Group-theoretical properties of nilpotent modular categories
GROUP-THEORETICAL PROPERTIES OF NILPOTENT
MODULAR CATEGORIES
VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK
To Yuri Ivanovich Manin on his 70th birthday
Abstract. We characterize a natural class of modular categories of prime
power Frobenius-Perron dimension as representation categories of twisted dou-
bles of finite p-groups. We also show that a nilpotent braided fusion category C
admits an analogue of the Sylow decomposition. If the simple objects of C have
integral Frobenius-Perron dimensions then C is group-theoretical in the sense
of [ENO]. As a consequence, we obtain that semisimple quasi-Hopf algebras
of prime power dimension are group-theoretical. Our arguments are based on
a reconstruction of twisted group doubles from Lagrangian subcategories of
modular categories (this is reminiscent to the characterization of doubles of
quasi-Lie bialgebras in terms of Manin pairs given in [Dr]).
1. introduction
In this paper we work over an algebraically closed field k of characteristic 0.
By a fusion category we mean a k-linear semisimple rigid tensor category C with
finitely many isomorphism classes of simple objects, finite dimensional spaces of
morphisms, and such that the unit object 1 of C is simple. We refer the reader to
[ENO] for a general theory of such categories. A fusion category is pointed if all its
simple objects are invertible. A pointed fusion category is equivalent to VecωG, i.e.,
the category of G-graded vector spaces with the associativity constraint given by
some cocycle ω ∈ Z3(G, k×) (here G is a finite group).
1.1. Main results.
Theorem 1.1. Any braided nilpotent fusion category has a unique decomposition
into a tensor product of braided fusion categories whose Frobenius-Perron dimen-
sions are powers of distinct primes.
The notion of nilpotent fusion category was introduced in [GN]; we recall it in
Subsection 2.2. Let us mention that the representation category Rep(G) of a finite
group G is nilpotent if and only if G is nilpotent. It is also known that fusion
categories of prime power Frobenius-Perron dimension are nilpotent [ENO]. On
the other hand, VecωG is nilpotent for any G and ω. Therefore it is not true that
any nilpotent fusion category is a tensor product of fusion categories of prime power
dimensions.
Theorem 1.2. A modular category C with integral dimensions of simple objects is
nilpotent if and only if there exists a pointed modular category M such that C⊠M
Date: March 31, 2007.
http://arxiv.org/abs/0704.0195v2
2 VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK
is equivalent, as a braided tensor category, to the center of a fusion category of the
form VecωG for a finite nilpotent group G.
We emphasize here that in general the equivalence in Theorem 1.2 does not
respect the spherical structures (equivalently, twists) of the categories involved and
thus is not an equivalence of modular categories. Fortunately, this is not a very
serious complication since the spherical structures on C are easy to classify: it is
well known that they are in bijection with the objects X ∈ C such that X⊗X = 1,
see [RT].
The categoryM in Theorem 1.2 is not uniquely determined by C. However, there
are canonical ways to choose M. In particular, one can always make a canonical
“minimal” choice for M such that dim(M) =
αp with αp ∈ {0, 1, 2} for odd p
and α2 ∈ {0, 1, 2, 3}, see Remark 6.11.
Theorem 1.3. A modular category C is braided equivalent to the center of a fusion
category of the form VecωG with G being a finite p-group if and only if it has the
following properties:
(i) the Frobenius-Perron dimension of C is p2n for some n ∈ Z+,
(ii) the dimension of every simple object of C is an integer,
(iii) the multiplicative central charge of C is 1.
See Subsection 2.6 for the definition of multiplicative central charge. In order
to avoid confusion we note that our definition of multiplicative central charge is
different from the definition of central charge of a modular functor from [BK, 5.7.10];
in fact, the central charge from [BK] equals to the square of our central charge.
Remark 1.4. If p 6= 2 then it is easy to see that (i) implies (ii) (see, e.g., [GN]).
1.2. Interpretation in terms of group-theoretical fusion categories and
semisimple quasi-Hopf algebras. The notion of group-theoretical fusion cate-
gory was introduced in [ENO, O1]. Group-theoretical categories form a large class
of well-understood fusion categories which can be explicitly constructed from finite
group data (which justifies the name). For example, as far as we know, all currently
known semisimple Hopf algebras have group-theoretical representation categories
(however, there are semisimple quasi-Hopf algebras whose representation categories
are not group-theoretical, see [ENO]).
Theorem 1.5. Let C be a fusion category such that all objects of C have integer
dimension and such that its center Z(C) is nilpotent. Then C is group-theoretical.
Remark 1.6. A consequence of this theorem is the following statement: every
semisimple (quasi-)Hopf algebra of prime power dimension is group-theoretical in
the sense of [ENO, Definition 8.40]. This provides a partial answer to a question
asked in [ENO].
1.3. Idea of the proof. We describe here the main steps in the proof of The-
orem 1.3. First we characterize centers of pointed fusion categories in terms of
Lagrangian subcategories and show that a modular category C is equivalent to the
representation category of a twisted group double if and only if it has a Lagrangian
(i.e., maximal isotropic) subcategory of dimension
dim(C). This result is remi-
niscent to the characterization of doubles of quasi-Lie bialgebras in terms of Manin
pairs [Dr, Section 2].
GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 3
Thus we need to show that a category satisfying the assumptions of Theorem 1.3
contains a Lagrangian subcategory. The proof is inspired by the following result
for nilpotent metric Lie algebras (i.e, Lie algebras with an invariant non-degenerate
scalar product) which can be derived from [KaO]: if g is a nilpotent metric Lie
algebra of even dimension then g contains an abelian ideal k, which is Lagrangian
(i.e., such that k⊥ = k). The relevance of metric Lie algebras to our considerations
is explained by the fact that they appear in [Dr] as classical limits of quasi-Hopf
algebras. In fact, our proof is a “categorification” of the proof of the above result.
Thus we need some categorical versions of linear algebra constructions involved in
this proof. Remarkably, the categorical counterparts exist for all notions required.
For example the notion of orthogonal complement in a metric Lie algebra is replaced
by the notion of centralizer in a modular tensor category introduced by M. Müger
[Mu2].
1.4. Organization of the paper. Section 2 is devoted to preliminaries on fusion
categories, which include nilpotent fusion categories, (pre)modular categories, cen-
tralizers, Gauss sums and central charge, and Deligne’s classification of symmetric
fusion categories.
In Section 3 we define the notions of isotropic and Lagrangian subcategories
of a premodular category C, generalizing the corresponding notions for a metric
group (which is, by definition, a finite abelian group with a quadratic form). We
then recall a construction, due to A. Bruguières [Br] and M. Müger [Mu1], which
associates to a premodular category C the “quotient” by its centralizer, called a
modularization. We prove in Theorem 3.4 an invariance property of the central
charge with respect to the modularization. This result will be crucial in the proof
of Theorem 6.5. We also study properties of subcategories of modular categories
and explain in Proposition 3.9 how one can use maximal isotropic subcategories of
a modular category C to canonically measure a failure of C to be hyperbolic (i.e.,
to contain a Lagrangian subcategory).
In Section 4 we characterize hyperbolic modular categories. More precisely, we
show in Theorem 4.5 that for a modular category C there is a bijection between La-
grangian subcategories of C and braided tensor equivalences C ∼−→ Z(VecωG) (where
G is a finite group, ω ∈ Z3(G,K×), and Z(VecωG) is the center of VecωG). Note that
the category Z(VecωG) is equivalent to Rep(Dω(G)) - the representation category of
the twisted double of G [DPR].
We then prove in Theorem 4.8 that if C is a modular category such that dim(C) =
n2, n ∈ Z+, the central charge of C equals 1, and C contains a symmetric subcate-
gory of dimension n, then either C is equivalent to the representation category of a
twisted double of a finite group or C contains an object with non-integer dimension.
We also give a criterion for a modular category C to be group-theoretical. Namely,
we show in Corollary 4.13 that C is group-theoretical if and only if there is an
isotropic subcategory E ⊂ C such that (E ′)ad ⊆ E .
In Section 5 we study pointed modular p-categories. We give a complete list of
such categories which do not contain non-trivial isotropic subcategories and analyze
the values of their central charges. We then prove in Proposition 5.3 that a nonde-
generate metric p-group (G, q) with central charge 1 such that |G| = p2n, n ∈ Z+,
contains a Lagrangian subgroup.
4 VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK
Section 6 is devoted to nilpotent modular categories. There we give proofs of our
main results stated in 1.1 above. They are contained in Theorem 6.5, Theorem 6.6,
Corollary 6.7, Theorem 6.10, and Theorem 6.12.
1.5. Acknowledgments. The research of V. Drinfeld was supported by NSF grant
DMS-0401164. The research of D. Nikshych was supported by the NSF grant DMS-
0200202 and the NSA grant H98230-07-1-0081. The research of V. Ostrik was sup-
ported by NSF grant DMS-0602263. S. Gelaki is grateful to the departments of
mathematics at the University of New Hampshire and MIT for their warm hospi-
tality during his Sabbatical. The authors are grateful to Pavel Etingof for useful
discussions.
2. Preliminaries
Throughout the paper we work over an algebraically closed field k of character-
istic 0. All categories considered in this paper are finite, abelian, semisimple, and
k-linear.
2.1. Fusion categories. For a fusion category C let O(C) denote the set of iso-
morphism classes of simple objects.
Let C be a fusion category. Its Grothendieck ring K0(C) is the free Z-module
generated by the isomorphism classes of simple objects of C with the multiplication
coming from the tensor product in C. The Frobenius-Perron dimensions of objects
in C (respectively, FPdim(C)) are defined as the Frobenius-Perron dimensions of
their images in the based ring K0(C) (respectively, as FPdim(K0(C))), see [ENO,
8.1]. For a semisimple quasi-Hopf algebra H one has FPdim(X) = dimk(X) for all
X in Rep(H), and so FPdim(Rep(H)) = dimk(H).
A fusion category is pointed if all its simple objects are invertible.
By a fusion subcategory of a fusion category C we understand a full tensor subcat-
egory of C. An example of a fusion subcategory is the maximal pointed subcategory
Cpt generated by the invertible objects of C.
A fusion category C is pseudo-unitary if its categorical dimension dim(C) coin-
cides with its Frobenius-Perron dimension, see [ENO] for details. In this case C
admits a canonical spherical structure (a tensor isomorphism between the iden-
tity functor of C and the second duality functor) with respect to which categori-
cal dimensions of objects coincide with their Frobenius-Perron dimensions [ENO,
Proposition 8.23]. The fact important for us in this paper is that a fusion category
of an integer Frobenius-Perron dimension is automatically pseudo-unitary [ENO,
Proposition 8.24].
Let C and D be fusion categories. Recall that for a tensor functor F : C → D its
image F (C) is the fusion subcategory of D generated by all simple objects Y in D
such that Y ⊆ F (X) for some simple X in C. The functor F is called surjective if
F (C) = D.
2.2. Nilpotent fusion categories. For a fusion category C let Cad be the trivial
component in the universal grading of C (see [GN]). Equivalently, Cad is the smallest
fusion subcategory of C which contains all the objects X ⊗X∗, X ∈ O(C).
For a fusion category C we define C(0) = C, C(1) = Cad, and C(n) = (C(n−1))ad for
every integer n ≥ 1. The non-increasing sequence of fusion subcategories of C
(1) C = C(0) ⊇ C(1) ⊇ · · · ⊇ C(n) ⊇ · · ·
GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 5
is called the upper central series of C. We say that a fusion category C is nilpotent if
every non-trivial subcategory of C has a non-trivial group grading, see [GN]. Equiv-
alently, C is nilpotent if its upper central series converges to Vec (the category of
finite dimensional k−vector spaces), i.e., C(n) = Vec for some n. The smallest such
n is called the nilpotency class of C. If C is nilpotent then every fusion subcategory
E ⊂ C is nilpotent, and if F : C → D is a surjective tensor functor, then D is
nilpotent (see [GN]).
Example 2.1. (1) Let G be a finite group and C = Rep(G). Then C is nilpo-
tent if and only if G is nilpotent.
(2) Pointed categories are precisely the nilpotent fusion categories of nilpotency
class 1. A typical example of a pointed category is VecωG, the category
of finite dimensional vector spaces graded by a finite group G with the
associativity constraint determined by ω ∈ Z3(G, k×).
In this paper we are especially interested in the following class of nilpotent fusion
categories.
Example 2.2. Let p be a prime number. Any category of dimension pn, n ∈ Z,
is nilpotent by [ENO, Theorem 8.28]. For representation categories of semisimple
Hopf algebras of dimension pn this follows from a result of A. Masuoka [Ma1].
By [GN], a nilpotent fusion category comes from a sequence of gradings, in
particular it has an integer Frobenius-Perron dimension. It follows from results of
[ENO] that a nilpotent fusion category C is pseudounitary.
2.3. Premodular categories and modular categories. Recall that a braided
tensor category C is a tensor category equipped with a natural isomorphism c :
⊗ ∼= ⊗rev satisfying the hexagon diagrams [JS]. Let cXY : X ⊗ Y ∼= Y ⊗X with
X,Y ∈ C denote the components of c.
A balancing transformation, or a twist, on a braided category C is a natural
automorphism θ : idC → idC satisfying θ1 = id1 and
(2) θX⊗Y = (θX ⊗ θY )cY XcXY .
A braided fusion category C is called premodular, or ribbon, if it has a twist θ
satisfying θ∗X = θX∗ for all objects X ∈ C.
The S-matrix of a premodular category C is S = {sXY }X,Y ∈O(C), where sXY is
the quantum trace of cYXcXY , see [T]. Equivalently, the S-matrix can be defined
as follows. For all X,Y, Z ∈ O(C) let NZXY be the multiplicity of Z in X ⊗ Y . For
every object X let d(X) denote its quantum dimension. Then
(3) sXY = θ
Z∈O(C)
NZXY θZd(Z).
The categorical dimension of C is defined by
(4) dim(C) =
X∈O(C)
d(X)2.
One has dim(C) 6= 0 [ENO, Theorem 2.3].
Note 2.3. Below we consider only fusion categories with integer Frobenius-Perron
dimensions of objects. Any such category C is pseudo-unitary (see 2.1). In par-
ticular, if C is braided then it has a canonical twist, which we will always assume
chosen.
6 VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK
A premodular category C is called modular if the S−matrix is invertible.
Example 2.4. For any fusion category C its center Z(C) is defined as the category
whose objects are pairs (X, cX,−), where X is an object of C and cX,− is a natural
family of isomorphisms cX,V : X ⊗ V ∼= V ⊗ X for all objects V in C satisfying
certain compatibility conditions (see e.g., [Kass]). It is known that the center of a
pseudounitary category is modular.
2.4. Pointed modular categories and metric groups. Let G be a finite abelian
group. Pointed premodular categories C with the group of simple objects isomorphic
to G (up to a braided equivalence) are in the natural bijection with quadratic forms
on G with values in the multiplicative group k∗ of the base field. Here a quadratic
form q : G → k∗ is a map such that q(g−1) = q(g) and b(g, h) := q(gh)
q(g)q(h)
a symmetric bilinear form, i.e., b(g1g2, h) = b(g1, h)b(g2, h) for all g1, g2, h ∈ G.
Namely, for g ∈ G the value of q(g) is the braiding automorphism of g ⊗ g (here
by abuse of notation g denotes the object of C corresponding to g ∈ G). See [Q,
Proposition 2.5.1] for a proof that if two categories C1 and C2 produce the same
quadratic form then they are braided equivalent (Quinn proves less canonical but
equivalent statement). We will denote the category corresponding to a group G
with quadratic form q by C(G, q) and call the pair (G, q) a metric group. The
category C(G, q) is pseudounitary and hence has a spherical structure such that
dimensions of all simple objects equal to 1; hence the categories C(G, q) always
have a canonical ribbon structure. The category C(G, q) is modular if and only if
the bilinear form b(g, h) associated with q is non-degenerate (in this case we will
say that the corresponding metric group is non-degenerate).
2.5. Centralizers. Let K be a fusion subcategory of a braided fusion category C.
In [Mu1, Mu2] M. Müger introduced the centralizer K′ of K, which is the fusion
subcategory of C consisting of all the objects Y satisfying
(5) cY XcXY = idX⊗Y for all objects X ∈ K.
If (5) holds we will say that objects X and Y centralize each other. In the case
of a ribbon category C, condition (5) is equivalent to sXY = d(X)d(Y ), see [Mu2,
Proposition 2.5]. Note that in the case of a pointed modular category the centralizer
corresponds to the orthogonal complement. The subcategory C′ of C is called the
transparent subcategory of C in [Br, Mu1].
For any fusion subcategory K ⊆ C of a braided fusion category C let Kco be
the commutator of K [GN], i.e., the fusion subcategory of C spanned by all simple
objects X ∈ C such that X ⊗ X∗ ∈ K. For example, if C = Rep(G), G a finite
group, then any fusion subcategory K of C is of the form K = Rep(G/N) for some
normal subgroup N of G, and Kco = Rep(G/[G,N ]) (see [GN]). It follows from the
definitions that (Kco)ad ⊆ K ⊆ (Kad)co.
Let K be a fusion subcategory of a pseudounitary modular category C. It was
shown in [GN] that
(6) (Kad)′ = (K′)co.
It was shown in [Mu2, Theorem 3.2] that for a fusion subcategory K of a modular
category C one has K′′ = K and
(7) dim(K) dim(K′) = dim(C).
GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 7
The subcategory K is symmetric if and only if K ⊆ K′. It is modular if and only if
K ∩ K′ = Vec, in which case K′ is also modular and there is a braided equivalence
C ∼= K ⊠K′.
Let C be a modular category. Then by [GN, Corollary 6.9], Cpt = (Cad)′.
2.6. Gauss sums and central charge in modular categories. Let C be a
modular category. For any subcategory K of C the Gauss sums of K are defined by
(8) τ±(K) =
X∈O(K)
θ±1X d(X)
Below we summarize some basic properties of twists and Gauss sums (see e.g.,
[BK, Section 3.1] for proofs).
Each θX , X ∈ O(C), is a root of unity (this statement is known as Vafa’s theo-
rem). The Gauss sums are multiplicative with respect to tensor product of modular
categories, i.e., if C1, C2 are modular categories then
(9) τ±(C1 ⊠ C2) = τ±(C1)τ±(C2).
We also have that
(10) τ+(C)τ−(C) = dim(C).
When k = C the multiplicative central charge ξ(C) is defined by
(11) ξ(C) = τ
dim(C)
where
dim(C) is the positive root. If dim(C) is a square of an integer, then
Formula (11) makes sense even if k 6= C. By Vafa’s theorem, ξ(C) is a root of unity.
Example 2.5. The center Z(C) of any fusion category C (see Example 2.4) is a
modular category with central charge 1 [Mu4, Theorem 1.2].
2.7. Symmetric fusion categories. The structure of symmetric fusion categories
is known, thanks to Deligne’s work [De]. Namely, let G be a finite group and let
z ∈ G be a central element such that z2 = 1. Consider the category Rep(G) with
its standard symmetric braiding σX,Y . Then the map σ
X,Y =
(1 + z|X + z|Y −
z|Xz|Y )σX,Y is also a symmetric braiding on the category Rep(G) (the meaning of
the factor 1
(1 + z|X + z|Y − z|Xz|Y ) is the following: if z|X or z|Y equals 1, then
this factor is 1; if z|X = z|Y = −1 then this factor is (−1)). We will denote by
Rep(G, z) the category Rep(G) with the commutativity constraint defined above.
Theorem 2.6. ([De]) Any symmetric fusion category is equivalent (as a braided
tensor category) to Rep(G, z) for uniquely defined G and z. The categorical dimen-
sion of X ∈ Rep(G, z) equals Tr(z|X) and dim(C) = FPdim(C) = |G|.
Now assume that the category Rep(G, z) is endowed with a twist θ such that the
dimension of any object is non-negative. It follows immediately from the theorem
that θX = z|X . We have
Corollary 2.7. Let C be a symmetric fusion category with the canonical spherical
structure (see 2.3).
(i) If dim(C) is odd then θX = idX for any X ∈ C.
8 VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK
(ii) In general either θX = idX for any X ∈ C, or C contains a fusion subcat-
egory C1 ⊂ C such that FPdim(C1) = 12FPdim(C) and θX = idX for any
X ∈ C1.
Proof. As for (i), it is clear that z = 1. For (ii) one takes C1 = Rep(G/〈z〉) ⊂
Rep(G). �
3. Isotropic subcategories and Bruguières-Müger modularization
3.1. Modularization.
Definition 3.1. Let C be a premodular category with braiding c and twist θ. A
fusion subcategory E of C is called isotropic if θ restricts to the identity on E , i.e.,
if θX = idX for all X ∈ E . An isotropic subcategory E is called Lagrangian if
E = E ′. The category C is called hyperbolic if it has a Lagrangian subcategory and
anisotropic if it has no non-trivial isotropic subcategories.
Remark 3.2. (a) When C = C(G, q) is a pointed modular category defined in
Example 2.4 then isotropic and Lagrangian subcategories of C correspond
to isotropic and Lagrangian subgroups of (G, q), respectively. We discuss
properties of pointed modular categories in Section 5.
(b) Let G be a finite group and let ω ∈ Z3(G, k×). Consider the pointed fusion
category VecωG. Its center C = Z(VecωG) is a modular category. It contains
a Lagrangian subcategory E ∼= Rep(G) formed by all objects in C which
are sent to multiples of the unit object of VecωG by the forgetful functor
Z(VecωG) → Vec
(c) It follows from the balancing axiom (2) that an isotropic subcategory E ⊆ C
is always symmetric. Conversely, if E is symmetric and dim(E) is odd then
E is isotropic, see 2.7. In particular, if dim(C) is odd then any symmetric
subcategory of C is isotropic.
(d) Recall that we assume that C is endowed with a canonical spherical struc-
ture, see 2.3. Any isotropic subcategory E ⊂ C is equivalent, as a symmetric
category, to Rep(G) for a canonically defined group G with its standard
braiding and identical twist, see 2.7. In particular, if E is Lagrangian then
dim(C) = dim(E)2 is a square of an integer.
Let C be a premodular category such that its centralizer C′ is isotropic and
dimensions of all objects X ∈ C′ are non-negative. Let us recall a construction,
due to A. Bruguières [Br] and M. Müger [Mu1], which associates to C a modular
category C̄ and a surjective braided tensor functor C → C̄.
Let G(C) be the unique (up to an isomorphism) group such that the category C′
is equivalent, as a premodular category, to Rep(G(C)) with its standard symmetric
braiding and identity twist.
Let A be the algebra of functions on G(C). The group G(C) acts on A via left
translations and so A is a commutative algebra in C′ and hence in C.
Consider the category C̄ := CA of right A-modules in the category C (see, e.g.,
[KiO, 1.2]). It was shown in [Br, KiO, Mu1] that C̄ is a braided fusion category and
that the “free module” functor
(12) F : C → C̄, X 7→ X ⊗A
is surjective and has a canonical structure of a braided tensor functor. One can
define a twist φ on C̄ in such a way that φY = θX for all Y ∈ O(C̄) and X ∈
GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 9
O(C) for which HomC(X,Y ) 6= 0. It follows that the category C̄ is modular, see
[Br, Mu1, KiO] for details. We will call the category C̄ a modularization of C.
Let d and d̄ denote the dimension functions in C and C̄, respectively. For any
object X in C̄ one has
(13) d̄(X) =
cf. [KiO, Theorem 3.5], [Br, Proposition 3.7].
Remark 3.3. Let E be an isotropic subcategory of a modular category C. Then
dim(Ē ′) = dim(C)/ dim(E)2 (see e.g. [KiO]).
3.2. Invariance of the central charge. In this subsection we prove an invariance
property of the central charge with respect to modularization, which will be crucial
in the sequel.
Theorem 3.4. Let C be a modular category and let E be an isotropic subcategory
of C. Let F : E ′ → Ē ′ be the canonical braided tensor functor from E ′ to its
modularization. Then ξ(Ē ′) = ξ(C).
Proof. Let A be the canonical commutative algebra in E . We have dim(E) = d(A).
By definition, Ē ′ is the category of left A-modules in E ′.
Let us compute the Gauss sums of Ē ′:
dim(E)τ±(Ē ′) = dim(E)
Y ∈O(Ē′)
φ±1Y d̄(Y )
Y ∈O(Ē′)
φ±1Y d(Y )d̄(Y )
Y ∈O(Ē′)
X∈O(C)
dimk HomC(X, Y )d(X)
 d̄(Y )
X∈O(C)
θ±1X d(X)
Y ∈O(Ē′)
dimk HomC(X, Y )d̄(Y )
X∈O(C)
θ±1X d(X)
Y ∈O(Ē′)
dimk HomĒ′(X ⊗A, Y )d̄(Y )
X∈O(C)
θ±1X d(X)d̄(F (X)) = τ
±(C),
where we used the relation (13) and the fact that F is an adjoint of the forgetful
functor from Ē ′ to E ′.
Combining this with the equation dim(Ē ′) = dim(C)/ dim(E)2 (see Remark 3.3)
we obtain the result. �
3.3. Maximal isotropic subcategories. Let C be a modular category and let L
be an isotropic subcategory of C which is maximal among isotropic subcategories
of C. Below we will show that the braided equivalence class of the modular category
L̄′ (the modularization of L′ by L) is independent of the choice of L.
Let C be a fusion category and let A and B be its fusion subcategories such that
X ⊗ Y ∼= Y ⊗ X for all X ∈ O(A) and Y ∈ O(B). Let A ∨ B denote the fusion
10 VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK
subcategory of C generated by A and B, i.e., consisting of all subobjects of X ⊗ Y ,
where X ∈ O(A) and Y ∈ O(B). Recall that the regular element of K0(C)⊗Z C is
X∈O(C) d(X)X . It is defined up to a scalar multiple by the property that
Y ⊗RC = d(Y )RC for all Y ∈ O(C) [ENO].
Lemma 3.5. Let C, A, B be as above. Then dim(A ∨ B) = dim(A) dim(B)
dim(A∩B) .
Proof. It is easy to see that
(14) RA ⊗RB = aRA∨B,
where the scalar a is equal to the multiplicity of the unit object 1 in RA ⊗ RB,
which is the same as the multiplicity of 1 in
Z∈O(A∩B) d(Z)
2Z ⊗ Z∗. Hence,
a = dim(A ∩ B). Taking dimensions of both sides of (14) we get the result. �
Let L(C) denote the lattice of fusion subcategories of a fusion category C. For
any two subcategories A and B their meet is their intersection and their joint is the
category A ∨ B.
Lemma 3.6. Let C be a fusion category such that X ⊗ Y ∼= Y ⊗X for all objects
X,Y in C. For all A, B, D ∈ L(C) such that D ⊆ A the following modular law
holds true:
(15) A ∩ (B ∨D) = (A ∩ B) ∨ D.
Proof. A classical theorem of Dedekind in lattice theory states that (15) is equiv-
alent to the following statement: for all A, B, D ∈ L(C) such that D ⊆ A, if
A∩ B = D ∩ B and A∨ B = D ∨ B then A = D (see e.g., [MMT]).
Let us prove the latter property. Take a simple object X ∈ A. Then X ∈
A ∨ B = D ∨ B so there are simple objects D ∈ D and B ∈ B such that X is
contained in D ⊗ B. Therefore, B is contained in D∗ ⊗ X and so B ∈ A. So
B ∈ A ∩ B = D ∩ B ⊆ D. Hence X ∈ D, as required. �
Remark 3.7. When C = Rep(G) is the representation category of a finite group
G, Lemma 3.6 gives a well-known property of the lattice of normal subgroups of G.
The next lemma gives an analogue of a diamond isomorphism for the “quotients
by isotropic subcategories.”
Lemma 3.8. Let C be a modular category, let D be an isotropic subcategory of C
and let B be a subcategory of D′. Let A, A0 be the canonical commutative algebras
in D and D ∩ B, respectively.
Then the category BA0 of A0-modules in B and the category (D ∨ B)A of A-
modules in D ∨ B are equivalent as braided tensor categories.
Proof. Note that
dim(BA0) =
dim(B)
dim(D ∩ B)
dim(D ∨ B)
dim(D)
= dim((D ∨ B)A)
by Lemma 3.5.
Define a functorH : BA0 → (D∨B)A byH(X) = X⊗A0A, X ∈ BA0 . ThenH has
a natural structure of a braided tensor functor. Note that for X = Y ⊗A0, Y ∈ B
we have H(X) = Y ⊗A, i.e., the composition of H with the free A0-module functor
is the free A-module functor. The latter functor is surjective and, hence, so is H .
Since a surjective functor between categories of equal dimension is necessarily
an equivalence (see [ENO, 5.7] or [EO, Proposition 2.20]) the result follows. �
GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 11
Proposition 3.9. Let C be a modular category and let L1, L2 be maximal among
isotropic subcategories of C. Then the modularization L̄′1 and L̄′2 are equivalent as
braided fusion categories.
Proof. Let D = L1 and B = L′1∩L′2. By maximality of L1,L2 we have L′1∩L2 ⊆ L1
and L1 ∩ L′2 ⊆ L2. Therefore, D ∩ B = L1 ∩ L2 and D ∨ B = L′1 ∩ (L1 ∨ L′2) = L′1
by Lemma 3.6.
Let A0 be the canonical commutative algebra in L1 ∩ L2. Applying Lemma 3.8
we see that L̄′1 is equivalent to the category (L′1 ∩L′2)A0 of A0-modules in L′1 ∩L′2.
The proposition now follows by interchanging L1 and L2. �
Remark 3.10. (i) We can call the modular category L̄′1 constructed in the
proof of Proposition 3.9 “the” canonical modularization corresponding to
C (it measures the failure of C to be hyperbolic). The above proof gives
a concrete equivalence L̄′1 ∼= L̄′2. But given another maximal isotropic
subcategory L3 ⊂ C the composition of equivalences L̄′1 ∼= L̄′2 and L̄′2 ∼= L̄′3
is not in general equal to the equivalence L̄′1 ∼= L̄′3. This is why we put
“the” above in quotation marks.
(ii) For a maximal isotropic subcategory L ⊂ C the corresponding modular-
ization does not have to be anisotropic, in contrast with the situation for
metric groups. Examples illustrating this phenomenon are, e.g., the cen-
ters of non-group theoretical Tambara-Yamagami categories considered in
[ENO, Remark 8.48].
4. Reconstruction of a twisted group double from a Lagrangian
subcategory
4.1. C-algebras. Let us recall the following definition from [KiO].
Definition 4.1. Let C be a ribbon fusion category. A C−algebra is a commutative
algebra A in C such that dimHom(1, A) = 1, the pairing A⊗A → A → 1 given by
the multiplication of A is non-degenerate, θA = idA and dim(A) 6= 0.
Let C be a modular category, let A be a C−algebra, and let CA be the fusion
category of right A−modules with the tensor product ⊗A. The free module functor
F : C → CA, X 7→ X ⊗A has an obvious structure of a central functor. By this we
mean that there is a natural family of isomorphisms F (X)⊗AY ∼= Y ⊗AF (X), X ∈
C, Y ∈ CA, satisfying an obvious multiplication compatibility, see e.g. [Be, 2.1].
Indeed, we have F (X) = X ⊗ A, and hence F (X) ⊗A Y = X ⊗ Y . Similarly,
Y ⊗A F (X) = Y ⊗ X . These two objects are isomorphic via the braiding of C
(one can check that the braiding gives an isomorphism of A-modules using the
commutativity of A).
Thus, the functor F extends to a functor F̃ : C → Z(CA) in such a way that F
is the composition of F̃ and the forgetful functor Z(CA) → CA.
Proposition 4.2. The functor F̃ : C → Z(CA) is injective (that is fully faithful).
Proof. Consider CA as a module category over C via F and over Z(CA) via F̃ .
We will prove the dual statement (see [ENO, Proposition 5.3]), namely that the
functor T : CA ⊠ CopA → C∗CA dual to F̃ is surjective (here and below the superscript
op refers to the tensor category with the opposite tensor product). Recall (see e.g.
[O1]) that the category C∗CA is identified with the category of A−bimodules. An
12 VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK
explicit description of the functor T is the following: by definition, any M ∈ CA
is a right A−module. Using the braiding and its inverse one can define on M
two structures of a left A−module: A ⊗ M
A,M−→ M ⊗ A → M . Both structures
make M into an A−bimodule, and we will denote the two results by M+ and
M−, respectively. Then we have T (M ⊠ N) = M+ ⊗A N−. In particular we
see that the functor C ⊠ Cop F⊠F−→ CA ⊠ CopA
T−→ C∗CA coincides with the functor
C ⊠ Cop ≃ Z(C) ≃ Z(C∗CA) → C
CA (see [O2]). Since the functor Z(C
CA) → C
CA is
surjective (see [EO, 3.39]) we see that the functor T is surjective. The proposition
is proved. �
Remark 4.3. Note that since C and Z(CA) are modular we have a factorization
Z(CA) = C ⊠D, where D is the centralizer of C in Z(CA). One observes that D is
identified with the category of “dyslectic” A−modules Rep0(A), see [KiO, P].
Corollary 4.4. Assume that dim(A) =
dim(C). Then the functors F̃ : C →
Z(CA) and T : CA ⊠ CopA → C∗CA are tensor equivalences.
Proof. We have already seen that dim(CA) = dim(C)dim(A) . Hence, dim(Z(CA)) =
dim(C)2
dim(A)2
= dim(C). Since F̃ is an injective functor between categories of equal
dimension, it is necessarily an equivalence by [EO, Proposition 2.19]. Hence the
dual functor T is also an equivalence. �
4.2. Hyperbolic modular categories as twisted group doubles. We are now
ready to state and prove our first main result which relates hyperbolic modular
categories and twisted doubles of finite groups.
Let C be a modular category. Consider the set of all triples (G,ω, F ), where
G is a finite group, ω ∈ Z3(G, k×), and F : C ∼−→ Z(VecωG) is a braided tensor
equivalence. Let us say that two triples (G1, ω1, F1) and (G2, ω2, F2) are equivalent
if there exists a tensor equivalence ι : Vecω1G1
∼−→ Vecω2G2 such that F2◦F2 = ι◦F1◦F1,
where Fi : Z(VecωiGi) → Vec
, i = 1, 2, are the canonical forgetful functors.
Let E(C) be the set of all equivalences classes of triples (G,ω, F ). Let Lagr(C)
be the set of all Lagrangian subcategories of C.
Theorem 4.5. For any modular category C there is a natural bijection
f : E(C) ∼−→ Lagr(C).
Proof. The map f is defined as follows. Note that each braided tensor equivalence
F : C ∼−→ Z(VecωG) gives rise to the Lagrangian subcategory f(G,ω, F ) of C formed
by all objects sent to multiples of the unit object 1 under the forgetful functor
Z(VecωG) → VecωG. This subcategory is clearly the same for all equivalent choices
of (G,ω, F ).
Conversely, given a Lagrangian subcategory E ⊆ C it follows from Deligne’s
theorem [De] that E = Rep(G) for a unique (up to isomorphism) finite group G.
Let A = Fun(G) ∈ Rep(G) = E ⊂ C. It is clear that A is a C−algebra and
dim(A) = dim(E) =
dim(C). Then by Corollary 4.4, the functor F̃ : C → Z(CA)
is an equivalence.
Finally, let us show that CA is pointed and K0(CA) = ZG. Note that there are
|G| non-isomorphic structures Ag, g ∈ G, of an invertible A-bimodule on A, since
the category of A-bimodules in E is equivalent to VecG. For each Ag there is a
GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 13
pair X,Y of simple objects in CA such that T (X ⊠ Y ) = Ag. Taking the forgetful
functor to CA we obtain Y = X∗ and X is invertible. Hence, for each g ∈ G there
is a unique invertible Xg ∈ CA such that T (Xg ⊠X∗g ) = Ag, and therefore g 7→ Xg
is an isomorphism of K0 rings. Thus, CA ∼= VecωG for some ω ∈ Z3(G, k×). We set
h(E) to be the class of the equivalence F̃ : C ∼−→ Z(CA).
Let show that the above constructions f and h are inverses of each other. Let E
be a Lagrangian subcategory of C and let A be the algebra defined in the previous
paragraph. The forgetful functor from C ∼= Z(CA) to CA is the free module functor,
and so f(h(E)) consists of all objects X in C such that X ⊗ A is a multiple of A.
Since A is the regular object of E , it follows that f(h(E)) = E and f ◦ h = id.
Proving that h ◦ f = id amounts to a verification of the following fact. Let G
be a finite group, let ω ∈ Z3(G, k×), and let A = Fun(G) be the canonical algebra
in Rep(G) ⊂ Z(VecωG). Then the category of A-modules in Z(VecωG) is equivalent
to VecωG and the functor of taking the free A-module coincides with the forgetful
functor from Z(VecωG) to Vec
G. This is straightforward and is left to the reader. �
Remark 4.6. Our reconstruction of the representation category of a twisted group
double from a Lagrangian subcategory can be viewed as a categorical analogue of
the following reconstruction of the double of a quasi-Lie bialgebra from a Manin
pair (i.e., a pair consisting of a metric Lie algebra and its Lagrangian subalgebra)
in the theory of quantum groups [Dr, Section 2].
Let g be a finite-dimensional metric Lie algebra (i.e., a Lie algebra on which a
nondegenerate invariant symmetric bilinear form is given). Let l be a Lagrangian
subalgebra of g. Then l has a structure of a quasi-Lie bialgebra and there is an
isomorphism between g and the double D(l) of l. The correspondence between
Lagrangian subalgebras of g and doubles isomorphic to g is bijective, see [Dr, Section
2] for details.
Remark 4.7. Given a hyperbolic modular category C there is no canonical way
to assign to it a pair (G, ω) such that C ∼= Z(VecωG) as a braided fusion category.
Indeed, it follows from [EG1] that there exist non-isomorphic finite groups G1, G2
such that Z(VecG1) ∼= Z(VecG2) as braided fusion categories. (See also [N].)
Theorem 4.8. Let C be a modular category such that dim(C) = n2, n ∈ Z+, and
such that ξ(C) = 1. Assume that C contains a symmetric subcategory V such that
dim(V) = n. Then either C is the center of a pointed category or it contains an
object with non-integer dimension.
Proof. Assume that V is not isotropic. Then V contains an isotropic subcategory K
such that dim(K) = 1
dim(V) (this follows from Deligne’s description of symmetric
categories, see 2.7). Hence the category K̄′ (modularization of K′) has dimension
4 and central charge 1. It follows from the explicit classification given in Example
5.1 (b),(d) that the category K̄′ contains an isotropic subcategory of dimension 2;
clearly this subcategory is equivalent to Rep(Z/2Z). Let A1 = Fun(Z/2Z) be the
commutative algebra of dimension 2 in this subcategory. Let I : K̄′ → K′ be the
right adjoint functor to the modularization functor F : K′ → K̄′.
We claim that the object A := I(A1) has a canonical structure of a C−algebra.
Indeed, we have a canonical morphism in Hom(F (A), A1) = Hom(A, I(A1)) =
Hom(A,A) ∋ id. Using this one can construct a multiplication on A via Hom(A1⊗
A1, A1) → Hom(F (A)⊗F (A), A1) = Hom(F (A⊗A), A1) = Hom(A⊗A,A). Since
14 VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK
the functor F is braided it follows from the commutativity of A1 that A is commu-
tative. Other conditions from Definition 4.1 are also easy to check. In particular
dim(A) = dim(K) dim(A1) = dim(V) =
dim(C). We also note that the category
RepK′(A) contains precisely two simple objects (actually, the functor M 7→ I(M) is
an equivalence of categories between RepK̄′(A1) and RepK′(A)); we will call these
two objects 1 (for A itself considered as an A−module) and δ. Clearly δ⊗A δ = 1.
By Corollary 4.4, we have an equivalence C∗CA = CA⊠C
A . Moreover, the forgetful
functor C∗CA → CA corresponds to the tensor product functor CA ⊠ C
A → CA. Now
consider the subcategory (K′)∗CA ⊂ C
CA (in other words A−bimodules in K
′); the
forgetful functor above restricts to S : (K′)∗CA → RepK′(A).
Let M ∈ (K′)∗CA be a simple object. We claim that there are three possibilities:
1) S(M) = 1, 2) S(M) = δ or 3) S(M) = 1⊕ δ. Indeed, M = X ⊠ Y ∈ CA ⊠ CopA
and S(M) = X ⊗ Y for some simple X,Y ∈ CA. Since 1 and δ are invertible the
result is clear.
Now, notice that if there exists M as in case 3) then we have X = Y ∗ and
dim(X) = dim(Y ) =
2. Thus the category CA contains an object with non-integer
dimension, which implies that the category C contains an object with non-integer
dimension (see e.g. [ENO, Corollary 8.36]), and the theorem is proved in this case.
Hence we will assume that for any M ∈ (K′)∗CA only 1) or 2) holds. This implies
that all objects of (K′)∗CA are invertible. Note that dim((K
′)∗CA) = dim(K
dim(C) and hence we have precisely 2
dim(C) simple objects. Consider all
objects M ∈ (K′)∗CA such that S(M) = 1; it is easy to see that there are precisely
dim(C) of those (indeed, X⊠Y 7→ X⊠ (Y ⊗A δ) gives a bijection between simple
bimodules M with S(M) = 1 and simple bimodules M with S(M) = δ). Let G
be the group of isomorphism classes of all objects M ∈ (K′)∗CA with S(M) = 1
(thus |G| =
dim(C)). Any object of this type is of the form Xg ⊠ (Xg)∗ for
some invertible Xg ∈ CA. Thus we already constructed
dim(C) invertible simple
objects in CA. Since dim(CA) =
dim(C) the objects Xg exhaust all simple objects
in CA. By Corollary 4.4, we are done. �
4.3. A criterion for a modular category to be group-theoretical. Let C be a
modular category. It is known that the entries of the S-matrix of C are cyclotomic
integers [CG, dBG]. Hence, we may identify them with complex numbers. In
particular, the notions of complex conjugation and absolute value of the elements
of the S-matrix make sense.
Remark 4.9. Let K ⊆ C be a fusion subcategory. Recall from [GN] that (Kad)′
is spanned by simple objects Y such that |sXY | = dXdY for all simple X in K. In
this case the ratio b(X,Y ) := sXY /(dXdY ) is a root of unity. Furthermore, for all
simple X ∈ K, Y1, Y2 ∈ K′ad and any simple subobject Z of Y1 ⊗ Y2 we have
(16) b(X,Y1)b(X,Y2) = b(X,Z),
as explained in [Mu2].
Lemma 4.10. Let C be a modular category and let K ⊆ C be a fusion subcategory
such that K ⊆ (Kad)′.
(1) There is a grading K = ⊕g∈G Kg such that K1 = K′ ∩ K.
(2) There is a non-degenerate symmetric bilinear form b on G such that b(g, h) =
sXY /(dXdY ) for all X ∈ Kg and Y ∈ Kh.
GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 15
(3) If K′ ∩ K is isotropic then there is a non-degenerate quadratic form q on
G such that q(g) = θX for all X ∈ Kg. In this case b is the bilinear form
corresponding to q.
Proof. Since Kad ⊆ K′ ∩ K ⊆ K the assertion (1) follows from [GN].
Let b(X,Y ) = sXY /(dXdY ) for all simple X,Y ∈ K. Clearly, b is symmetric
and b(X,Y ) = 1 for all simple X in K if and only if Y ∈ K′ ∩ K = K1. To prove
(2) it suffices to check that b depends only on h ∈ G such that Y ∈ Kh (then the
G-linear property follows from (16)). Let Y1, Y2 be simple objects in Kh. Then
Y1 ⊗ Y ∗2 ∈ K′ ∩ K and so b(X,Y1)b(X,Y ∗2 ) = 1, whence b(X,Y1) = b(X,Y2), as
desired.
Finally, (3) is a direct consequence of our discussion in Section 3.1. �
For a subcategory K ⊆ C satisfying the hypothesis of Lemma 4.10 let (GK, bK)
be the corresponding abelian grading group and bilinear form. Note that if such
K is considered as a subcategory of Crev then the corresponding bilinear form is
(GK, b
Theorem 4.11. Let C be a modular category. Then symmetric subcategories of
Z(C) ∼= C ⊠ Crev of dimension dim(C) are in bijection with triples (L, R, ι), where
L ⊆ C, R ⊆ Crev are symmetric subcategories such that (L′)ad ⊆ L, (R′)ad ⊆ R,
and ι : (GL′ , bL′) ∼= (GR′ , bR′) is an isomorphism of bilinear forms.
Namely, any such subcategory is of the form
(17) DL,R,ι = ⊕g∈G
Lg ⊠Rι(g).
Proof. Let X1 ⊠ Y1 and X2 ⊠ Y2 be two simple objects of C⊠ Crev. They centralize
each other if and only if
|sX1X2 | = dX1dX2 ,(18)
|sY1Y2 | = dY1dY2 , and(19)
sX1X2
dX1dX2
sY1Y2
dY1dY2
= 1.(20)
Let D be a symmetric subcategory of C ⊠ Crev and let L (respectively, R) be
the centralizers of fusion subcategories of C (respectively, Crev) formed by left (re-
spectively, right) tensor factors of simple objects in D. By conditions (18), (19),
and Remark 4.9 we must have L′ad ⊆ L and R′ad ⊆ R. Hence, Lemma 4.10
gives gradings L′ = ⊕g∈GL (L′)g with (L′)1 = L′ ∩ L and R′ = ⊕g∈GR (R′)g
with (R′)1 = R′ ∩ R. The condition (20) gives an isomorphism of bilinear forms
ι : (GL′ , bL′) ∼= (GR′ , bR′) which is well-defined be the property that whenever
X ∈ (L′)g and Y ∈ R′ are simple objects such that X ⊠ Y ∈ D then Y ∈ (R′)ι(g).
Note that
(21) D ⊆ ⊕g∈G
Lg ⊠Rι(g),
and hence
dim(D) ≤ dim(L ∩ L′) dim(R∩R′)|GL′ | = dim(L′) dim(R∩R′).
The same inequality holds with L and R interchanged. Therefore,
dim(C)2 = dim(D)2 ≤ dim(L′) dim(L′ ∩ L) dim(R′) dim(R∩R′) ≤ dim(C)2.
16 VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK
Here the first inequality becomes equality if and only if the inclusion in (21) is an
equality and the second inequality becomes equality if and only if L′ ∩ L = L and
R′ ∩R = R, i.e., when L and R are symmetric. �
Remark 4.12. The subcategory DL,R,ι constructed in Theorem 4.11 is Lagrangian
if and only if L and R are isotropic subcategories of C and ι is an isomorphism of
metric groups.
Corollary 4.13. Let C be a modular category. The following conditions are equiv-
alent:
(i) C is group-theoretical.
(ii) There is a finite group G and a 3-cocycle ω ∈ Z3(G, k×) such that Z(C) ∼=
Z(V ecωG) as a braided fusion category.
(iii) C ⊠ Crev contains a Lagrangian subcategory.
(iv) There is an isotropic subcategory E ⊂ C such that (E ′)ad ⊆ E.
Proof. The equivalence (i)⇔(ii) is a consequence of [ENO], (ii)⇔(iii) follows from
Theorem 4.5, and (iii)⇔(iv) follows from taking E = R = L and ι = idG
Theorem 4.11, cf. Remark 4.12. �
Combining the above criterion with Theorem 4.8 we obtain the following useful
characterization of group-theoretical modular categories.
Corollary 4.14. A modular category C is group-theoretical if and only if simple
objects of C have integral dimension and there is a symmetric subcategory L ⊂ C
such that (L′)ad ⊆ L.
5. Pointed modular categories
In this section we analyze the structure of pointed modular categories, their
central charges, and Lagrangian subgroups. Recall that such categories canonically
correspond to metric groups [Q].
Let G = Z/nZ. The corresponding braided categories of the form C(Z/nZ, q) are
completely classified by numbers σ = q(1) such that σn = 1 (n is odd) or σ2n = 1
(n is even). Then the braiding of objects corresponding to 0 ≤ a, b < n is the
multiplication by σab and the twist of the object a is the multiplication by σa
[Q]). We will denote the category corresponding to σ by C(Z/nZ, σ).
Example 5.1. (a) Let G = Z/2Z. There are 4 possible values of σ: ±1,±i.
The categories C(Z/2Z,±i) are modular with central charge 1±i√
and the categories
C(Z/2Z,±1) are symmetric. The category C(Z/2Z, 1) is isotropic and the category
C(Z/2Z,−1) is not.
(b) Let G = Z/4Z. The twist of the object 2 ∈ Z/4Z is σ4 = ±1. If this twist is
-1 then σ is a primitive 8th root of 1 and the corresponding category is modular;
its Gauss sum is 1 + σ + σ4 + σ9 = 2σ and the central charge is σ. Note that if
σ4 = 1 then the category C(Z/4Z, σ) contains a nontrivial isotropic subcategory.
(c) Let G = Z/2kZ with k ≥ 3. Since the twist of the object 2k−1 is σ22k−2 = 1,
the category C(Z/2kZ, σ) always contains a nontrivial isotropic subcategory.
(d) Let G = Z/2Z × Z/2Z. There are five modular categories with this group.
We give for each of them the list of values of q on nontrivial elements of G:
(1) C(Z/2Z× Z/2Z, i): the values of q are i, i,−1, and the central charge is i.
GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 17
(2) C(Z/2Z× Z/2Z,−i): the values of q are −i,−i,−1, and the central charge
is −i.
(3) C(Z/2Z×Z/2Z,−1): the values of q are −1,−1,−1, and the central charge
is −1.
(4) C(Z/2Z× Z/2Z, 1): the values of q are i,−i, 1, and the central charge 1.
(5) The double of Z/2Z: the values of q are 1, 1,−1, and the central charge 1.
In this list, each category of central charge 1 contains a nontrivial isotropic subcat-
egory while the others contain a nontrivial symmetric (but not isotropic) subcate-
gory.
(e) Let G = Z/2Z × Z/4Z. Assume that the category C(G, q) does not contain
a nontrivial isotropic subcategory. Then C(G, q) is equivalent to C(Z/4Z, σ) ⊠
C(Z/2Z,±i) where σ is a primitive 8th root of 1. The possible central charges are
±1 and ±i.
(f) Let G = Z/2Z × Z/2Z × Z/2Z. Assume that the category C(G, q) does not
contain a nontrivial isotropic subcategory. Then C(G, q) is equivalent to C(Z/2Z×
Z/2Z, σ)⊠ C(Z/2Z, σ′), where σ′ = ±i and σ 6= 1,−σ′.
Example 5.2. Let p be an odd prime.
(a) Let G = Z/pZ. The category C(Z/pZ, σ) is modular for σ 6= 1 and is isotropic
for σ = 1. The central charge of the modular category C(Z/pZ, σ) is ±1 for p = 1
mod 4 and ±i for p = 3 mod 4.
(b) Let G = Z/pZ × Z/pZ. There are two modular pointed categories with
underlying group G. One has central charge 1 (and is equivalent to the center of
Z/pZ), and the other one has central charge -1.
Recall that for a metric group (G, q) its Gauss sum is τ±(G, q) =
a∈G q(a)
A subgroup H of G is called isotropic if q|H = 1. An isotropic subgroup is called
Lagrangian if H⊥ = H .
The following proposition is well known.
Proposition 5.3. Let (G, q) be a non-degenerate metric group such that |G| = p2n
where p is a prime number and n ∈ Z+. Suppose that τ±(G, q) =
|G| (i.e., the
central charge of G is 1). Then G contains a Lagrangian subgroup.
Proof. It suffices to prove that G contains a non-trivial isotropic subgroup H , then
one can pass to H⊥/H and use induction.
Assume that p is odd. Assume that G contains a direct summand Z/pkZ with
k > 1. Then the subgroup Z/pZ ⊂ Z/pkZ is isotropic, since otherwise it is a non-
degenerate metric subgroup of G and hence can be factored. Thus we are reduced
to the case when G is a direct sum of k copies of Z/pZ. When k > 2, the quadratic
form on G is isotropic (by the Chevalley - Waring theorem). Thus we are reduced
to the case k = 2, which is easy (see Example 5.2 (b)).
Assume now that p = 2. Again assume that G contains a direct summand Z/2kZ
with k > 1. Again the subgroup Z/2Z ⊂ Z/2kZ is inside its orthogonal complement;
moreover it is isotropic if k ≥ 3. If k = 2 and the subgroup Z/2Z ⊂ Z/4Z is not
isotropic then the subgroup Z/4Z is a non-degenerate metric subgroup and hence
factors out; let G = G1 ⊕ Z/4Z be the corresponding decomposition of G. If G1
contains Z/2Z such that Z/2Z ⊆ Z/2Z⊥ then we are done: if this subgroup is
not isotropic then the diagonal subgroup Z/2Z ⊂ Z/2Z ⊕ Z/2Z ⊂ G1 ⊕ Z/4Z is
isotropic. Thus G1 is a sum of Z/2Z’s and each summand is non-degenerate. But
note that the central charge of a non-degenerate metric group Z/4Z is a primitive
18 VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK
eighth root of 1 (see Example 5.1 (b)) which is also the central charge of a non-
degenerate metric Z/2Z (see Example 5.1 (a)). This implies that the number of
Z/2Z summands in G1 is odd which is impossible since the order of G is a square.
Thus we are reduced to the case when G is a sum of k copies of Z/2Z. In this case
all possible values of the quadratic form q are ±1,±i and since τ+(G, q) = 2k/2,
there is at least one non-identity a ∈ G with q(a) = 1. So the subgroup generated
by a is isotropic. The proposition is proved. �
6. Nilpotent modular categories
In this section we prove our main results, stated in 1.1, and derive a few corol-
laries.
Recall the definitions of Kad and Kco from 2.5.
Proposition 6.1. Let C be a nilpotent modular category. Then for any maximal
symmetric subcategory K of C one has (K′)ad ⊆ K. Equivalently, there is a grading
of K′ such that K is the trivial component:
(22) K′ = ⊕g∈GK′g, K′1 = K.
Proof. The two conditions are equivalent since by [GN] the adjoint subcategory is
the trivial component of the universal grading.
Let K be a symmetric subcategory of C, i.e., such that K ⊆ K′. Assume that
(K′)ad is not contained in K. It suffices to show that K is not maximal.
Let E = (Kco ∩ (K′)ad) ∨ K. Clearly, K ⊆ E ⊆ K′. We have
E ′ = ((Kco ∩ (K′)ad) ∨ K)′
= K′ ∩ ((Kco)′ ∨ ((K′)ad)′)
= K′ ∩ ((K′)ad ∨ Kco)
= (K′ ∩ Kco) ∨ (K′)ad,
where we used the modular law of the lattice L(C) from Lemma 3.6. Since K ⊆
K′ ∩ Kco and Kco ∩ (K′)ad ⊆ (K′)ad we see that E ⊆ E ′, i.e., E is symmetric.
Let n be the largest positive integer such that (K′)(n) 6⊆ K. Such n exists by our
assumption and the nilpotency of K′. We claim that (K′)(n) ⊆ Kco. Indeed,
(K′)(n) ⊆ ((K′)(n+1))co ⊆ Kco
since D ⊆ (Dad)co for every subcategory D ⊆ C. Therefore, Kco ∩ (K′)(n) = (K′)(n)
is not contained in K and
K ( (Kco ∩ (K′)(n)) ∨K ⊆ (Kco ∩ (K′)ad) ∨ K = E ,
which completes the proof. �
Recall that in a fusion category whose dimension is an odd integer the dimensions
of all objects are automatically integers [GN, Corollary 3.11].
Corollary 6.2. A nilpotent modular category C with integral dimensions of simple
objects is group-theoretical.
Proof. This follows immediately from Corollary 4.14 and Proposition 6.1. �
GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 19
Remark 6.3. It follows from Corollary 4.13 that a nilpotent modular category C
with integral dimensions of simple objects contains an isotropic subcategory E such
that (E ′)ad ⊆ E . The corresponding grading
(23) E ′ = ⊕h∈H E ′h, E ′1 = E ,
gives rise to a non-degenerate quadratic form q on H defined by q(h) = θV for any
non-zero V ∈ Ch. We have a braided equivalence Ē ′ ∼= C(H, q).
We may assume that E is maximal among isotropic subcategories of C. In this
case, Proposition 3.9 implies that the isomorphism class of the above metric group
(H, q) does not depend on the choice of the maximal isotropic subcategory E .
Corollary 6.4. The central charge of a modular nilpotent category with integer
dimensions of objects is always an 8th root of 1. Moreover, the central charge of a
modular p−category is ±1 if p = 1mod 4 and ±1, ±i if p = 3mod 4. The central
charge of a modular p−category of dimension p2k, k ∈ Z+ with odd p is ±1.
Proof. By Remark 6.3 and Theorem 3.4 the central charge always equals the central
charge of some pointed category, so the first claim follows from Examples 5.1-5.2.
The second and third claims follow from Example 5.2. �
Theorem 6.5. Let C be a modular category with integral dimensions of simple
objects. Then C is nilpotent if and only if there exists a pointed modular category
M such that C⊠M is equivalent (as a braided fusion category) to Z(VecωG), where
G is a nilpotent group.
Proof. Note that for a nilpotent group G the category Z(VecωG) is a tensor product
of modular p-categories and, hence, is nilpotent. So if C ⊠M ∼= Z(VecωG) then C is
nilpotent (as a subcategory of a nilpotent category).
Let us prove the converse implication. Pick an isotropic subcategory E ⊂ C such
that (E ′)ad ⊆ E (such a subcategory exists by Remark 6.3). There is a metric
group (H, q) such that Ē ′ ∼= C(H, q). Let E ′ = ⊕h∈H E ′h , where E1 = E be the
corresponding grading from (23).
Let M be the reversed category of Ē ′ (i.e., with the opposite braiding and twist).
Then M ∼= C(H, q−1) and ξ(M) = ξ(C(H, q))−1 = ξ(C)−1 by Theorem 3.4.
The modular category Cnew = C ⊠ M is nilpotent and ξ(Cnew) = 1. The cate-
gory Enew := ⊕h∈H Eh ⊠ h is a Lagrangian subcategory of Cnew and the required
statement follows from Theorem 4.5. �
Let p be a prime number.
Theorem 6.6. A modular category C is equivalent to the center of a fusion category
of the form VecωG with G being a p-group if and only if it has the following properties:
(i) the Frobenius-Perron dimension of C is p2n for some n ∈ Z+,
(ii) the dimension of every simple object of C is an integer,
(iii) the multiplicative central charge of C is 1.
Proof. It is clear that for any finite p-group G and ω ∈ Z3(G, k×) the modular
category Z(VecωG) satisfies properties (i) and (ii). The central charge of Z(Vec
equals 1 by [Mu4, Theorem 1.2].
Let us prove the converse. Suppose that C satisfies conditions (i), (ii), and (iii).
Let E be an isotropic subcategory of C such that (E ′)ad ⊆ E (such an E exists by
Remark 6.3). There is a grading E ′ = ⊕h∈H E ′h with E ′1 = E and θ being constant
20 VLADIMIR DRINFELD, SHLOMO GELAKI, DMITRI NIKSHYCH, AND VICTOR OSTRIK
on each E ′h, h ∈ H . Note that H is a metric p-group whose order is a square.
By Proposition 5.3 it contains a Lagrangian subgroup H0, whence ⊕h∈H0 E ′h is a
Lagrangian subcategory of C.
Thus, C ∼= VecωG for some G and ω by Theorem 4.5. Since |G|2 = dim(Vec
dim(C) it follows that G is a p-group. �
Finally, we apply our results to show that certain fusion categories (more pre-
cisely, representation categories of certain semisimple quasi-Hopf algebras) are group-
theoretical and to obtain a categorical analogue of the Sylow decomposition of
nilpotent groups.
Corollary 6.7. Let C be a fusion category with integral dimensions of simple objects
and such that Z(C) is nilpotent. Then C is group-theoretical.
Proof. By Corollary 6.2 the category Z(C) is group-theoretical. Hence, C ⊠ Crev is
group theoretical (as a dual category of Z(C), see [ENO]). Therefore, C is group-
theoretical (as a fusion subcategory of C ⊠ Crev). �
Corollary 6.8. Let C be a fusion category of dimension pn, n ∈ Z+, such that
all objects of C have integer dimension (this is automatic if p > 2). Then C is
group-theoretical.
In other words, semisimple quasi-Hopf algebras of dimension pn are group-
theoretical.
Remark 6.9. Semisimple Hopf algebras of dimension pn were studied by several
authors, see e.g., [EG2], [Kash], [Ma1], [Ma2], [MW], [Z].
From Corollary 6.2 we obtain the following Sylow decomposition.
Theorem 6.10. Let C be a braided nilpotent fusion category such that all objects
of C have integer dimension. Then C is group-theoretical and has a decomposition
into a tensor product of braided fusion categories of prime power dimension. If the
factors are chosen in such a way that their dimensions are relatively prime, then
such a decomposition is unique up to a permutation of factors.
Proof. It was shown in [GN, Theorem 6.11] that the center of a braided nilpotent
fusion category is nilpotent. Hence, Z(C) is group-theoretical by Corollary 6.2.
Since C is equivalent to a subcategory of Z(C), it is group-theoretical by [ENO,
Proposition 8.44]. This means that there is a group G and ω ∈ Z3(G, k∗) such
that C is dual to VecωG with respect to some indecomposable module category. The
group G is necessarily nilpotent since Rep(G) ⊆ Z(VecωG) ∼= Z(C). Hence, G is
isomorphic to a direct product of its Sylow p-subgroups, G = G1 × · · · ×Gn, and
so VecωG is equivalent to a tensor product of p-categories. It follows from [ENO,
Proposition 8.55] that the dual category C is also a product of fusion p-categories,
as desired.
Now suppose that C is decomposed into factors of prime power Frobenius-Perron
dimension, C ≃ ⊠pCp. It is easy to see that the objects from Cp ⊂ C are characterized
by the following property:
(24) X ∈ Cp if and only if there exists k ∈ Z+ such that Hom(1, X⊗
) 6= 0.
This shows that the decomposition in question is unique. �
GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 21
Remark 6.11. Let C be a nilpotent modular category with integral dimensions
of simple objects. We already mentioned in the introduction that the choice of a
tensor complement M satisfying C ⊠ M ∼= VecωG is not unique. In the proof of
Theorem 6.5 such M can be chosen canonically as the category opposite to the
canonical modularization corresponding to a maximal isotropic subcategory of C,
see Proposition 3.9.
Another canonical way is to choose an M of minimal possible dimension. This
is done as follows. By Theorem 6.10, we have C = ⊠p Cp and M = ⊠p Mp,
where Cp,Mp are modular p-categories. By Theorem 6.6, Mp has to be chosen in
such a way that dim(Cp) dim(Mp) is a square and ξ(Mp) = ξ(Mp)−1. It follows
from Examples 5.1, 5.2 and Corollary 6.4 that there is a unique such choice of
Mp with minimal dim(Mp), in which case dim(Mp) ∈ {1, p, p2} for odd p and
dim(M2) ∈ {1, 2, 4, 8}.
Theorem 6.12. Let C be a braided nilpotent fusion category. Then C has a unique
decomposition into a tensor product of braided fusion categories of prime power
dimension.
Proof. According to Theorem 6.10 the result is true if the dimensions of simple
objects of C are integers. In general, define subcategories Cp ⊂ C by condition (24)
above. For a simple object X ∈ C it is known (see [GN]) that FPdim(X) =
N ∈ N. Thus X⊠X ∈ C⊠C has an integer dimension. The category C⊠C contains
a fusion subcategory (C⊠ C)int consisting of all objects with integer dimension, see
[GN]. We can apply Theorem 6.10 to the category (C ⊠ C)int and obtain a unique
decomposition X = ⊗pXp with Xp ∈ Cp. The theorem is proved. �
Corollary 6.13. Let C be a braided nilpotent fusion category. Assume that X ∈ C
is simple and its dimension is not integer. Then FPdim(X) ∈
Proof. This follows immediately from Theorem 6.12 since if a category of prime
power Frobenius-Perron dimension pk contains an object of a non-integer dimension
then p = 2, see [ENO]. �
Example 6.14. It is easy to see that the Tambara-Yamagami categories from [TY]
are nilpotent and indecomposable into a tensor product. Thus Theorem 6.12 implies
that if such a category admits a braiding, then its dimension should be a power of
2 (since the dimension of a Tambara-Yamagami category is always divisible by 2).
A stronger result is contained in [S].
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GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES 23
V.D.: Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
E-mail address: drinfeld@math.uchicago.edu
S.G.: Department of Mathematics, Technion-Israel Institute of Technology, Haifa
32000, Israel
E-mail address: gelaki@math.technion.ac.il
D.N.: Department of Mathematics and Statistics, University of New Hampshire,
Durham, NH 03824, USA
E-mail address: nikshych@math.unh.edu
V.O.: Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
E-mail address: vostrik@math.uoregon.edu
	1. introduction
	1.1. Main results
	1.2. Interpretation in terms of group-theoretical fusion categories and semisimple quasi-Hopf algebras.
	1.3. Idea of the proof
	1.4. Organization of the paper
	1.5. Acknowledgments
	2. Preliminaries
	2.1. Fusion categories
	2.2. Nilpotent fusion categories
	2.3. Premodular categories and modular categories
	2.4. Pointed modular categories and metric groups
	2.5. Centralizers
	2.6. Gauss sums and central charge in modular categories
	2.7. Symmetric fusion categories
	3. Isotropic subcategories and Bruguières-Müger modularization
	3.1. Modularization
	3.2. Invariance of the central charge
	3.3. Maximal isotropic subcategories
	4. Reconstruction of a twisted group double from a Lagrangian subcategory
	4.1. C-algebras
	4.2. Hyperbolic modular categories as twisted group doubles
	4.3. A criterion for a modular category to be group-theoretical
	5. Pointed modular categories
	6. Nilpotent modular categories
	References
