the following commutation rules
I
L̃
(ε)
1
J
L̃
(ε)
2 =
I⊗J
L̃
(ε)
12 ,
I
L̃
(ε)
1
J
L
(σ)
2 =
J
L
(σ)
2
I
L̃
(ε)
1 ,
IJ
R
(ε)
12
I
L̃
(ε)
1
J
T 2 =
J
T 2
I
L̃
(ε)
1 , (4.9)
IJ
R
(ε)
12
I
L̃
(ε)
1
J
L̃
(σ)
2 =
J
L̃
(σ)
2
I
L̃
(ε)
1
IJ
R
(ε)
12 ∀ ε, σ ∈ {±},
IJ
R
(ε)
12
I
L̃
(σ)
1
J
L̃
(σ)
2 =
J
L̃
(σ)
2
I
L̃
(σ)
1
IJ
R
(ε)
12 ∀ ε, σ ∈ {±}.
For instance, here is a proof of the third equality with ε = +:
IJ
R
(+)
12
I
L̃
(+)
1
J
T 2 =
IJ
R
(+)
12
( I
ai
)
1
b̃i
J
T2 =
IJ
R
(+)
12
( I
ai
)
1
J
T
(
S−1(b′′i )?
)
2
b̃′i =
IJ
R
(+)
12
( I
ai
I
aj
)
1
( J
S−1(bi)
)
2
J
T2b̃j
Modular Group Representations in Combinatorial Quantization 17
=
IJ
R
(+)
12
( I
ai
)
1
( J
S−1(bi)
)
2
J
T2
( I
aj
)
b̃j =
J
T2
I
L̃
(+)
1 ,
since ai ⊗ S−1(bi) = R(+)−1. Now, let
I
C(±) = Ψ−11,0
( I
L(±)
I
L̃(±)) ∈ Matdim(I)(L1,0(H)) .
Thanks to (2.5) and (4.9), it is easy to see that
I⊗J
C
(±)
12 =
I
C
(±)
1
J
C
(±)
2 .
Lemma 4.11.
1) An element x ∈ L1,0(H) is invariant under the right action of H if, and only if, x
I
C(±) =
I
C(±)x for all I.
2) A linear form ψ ∈ H∗ is symmetric if, and only if,
I
C(±) . ψ = ψIdim(I) for all I.
Proof. 1) Let U = A or B, then by (2.4), (4.2) and (2.1) we get
J
U2 · S−1
( I
L
(±)
1
)
=
J
U2 · S−1
(
b
(±)
i
)( I
a
(±)
i
)
1
=
J
S−1
(
b
(±)
i
′′ )
2
J
U2
( J
b
(±)
i
′)
2
( I
a
(±)
i
)
1
=
J
S−1
(
b
(±)
i
)
2
J
U2
( J
b
(±)
j
)
2
( I
a
(±)
i
I
a
(±)
j
)
1
=
( I
a
(±)
i
)
1
J
S−1
(
b
(±)
i
)
2
J
U2
( I
a
(±)
j
)
1
( J
b
(±)
j
)
2
=
IJ
R
(±)−1
12
J
U2
IJ
R
(±)
12
with R(±) = a
(±)
i ⊗ b(±)i . Second, using (2.5) and (4.9) we get
I
C
(±)
1
J
U2
I
C
(±)−1
1 =
IJ
R
(±)−1
12
J
U2
IJ
R
(±)
12 .
For instance (with the shortened notation)
Ψ1,0
(
C
(+)
1 A2C
(+)−1
1
)
= L
(+)
1 L̃
(+)
1 L
(+)
2 L
(−)−1
2 L̃
(+)−1
1 L
(+)−1
1 = L
(+)
1 L
(+)
2 L
(−)−1
2 L
(+)−1
1
= R−112 L
(+)
2 L
(+)
1 R12L
(−)−1
2 L
(+)−1
1 = R−112 L
(+)
2 L
(−)−1
2 R12
= Ψ1,0
(
R−112 A2R12
)
and we have equality since Ψ1,0 is an isomorphism; the others cases are similar. It follows
that
J
U2 · S−1
( I
L
(±)
1
)
=
I
C
(±)
1
J
U2
I
C
(±)−1
1 or in other words
J
U cd · S−1
( I
L(±)a
b
)
=
I
C(±)a
i
J
U cd
I
C(±)−1i
b,
which means that
J
U cd is invariant under the action of S−1
( I
L(±)a
b
)
if, and only if, it commutes
with
I
C(±)a
b . Since the elements
J
U cd (resp. S−1
( I
L(±)a
b
)
) generate L1,0(H) (resp. H) as an algebra,
we get that an element is invariant if, and only if, it commutes with all the coefficients of the
matrices
I
C(±), as desired.
2) Consider the left action � of H on H∗ given by h � ψ = ψ
(
S−1(h′)?h′′
)
. It is easy to see
that ψ is symmetric if, and only if, it is invariant under � (namely h �ψ = ε(h)ψ for all h ∈ H).
The definitions and (2.1) yields
I
C(±) . ψ =
( I
a
(±)
i
I
a
(±)
j
)
bib̃j . ψ =
( I
a
(±)
i
I
a
(±)
j
)
ψ
(
S−1(bj)?bi
)
=
( I
a
(±)
i
)
ψ
(
S−1(b′i)?b
′′
i
)
=
I
L(±) � ψ.
Since the coefficients of the matrices
I
L(±) generate H as an algebra, we get the result. �
18 M. Faitg
End of the proof of Theorem 4.10. Let x ∈ Linv1,0(H) and ψ ∈ SLF(H). We apply the
previous lemma
J
C(±) . (x . ψ) =
( J
C(±)x
)
. ψ =
(
x
J
C(±)) . ψ = x .
( J
C(±) . ψ
)
= (x . ψ)Idim(J).
Then x . ψ ∈ SLF(H), as desired. �
Remark 4.12. Let
I
C =
I
v2
I
B
I
A−1
I
B−1
I
A. It can be shown that
I
C satisfies the fusion relation
of L0,1(H), that
I
C =
I
C(+)
I
C(−)−1, that x ∈ L1,0(H) is invariant if, and only if, x
I
C =
I
Cx and
that ψ ∈ H∗ is symmetric if, and only if,
I
C . ψ = ψIdim(I). The details in our general setting
are given in [22] for arbitrary genus. Observe that geometrically,
I
C is the boundary of the
surface Σ1,0\D, see Fig. 1.
We now need to determine explicit formulas for the representation of particular types of
invariants that will appear in the proof of the modular identities in Section 5. If ψ ∈ H∗ and
a ∈ H, we define
ψa = ψ(a?),
where ψ(a?) : x 7! ψ(ax). This defines a right representation of H on H∗. Obviously, if z ∈ Z(H)
and ψ ∈ SLF(H) then ψz ∈ SLF(H).
Recall that zA = jA(z) (resp. zB = jB(z)) is the image of z ∈ L0,1(H) by the map jA(
I
M) =
I
A
(resp. jB(
I
M) =
I
B). See (4.4) for the general definition.
Proposition 4.13. Let z ∈ Linv0,1(H) = Z(H) and let ψ ∈ SLF(H). Then
zA . ψ = ψz and zB . ψ =
(
D−1(z)ψv
)v−1
,
where D is the isomorphism defined in (3.7).
Proof. The first relation is obvious. For the second formula, we write zB =
∑
I tr
(
ΛI
I
g
I
B
)
with
D−1(z) =
∑
I tr
(
ΛI
I
T
)
∈ SLF(H) by Lemma 3.9. We also write ψ =
∑
J tr
(
ΘJ
J
T
)
. Then, thanks
to (4.8):
zB . ψ =
∑
I,J
tr12
(
(ΛI)1(ΘJ)2
I
g1
I
B1 .
J
T 2
)
=
∑
I,J
tr12
(
(ΛI)1(ΘJ)2
I
g1
I
(aiaj)1
I⊗J
T12
I
(bj)1
J
(bi)2
IJ
(R′)12
)
= D−1(z)(gaiaj?bjbk)ψ(?biak) = D−1(z)
(
?bjbkS
2(aiaj)g
)
ψ(?biak)
with tr12 = tr⊗ tr, R = ai ⊗ bi. Thanks to the Yang–Baxter equation, we have
bjbk ⊗ aiaj ⊗ biak = R23R21R31 = R31R21R23 = bibj ⊗ ajak ⊗ aibk.
It follows that
zB . ψ = D−1(z)
(
?bibjS
2(ajak)g
)
ψ(?aibk)
= D−1(z)
(
?v−1biak
)
ψ(?aibk) = D−1(z)
(
?
(
v−1
)′)
ψ
(
?v
(
v−1
)′′)
,
Modular Group Representations in Combinatorial Quantization 19
where we used (2.6), (2.9) and (2.2). Hence for x ∈ H:
(zB . ψ)(x) = D−1(z)
((
v−1
)′
x′
)
ψ
(
v
(
v−1
)′′
x′′
)
=
(
D−1(z)ψv
)(
v−1x
)
=
(
D−1(z)ψv
)v−1
(x)
as desired. �
Lemma 4.14. Let z ∈ Linv0,1(H) = Z(H) and let ψ ∈ SLF(H). Then
zB−1 . ψ =
(
S
(
D−1(z)
)
ψv
)v−1
.
It follows that if S(ψ) = ψ for all ψ ∈ SLF(H), then ρSLF(zB−1) = ρSLF(zB).
Proof. This proof is quite similar to that of the previous proposition. Due to the fact that
Ψ1,0
( I
B−1
)
=
I
L(−)S(
I
T )
I
L(+)−1 together with Lemma 4.5 and formulas (2.11), (2.2) and (2.1), it
is not too difficult to show that
I
B−11 .
J
T2 =
t⊗id( I∗
(ai)1
I∗⊗J
T12
I∗(
ajS
−2(bjbk)
)
1
J
(akbi)2
)
,
where t⊗id means transpose on the first tensorand. Write zB−1 =
∑
I tr
(
ΛI
I
g
I
B−1
)
with D−1(z) =∑
I tr
(
ΛI
I
T
)
∈ SLF(H) by Lemma 3.9, and ψ =
∑
J tr
(
ΘJ
J
T
)
. Thanks to (2.11), observe that
S
(
D−1(z)
)
=
∑
I
tr
(
ΛIS(
I
T )
)
=
∑
I
tr
(
tΛI
I∗
T
)
.
Using the fact that S(g) = g−1 and (2.11), we thus get
zB−1 . ψ =
∑
I,J
tr12
(
(ΛI
I
g)1(ΘJ)2
t⊗id( I∗
(ai)1
I∗⊗J
T12
I∗
(ajS
−2(bjbk))1
J
(akbi)2
))
=
∑
I,J
tr12
(
(tΛI)1(ΘJ)2
I∗
(ai)1
I∗⊗J
T12
I∗
(ajS
−2(bjbk)g
−1)1
J
(akbi)2
)
= S
(
D−1(z)
)(
ai?ajS
−2(bjbk)g
−1)ψ(?akbi)
= S
(
D−1(z)
)(
?ajS
−2(bjbk)g
−1ai
)
ψ(?akbi) = S
(
D−1(z)
)(
?
(
v−1
)′)
ψ
(
?v
(
v−1
)′′)
.
For the last equality we used (2.6), (2.9) and (2.2). Hence we get as in the previous proof
zB−1 . ψ =
(
S
(
D−1(z)
)
ψv
)v−1
. �
5 Projective representation of SL2(Z)
As previously, H is a finite-dimensional factorizable ribbon Hopf algebra.
5.1 Mapping class group of the torus
Let Σg,n be a compact oriented surface of genus g with n open disks removed. Recall that
the mapping class group MCG(Σg,n) is the group of all isotopy classes of orientation-preserving
homeomorphisms which fix the boundary pointwise, see [23].
Let us put the base point of π1(Σg,n\D) on the boundary circle c. Since c is pointwise fixed,
we can consider the action of MCG(Σg,n\D) on π1(Σg,n\D), obviously defined by
∀ [f ] ∈ MCG(Σg,n), ∀ [γ] ∈ π1(Σg,n\D), [f ] · [γ] = f∗([γ]) = [f ◦ γ].
Until now, we identify f with its isotopy class [f ] and γ with its homotopy class [γ].
20 M. Faitg
Here we focus on the torus Σ1,0 = S1 × S1. Consider Σ1,0\D, where D is an embedded
open disk. The surface Σ1,0\D is represented as a ribbon graph in Fig. 1 together with the
canonical curves a and b. The groups MCG(Σ1,0\D) and MCG(Σ1,0) are generated by the Dehn
twists τa, τb about the free homotopy class of the curves a and b. It is well-known (see [23]) that
MCG(Σ1,0) = SL2(Z) =
〈
τa, τb | τaτbτa = τbτaτb, (τaτb)
6 = 1
〉
.
This presentation is not the usual one of SL2(Z), which is
SL2(Z) =
〈
s, t | (st)3 = s2, s4 = 1
〉
.
The link between the two presentations is s = τ−1a τ−1b τ−1a , t = τa.
The action of the Dehn twists τa and τb on π1(Σ1,0\D) is given by
(τa)∗(a) = a, (τa)∗(b) = ba and (τb)∗(a) = b−1a, (τb)∗(b) = b. (5.1)
For instance, the action of τa on b is depicted by
b ba
5.2 Automorphisms α and β
The fundamental idea, proposed in [5] and [6], is to mimic the action of the Dehn twists of
Σg,n\D on π1(Σg,n\D) at the level of the algebra Lg,n(H). Let us be more precise. We focus on
the case (g, n) = (1, 0).
In π1(Σ1,0\D) we have the two canonical curves a and b, while in L1,0(H) we have the matri-
ces
I
A and
I
B. In view of (5.1), let us try to define two morphisms τ̃a, τ̃b : L1,0(H)! L1,0(H) by
the same formulas
τ̃a
( I
A
)
=
I
A, τ̃a
( I
B
)
=
I
B
I
A,
τ̃b
( I
A
)
=
I
B−1
I
A, τ̃b
( I
B
)
=
I
B.
Let us see the behavior of these mappings under the fusion and exchange relations. For the
exchange relation, no problem arises
R12τ̃a(B)1R21τ̃a(A)2 = R12B1A1R21A2 (definition)
= R12B1R21A2R12A1R
−1
12 (equation (3.3))
= A2R12B1A1R
−1
12 (equation(4.1))
= τ̃a(A)2R12τ̃a(B)1R
−1
12 (definition),
and a similar computation holds for τ̃b. The fusion relation is almost satisfied
τ̃a(B)12 = B12A12 (definition)
= ∆(v)12B12v
−1
1 v−12 R21R12A12 (trick based on (2.2))
= ∆(v)12B1R21B2R
−1
21 v
−1
1 v−12 R21R12A1R21A2R
−1
21 (equation (3.1))
Modular Group Representations in Combinatorial Quantization 21
= ∆(v)12v
−1
1 v−12 B1R21B2R12A1R21A2R
−1
21 (v is central)
= ∆(v)12v
−1
1 v−12 B1A1R21B2A2R
−1
21 (equation (4.1))
= ∆(v)12v
−1
1 v−12 τ̃a(B)1R21τ̃a(B)2R
−1
21 (definition),
and we get similarly
τ̃b(A)12 = B−112 A12 = ∆
(
v−1
)
12
v1v2τ̃b(A)1R21τ̃b(A)2R
−1
21 .
From this we conclude that the elements
I
v−1
I
B
I
A and
I
v
I
B−1
I
A satisfy the relation (3.1). Since v is
central, we see that the exchange relation still holds with these elements. We thus have found
the morphisms which mimic τa and τb. We denote them by α and β respectively, and we have
the following proposition (the maps α, β and the fact that they are automorphisms were already
in [5, Lemma 2] and [6, equations (4.1) and (4.2)]).
Proposition 5.1. We have two automorphisms α, β of L1,0(H) defined by
α(
I
A) =
I
A, α(
I
B) =
I
v−1
I
B
I
A,
β(
I
A) =
I
v
I
B−1
I
A, β(
I
B) =
I
B.
Moreover, these automorphisms are inner: there exist α̂, β̂ ∈ L1,0(H) unique up to scalar
such that
∀x ∈ L1,0(H), α(x) = α̂xα̂−1, β(x) = β̂xβ̂−1.
Proof. By Theorem 4.8, L1,0(H) is a matrix algebra. Hence, by the Skolem–Noether theorem,
every automorphism of L1,0(H) is inner. �
A natural question is then to find explicitly the elements α̂, β̂. The answer is amazingly
simple (it has been provided in [5] and [38] for the semisimple case). Recall the notation (4.4).
Theorem 5.2. Up to scalar, α̂ = v−1A and β̂ = v−1B .
Proof. We must show that
v−1A
I
A =
I
Av−1A , v−1A
I
B =
I
v−1
I
B
I
Av−1A and v−1B
I
A =
I
v
I
B−1
I
Av−1B , v−1B
I
B =
I
Bv−1B .
It is obvious that v−1A (resp. v−1B ) commutes with the matrices
I
A (resp.
I
B) since it is central in
jA(L0,1(H)) (resp. in jB(L0,1(H))). Let us show the other commutation relation for v−1A . We
use the isomorphism Ψ1,0. Observe that Ψ1,0(xA) = x for all x ∈ H. Hence, using the exchange
relation of Definition 4.4 and (2.2), we have
Ψ1,0
(
v−1A
I
B
)
=
I
L(+)v−1
I
T
I
L(−)−1 =
I
L(+)
I
T
(
?v′−1
)
v′′−1
I
L(−)−1 =
I
L(+)
I
T
I
(v′)−1v′′−1
I
L(−)−1
=
I
L(+)
I
T
I
v−1
I
bi
I
ajv
−1aibj
I
L(−)−1 =
I
v−1
I
L(+)
I
T
I
biai
I
ajbj
I
L(−)−1v−1
=
I
v−1
I
L(+)
I
T
I
L(−)−1
I
L(+)
I
L(−)−1v−1 = Ψ1,0
(I
v−1
I
B
I
Av−1A
)
as desired. We now apply the morphism α−1 ◦ β−1 to the equality v−1A
I
B =
I
v−1
I
B
I
Av−1A :
α−1 ◦ β−1
(
v−1A
I
B
)
=
I
vv−1B
I
B
I
A−1 = α−1 ◦ β−1
(I
v
−1 I
B
I
Av−1A
)
=
I
B
I
A−1
I
Bv−1B .
Using that vB and
I
B commute, we easily get the desired equality. �
22 M. Faitg
5.3 Projective representation of SL2(Z) on SLF(H)
We now show that the elements v−1A , v−1B give rise to a projective representation of MCG(Σ1,0)
on SLF(H) via the following assignment
τa 7! ρSLF
(
v−1A
)
, τb 7! ρSLF
(
v−1B
)
.
We must then check that
ρSLF
(
v−1A v−1B v−1A
)
∼ ρSLF
(
v−1B v−1A v−1B
)
, ρSLF
(
v−1A v−1B
)6 ∼ 1,
where ∼ means equality up to scalar. As we will see, it turns out that the braid relation holds
in the algebra L1,0(H) itself (the scalar being 1), while the relation
(
v−1A v−1B
)6 ∼ 1 only holds in
the representation SLF(H). This is because the relation (τa)∗(τb)∗(τa)∗ = (τb)∗(τa)∗(τb)∗ holds
on π1(Σ1,0\D), while the relation ((τa)∗(τb)∗)
6 = 1 only holds on π1(Σ1,0). As pointed out in
Remark 4.12, the matrices
I
C corresponding to the boundary circle vanishes on SLF(H). Thus
applying ρSLF amounts to gluing back the disk D. See [22] for the generalization of this fact to
higher genus.
Integrals on H will play an important role. Recall that a left integral (resp. right integral) is
a non-zero linear form µl (resp. µr) on H which satisfies
∀x ∈ H,
(
id⊗ µl
)
◦∆(x) = µl(x)1
(
resp.
(
µr ⊗ id
)
◦∆(x) = µr(x)1
)
. (5.2)
Since H is finite-dimensional, this is equivalent to:
∀ψ ∈ O(H), ψµl = ε(ψ)µl
(
resp. µrψ = ε(ψ)µr
)
. (5.3)
It is well-known that left and right integrals always exist if H is finite-dimensional. Moreover,
they are unique up to scalar. We fix µl. Then µl ◦ S−1 is a right integral, and we choose
µr = µl ◦ S−1. (5.4)
The following proposition is important for the sequel (points 2 and 3 are well-known and can
be deduced from results of Radford, see [36]).
Proposition 5.3. Let ϕv, ϕv−1 ∈ H∗ defined by
ϕv = µl
(
v−1
)−1
µl
(
g−1v−1?
)
, ϕv−1 = µl(v)−1µl
(
g−1v?
)
.
Then
D(ϕv) = v, D
(
ϕv−1
)
= v−1.
It follows that
1) ϕv, ϕv−1 ∈ SLF(H),
2) µl
(
g−1?
)
, µr(g?) ∈ SLF(H),
3) ∀x, y ∈ H, µl(xy) = µl
(
yS2(x)
)
, µr(xy) = µr
(
S2(y)x
)
.
Proof. Consider the following computation, where we use (2.2) and (5.2):
D
(
µl
(
g−1v−1?
))
=
〈
µl
(
g−1v−1?
)
⊗ id, (g ⊗ 1)RR′
〉
=
〈
µl
(
g−1v−1?
)
⊗ id, gv
(
v−1
)′′ ⊗ v(v−1)′〉
= µl
((
v−1
)′′)
v
(
v−1
)′
= µl
(
v−1
)
v.
Modular Group Representations in Combinatorial Quantization 23
Since H is factorizable, the map D is an isomorphism of vector spaces. The left integral µl
is non-zero, so µl
(
g−1v−1?
)
is non-zero either. Since D is an isomorphism, it follows that
D
(
µl
(
g−1v−1?
))
= µl
(
v−1
)
v 6= 0, and thus µl
(
v−1
)
6= 0. Hence the formula for ϕv is well
defined. Moreover, we have the restriction D : SLF(H)
∼
! Z(H), so since v ∈ Z(H), we get
that ϕv ∈ SLF(H). This allows us to deduce the properties stated about µl. Using (5.4), we
obtain the properties 1), 2) and 3) for µr. We can now proceed with the computation for ϕv−1 :
D
(
µl
(
g−1v?
))
=
〈
µl
(
g−1v?
)
⊗ id, (g ⊗ 1)RR′
〉
= µl(vaibj)biaj
= µr(vS(bj)S(ai))biaj = µr
(
vS(ai)S
−1(bj)
)
biaj
=
〈
µr ⊗ id, (v ⊗ 1)(R′R)−1
〉
= µr(v′)v′′v−1 = µr(v)v−1 = µl(v)v−1,
where we used (5.4), the property 3) previously shown and (2.2). We conclude as before. �
Since D is an isomorphism of algebras, we have ϕ−1
v−1 = ϕv, and thus
ϕv−1ϕv
−2
v−1 =
µl
(
v−1
)
µl(v)
ε. (5.5)
By Proposition 4.13, the actions of v−1A and v−1B on SLF(H) are
∀ψ ∈ SLF(H), v−1A . ψ = ψv
−1
= ψ
(
v−1?
)
and v−1B . ψ =
(
ϕv−1ψv
)v−1
. (5.6)
Lemma 5.4. ϕv−1ϕv
−1
v−1 = ϕv
−1
v−1.
Proof. For x ∈ H〈
ϕv−1ϕv
−1
v−1 , x
〉
= µl(v)−2µl
(
vg−1x′
)
µl
(
g−1x′′
)
= µl(v)−2
〈
µl(v?)µl, g−1x
〉
= µl(v)−1µl(g−1x) = ϕv
−1
v−1(x).
We simply used (5.3). �
This lemma has an important consequence.
Proposition 5.5. The following braid relation holds in L1,0(H):
v−1A v−1B v−1A = v−1B v−1A v−1B .
Proof. The morphisms α and β satisfy the braid relation αβα = βαβ. Hence by Theorem 5.2
and since Z(L1,0(H)) ∼= C, we have λv−1A v−1B v−1A = v−1B v−1A v−1B for some λ ∈ C. We evaluate on
the counit
λv−1A v−1B v−1A . ε = λv−1A v−1B . ε = λv−1A . ϕv
−1
v−1 = λϕv
−2
v−1 ,
v−1B v−1A v−1B . ε = v−1B v−1A . ϕv
−1
v−1 = v−1B . ϕv
−2
v−1 =
(
ϕv−1ϕv
−1
v−1
)v−1
=
(
ϕv
−1
v−1
)v−1
= ϕv
−2
v−1 .
We used ε
(
v−1?
)
= ε
(
v−1
)
ε = ε and Lemma 5.4. It follows that λ = 1. �
Observe that (αβ)6 6= id, thus the other relation of MCG(Σ1,0) does not hold in L1,0(H). In
order to show it in the representation, we begin with a technical lemma, in which we use the
notation of (4.4).
Lemma 5.6. For all z ∈ Linv0,1(H) = Z(H), we have zv2A−1B−1A = zB−1.
24 M. Faitg
Proof. Write as usual z =
∑
I tr
(
ΛI
I
g
I
M
)
(Lemma 3.9). We first show that zv−1AB−1 = zvB−1A
in the Heisenberg double. We have
Ψ1,0
(
zv−1AB−1
)
=
∑
I
tr
(
ΛI
I
g
I
v−1
I
L(+)
I
T−1
I
L(+)−1) =
∑
I
tr
(
ΛI
I
g
I
v−1
I
aibiS(
I
T )
I
S(aj)bj
)
.
Using the defining relation of H(O(H)) together with (2.1) and (2.6), we get
I
aibiS
( I
T
)
=
( I
ai
I
ak
I
S(bk)
)
S
( I
T
)
bi =
( I
ai
I
g−1
I
v
)
S
( I
T
)
bi.
It follows that
Ψ1,0(zv−1AB−1) =
∑
I
tr
(
ΛI
I
S2(ai)S(
I
T )
I
S(aj)bibj
)
= D−1(z)
(
S2(ai)S(?)S(aj)
)
bibj
= S
(
D−1(z)
)
(S(ai)aj?)bibj = S
(
D−1(z)
)
∈ O(H)⊗ 1.
A similar computation shows that Ψ1,0zvB−1A) = S
(
D−1(z)
)
. Hence zv−1AB−1 = zvB−1A. Ap-
plying the morphism α to this equality, we find
α(zv−1AB−1) = zB−1 = α(zvB−1A) = zv2A−1B−1A
as desired. �
Consider ω̂ = (vAvBvA)−1 ∈ L1,0(H), which implements the automorphism ω = αβα: ω(x) =
ω̂xω̂−1. The key observation is the following lemma.
Lemma 5.7. For all ψ ∈ SLF(H):
ω̂2 . ψ =
µl
(
v−1
)
µl(v)
S(ψ).
Proof. First, we show the formula for ψ = ε:
ω̂2 . ε =
(
v−1A v−1B v−1A
)2
. ε = v−1A v−1B v−1A . ϕv
−2
v−1 =
(
ϕv−1ϕv
−2
v−1
)v−2
=
µl
(
v−1
)
µl(v)
εv
−2
=
µl
(
v−1
)
µl(v)
ε,
where we used (5.6) and (5.5). Second, note that ω(
I
A) =
I
v2
I
A−1
I
B−1
I
A and ω(
I
B) =
I
A. Then,
thanks to Lemma 5.6, it follows that for every z ∈ Z(H) = Linv0,1(H),
ω2(zB) = ω(zA) = zv2A−1B−1A = zB−1 .
Hence ω̂2zB = zB−1ω̂2. Finally, observe that by Proposition 4.13
∀ψ ∈ SLF(H), D
(
ψv
)
B
. ε = ψ.
These three facts together with Lemma 4.14 yield
ω̂2 . ψ = ω̂2D
(
ψv
)
B
. ε = D
(
ψv
)
B−1ω̂
2 . ε =
µl
(
v−1
)
µl(v)
D
(
ψv
)
B−1 . ε
=
µl
(
v−1
)
µl(v)
S
(
ψv
)v−1
=
µl
(
v−1
)
µl(v)
S(ψ)
as desired. �
Modular Group Representations in Combinatorial Quantization 25
Recall that PSL2(Z) = SL2(Z)/{±I2} admits the following presentations
PSL2(Z) =
〈
τa, τb | τaτbτa = τbτaτb, (τaτb)
3 = 1
〉
=
〈
s, t | (st)3 = 1, s2 = 1
〉
.
Theorem 5.8. Recall that we assume that H is a finite-dimensional factorizable ribbon Hopf
algebra. The following assignment defines a projective representation of MCG(Σ1,0) = SL2(Z)
on SLF(H):
τa 7! ρSLF
(
v−1A
)
, τb 7! ρSLF
(
v−1B
)
.
If moreover S(ψ) = ψ for all ψ ∈ SLF(H), then this defines actually a projective representation
of PSL2(Z).
We denote this projective representation by θSLF.
Proof. By Proposition 5.5, we know that the braid relation is satisfied in L1,0(H). By Lem-
ma 5.7, we have
(v−1A v−1B )3 . ψ = ω̂2 . ψ =
µl
(
v−1
)
µl(v)
S(ψ).
If S|SLF(H) = id, then
ρSLF
(
v−1A v−1B
)3
=
µl
(
v−1
)
µl(v)
id.
Otherwise,
(
v−1A v−1B
)6
. ψ =
µl
(
v−1
)
µl(v)
ω̂2 . S(ψ) =
µl
(
v−1
)2
µl(v)2
S2(ψ)
=
µl
(
v−1
)2
µl(v)2
ψ
(
g?g−1
)
=
µl
(
v−1
)2
µl(v)2
ψ. �
Observe that the quantity µl(v−1)
µl(v)
does not depend on the choice up to scalar of µl.
5.4 Equivalence with the Lyubashenko–Majid representation
Recall that H is a finite-dimensional factorizable ribbon Hopf algebra. Under this assumption,
two operators S, T : H ! H are defined in [33]:
S(x) =
(
id⊗ µl
)(
R−1(1⊗ x)R′−1
)
, T (x) = v−1x.
It is shown that they are invertible and satisfy (ST )3 = λS2, S2 = S−1, with λ ∈ C\{0}. We
warn the reader that in [33], they consider the inverse of the ribbon element (see the bottom of
the third page of their paper). That is why there is v−1 in the formula for T .
Now we introduce two maps. The first is
χ : H∗ ! H,
β 7! (β ⊗ id)(R′R),
while the second is
γ : H ! H∗,
x 7! µr(S(x)?).
26 M. Faitg
The map χ is another variant of the map Ψ of Section 2.2 and is called Drinfeld morphism
in [24]. The map γ is denoted φ̂−1 in [24]. Under our assumptions, the inverse of γ exists (this
is a result of Radford [36]), but we do not use it. Consider the space of left q-characters
Chl(H) =
{
β ∈ H∗ | ∀x, y ∈ H, β(xy) = β
(
S2(y)x
)}
.
These maps satisfy the following restrictions
χ : Chl(H) −! Z(H), γ : Z(H) −! Chl(H).
This is due to the fact that they intertwine the adjoint and the coadjoint actions (for the first the
computation is analogous to that of the proof of Proposition 3.5, while the second is immediate
by Proposition 5.3).
It is not too difficult to show (see, e.g., [29, Remark IV.1.2]) that
∀ z ∈ Z(H), S(z) = χ ◦ γ(z).
It follows that Z(H) is stable under S and T . But since S2 is inner, we have S4(z) = S−2(z) = z
for each z ∈ Z(H). Thus there exists a projective representation θLM of SL2(Z) on Z(H), defined
by
θLM(s) = S|Z(H), θLM(t) = T|Z(H).
We will need the following relation between left and right integrals.
Lemma 5.9. Under our assumptions H is unibalanced, which means that µl = µr
(
g2?
)
.
Proof. The terminology “unibalanced” is picked from [11], where some facts about integrals
and cointegrals are recalled. Recall (see, e.g., [20, Proposition 8.10.10]) that a finite-dimensional
factorizable Hopf algebra is unimodular, which means that there exists c ∈ H, called two-sided
cointegral, such that xc = cx = ε(x)c for all x ∈ H. Let a ∈ H be the comodulus of µr:
ψµr = ψ(a)µr for all ψ ∈ O(H) (see, e.g., [11, equation (4.9)]). By a result of Drinfeld (see [35,
Proposition 10.1.14], but be aware that in this book the notations and conventions for a and g
are different from those we use), we know that
uS(u)−1 = a(a⊗ id)(R),
where a ∈ H∗ is the modulus of the left cointegral cl of H. Here, since c = cl is two-sided, we
have a = ε. Thus g2 = u2v−2 = uS(u)−1 = a thanks to (2.2) and (2.9). We deduce that
µl = µr ◦ S = µr(a?) = µr
(
g2?
)
,
where the second equality is [11, Proposition 4.7]. �
The left q-characters are just shifted symmetric linear forms. More precisely, we have an
isomorphism of algebras:(
g−1
)∗
: SLF(H) ! Chl(H),
ψ 7! ψ
(
g−1?
)
.
Let us define shifted versions of χ and of γ:
χg−1 = χ ◦
(
g−1
)∗
: SLF(H)
∼=−! Z(H), γg = g∗ ◦ γ : Z(H)
∼=−! SLF(H).
The equality S = χg−1 ◦ γg still holds, but we have now SLF(H) instead of Chl(H).
In order to show the equivalence of θSLF and θLM, we begin with two technical lemmas.
Modular Group Representations in Combinatorial Quantization 27
Lemma 5.10.
1.
(
v−1
)′ ⊗ S−1((v−1)′′) = S
((
v−1
)′′)⊗ (v−1)′.
2. ∀ψ ∈ O(H), ∀h ∈ H, µr(h?)ψ = µr(h′?)ψ
(
S−1(h′′)
)
.
Proof. The first equality follows from S
(
v−1
)
= v−1. For the second one〈
µr(h?)ψ, x
〉
= µr(hx′)ψ(x′′) = µr(h′x′)ψ
(
S−1(h′′′)h′′x′′
)
= ψ
(
S−1(h′′′)µr(h′x′)h′′x′′
)
= ψ
(
S−1(h′′)µr((h′x)′)(h′x)′′
)
= µr(h′x)ψ
(
S−1(h′′)
)
,
where we simply used the defining property (5.2) of µr. �
We will employ an immediate consequence of Lemma 5.9, namely (see Proposition 5.3)
ϕv = µl
(
v−1
)−1
µr
(
gv−1?
)
. (5.7)
Lemma 5.11. It holds
ρSLF
(
v2AvB
)
= µl
(
v−1
)−1
γg ◦ χg−1 .
Proof. We compute each side of the equality. On the one hand:
γg ◦ χg−1(ψ) = γg
(
(ψ ⊗ id)
(
g−1
(
v−1
)′
v ⊗
(
v−1
)′′
v
))
= ψ
(
g−1
(
v−1
)′
v
)
µr
(
gS
((
v−1
)′′)
v?
)
,
whereas on the other hand
v2AvB . ψ =
(
ϕvψ
v
)v
= µl
(
v−1
)−1(
µr
(
gv−1?
)
ψv
)v
= µl
(
v−1
)−1[
µr
(
g
(
v−1
)′
?
)
ψ
(
vg−1S−1
((
v−1
)′′))]v
= µl
(
v−1
)−1
µr
(
gvS
((
v−1
)′′)
?
)
ψ
(
vg−1
(
v−1
)′)
as desired. We used that vA . ψ = ψv, vB . ψ =
(
ϕvψ
v
)v−1
, (5.7) and Lemma 5.10. �
The link between the two presentations of SL2(Z) is s = τ−1a τ−1b τ−1a , t = τa. Hence we define
two operators S ′, T ′ : SLF(H)! SLF(H) by
S ′ = θSLF
(
τ−1a τ−1b τ−1a
)
= ρSLF(vAvBvA), T ′ = θSLF(τa) = ρSLF
(
v−1A
)
.
Theorem 5.12. Recall that we assume that H is a finite-dimensional factorizable ribbon Hopf
algebra. Then the projective representation θSLF of Theorem 5.8 is equivalent to θLM.
Proof. Consider the following isomorphism of vector spaces
f = ρSLF
(
v−1A
)
◦ γg : Z(H) ! SLF(H),
z 7! γg(z)
v−1
= µr
(
gv−1S(z)?
)
.
By Lemma 5.11,
S ′ = µl
(
v−1
)−1
ρSLF
(
v−1A
)
◦ γg ◦ χg−1 ◦ ρSLF(vA).
Thus
f ◦ S = ρSLF
(
v−1A
)
◦ γg ◦ χg−1 ◦ γg = µl
(
v−1
)
S ′ ◦ f.
Next,
f ◦ T (z) = f
(
v−1z
)
= γg(z)
v−2
= ρSLF(v−1A )
(
γg(z)
v−1)
= T ′ ◦ f(z).
Then f is an intertwiner of projective representations. �
28 M. Faitg
6 The example of H = U q(sl(2))
Let q be a primitive root of unity of order 2p, with p ≥ 2. We now work in some detail the case of
H = U q(sl(2)), the restricted quantum group associated to sl(2,C), which will be denoted U q in
the sequel. For the definitions and main properties about U q, Z
(
U q
)
, SLF
(
U q
)
, their canonical
bases and the U q-modules, see [7, 21, 24, 27, 31, 39]. Here we take back the notations and
conventions of [21].
To explicitly describe the representation of SL2(Z), we use the GTA basis of SLF
(
U q
)
which
is studied in detail in [21], and which has been introduced in [27] and [7].
Even though U q is not braided, we consider this example because this Hopf algebra is well-
studied and is related to certain models in logarithmic conformal field theory [24]. As we
shall recall below, it is very close from being braided (it suffices to add a square root of the
generator K) and its important properties for our purposes are that it is factorizable (by abuse
of terminology since it is not braided) and that it contains a ribbon element. It will follow that
the defining relations of L0,1(H) and L1,0(H) still make sense in this context and that all the
previous results remain true.
6.1 The braided extension of Uq
Recall that U q is not braided [31, Proposition 3.7.3]. But its extension by a square root of K is
braided, as shown in [24]. Let U q
1/2 be this extension and R ∈ U q1/2 ⊗ U q1/2 be the universal
R-matrix, given by
R = qH⊗H/2
p−1∑
m=0
q̂m
[m]!
qm(m−1)/2Em ⊗ Fm,
with
qH⊗H/2 =
1
4p
4p−1∑
n,j=0
q−nj/2Kn/2 ⊗Kj/2,
where q1/2 is a fixed square root of q. We use the notation qH⊗H/2 because qH⊗H/2v⊗w = qab/2
if K1/2v = qa/2v and K1/2w = qb/2w; also recall the notation q̂ = q − q−1. There is a ribbon
element v associated to this R-matrix (we choose g = Kp+1 as pivotal element, and by (2.9) this
fixes the choice of v). The formulas for RR′ and v are in [24]; it is important to note that K1/2
does not appear in the expression of these elements
RR′ ∈ U q ⊗ U q and v ∈ U q.
Moreover, thanks to the formula for RR′, it is obvious that the map
Ψ: U
∗
q ! U q,
β 7! (β ⊗ id)(RR′)
is an isomorphism of vector spaces. Hence, even if U q is not braided, it can be thought of as
a factorizable Hopf algebra.
Let I be a U q
1/2-module. Since U q ⊂ U q1/2, I determines a U q-module, which we denote I|Uq .
We say that a U q-module J is liftable if there exists a U q
1/2-module J̃ such that J̃|Uq = J . Not
every U q-module is liftable, see [31]. But the simple modules and the PIMs are liftable, which
is enough for us. Indeed, it suffices to define the action of K1/2 on these modules. Take back
Modular Group Representations in Combinatorial Quantization 29
the notations of [21] for the canonical basis of modules. For the simple module X ε(s) (ε ∈ {±}),
there are two choices for ε1/2, and so the two possible liftings are defined by
K1/2vj = ε1/2q(s−1−2j)/2vj
and the action of E and F is unchanged. The two possible liftings of the PIM Pε(s) are defined
by
K1/2b0 = ε1/2q(s−1)/2b0, K1/2x0 =
(
ε1/2qp/2
)
q(p−s−1)/2x0,
K1/2y0 =
(
−ε1/2qp/2
)
q(p−s−1)/2y0, K1/2a0 = ε1/2q(s−1)/2a0
and the action of E and F is unchanged.
Let C̃− be the 1-dimensional U q
1/2-module with basis v defined by Ev = Fv = 0, K1/2v = −v
(which is a lifting of X+(1) = C). If Ĩ is a lifting of a simple module or a PIM I, then we have
seen that the only possible liftings of I are Ĩ+ = Ĩ and Ĩ− = Ĩ ⊗ C̃−. Moreover, using (2.1), we
get equalities which will be used in the next section
(Ĩ−J̃
R
)
12
=
(Ĩ+J̃
R
)
12
( J̃
Kp
)
2
,
(ĨJ̃−
R
)
12
=
(ĨJ̃+
R
)
12
( Ĩ
Kp
)
1
,(Ĩ−J̃
R′
)
12
=
(Ĩ+J̃
R′
)
12
( J̃
Kp
)
2
,
(ĨJ̃−
R′
)
12
=
(ĨJ̃+
R′
)
12
( Ĩ
Kp
)
1
. (6.1)
6.2 L0,1
(
Uq
)
and L1,0
(
Uq
)
We define L0,1
(
U q
)
as the quotient of T(U q
∗) by the fusion relation
I⊗J
M 12 =
I
M1
ĨJ̃
(R′)12
J
M2
ĨJ̃(
R′−1
)
12
,
where I, J are simple modules or PIMs and Ĩ, J̃ are liftings of I and J . From (6.1) and the
fact that Kp is central, we see that this does not depend on the choice of Ĩ and J̃ . As we saw
in Section 2.1, the matrix coefficients of the PIMs linearly span L0,1(H), thus we can restrict
to them in the definition. However, the simple modules are added for convenience. All the
results of Section 3 remain true for L0,1
(
U q
)
. In particular, Ψ0,1 is an isomorphism since U q is
factorizable (in the generalized sense explained in Section 6.1).
We now describe L0,1
(
U q
)
by generators and relations. Let
M =
X+(2)
M =
(
a b
c d
)
and R̃ =
X̃+(2)X̃+(2)
R = q−1/2
q 0 0 0
0 1 q̂ 0
0 0 1 0
0 0 0 q
,
where X̃+(2) is the lifting of X+(2) defined by K1/2v0 = q1/2v0. By the decomposition rules
of tensor products (see [39] and also [31]), every PIM (and every simple module) is a direct
summand of some tensor power X+(2)⊗n. Thus every matrix coefficient of a PIM is a matrix
coefficient of some X+(2)⊗n (with n ≥ p). It follows from the fusion relation that a, b, c, d
generate L0,1
(
U q
)
. Let us determine relations between these elements. First, we have the
reflection equation:
R̃12M1R̃21M2 = M2R̃12M1R̃21.
This equation is equivalent to the following exchange relations
da = ad, db = q2bd, dc = q−2cd,
30 M. Faitg
ba = ab+ q−1q̂bd, cb = bc+ q−1q̂
(
da− d2
)
, ca = ac− q−1q̂dc.
Second, since X+(2)⊗2 ∼= X+(1) ⊕ X+(3),1 there exists a unique (up to scalar) morphism
Φ: X+(1)! X+(2)⊗2. It is easily computed
Φ(1) = qv0 ⊗ v1 − v1 ⊗ v0.
By fusion, we have
X+(2)⊗2
M 12Φ = M1R̃21M2R̃
−1
21 Φ = Φ.
This gives just one new relation, which is the analogue of the quantum determinant
ad− q2bc = 1.
Finally, let us compute the RSD isomorphism on M
Ψ0,1
(
a b
c d
)
=
X̃+(2)
L(+)
X̃+(2)
L(−)−1 =
(
K1/2 q̂K1/2F
0 K−1/2
)(
K1/2 0
q̂K−1/2E K−1/2
)
=
(
K + q−1q̂2FE q−1q̂F
q̂K−1E K−1
)
.
We deduce the relations bp = cp = 0 and d2p = 1 from the defining relations of U q.
Theorem 6.1. L0,1
(
U q
)
admits the following presentation〈
a, b, c, d
∣∣∣∣∣
da = ad, db = q2bd, dc = q−2cd
ba = ab+ q−1q̂bd, cb = bc+ q−1q̂
(
da− d2
)
, ca = ac− q−1q̂dc
ad− q2bc = 1, bp = cp = 0, d2p = 1
〉
.
A basis is given by the monomials bicjdk with 0 ≤ i, j ≤ p− 1, 0 ≤ k ≤ 2p− 1.
Proof. Let A be the algebra defined by this presentation. It is readily seen that a = d−1 +
q2bcd−1 and that the monomials bicjdk with 0 ≤ i, j ≤ p − 1, 0 ≤ k ≤ 2p − 1 linearly span A.
Thus dim(A) ≤ 2p3. But we know that 2p3 = dim
(
U q
)
= dim
(
L0,1
(
U q
))
since the monomials
EiF jK` with 0 ≤ i, j ≤ p − 1, 0 ≤ k ≤ 2p − 1 form the PBW basis of U q. It follows that
dim(A) ≤ dim
(
L0,1
(
U q
))
. Since these relations are satisfied in L0,1
(
U q
)
, there exists a surjection
p : A! L0,1
(
U q
)
. Thus dim(A) ≥ dim
(
L0,1
(
U q
))
, and the theorem is proved. �
Remark 6.2. A consequence of this theorem is that L0,1
(
U q
)
is a restricted version (i.e., a finite-
dimensional quotient by monomial central elements) of L0,1(Uq)spe, the specialization at our root
of unity q of the algebra L0,1(Uq). A complete study of the algebra L0,1(Uq)spe will appear in [9].
Applying the isomorphism of algebras D defined in (3.7) to the GTA basis of SLF
(
U q
)
, we
get a basis of Z
(
U q
)
. We introduce notations for these basis elements
X ε(s)
W = D(χεs), Hs′ = D(Gs′) (6.2)
with 1 ≤ s ≤ p, ε ∈ {±} and 1 ≤ s′ ≤ p− 1. They satisfy the same multiplication rules than the
elements of the GTA basis, see [21, Section 5] or [27] (the elements χ(s) defined in [27] correspond
1This decomposition does not hold if p = 2: in that case, we have X+(2)⊗2 ∼= P+(1). The morphism Φ remains
valid and corresponds to sending C = X+(1) in Soc
(
P+(1)
)
.
Modular Group Representations in Combinatorial Quantization 31
to [s]Hs here). Let us mention that under the identification L0,1
(
U q
)
= U q via Ψ0,1, it holds
by definition
X ε(s)
W = tr(
X ε(s)
Kp+1
X ε(s)
M ), since we choose Kp+1 as pivotal element. In particular,
X+(2)
W = −qa− q−1d = −q̂2FE − qK − q−1K−1 = −q̂2C, (6.3)
where C is the standard Casimir element of U q.
Similarly, we define L1,0
(
U q
)
as the quotient of L0,1
(
U q
)
∗L0,1
(
U q
)
by the exchange relations
ĨJ̃
R12
I
B1
ĨJ̃
(R′)12
J
A2 =
J
A2
ĨJ̃
R12
I
B1
ĨJ̃
(R−1)12,
where I, J are simple modules or PIMs and Ĩ, J̃ are liftings of I and J . From (6.1), we see
again that this does not depend on the choice of Ĩ and J̃ . The coefficients of
X+(2)
A and of
X+(2)
B
generate L1,0
(
U q
)
. The morphism Ψ1,0 : L1,0
(
U q
)
! H
(
O
(
U q
))
is well-defined (the square root
of K does not appear). Indeed, the matrix Ψ1,0
(X+(2)
A
)
belongs to 1 ⊗ U q ⊂ H
(
O
(
U q
))
and is
the same as the image of the matrix M under the morphism Ψ0,1 above; moreover, thanks to
the commutation relations of the Heisenberg double we get
Ψ1,0
(X+(2)
B
)
=
X̃+(2)
L(+)
X̃+(2)
T
X̃+(2)
L(−)−1 =
(
K1/2 q̂K1/2F
0 K−1/2
)(
α β
γ δ
)(
K1/2 0
q̂K−1/2E K−1/2
)
=
(
q1/2αK + q−1/2q̂δK + q1/2q̂γKF + q−1/2q̂βE + q−1/2q̂2δFE q−1/2β + q−1/2q̂δF
q−1/2γ + q1/2q̂δK−1E q1/2δK−1
)
.
From these formulas we see that Ψ1,0 is surjective: indeed, thanks to the coefficients of Ψ1,0
(X+(2)
A
)
we obtain E, F , K and then thanks to the coefficients of Ψ1,0
(X+(2)
B
)
we obtain α, β, γ, δ.
Moreover, by the exchange relation in L1,0
(
U q
)
and the fact that L0,1
(
U q
) ∼= U q, we know that
dim
(
L1,0
(
U q
))
≤ dim
(
U q
)2
= dim
(
H
(
O
(
U q
)))
, and hence Ψ1,0 is an isomorphism of algebras.
In order to get a presentation of L1,0
(
U q
)
, one can again restrict to I = J = X+(2) and write
down the corresponding exchange relations. We do not give this presentation of L1,0
(
U q
)
since
we will not use it in this work.
With these definitions of L0,1
(
U q
)
and L1,0
(
U q
)
, all the computations made under the general
assumptions before remain valid with Ūq and we have a projective SL2(Z)-representation on
SLF
(
U q
)
.
6.3 Explicit description of the SL2(Z)-projective representation
Note that it can be shown directly that U q is unimodular and unibalanced, see for instance [29,
Corollary II.2.8] (also note that in [11] it is shown that all the simply laced restricted quantum
groups at roots of unity are unibalanced).
Proposition 6.3. For all z ∈ Z
(
U q
)
, S(z) = z and for all ψ ∈ SLF
(
U q
)
, S(ψ) = ψ. It follows
that in the case of U q, θSLF is in fact a projective representation of PSL2(Z).
32 M. Faitg
Proof. By [24, Appendix D], the canonical central elements are expressed as es = Ps(C),
w±s = π±s Qs(C) where Ps and Qs are polynomials, C is the Casimir element and π±s are discrete
Fourier transforms of
(
Kj
)
0≤j≤2p−1. It is easy to check that S(C) = C and that S
(
π±s
)
= π±s ,
thus S(es) = es and S
(
w±s
)
= w±s . Next, let ψ ∈ SLF
(
U q
)
. Since γg is an isomorphism, we can
write ψ = γg(z) = µr(gS(z)?) with z ∈ Z
(
U q
)
. Then
S(ψ) = S
(
µr(gz?)
)
= µr ◦ S
(
?zg−1
)
= µl
(
g−1z?
)
= µr(gz?) = ψ.
We used that S(z) = z, Proposition 5.3 and the fact that U q is unibalanced. �
Recall that the action of the elements v−1A , v−1B on SLF(H) implements the SL2(Z)-repre-
sentation. We now determine this action on the GTA basis. Some preliminaries are in order;
first recall (see [24]) that the expression of the ribbon element v in the canonical basis
(
es, w
±
s
)
of Z
(
U q
)
is
v =
p∑
s=0
vX+(s)es + q̂
p−1∑
s=1
vX+(s)
(
p− s
[s]
w+
s −
s
[s]
w−s
)
(6.4)
with q̂ = q − q−1 and
vX+(s) = vX−(p−s) = (−1)s−1q
−(s2−1)
2 .
Note that
X ε(s)
v = vX ε(s)id and vX+(0) is just a notation for vX−(p) used to unify the formula.
Expressing v−1 in this basis is obvious. Second, it is easy to see that the action of Z
(
U q
)
on
SLF
(
U q
)
is(
χ+
s
)et = δs,tχ
+
s ,
(
χ−s
)et = δp−s,tχ
−
s , Gets = δs,tGs(
χ+
s
)w±t = 0,
(
χ−s
)w±t = 0, G
w+
t
s = δs,tχ
+
s , G
w−t
s = δs,tχ
−
p−s, (6.5)
where ϕz = ϕ(z?). Finally, we have the following lemma.
Lemma 6.4. Let z ∈ Linv0,1(H) = Z(H) and let ψ ∈ SLF(H). Then
zvB−1A . ψ = S
(
D−1(z)
)
ψ.
Proof. The proof is analogous to those of the two similar results in Section 4.3 and is thus left
to the reader. Note that this lemma is not specific to the case of U q. �
Theorem 6.5. The actions of v−1A and v−1B on the GTA basis are given by
v−1A . χεs = v−1X ε(s)χ
ε
s, v−1A . Gs′ = v−1X+(s′)Gs′ − v
−1
X+(s′)q̂
(
p− s′
[s′]
χ+
s′ −
s′
[s′]
χ−p−s′
)
and
v−1B . χεs = ξε(−ε)p−1sq−(s2−1)
(
p−1∑
`=1
(−1)s(−ε)p−`
(
q`s + q−`s
)(
χ+
` + χ−p−`
)
+ χ+
p
+ (−ε)p(−1)sχ−p
)
+ ξε(−1)sq−(s
2−1)
p−1∑
j=1
(−ε)j+1[j][js]Gj ,
v−1B . Gs′ = ξ(−1)s
′
q−(s
′2−1) q̂p
[s′]
p−1∑
j=1
(−1)j+1[j][js′]
(
2Gj − q̂
p− j
[j]
χ+
j + q̂
j
[j]
χ−p−j
)
,
with ε ∈ {±}, 0 ≤ s ≤ p, 1 ≤ s′ ≤ p− 1 and ξ−1 = 1−i
2
√
p
q̂p−1
[p−1]!(−1)pq−(p−3)/2.
Modular Group Representations in Combinatorial Quantization 33
Proof. The formulas for v−1A are easily deduced from Proposition 4.13, (6.4) and (6.5). Com-
puting the action of v−1B is more difficult. We will use the commutation relations of v−1B with
the A,B-matrices (which follow from Theorem 5.2), namely
v−1B
I
A =
I
v
I
B−1
I
Av−1B , v−1B
I
B =
I
Bv−1B (6.6)
to compute the action of v−1B by induction. The multiplication rules of the GTA basis (see [21,
Section 5]) will be used several times. Let us denote
v−1B . χεs =
∑
σ∈{±}
p∑
`=1
λσ` (ε, s)χσ` +
p−1∑
j=1
δj(ε, s)Gj .
Relation (6.6) implies v−1B
X+(2)
WA =
X+(2)
WvB−1Av
−1
B (recall (6.2)). On the one hand, we obtain
by (6.3)
v−1B
X+(2)
WA . χ
ε
s = v−1B . χεs
(
−q̂2C?
)
=
p∑
`=1
σ∈{±}
−ε
(
qs + q−s
)
λσ` (ε, s)χσ` +
p−1∑
j=1
−ε
(
qs + q−s
)
δj(ε, s)Gj .
On the other hand, we use Lemma 6.4 and the multiplication rules
X+(2)
WvB−1Av
−1
B . χεs =
∑
σ∈{±}
p∑
`=1
λσ` (ε, s)χ+
2 χ
σ
` +
p−1∑
j=1
δj(ε, s)χ
+
2 Gj
=
∑
σ∈{±}
(
λσ2 (ε, s) + 2λ−σp (ε, s)
)
χσ1 +
p−2∑
`=2
(
λσs−1(ε, s) + λσs+1(ε, s)
)
χσ`
+
(
λσp−2(ε, s) + 2λσp (ε, s)
)
χσp−1 + λσp−1(ε, s)χ
σ
p +
δ2(ε, s)
[2]
G1
+
p−2∑
j=2
[j]
(
δj−1(ε, s)
[j − 1]
+
δj+1(ε, s)
[j + 1]
)
Gj +
δp−2(ε, s)
[2]
Gp−1.
This gives recurrence equations between the coefficients which are easily solved
v−1B . χεs = λ(ε, s)
(
p−1∑
`=1
(−1)s(−ε)p−`
(
q`s + q−`s
)(
χ+
` + χ−p−`
)
+ χ+
p + (−ε)p(−1)sχ−p
)
+ δ(ε, s)
p−1∑
j=1
(−ε)j+1 [j][js]
[s]
Gj .
The coefficients λ(ε, s) = λ+p (ε, s) and δ(ε, s) = δ1(ε, s) are still unknown. In order to compute
them by induction, we use the relation v−1B
X+(2)
WB =
X+(2)
WBv
−1
B , which is another consequence
of (6.6). Before, note that
X+(2)
WB . χ
ε
s =
(
χ+
2 (χεs)
v
)v−1
=
vX ε(s)
vX ε(s−1)
χεs−1 +
vX ε(s)
vX ε(s+1)
χεs+1 = −εq−s+
1
2χεs−1 − εqs+
1
2χεs+1,
34 M. Faitg
with 1 ≤ s ≤ p− 1 and the convention that χ±0 = 0. It follows that
v−1B . χεs+1 = −εq−s−
1
2
X+(2)
WB .
(
v−1B . χεs
)
− q−2sv−1B . χεs−1. (6.7)
Due to (6.4), (6.5) and the multiplication rules, we have
X+(2)
WB .
(
v−1B . χεs
)
=
(
χ+
2
(
v−1B . χεs
)v)v−1
=
vX+(p−1)
vX+(p)
(
λ+p−1(ε, s) + q̂δp−1(ε, s)
)
χ+
p +
vX+(2)
[2]
δ2(ε, s)G1 + · · · ,
where the dots (· · · ) mean the remaining of the linear combination in the GTA basis. After
replacement by the values found previously and insertion in relation (6.7), we obtain
λ(ε, s+ 1)χ+
p + δ(ε, s+ 1)G1 + · · ·
=
(
q−(s+1)
(
qs + q−s
)
λ(ε, s)− q−2sλ(ε, s− 1) + (−ε)p−1(−1)s−1q̂q−(s+1)δ(ε, s)
)
χ+
p
+
(
−q−(s+2)
(
qs + q−s
)
δ(ε, s)− q−2sδ(ε, s− 1)
)
G1 + · · · .
These are recurrence equations. It just remains to determine the first values λ(ε, 1), δ(ε, 1).
Observe that, since U q is unibalanced:
v−1B . χ+
1 =
(
ϕv−1
(
χ+
1
)v)v−1
= µl(v)−1µl
(
Kp−1v?
)v−1
= µl(v)−1µr
(
Kp+1?
)
. (6.8)
In [21, Section 4.3] the decomposition of µr(Kp+1?) in the GTA basis has been found (when µr
is suitably normalized). Thanks to this, we obtain
v−1B . χ+
1 = λ(+, 1)χ+
p + δ(+, 1)G1 + · · · = ξ(−1)p−1χ+
p − ξG1 + · · ·
and
v−1B . χ−1 = vX−(1)v
−1
B
X−(1)
WB . χ
+
1 = vX−(1)
X−(1)
WB . v
−1
B . χ+
1 = vX−(1)
(
χ−1
(
v−1B . χ+
1
)v)v−1
= −ξχ+
p + ξG1 + · · · = λ(−, 1)χ+
p + δ(−, 1)G1 + · · · .
The scalar ξ does not depend on the choice of µr thanks to the factor µl(v)−1 = µl ◦ S(v)−1 =
µr(v)−1 in (6.8). Thanks to the formulas [24] for µr and v in the PBW basis we compute µr(v)
and this gives the value of ξ. We are now in position to solve the recurrence equations. It is
easy to check that the solutions are
δ(ε, s) = ξε(−1)sq−(s
2−1)[s], λ(ε, s) = ξε(−ε)p−1sq−(s2−1).
We now proceed with the proof of the formula for Gs′ . Relation (6.6) implies v−1B H1
B = H1
Bv
−1
B
(recall (6.2)). By (6.4), (6.5) and the multiplication rules, we have on the one hand
v−1B H1
B . χ
+
s = v−1B .
(
G1
(
χ+
s
)v)v−1
= [s]v−1B . Gs − q̂(p− s)v−1B . χ+
s + q̂sv−1B . χ−p−s,
whereas on the other hand
H1
Bv
−1
B . χ+
s =
(
G1
(
v−1B . χ+
s
)v)v−1
= q̂p
p−1∑
j=1
δj(+, s)
(
Gj − q̂
p− j
[j]
χ+
j + q̂
j
[j]
χ−p−j
)
.
Equalizing both sides and inserting the previously found values, we obtain the desired for-
mula. �
Modular Group Representations in Combinatorial Quantization 35
Remark 6.6. The guiding principle of the previous computations was that the mutiplication of
two symmetric linear forms in the GTA basis is easy when one of them is χ+
2 , χ−1 or G1 (see [21,
Section 5]), and that all the formulas can be derived from v−1B . χ+
1 using only such products.
Recall that the standard representation C2 of SL2(Z) = MCG(Σ1,0) is defined by
τa 7!
(
1 0
−1 1
)
, τb 7!
(
1 1
0 1
)
.
Lemma 6.7. Let V be a (projective) representation of SL2(Z) which admits a basis (xs, ys) such
that
τaxs =
∑
`
a`(s)x`, τbxs =
∑
`
b`(s)(x` + y`),
τays =
∑
`
a`(s)(y` − x`), τbys =
∑
`
b`(s)y`.
Then there exists a (projective) representation W of SL2(Z) such that V ∼= C2 ⊗ W . More
precisely, W admits a basis (ws) such that
τaws =
∑
`
a`(s)w`, τbws =
∑
`
b`(s)w`.
Proof. It is easy to check that the formulas for τaws and τbws indeed define a SL2(Z)-repre-
sentation on W . Let (e1, e2) be the canonical basis of C2. Then
e1 ⊗ ws 7! ys, e2 ⊗ ws 7! xs
is an isomorphism which intertwines the SL2(Z)-action. �
Theorem 6.8. The (p + 1)-dimensional subspace V = vect
(
χ+
s + χ−p−s, χ
±
p
)
1≤s≤p−1 is stable
under the SL2(Z)-action of Theorem 6.5. Moreover, there exists a (p−1)-dimensional projective
representation W of SL2(Z) such that
SLF
(
U q
)
= V ⊕
(
C2 ⊗W
)
.
Proof. By [21, Corollary 5.1], V is an ideal of SLF
(
U q
)
. It is easy to see that V is moreover
stable under the action (6.5) of Z
(
U q
)
. Thus we deduce without any computation that V is
SL2(Z)-stable. Next, in view of the formulas in Theorem 6.5, it is natural to define
xs = q̂
p− s
[s]
χ+
s − q̂
s
[s]
χ−p−s, ys = Gs − xs.
Then
v−1A . xs = v−1X+(s)
xs, v−1B . xs = ξ(−1)sq−(s
2−1) q̂p
[s]
p−1∑
j=1
(−1)j+1[j][js](xj + yj) ,
v−1A . ys = v−1X+(s)
(ys − xs), v−1B . ys = ξ(−1)sq−(s
2−1) q̂p
[s]
p−1∑
j=1
(−1)j+1[j][js]yj .
The result follows from Lemma 6.7. �
36 M. Faitg
The structure of the Lyubashenko–Majid representation on Z
(
U q
)
has been described in [24].
By Theorem 5.12, this projective representation is equivalent to the one constructed here and
Theorem 6.8 is in perfect agreement with the decomposition given in [24].
Note that the subspace V is generated by the characters of the finite-dimensional projective
U q-modules: V = vect
(
χP
)
P∈Proj
(
Uq
). We precise that, explicitly, the projective representa-
tion W has a basis (ws)1≤s≤p−1 such that the action of τa, τb is
τaws = v−1X+(s)
ws, τbws = ξ(−1)sq−(s
2−1) q̂p
[s]
p−1∑
j=1
(−1)j+1[j][js]wj .
6.4 A conjecture about the representation of Linv
1,0
(
Uq
)
on SLF
(
Uq
)
Another natural (but harder) question is to determine the structure of SLF
(
U q
)
under the
action of Linv1,0
(
U q
)
. As mentioned in the proof of Theorem 6.8, the subspace
V = vect
(
χ+
s + χ−p−s, χ
±
p
)
1≤s≤p−1
is quite “stable”. We propose the following conjecture.
Conjecture 6.9. V is a Linv1,0
(
U q
)
-submodule of SLF
(
U q
)
.
In order to prove this conjecture one needs to find a basis or a generating set of Linv1,0
(
U q
)
,
and then to show that V is stable under the action of the basis elements (or of the generating
elements). Both tasks are difficult.
Let us mention that since V is an ideal of SLF
(
U q
)
, it is stable under the action of zA, zB
and zB−1A for all z ∈ Z
(
U q
)
= Linv0,1
(
U q
)
(see Proposition 4.13 and Lemma 6.4). Also recall the
wide family of invariants given in (4.3); we can try to test the conjecture with them. A long
computation (which is not specific to U q) shows that
tr12
(
I⊗J
g12Φ12
I
A1
IJ
(R′)12
J
B2
IJ
R12
)
. χK = vJtr13
(I⊗K
T13
I⊗K
v −113 sIJ,K(Φ)13
)
,
where χK is the character of K,
J
v = vJ id (note that we may assume that I, J , K are simple
modules) and
sIJ,K(Φ) = tr2
(
J
g2
JK
R 23Φ12
JK
(R′)23
)
.
Proving that V is stable under the action of these invariants amounts to show symmetry pro-
perties between sIJ,X+(s) and sIJ,X−(p−s) for all simple U q-modules I, J . We have checked that
it is true if Φ = idI⊗J (in this case sIJ,K(idI⊗J) = sJ,K idI⊗K , where sJ,K is the usual S-matrix)
for all simple modules I, J , and also that it holds for I = J = X+(2) with every Φ.
Proposition 6.10.
1) SLF
(
U q
)
is indecomposable as a Linv1,0
(
U q
)
-module.
2) Assume that the Conjecture holds. Then the Linv1,0
(
U q
)
-modules V and SLF
(
U q
)
/V are
simple. It follows that SLF
(
U q
)
has length 2 as a Linv1,0
(
U q
)
-module.
Proof. These are basically consequences of (6.5) and of the multiplication rules in the GTA
basis [21, Section 5]. To avoid particular cases, let χε0 = 0, χεp+1 = χ−ε1 , χε−1 = χ−εp−1 and
e−1 = ep+1 = 0.
Modular Group Representations in Combinatorial Quantization 37
1) Observe that SLF
(
U q
)
is generated by χ+
1 = ε as a Linv1,0
(
U q
)
-module: this is a general
fact which follows immediately from Lemma 6.4. Explicitly (see (5.2))
X ε(s)
WvB−1A . χ
+
1 = χεsχ
+
1 = χεs, Hs
vB−1A . χ
+
1 = Gsχ
+
1 = Gs.
Write SLF
(
U q
)
= U1 ⊕ U2. At least one of the two subspaces U1, U2 necessarily contains an
element of the form ϕ = G1 +
∑
i 6=1 λiGi +
∑
j,ε η
ε
jχ
ε
j ; assume that it is U1. Then
(
w+
1
)
A
. ϕ =
ϕw
+
1 = χ+
1 ∈ U1 thanks to (6.5). It follows that U1 = SLF
(
U q
)
and U2 = {0}, as desired.
2) Let 0 6= U ⊂ V be a submodule, and let ψ =
∑p
j=0 λj
(
χ+
j + χ−p−j
)
∈ U with λs 6= 0 for
some s. Then using Proposition 4.13 and (6.5), we get (es)A . ψ = ψes = λs
(
χ+
s + χ−p−s
)
, and
thus χ+
s + χ−p−s ∈ U . Apply
X+(2)
WvB−1A (we use Lemma 6.4 and Proposition 6.3)
X+(2)
WvB−1A .
(
χ+
s + χ−p−s
)
= χ+
2
(
χ+
s + χ−p−s
)
=
(
χ+
s−1 + χ−p−s+1
)
+
(
χ+
s+1 + χ−p−s−1
)
.
Hence
(es−1)A
X+(2)
WvB−1A .
(
χ+
s + χ−p−s
)
= χ+
s−1 + χ−p−s+1,
(es+1)A
X+(2)
WvB−1A .
(
χ+
s + χ−p−s
)
= χ+
s+1 + χ−p−s−1.
It follows that χ+
s−1 + χ−p−s+1, χ
+
s+1 + χ−p−s−1 ∈ U . Continuing like this, one gets step by step
that all the basis vectors belong to U , hence U = V.
Next, let Gs and χ+
s be the classes of Gs and χ+
s modulo V (with χ+
0 = χ+
p = 0). Let
0 6= U ⊂ SLF
(
U q
)
/V be a submodule and ω =
∑p−1
j=1 νjGj + σjχ
+
j ∈ U be non-zero. If all the
νj are 0, then there exists σs 6= 0 and (es)A .ω = σsχ
+
s ∈ U . If one of the νj , say νs, is non-zero,
then (w+
s )A . ω = νsχ
+
s ∈ U . In both cases we get χ+
s ∈ U . Now we proceed as previously
(es−1)A
X+(2)
WvB−1A . χ
+
s = χ+
s−1, (es+1)A
X+(2)
WvB−1A . χ
+
s = χ+
s+1.
Thus we get step by step that χ+
j ∈ U for all j. Apply H1
vB−1A
H1
vB−1A . χ
+
j = G1χ
+
j + V = [j]Gj + P.
It follows that Gj ∈ U for all j, and thus U = SLF
(
U q
)
/V as desired. �
In order to determine the structure of SLF
(
U q
)
if the Conjecture is true, it will remain to
determine whether V is a direct summand of SLF
(
U q
)
or not.
Acknowledgements
I am grateful to my advisors, Stéphane Baseilhac and Philippe Roche, for their regular support
and their useful remarks. I thank the referees for carefully reading the manuscript and for many
valuable comments which improved the paper.
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https://doi.org/10.1007/BF02102012
1 Introduction
2 Some basic facts
2.1 Dual of a finite-dimensional algebra
2.2 Braided Hopf algebras, factorizability, ribbon element
2.3 Dual Hopf algebra O(H)
3 The loop algebra L0,1(H)
3.1 Definition of L0,1(H) and H-module-algebra structure
3.2 Isomorphism L0,1(H) .5-.5.5-.5.5-.5.5-.5H
4 The handle algebra L1,0(H)
4.1 Definition of L1,0(H) and H-module-algebra structure
4.2 Isomorphism L1,0(H) .5-.5.5-.5.5-.5.5-.5H(O(H))
4.3 Representation of L1,0inv(H) on `39`42`"613A``45`47`"603ASLF(H)
5 Projective representation of SL2(Z)
5.1 Mapping class group of the torus
5.2 Automorphisms and
5.3 Projective representation of SL2(Z) on `39`42`"613A``45`47`"603ASLF(H)
5.4 Equivalence with the Lyubashenko–Majid representation
6 The example of H = Uq(sl(2))
6.1 The braided extension of Uq
6.2 L0,1(to.Uq)to. and L1,0(to.Uq)to.
6.3 Explicit description of the SL2(Z)-projective representation
6.4 A conjecture about the representation of L1,0inv(to.Uq)to. on `39`42`"613A``45`47`"603ASLF(to.Uq)to.
References
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| id | nasplib_isofts_kiev_ua-123456789-210311 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:25:06Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Faitg, M. 2025-12-05T09:32:54Z 2019 Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras / M. Faitg // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 39 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 16T05; 81R05 arXiv: 1805.00924 https://nasplib.isofts.kiev.ua/handle/123456789/210311 https://doi.org/10.3842/SIGMA.2019.077 Let Σg,n be a compact oriented surface of genus g with n open disks removed. The algebra Lg,n(H) was introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche and is a combinatorial quantization of the moduli space of flat connections on Σg,n. Here we focus on the two building blocks L₀,₁(H) and L₁,₀(H) under the assumption that the gauge Hopf algebra H is finite-dimensional, factorizable, and ribbon, but not necessarily semisimple. We construct a projective representation of SL₂(Z), the mapping class group of the torus, based on L₁,₀(H), and we study it explicitly for H = Ūq(sl(2)). We also show that it is equivalent to the representation constructed by Lyubashenko and Majid. I am grateful to my advisors, Stéphane Baseilhac and Philippe Roche, for their regular support and their useful remarks. I thank the referees for carefully reading the manuscript and for many valuable comments, which improved the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras Article published earlier |
| spellingShingle | Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras Faitg, M. |
| title | Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras |
| title_full | Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras |
| title_fullStr | Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras |
| title_full_unstemmed | Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras |
| title_short | Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras |
| title_sort | modular group representations in combinatorial quantization with non-semisimple hopf algebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210311 |
| work_keys_str_mv | AT faitgm modulargrouprepresentationsincombinatorialquantizationwithnonsemisimplehopfalgebras |