On the Equivalence of Module Categories over a Group-Theoretical Fusion Category
We give a necessary and sufficient condition in terms of group cohomology for two indecomposable module categories over a group-theoretical fusion category C to be equivalent. This concludes the classification of such module categories.
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irk-123456789-1486422019-02-19T01:31:50Z On the Equivalence of Module Categories over a Group-Theoretical Fusion Category Natale, S. We give a necessary and sufficient condition in terms of group cohomology for two indecomposable module categories over a group-theoretical fusion category C to be equivalent. This concludes the classification of such module categories. 2017 Article On the Equivalence of Module Categories over a Group-Theoretical Fusion Category / S. Natale // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 9 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 18D10; 16T05 DOI:10.3842/SIGMA.2017.042 http://dspace.nbuv.gov.ua/handle/123456789/148642 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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We give a necessary and sufficient condition in terms of group cohomology for two indecomposable module categories over a group-theoretical fusion category C to be equivalent. This concludes the classification of such module categories. |
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Natale, S. |
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Natale, S. On the Equivalence of Module Categories over a Group-Theoretical Fusion Category Symmetry, Integrability and Geometry: Methods and Applications |
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Natale, S. |
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Natale, S. |
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On the Equivalence of Module Categories over a Group-Theoretical Fusion Category |
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On the Equivalence of Module Categories over a Group-Theoretical Fusion Category |
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On the Equivalence of Module Categories over a Group-Theoretical Fusion Category |
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On the Equivalence of Module Categories over a Group-Theoretical Fusion Category |
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On the Equivalence of Module Categories over a Group-Theoretical Fusion Category |
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on the equivalence of module categories over a group-theoretical fusion category |
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Інститут математики НАН України |
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2017 |
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On the Equivalence of Module Categories over a Group-Theoretical Fusion Category / S. Natale // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 9 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT natales ontheequivalenceofmodulecategoriesoveragrouptheoreticalfusioncategory |
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2025-07-12T19:51:38Z |
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1837472056491376640 |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 042, 9 pages
On the Equivalence of Module Categories
over a Group-Theoretical Fusion Category
Sonia NATALE
Facultad de Matemática, Astronomı́a, F́ısica y Computación, Universidad Nacional de Córdoba,
CIEM-CONICET, Córdoba, Argentina
E-mail: natale@famaf.unc.edu.ar
URL: http://www.famaf.unc.edu.ar/~natale/
Received April 28, 2017, in final form June 14, 2017; Published online June 17, 2017
https://doi.org/10.3842/SIGMA.2017.042
Abstract. We give a necessary and sufficient condition in terms of group cohomology
for two indecomposable module categories over a group-theoretical fusion category C to be
equivalent. This concludes the classification of such module categories.
Key words: fusion category; module category; group-theoretical fusion category
2010 Mathematics Subject Classification: 18D10; 16T05
1 Introduction
Throughout this paper we shall work over an algebraically closed field k of characteristic zero.
Let C be a fusion category over k. The notion of a C-module category provides a natural
categorification of the notion of representation of a group. The problem of classifying module
categories plays a fundamental role in the theory of tensor categories.
Two fusion categories C and D are called categorically Morita equivalent if there exists an
indecomposable C-module category M such that Dop is equivalent as a fusion category to the
category FunC(M,M) of C-module endofunctors of M. This defines an equivalence relation in
the class of all fusion categories.
Recall that a fusion category C is called pointed if every simple object of C is invertible. A basic
class of fusion categories consists of those which are categorically Morita equivalent to a pointed
fusion category; a fusion category in this class is called group-theoretical. Group-theoretical
fusion categories can be described in terms of finite groups and their cohomology.
The purpose of this note is to give a necessary and sufficient condition in terms of group
cohomology for two indecomposable module categories over a group-theoretical fusion category
to be equivalent. For this, it is enough to solve the same problem for indecomposable module
categories over pointed fusion categories.
Let C be a pointed fusion category. Then there exist a finite group G and a 3-cocycle ω on G
such that C ∼= C(G,ω), where C(G,ω) is the category of finite-dimensionalG-graded vector spaces
with associativity constraint defined by ω (see Section 2.3 for a precise definition). LetM be an
indecomposable right C-module category. Then there exists a subgroup H of G and a 2-cochain
ψ ∈ C2(H, k×) satisfying
dψ = ω|H×H×H , (1.1)
such that M is equivalent as a C-module category to the category M0(H,ψ) of left A(H,ψ)-
modules in C, where A(H,ψ) = kψH is the group algebra of H with multiplication twisted
by ψ [8], [1, Example 9.7.2].
The main result of this paper is the following theorem.
mailto:natale@famaf.unc.edu.ar
http://www.famaf.unc.edu.ar/~natale/
https://doi.org/10.3842/SIGMA.2017.042
2 S. Natale
Theorem 1.1. Let H,L be subgroups of G and let ψ ∈ C2(H, k×) and ξ ∈ C2(L, k×) be 2-
cochains satisfying condition (1.1). Then M0(H,ψ) and M0(L, ξ) are equivalent as C-module
categories if and only if there exists an element g ∈ G such that H = gL and the class of the
2-cocycle
ξ−1ψgΩg|L×L (1.2)
is trivial in H2(L, k×).
Here we use the notation gx = gxg−1 and gL = {gx : x ∈ L}. The 2-cochain ψg ∈ C2(L, k×)
is defined by ψg(g1, g2) = ψ(gg1,
gg2), for all g1, g2 ∈ L, and Ωg : G×G→ k× is given by
Ωg(g1, g2) =
ω(gg1,
gg2, g)ω(g, g1, g2)
ω(gg1, g, g2)
.
Observe that [8, Theorem 3.1] states that the indecomposable module categories considered in
Theorem 1.1 are parameterized by conjugacy classes of pairs (H,ψ). However, this conjugation
relation is not described loc. cit. (compare also with [7] and [1, Section 9.7]).
Consider for instance the case where C is the category of finite-dimensional representa-
tions of the 8-dimensional Kac Paljutkin Hopf algebra. Then C is group-theoretical. In fact,
C ∼= C(G,ω,C, 1), where G ∼= D8 is a semidirect product of the group L = Z2 × Z2 by C = Z2
and ω is a certain (nontrivial) 3-cocycle on G [9]. Let ξ represent a nontrivial cohomology class
in H2(L, k×). According to the usual conjugation relation among pairs (L,ψ), the result in
[8, Theorem 3.1] would imply that the pairs (L, 1) and (L, ξ), not being conjugated under the
adjoint action of G, give rise to two inequivalent C-module categories. These module categories
both have rank one, whence they give rise to non-isomorphic fiber functors on C. However, it
follows from [4, Theorem 4.8(1)] that the category C has a unique fiber functor up to isomor-
phism. In fact, in this example there exists g ∈ G such that Ωg|L×L is a 2-cocycle cohomologous
to ξ. See Example 3.6.
Certainly, the condition given in Theorem 1.1 and the usual conjugacy relation agree in the
case where the 3-cocycle ω is trivial, and it reduces to the conjugation relation among subgroups
when they happen to be cyclic.
As explained in Section 3.1, condition (1.2) is equivalent to the condition that A(L, ξ) and
gA(H,ψ) be isomorphic as algebras in C, where G→ Aut⊗C, g 7→ g( ), is the adjoint action of G
on C (see Lemma 3.2).
Theorem 1.1 can be reformulated as follows.
Theorem 1.2. Two C-module categories M0(H,ψ) and M0(L, ξ) are equivalent if and only if
the algebras A(H,ψ) and A(L, ξ) are conjugated under the adjoint action of G on C.
Theorem 1.1 is proved in Section 3.3. Our proof relies on the fact that, as happens with group
actions on vector spaces, the adjoint action of the group G in the set of equivalence classes of
C-module categories is trivial (Lemma 3.1). In the course of the proof we establish a relation
between the 2-cocycle in (1.2) and a 2-cocycle attached to g, ψ and ξ in [8] (Remark 3.4 and
Lemma 3.5).
We refer the reader to [1] for the main notions on fusion categories and their module categories
used throughout.
On the Equivalence of Module Categories over a Group-Theoretical Fusion Category 3
2 Preliminaries and notation
2.1
Let C be a fusion category over k. A (right) C-module category is a finite semisimple k-linear
abelian category M equipped with a bifunctor ⊗̄ : M×C →M and natural isomorphisms
µM,X,Y : M⊗̄(X ⊗ Y )→ (M⊗̄X)⊗̄Y, rM : M⊗̄1→M,
X, Y ∈ C, M ∈M, satisfying the following conditions:
µM⊗̄X,Y,ZµM,X,Y⊗Z(idM ⊗̄aX,Y,Z) = (µM,X,Y ⊗ idZ)µM,X⊗Y,Z , (2.1)
(rM ⊗ idY )µM,1,Y = idM ⊗̄lY , (2.2)
for all M ∈M, X,Y ∈ C, where a : ⊗ ◦(⊗× idC)→ ⊗◦ (idC ×⊗) and l : 1⊗?→ idC , denote the
associativity and left unit constraints in C, respectively.
Let A be an algebra in C. Then the category AC of left A-modules in C is a right C-module
category with action ⊗̄ : AC×C → CA, given by M⊗̄X = M⊗X endowed with the left A-module
structure (mM ⊗ idX)a−1
A,M,X : A ⊗ (M ⊗ X) → M ⊗ X, where mM : A ⊗M → M is the A-
module structure in M . The associativity constraint of AC is given by a−1
M,X,Y : M⊗̄(X ⊗ Y )→
(M⊗̄X)⊗̄Y , for all M ∈ AC, X,Y ∈ C.
A C-module functor M → M′ between right C-module categories (M, ⊗̄) and (M′, ⊗̄′) is
a pair (F, ζ), where F : M →M′ is a functor and ζM,X : F (M⊗̄X) → F (M)⊗̄′X is a natural
isomorphism satisfying
(ζM,X ⊗ idY )ζM⊗̄X,Y F (µM,X,Y ) = µ′F (M),X,Y ζM,X⊗Y , (2.3)
r′F (M)ζM,1 = F (rM ), (2.4)
for all M ∈M, X,Y ∈ C.
Let M and M′ be C-module categories. An equivalence of C-module categories M → M′
is a C-module functor (F, ζ) : M→M′ such that F is an equivalence of categories. If such an
equivalence exists, M and M′ are called equivalent C-module categories. A C-module category
is called indecomposable if it is not equivalent to a direct sum of two nontrivial C-submodule
categories.
Let M,M′ be indecomposable C-module categories. Then FunC(M,M) is a fusion cate-
gory with tensor product given by composition of functors and the category FunC(M,M′) is
an indecomposable module category over FunC(M,M) in a natural way. If A and B are inde-
composable algebras in C such that M ∼=A C and M′ ∼= BC, then FunC(M,M)op is equivalent
to the fusion category ACA of (A,A)-bimodules in C and there is an equivalence of ACA-module
categories BCA ∼= FunC(M,M′), where BCA is the category of (B,A)-bimodules in C.
2.2
Let M be a C-module category. Every tensor autoequivalence ρ : C → C induces a C-module
category structure Mρ on M in the form M⊗̄ρX = M⊗̄ρ(X), with associativity constraint
µρM,X,Y = µM,ρ(X)⊗ρ(Y )
(
idM ⊗̄ρ2
X,Y
−1)
: M⊗̄ρ(X ⊗ Y )→ (M⊗̄ρ(X))⊗̄ρ(Y ),
for all M ∈M, X,Y ∈ C, where ρ2
X,Y : ρ(X)⊗ ρ(Y )→ ρ(X ⊗Y ) is the monoidal structure of ρ.
See [7, Section 3.2].
Suppose that A is an algebra in C. Then ρ(A) is an algebra in C with multiplication
mρ(A) = ρ(mA)ρ2
A,A : ρ(A)⊗ ρ(A)→ ρ(A).
4 S. Natale
The functor ρ induces an equivalence of C-module categories ρ(A)C → (AC)ρ with intertwining
isomorphisms
ρ2
M,X
−1
: ρ(M⊗̄X)→ ρ(M)⊗̄ρX.
2.3
Let G be a finite group. Let X be a G-module. Given an n-cochain f ∈ Cn(G,X) (where
C0(G,M) = M), the coboundary of f is the (n+ 1)-cochain df = dnf ∈ Cn+1(G,X) defined by
dnf(g1, . . . , gn+1) = g1.f(g2, . . . , gn+1) +
n∑
i=1
f(g1, . . . , gigi+1, . . . , gn)
+ (−1)n+1f(g1, . . . , gn),
for all g1, . . . , gn+1 ∈ G. The kernel of dn is denoted Zn(G,M); an element of Zn(G,M) is an
n-cocycle. We have dndn−1 = 0, for all n ≥ 1. The nth cohomology group of G with coefficients
in M is Hn(G,M) = Zn(G,M)/dn−1(Cn−1(G,M)). We shall write f ≡ f ′ when the cochains
f, f ′ ∈ Cn(G, k×) differ by a coboundary.
We shall assume that every cochain f is normalized, that is, f(g1, . . . , gn) = 1, whenever one
of the arguments g1, . . . , gn is the identity. If H is a subgroup of G and f ∈ Cn(H, k×), we shall
indicate by fg the n-cochain in g−1
H given by fg(h1, . . . , hn) = f(gh1, . . . ,
ghn), h1, . . . , hn ∈ H.
Let ω : G × G × G → k× be a 3-cocycle on G. Let C(G,ω) denote the fusion category of
finite-dimensional G-graded vector spaces with associativity constraint defined, for all U, V,W ∈
C(G,ω), as
aX,Y,Z((u⊗ v)⊗ w) = ω−1(g1, g2, g3)u⊗ (v ⊗ w),
for all homogeneous vectors u ∈ Ug1 , v ∈ Vg2 , w ∈ Wg3 , g1, g2, g3 ∈ G. Any pointed fusion
category is equivalent to a category of the form C(G,ω).
A fusion category C is called group-theoretical if it is categorically Morita equivalent to
a pointed fusion category. Equivalently, C is group-theoretical if and only if there exist a finite
group G and a 3-cocycle ω : G × G × G → k× such that C is equivalent to the fusion category
C(G,ω,H, ψ) =A(H,ψ) C(G,ω)A(H,ψ), where H is a subgroup of G such that the class of ω|H×H×H
is trivial and ψ : H ×H → k× is a 2-cochain on H satisfying condition (1.1).
Let C(G,ω,H, ψ) ∼= C(G,ω)∗M0(H,ψ) be a group-theoretical fusion category. Then there is
a bijective correspondence between equivalence classes of indecomposable C(G,ω,H, ψ)-module
categories and equivalence classes of indecomposable C(G,ω)-module categories. This correspon-
dence attaches to every indecomposable C(G,ω)-module category M the C(G,ω,H, ψ)-module
category
M(H,ψ) = FunC(G,ω)(M0(H,ψ),M).
3 Indecomposable module categories over C(G,ω)
Throughout this section G is a finite group and ω : G×G×G→ k× is a 3-cocycle on G.
3.1
Let g ∈ G. Consider the 2-cochain Ωg : G×G→ k× given by
Ωg(g1, g2) =
ω(gg1,
gg2, g)ω(g, g1, g2)
ω(gg1, g, g2)
.
On the Equivalence of Module Categories over a Group-Theoretical Fusion Category 5
For all g ∈ G we have the relation
dΩg =
ω
ωg
. (3.1)
Let C = C(G,ω) and let g ∈ G. For every object V of C let gV be the object of C such that
gV = V as a vector space with G-grading defined as (gV )x = Vgx, x ∈ G. For every g ∈ G,
we have a functor adg : C → C, given by adg(V ) = gV and adg(f) = f , for every object V and
morphism f of C. Relation (3.1) implies that adg is a tensor functor with monoidal structure
defined by(
ad2
g
)
U,V
: gU ⊗ gV → g(U ⊗ V ),
(
ad2
g
)
U,V
(u⊗ v) = Ωg(h, h
′)−1u⊗ v,
for all h, h′ ∈ G, and for all homogeneous vectors u ∈ Uh, v ∈ Vh′ .
For every g, g1, g2 ∈ G, let γ(g1, g2) : G→ k× be the map defined in the form
γ(g1, g2)(g) =
ω(g1, g2, g)ω(g1g2g, g1, g2)
ω(g1, g2g, g2)
.
The following relation holds, for all g1, g2 ∈ G:
Ωg1g2 = Ωg2
g1Ωg2dγ(g1, g2). (3.2)
In this way, ad: G → Aut⊗C, ad(g) =
(
adg, ad2
g
)
, gives rise to an action by tensor autoe-
quivalences of G on C where, for every g, x ∈ G, V ∈ C(G,ω), the monoidal isomorphisms
ad2
V : g(g
′
V )→ gg′V are given by
ad2
V (v) = γ(g, g′)(x)v,
for all homogeneous vectors v ∈ Vx, h ∈ G. The equivariantization CG with respect to this
action is equivalent to the category of finite-dimensional representations of the twisted quantum
double DωG (see [5, Lemma 6.3]).
For each g ∈ G, and for each C-module category M, let Mg denote the module category
induced by the functor adg as in Section 2.2. Recall that the action of C on Mg is defined by
M⊗̄gV = M⊗̄(gV ), for all objects V of C.
Lemma 3.1. Let g ∈ G and let M be a C-module category. Then Mg ∼= M as C-module
categories.
Proof. For each g ∈ G, let {g} denote the object of C such that {g} = k with degree g. In
what follows, by abuse of notation, we identify {g} ⊗ {h} and {gh}, g, h ∈ G, by means of the
canonical isomorphisms of vector spaces.
Let Rg : Mg → M be the functor defined by the right action of {g}: Rg(M) = M⊗̄{g}.
Consider the natural isomorphism ζ : Rg ◦ ⊗̄g → ⊗̄ ◦ (Rg × idC), defined as
ζM,V = µM,{g},V µ
−1
M,gV,{g} : Rg(M⊗̄gV )→ Rg(M)⊗̄V,
for all objects M of M and V of C, where µ is the associativity constraint of M.
The functor Rg is an equivalence of categories with quasi-inverse given by the functor Rg−1 :
M→Mg.
A direct calculation, using the coherence conditions (2.1) and (2.2) for the module cate-
gory M, shows that ζ satisfies conditions (2.3) and (2.4). Hence (Rg, ζ) is a C-module functor.
Therefore Mg ∼=M as C-module categories, as claimed. �
Lemma 3.2. Let H be a subgroup of G and let ψ be a 2-cochain on H satisfying (1.1). Let
A(H,ψ) denote the corresponding indecomposable algebra in C. Then, for all g ∈ G, gA(H,ψ) ∼=
A(gH,ψg
−1
Ωg−1) as algebras in C.
Proof. By definition, gA(H,ψ) = A
(
gH,ψg
−1(
Ωg−1
g
)−1)
. It follows from formula (3.2) that(
Ωg−1
g
)−1
and Ωg−1 differ by a coboundary. This implies the lemma. �
6 S. Natale
3.2
Let H, L be subgroups of G and let ψ ∈ C2(H, k×), ξ ∈ C2(L, k×), be 2-cochains such that
ω|H×H×H = dψ and ω|L×L×L = dξ.
Let B be an object of the category A(H,ψ)CA(L,ξ) of (A(H,ψ), A(L, ξ))-bimodules in C. For
each z ∈ G, let πl(h) : Bz → Bhz and πr(s) : Bz → Bzs, denote the linear maps induced by the
actions of h ∈ H and s ∈ L, respectively. Then the following relations hold, for all h, h′ ∈ H,
s, s′ ∈ L:
πl(h)πl(h
′) = ω(h, h′, z)ψ(h, h′)πl(hh
′), (3.3)
πr(s
′)πr(s) = ω(z, s, s′)−1ξ(s, s′)πr(ss
′), (3.4)
πl(h)πr(s) = ω(h, z, s)πr(s)πl(h). (3.5)
Lemma 3.3. Let g ∈ G and let Bg denote the homogeneous component of degree g of B. Then
the map π : H ∩ gL→ GL(Bg), defined as π(x) = πr
(
g−1
x
)−1
πl(x) is a projective representation
of H ∩ gL with cocycle αg given, for all x, y ∈ H ∩ gL, as follows:
αg(x, y) = ψ(x, y)ξ−1
(
g−1
x, g
−1
y
)ω(x, y, g)ω
(
x, yg, g
−1(
y−1
))
ω
(
xyg, g−1
(
y−1
)
, g−1
(
x−1
))dug(x, y)
×
ω
(
g−1
y, g
−1(
y−1
)
, g
−1(
x−1
))
ω
(
g−1x, g−1y, g−1
(
y−1x−1
)) ,
where the 1-cochain ug is defined as ug(x) = ω
(
xg, g
−1
x, g
−1(
x−1
))
.
Proof. It follows from (3.4) that πr(s)
−1 = ω
(
z, s, s−1
)
ξ
(
s, s−1
)−1
πr
(
s−1
)
, for all z ∈ G, s ∈ L.
In addition, for all h, h′ ∈ L, we have the following relation:
ξ
(
h′
−1
, h−1
)
ξ(h, h′) = df(h, h′)
ω
(
h′, h′−1, h−1
)
ω
(
h, h′, h′−1h−1
) ,
where f is the 1-cochain given by f(h) = ξ
(
h, h−1
)
. A straightforward computation, using
this relation and conditions (3.3), (3.4) and (3.5), shows that π(x)π(y) = αg(x, y)π(xy), for all
x, y ∈ H ∩ gL. This proves the lemma. �
Remark 3.4. Lemma 3.3 is a version of [8, Proposition 3.2], where it is shown that B is
a simple object of A(H,ψ)CA(L,ξ) if and only if B is supported on a single double coset HgL and
the projective representation π in the component Bg is irreducible.
For all g ∈ G, ψgΩg is a 2-cochain in g−1
H such that ω|g−1H×g−1H×g−1H
= d(ψgΩg). Then
the product ξ−1ψgΩg defines a 2-cocycle of g
−1
H ∩ L.
Lemma 3.5. The class of the 2-cocycle
(
ξ−1ψgΩg
)g−1
in H2(H ∩ gL, k×) coincides with the
class of the 2-cocycle αg in Lemma 3.3.
Proof. A direct calculation shows that for all x, y ∈ G,
ω
(
y, y−1, x−1
)
ω
(
x, y, y−1x−1
) ω(gx, gy, g)ω(gx, gyg, y−1
)
ω
(
gxgyg, y−1, x−1
) = Ωg(x, y)dθg(x, y),
where the 1-cochain θg is defined as θg(x) = ω
(
g, x, x−1
)−1
. This implies that αgg ≡ ξ−1ψgΩg,
as was to be proved. �
On the Equivalence of Module Categories over a Group-Theoretical Fusion Category 7
3.3
In this subsection we give a proof of the main result of this paper.
Proof of Theorem 1.1. Let H,L be subgroups of G and let ψ ∈ C2(H, k×) and ξ ∈ C2(L, k×)
be 2-cochains satisfying condition (1.1). Let A(H,ψ), A(L, ξ) be the associated algebras in C
and let M0(H,ψ), M0(L, ξ) be the corresponding C-module categories.
Let M = M0(L, ξ). For every g ∈ G, let Mg denote the module category induced by the
autoequivalence adg : C → C. The C-module category Mg is equivalent to gA(L,ξ)C. Hence, by
Lemma 3.2, Mg ∼=M0
(
gL, ξg
−1
Ωg−1
)
.
Suppose that there exists an element g ∈ G such that H = gL and the class of the cocy-
cle ξ−1ψgΩg is trivial on L. Relation (3.2) implies that Ωg−1
g = Ω−1
g−1 , and thus the class of
ψ−1ξg
−1
Ωg−1 is trivial on H. Then ψ = ξg
−1
Ωg−1df , for some 1-cochain f ∈ C1(H, k×). There-
fore gA(L, ξ) = A
(
H, ξg
−1
Ωg−1
) ∼= A(H,ψ) as algebras in C. Thus we obtain equivalences of
C-module categories
M0(L, ξ) ∼=M0(L, ξ)g ∼= gA(L,ξ)C ∼=M0(H,ψ),
where the first equivalence is deduced from Lemma 3.1.
Conversely, suppose that F : M0(L, ξ)→M0(H,ψ) is an equivalence of C-module categories.
Recall that there is an equivalence
FunC (M0(L, ξ),M0(H,ψ)) ∼= A(H,ψ)CA(L,ξ).
Under this equivalence, the functor F corresponds to an object B of A(H,ψ)CA(L,ξ) such that
there exists an object B′ of A(L,ξ)CA(H,ψ) satisfying
B ⊗A(L,ξ) B
′ ∼= A(H,ψ), (3.6)
as A(H,ψ)-bimodules in C, and
B′ ⊗A(H,ψ) B ∼= A(L, ξ), (3.7)
as A(L, ξ)-bimodules in C.
Let FPdimA(H,ψ)M denote the Frobenius–Perron dimension of an object M of A(H,ψ)CA(H,ψ).
Then we have
dimM = dimA(H,ψ) FPdimA(H,ψ)M = |H|FPdimA(H,ψ)M.
Taking Frobenius–Perron dimensions in both sides of (3.6) and using this relation we obtain
that dim
(
B ⊗A(L,ξ) B
′) = |H|.
On the other hand, dim(B ⊗A(H,ψ) B
′) = dimB dimB′
dimA(L,ξ) = dimB dimB′
|L| . Thus
dimB dimB′ = |H||L|. (3.8)
Since A(H,ψ) is an indecomposable algebra in C, then it is a simple object of A(H,ψ)CA(H,ψ).
Then (3.7) implies that B is a simple object of A(H,ψ)CA(L,ξ) and B′ is a simple object of
A(L,ξ)CA(H,ψ).
In view of [8, Proposition 3.2], the support of B is a two sided (H,L)-double coset, that is,
B =
⊕
(h,h′)∈H×LBhgh′ , where g ∈ G is a representative of the double coset that supports B.
Moreover, the homogeneous component Bg is an irreducible αg-projective representation of
the group gL ∩ H, where the 2-cocycle αg satisfies αg ≡
(
ξ−1ψgΩg
)g−1
; see Remark 3.4 and
Lemmas 3.3 and 3.5.
8 S. Natale
Notice that the actions of h ∈ H and h′ ∈ L induce isomorphisms of vector spaces Bg ∼= Bhg
and Bg ∼= Bgh′ . Hence
dimB = |HgL| dimBg =
|H||L|
|H ∩ gL|
dimBg = [H : H ∩ gL]|L| dimBg. (3.9)
In particular, dimB ≥ |L|. Reversing the roles of H and L, the same argument implies that
dimB′ ≥ |H|. Combined with relations (3.8) and (3.9) this implies
|H||L| = dimB dimB′ ≥ |H|[H : H ∩ gL]|L| dimBg.
Hence [H : H ∩ gL] dimBg = 1, and therefore [H : H ∩ gL] = 1 and dimBg = 1. The first
condition means that H ⊆ gL, while the second condition implies that the class of αg is trivial
in H2(H ∩ gL, k×). Since the rank of M0(H,ψ) equals the index [G : H] and the rank of
M0(H, ξ) equals the index [G : L], then |H| = |L|. Thus we get that H = gL and that the class
of the 2-cocycle (1.2) is trivial in H2(L, k×). This finishes the proof of the theorem. �
Example 3.6. Let B8 be the 8-dimensional Kac Paljutkin Hopf algebra. The Hopf algebra B8
fits into an exact sequence
k −→ kC −→ B8 −→ kL −→ k,
where C = Z2 and L = Z2 × Z2. See [3]. This exact sequence gives rise to mutual actions by
permutations
C
C←− C × L B−→ L,
and compatible cocycles τ : L × L →
(
kC
)×
, σ : C × C → (kL)×, such that B8 is isomorphic
to the bicrossed product kCτ#σkL. The data �, �, σ and τ are explicitly determined in [4,
Proposition 3.11] as follows. Let C = 〈x : x2 = 1〉, L = 〈z, t : z2 = t2 = ztz−1t−1 = 1〉. Then
� : C ×L→ C is the trivial action of L on C, � : C ×L→ L is the action defined by x� z = z
and x� t = zt,
τxn
(
zitj , zi
′
tj
′)
= (−1)nji
′
,
for all 0 ≤ n, i, i′, j, j′ ≤ 1, and
σzitj
(
xn, xn
′)
= (
√
−1)j
(
n+n′−〈n+n′〉
2
)
,
for all 0 ≤ i, j, n, n′ ≤ 1, where 〈n + n′〉 denotes the remainder of n + n′ in the division by 2.
Here we use the notation τ(a, a′)(y) =: τy(a, a
′) and, similarly, σ(y, y′)(a) =: σa(y, y
′), a, a′ ∈ L,
y, y′ ∈ C.
In view of [9, Theorem 3.3.5] (see [6, Proposition 4.3]), the fusion category of finite-dimen-
sional representations of Bop
8
∼= B8 is equivalent to the category C(G,ω,L, 1), where G = LoC
is the semidirect product with respect to the action �, and ω is the 3-cocycle arising from the
pair (τ, σ) under one of the maps of the so-called Kac exact sequence associated to the matched
pair.
In this example G is isomorphic to the dihedral group D8 of order 8. The 3-cocycle ω is
determined by the formula
ω
(
xnzitj , xn
′
zi
′
tj
′
, xn
′′
zi
′′
tj
′′)
= τxn
(
zi
′
tj
′
, xn
′
� zi
′′
tj
′′)
σzi′′ tj′′
(
xn, xn
′)
, (3.10)
for all 0 ≤ i, j, i′, j′, i′′, j′′, n, n′, n′′ ≤ 1.
On the Equivalence of Module Categories over a Group-Theoretical Fusion Category 9
Notice that ω|L×L×L = 1. Hence, for every 2-cocycle ξ on L, the pair (L, ξ) gives rise to an
indecomposable C-module category M(L, ξ). Formula (3.10) implies that Ωx|L×L is given by
Ωx
(
zitj , zi
′
tj
′)
= (−1)ji
′
, 0 ≤ i, i′, j, j′ ≤ 1.
Then Ωx is a 2-cocycle representing the unique nontrivial cohomology class in H2(L, k×). By
Theorem 1.1, for any 2-cocycle ξ on L,M0(L, 1) andM0(L, ξ) are equivalent as C(G,ω)-module
categories, and therefore so are the corresponding C-module categories M(L, 1) and M(L, ξ).
This implies that indecomposable C-module categories are in this example parameterized by
conjugacy classes of subgroups of D8 on which ω has trivial restriction, as claimed in [2, Sec-
tion 6.4].
Acknowledgements
This research was partially supported by CONICET and SeCyT - Universidad Nacional de
Córdoba, Argentina.
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1 Introduction
2 Preliminaries and notation
2.1
2.2
2.3
3 Indecomposable module categories over C(G, )
3.1
3.2
3.3
References
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