On fusion rules and solvability of a fusion category

Melisa Escañuela González; Sonia Natale

doi:10.1515/jgth-2016-0020

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On fusion rules and solvability of a fusion category

Melisa Escañuela González and Sonia Natale

Published/Copyright: May 18, 2016

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From the journal Journal of Group Theory Volume 20 Issue 1

Abstract

We address the question whether or not the condition on a fusion category being solvable is determined by its fusion rules. We prove that the answer is affirmative for some families of non-solvable examples arising from representations of semisimple Hopf algebras associated to exact factorizations of the symmetric and alternating groups. In the context of spherical fusion categories, we also consider the invariant provided by the S-matrix of the Drinfeld center and show that this invariant does determine the solvability of a fusion category provided it is group-theoretical.

1 Introduction

Throughout this paper we shall work over an algebraically closed field k of characteristic zero. Let G be a finite group. An important invariant of G is given by its character table, defined as the collection {χi⁢(gj)}0≤i,j≤n, where ϵ=χ0,…,χn are the irreducible characters of G over k and e=g0,…,gn are representatives of the conjugacy classes of G. Several structural properties of G can be read off from its character table. For instance, the character table of G allows one to determine the lattice of normal subgroups of G and to decide if the group G is nilpotent or solvable. See [16, p. 23]. It is known, however, that the character table of a finite solvable group G does not determine its derived length [19, 20].

In particular, if G and Γ are finite groups with the same character table, then G is solvable if and only if Γ is solvable. In addition, the knowledge of the character table of a finite group G is equivalent to the knowledge of the structure constants, in the canonical basis consisting of isomorphism classes of irreducible representations, of the Grothendieck ring of the fusion category Rep⁡G of finite-dimensional representations of G over k, the so-called the fusion rules of Rep⁡G.

The notions of nilpotency and solvability of a group G have been extended to general fusion categories in [15, 12]. Let 𝒞 be a fusion category over k. Then 𝒞 is nilpotent if there exists a series of fusion subcategories

(1.1)Vect=𝒞0⊆𝒞1⊆…⊆𝒞n=𝒞,

and a series of finite groups G1,…,Gn, such that 𝒞i is a Gi-extension of 𝒞i-1, for all i=1,…,n. On the other side, 𝒞 is solvable if there exists a sequence of fusion categories Vect=𝒞0,…,𝒞n=𝒞, n≥0, and a sequence of cyclic groups of prime order G1,…,Gn, such that, for all 1≤i≤n, 𝒞i is a Gi-equivariantization or a Gi-extension of 𝒞i-1. See Section 2.2. Some features related to nilpotency and solvability have also been extended from the context of finite groups to that of fusion categories; remarkably, an analogue of Burnside’s pa⁢qb-theorem was established for fusion categories in [12].

It is apparent from the definition of nilpotency of a fusion category 𝒞 given in [15] that this property depends only upon the Grothendieck ring of 𝒞, that is, it is determined by its fusion rules. In this paper we address the question of whether or not the solvability of a fusion category 𝒞 is also determined by its fusion rules.

Since a solvable fusion category has nontrivial invertible objects and a simple group has no nontrivial one-dimensional representation, it follows that no solvable fusion category can have the same fusion rules as a simple finite group. We show that if 𝒞 is a fusion category with the same fusion rules as a dihedral group, then 𝒞 is solvable. On the other hand, if 𝒞 has the fusion rules of a symmetric group 𝕊n, n≥5, then 𝒞 is not solvable; Theorem 3.4 and Corollary 3.7.

We study some families of examples of non-solvable fusion categories arising from representations of semisimple Hopf algebras associated to exact factorizations of the symmetric group 𝕊n and the alternating group 𝔸n. For a wide class of such fusion categories 𝒞, we show that 𝒞 cannot have the fusion rules of any solvable fusion category. See Theorems 4.14, 4.18, 4.20 and 4.22.

In the context of braided fusion categories, the solvability of a fusion category 𝒞 is related to the existence of Tannakian subcategories of 𝒞; it is known that if 𝒞 is a non-pointed integral solvable braided fusion category, then it must contain a nontrivial Tannakian subcategory [30, Lemma 5.1].

We show that if 𝒞~ is a non-pointed braided fusion category which has the same fusion rules as a solvable fusion category 𝒞, then 𝒞~ contains a nontrivial Tannakian subcategory. See Theorem 5.1.

For a spherical fusion category 𝒞 we study a somehow stronger invariant, analogous to the character table of a finite group, consisting of the S-matrix of the Drinfeld center 𝒵⁢(𝒞) of 𝒞. Indeed, the S-matrix of a modular category 𝒟 is usually named the ‘character table’ of 𝒟 in the literature; see for instance [13]. A celebrated formula due to Verlinde, and valid for any modular category, implies that the S-matrix of 𝒵⁢(𝒞) determines its fusion rules. We call two spherical fusion categories S-equivalent if their Drinfeld centers have ‘the same’ S-matrix; see Section 6.3.

We prove in Theorem 6.5 that the S-matrix of the Drinfeld center does determine the solvability of a group-theoretical fusion category. That is, if 𝒞 and 𝒟 are S-equivalent spherical fusion categories and 𝒞 is group-theoretical, then 𝒞 is solvable if and only if 𝒟 is solvable. We also show that being group-theoretical is a property invariant under S-equivalence, that is, it is a property determined by the S-matrix of the Drinfeld center; see Theorem 6.3.

The paper is organized as follows. Section 2 contains the main notions and facts on fusion categories that will be needed in the rest of the paper. In Section 3 we study the notion of Grothendieck equivalence of fusion categories and its connection with solvability, and prove some results on the fusion rules of dihedral and symmetric groups. In Section 4 we consider examples of non-solvable fusion categories arising from exact factorizations of the symmetric and alternating groups. The case of braided fusion categories is studied in Section 5. Finally, in Section 6 we study the notion of S-equivalence of spherical fusion categories.

2 Preliminaries

The category of finite-dimensional vector spaces over k will be denoted by Vect. A fusion category over k is a semisimple rigid monoidal category over k with finitely many isomorphism classes of simple objects, finite-dimensional Hom spaces, and such that the unit object 1 is simple. Unless otherwise stated, all tensor categories will be assumed to be strict. We refer the reader to [11, 10] for the main notions on fusion categories used throughout.

2.1 Fusion categories

Let 𝒞 be a fusion category over k. The Grothendieck group K0⁢(𝒞) is a free abelian group with basis Irr⁡(𝒞) consisting of isomorphism classes of simple objects of 𝒞. For an object X of 𝒞, let us denote by [X] its class in K0⁢(𝒞).

The tensor product of 𝒞 endows K0⁢(𝒞) with a ring structure with unit element [𝟏] and such that, for all objects X and Y of 𝒞,

[X]⁢[Y]=[X⊗Y].

Let X,Y∈Irr⁡(𝒞). Then one can write

X⁢Y=∑Z∈Irr⁡(𝒞)NX,YZ⁢Z,

where NX,YZ are non-negative integers, for all X,Y,Z∈Irr⁡(𝒞). The collection of numbers {NX,YZ}X,Y,Z are called the fusion rules of 𝒞 and they determine the ring structure of K0⁢(𝒞). They are given by the formula

NX,YZ=dim⁡Hom𝒞⁡(Z,X⊗Y) for all X,Y,Z∈Irr⁡(𝒞).

In the terminology of [15, Section 2.1], the pair (K0⁢(𝒞),Irr⁡(𝒞)) is a unital based ring.

A fusion subcategory of 𝒞 is a full tensor subcategory 𝒟 such that 𝒟 is replete and stable under direct summands. Fusion subcategories of 𝒞 are in bijective correspondence with subrings of K0⁢(𝒞) spanned by a subset of Irr⁡(𝒞), that is, based subrings of K0⁢(𝒞).

The Frobenius–Perron dimension of a simple object X∈𝒞 is, by definition, the Frobenius–Perron eigenvalue of the matrix of left multiplication by the class of X in the basis Irr⁡(𝒞) of the Grothendieck ring of 𝒞 consisting of isomorphism classes of simple objects. The Frobenius–Perron dimension of 𝒞 is the number

FPdim⁡𝒞=∑X∈Irr⁡(𝒞)(FPdim⁡X)2.

We shall indicate by cd⁢(𝒞) the set of Frobenius–Perron dimensions of simple objects of 𝒞. If 1=d0,d1,…,dr are distinct positive real numbers and n1,…,nr are natural numbers, we shall say that 𝒞 is of type(d0,n0;d1,n1;…;dr,nr) if 𝒞 has ni isomorphism classes of simple objects of Frobenius–Perron dimension di, for all i=0,…,r.

The group of invertible objects of 𝒞 will be denoted by G⁢(𝒞). Thus G⁢(𝒞) coincides with the subset of elements Y of Irr⁡(𝒞) such that FPdim⁡Y=1. Thus, if 𝒞 is of type (1,n0;d1,n1;…;dr,nr), then n0=|G⁢(𝒞)|. The category 𝒞 is called integral if FPdim⁡X∈ℤ, for all simple object X∈𝒞, and it is called weakly integral if FPdim⁡𝒞∈ℤ.

Recall that a right module category over a fusion category 𝒞 is a finite semisimple k-linear abelian category ℳ endowed with a bifunctor ⊗:ℳ×𝒞→ℳ satisfying the associativity and unit axioms for an action, up to coherent natural isomorphisms. The module category ℳ is called indecomposable if it is not equivalent as a module category to a direct sum of nontrivial module categories. If ℳ is an indecomposable module category over 𝒞, then the category 𝒞ℳ* of 𝒞-module endofunctors of ℳ is also a fusion category.

Two fusion categories 𝒞 and 𝒟 are Morita equivalent if 𝒟 is equivalent to 𝒞ℳ* for some indecomposable module category ℳ. If 𝒞 and 𝒟 are Morita equivalent fusion categories, then FPdim⁡𝒞=FPdim⁡𝒟.

By [12, Theorem 3.1], the fusion categories 𝒞 and 𝒟 are Morita equivalent if and only if their Drinfeld centers are equivalent as braided fusion categories.

A fusion category 𝒞 is pointed if all its simple objects are invertible. If 𝒞 is a pointed fusion category, then there exist a finite group G and a 3-cocycle ω on G such that 𝒞 is equivalent to the category 𝒞⁢(G,ω) of finite-dimensional G-graded vector spaces with associativity constraint defined by ω. A fusion category Morita equivalent to a pointed fusion category is called group-theoretical.

2.2 Nilpotent and solvable fusion categories

Let G be a finite group. A G-grading on a fusion category 𝒞 is a decomposition 𝒞=⊕g∈G𝒞g, such that 𝒞g⊗𝒞h⊆𝒞g⁢h and 𝒞g*⊆𝒞g-1, for all g,h∈G. A G-grading is faithful if 𝒞g≠0, for all g∈G. The fusion category 𝒞 is called a G-extension of a fusion category 𝒟 if there is a faithful grading 𝒞=⊕g∈G𝒞g with neutral component 𝒞e≅𝒟.

If 𝒞 is any fusion category, there exist a finite group U⁢(𝒞), called the universal grading group of 𝒞, and a canonical faithful grading 𝒞=⊕g∈U⁢(𝒞)𝒞g, with neutral component 𝒞e=𝒞ad, where 𝒞ad is the adjoint subcategory of 𝒞, that is, the fusion subcategory generated by X⊗X*, X∈Irr⁡(𝒞).

In fact, K0⁢(𝒞)ad=K0⁢(𝒞ad) is a based subring of K0⁢(𝒞) and K0⁢(𝒞) decomposes into a direct sum of indecomposable based K0⁢(𝒞)ad-bimodules

K0⁢(𝒞)=⊕g∈U⁢(𝒞)K0⁢(𝒞)g,with K0⁢(𝒞)e=K0⁢(𝒞)ad.

Then the group structure on U⁢(𝒞):=U⁢(K0⁢(𝒞)) is defined by the following property: g⁢h=t if and only if Xg⁢Xh∈K0⁢(𝒞)t, for all Xg∈K0⁢(𝒞)g, Xh∈K0⁢(𝒞)h, g,h,t∈U⁢(𝒞); see [15, Theorem 3.5].

A fusion category 𝒞 is called (cyclically) nilpotent if there exists a sequence of fusion categories Vect=𝒞0⊆𝒞1⊆…⊆𝒞n=𝒞, and finite (cyclic) groups G1,…,Gn, such that for all i=1,…,n, 𝒞i is a Gi-extension of 𝒞i-1.

On the other side, 𝒞 is solvable if it is Morita equivalent to a cyclically nilpotent fusion category, that is, if there exists a cyclically nilpotent fusion category 𝒟 and an indecomposable right module category ℳ over 𝒟 such that 𝒞 is equivalent to the fusion category 𝒟ℳ* of 𝒟-linear endofunctors of ℳ.

Consider an action of a finite group G on a fusion category 𝒞 by tensor autoequivalences ρ:G¯→Aut¯⊗⁢𝒞. The equivariantization of 𝒞 with respect to the action ρ, denoted 𝒞G, is a fusion category whose objects are pairs (X,μ), such that X is an object of 𝒞 and μ=(μg)g∈G, is a collection of isomorphisms μg:ρg⁢X→X, g∈G, satisfying appropriate compatibility conditions.

The forgetful functor F:𝒞G→𝒞, F⁢(X,μ)=X, is a dominant tensor functor that gives rise to a central exact sequence of fusion categories Rep⁡G→𝒞G→𝒞 (see [4]), where Rep⁡G is the category of finite-dimensional representations of G.

The category 𝒞G is integral (respectively, weakly integral) if and only if so is 𝒞. See [3, Proposition 4.9], [4, Proposition 2.12].

According to [12, Definition 1.2], a fusion category 𝒞 is solvable if and only if there exists a sequence of fusion categories Vect=𝒞0,…,𝒞n=𝒞, n≥0, and a sequence of cyclic groups of prime order G1,…,Gn, such that, for all 1≤i≤n, 𝒞i is a Gi-equivariantization or a Gi-extension of 𝒞i-1.

It is shown in [12, Proposition 4.1] that the class of solvable fusion categories is stable under taking extensions and equivariantizations by solvable groups, Morita equivalent categories, tensor products, Drinfeld center, fusion subcategories and components of quotient categories.

In view of [12, Proposition 4.5 (iv)], every nontrivial solvable fusion category has nontrivial invertible objects.

Suppose that the finite group G acts on the fusion category 𝒞 by tensor autoequivalences. Let Y∈Irr⁡𝒞. The stabilizer of Y is the subgroup

GY={g∈G:ρg⁢(Y)≅Y}.

Let αY:GY×GY→k* be the 2-cocycle defined by the relation

(2.1)αY⁢(g,h)-1⁢idY=cg⁢ρg⁢(ch)⁢(ρ2Yg,h)-1⁢(cg⁢h)-1:Y→Y,

where, for all g∈GY, cg:ρg⁢(Y)→Y is a fixed isomorphism [5, Section 2.3].

Then the simple objects of 𝒞G are parameterized by pairs (Y,U), where Y runs over the G-orbits on Irr⁡(𝒞) and U is an equivalence class of an irreducible αY-projective representation of GY. We shall use the notation SY,U to indicate the isomorphism class of the simple object corresponding to the pair (Y,U). The dimension of SY,U is given by the formula

(2.2)FPdimSY,U=[G:GY]dimUFPdimY.

Lemma 2.1

Let p be a prime number. Suppose that the group Zp acts on a fusion category C by tensor autoequivalences. Assume in addition that G⁢(CZp) is of order p and G⁢(C)≠{𝟏}. Then CZp has a simple object of Frobenius–Perron dimension p.

Proof.

Let Y be an invertible object of 𝒞 and let U be an irreducible αY-projective representation of the subgroup GY⊆ℤp. Since GY is cyclic, then αY=1 in H2⁢(GY,k*) and dim⁡U=1. Then the Frobenius–Perron dimension of the simple object SY,U is given by

FPdimSY,U=[G:GY]FPdimY=[G:GY].

Moreover, if Y=𝟏, then GY=ℤp. Therefore, letting U0=ϵ,U1,…,Up-1 the non-isomorphic representations of ℤp, we get that 𝟏=S𝟏,U0,S𝟏,U1,…,S𝟏,Up-1 are all the non-isomorphic invertible objects of 𝒞ℤp. Hence for all invertible object Y≠𝟏 of 𝒞, we must have [G:GY]=p and the simple object SY,U has Frobenius–Perron dimension p. This proves the lemma. ∎

Proposition 2.2

Let p be a prime number. Suppose that C is a solvable fusion category such that G⁢(C)≅Zp and C has no simple objects of Frobenius–Perron dimension p. Then C is cyclically nilpotent.

Proof.

The proof is by induction on FPdim⁡𝒞≥p. If FPdim⁡𝒞=p, then there is nothing to prove. Suppose FPdim⁡𝒞>p. Since 𝒞 is solvable then, for some prime number q, 𝒞 must be a ℤq-extension or a ℤq-equivariantization of a fusion category 𝒟. If the second possibility holds, then the assumption that G⁢(𝒞)≅ℤp implies that q=p. Moreover, since 𝒟 is also solvable, then 𝒟pt≠Vect. By Lemma 2.1, 𝒞 must have a simple object of dimension p, which contradicts the assumption.

Therefore 𝒞 must be a ℤq-extension of a fusion subcategory 𝒟. In particular, 𝒟 cannot have simple objects of dimension p and since 𝒟 is solvable, then 𝒟pt≠Vect, whence G⁢(𝒟)=G⁢(𝒞)≅ℤp. By induction, 𝒟 and then also 𝒞, is cyclically nilpotent. This finishes the proof of the proposition. ∎

Lemma 2.3

Let C be a fusion category and let G be a finite group acting on C by tensor autoequivalences. Then the forgetful functor U:Z⁢(CG)→CG induces an injective ring homomorphism K0⁢(G)→Z⁢(K0⁢(CG)). In particular, the group G^ is isomorphic to a subgroup of the center of G⁢(CG).

Proof.

By [12, Proposition 2.10], the Drinfeld center 𝒵⁢(𝒞) contains a Tannakian subcategory ℰ≅Rep⁡G such that ℰ embeds into 𝒞 under the forgetful functor U:𝒵⁢(𝒞)→𝒞. As a consequence we obtain the lemma. ∎

2.3 Braided fusion categories

A braided fusion category is a fusion category 𝒞 endowed with a braiding, that is, a natural isomorphism cX,Y:X⊗Y→Y⊗X, X,Y∈𝒞, subject to the so-called hexagon axioms.

If 𝒟 is a fusion subcategory of a braided fusion category 𝒞, the Müger centralizer of 𝒟 in 𝒞 will be denoted by 𝒟′. Thus 𝒟′ is the full fusion subcategory generated by all objects X∈𝒞 such that cY,X⁢cX,Y=idX⊗Y, for all objects Y∈𝒟.

The centralizer 𝒞′ of 𝒞 is called the Müger (or symmetric) center of 𝒞. The category 𝒞 is called symmetric if 𝒞′=𝒞. If 𝒞 is any braided fusion category, its Müger center 𝒞′ is a symmetric fusion subcategory of 𝒞. The category 𝒞 is called non-degenerate (respectively, slightly degenerate) if 𝒞′≅Vect (respectively, if 𝒞′≅sVect, where sVect denotes the category of super-vector spaces).

For a fusion category 𝒞, the Drinfeld center of 𝒞 will be denoted 𝒵⁢(𝒞). It is known that 𝒵⁢(𝒞) is a braided non-degenerate fusion category of Frobenius–Perron dimension FPdim⁡𝒵⁢(𝒞)=(FPdim⁡𝒞)2.

Let G be a finite group. The fusion category Rep⁡G of finite-dimensional representations of G is a symmetric fusion category with respect to the canonical braiding. A braided fusion category ℰ is called Tannakian, if ℰ≅Rep⁡G for some finite group G as braided fusion categories.

Every symmetric fusion category is equivalent, as a braided fusion category, to the category Rep⁡(G,u) of representations of a finite group G on finite-dimensional super-vector spaces, where u∈G is a central element of order 2 which acts as the parity operator [6]. In particular, if 𝒞 is symmetric, then it is equivalent to the category of representations of a finite group as a fusion category.

Let G be a finite group. A G-crossed braided fusion category is a fusion category 𝒟 endowed with a G-grading 𝒟=⊕g∈G𝒟g and an action of G by tensor autoequivalences ρ:G¯→Aut¯⊗⁢𝒟, such that ρg⁢(𝒟h)⊆𝒟g⁢h⁢g-1, for all g,h∈G, and a G-braiding c:X⊗Y→ρg⁢(Y)⊗X, g∈G, X∈𝒟g, Y∈𝒟, subject to compatibility conditions. The G-braiding c restricts to a braiding in the neutral component 𝒟e.

If 𝒟 is a G-crossed braided fusion category, then the equivariantization 𝒟G under the action of G is a braided fusion category containing Rep⁡G as a Tannakian subcategory. Furthermore, the group G acts by restriction on 𝒟e by braided tensor autoequivalences. The equivariantization 𝒟eG coincides with the centralizer ℰ′ of the Tannakian subcategory ℰ in 𝒟G. See [23].

Let ℰ be Tannakian subcategory of a braided fusion category 𝒞 and let G be a finite group such that ℰ≅Rep⁡G as symmetric categories. Furthermore, let A∈𝒞 be the algebra corresponding to the algebra kG∈Rep⁡G of functions on G with the regular action. The de-equivariantization 𝒞G of 𝒞 with respect to Rep⁡G is the fusion category 𝒞A of right A-modules in 𝒞. This is a G-crossed braided fusion category such that 𝒞≅(𝒞G)G. The neutral component of 𝒞G with respect to the associated G-grading, denoted by 𝒞G0, coincides with the de-equivariantization ℰ′G of the centralizer of ℰ by the group G.

Remark 2.4

Let us recall from [30, Proposition 4.1] that if ℰ≅Rep⁡G⊆𝒞 is a Tannakian subcategory, then the fusion category 𝒞 is weakly integral (respectively, weakly group-theoretical) if and only if 𝒞G0 is weakly integral (respectively, weakly group-theoretical). In addition, 𝒞 is solvable if and only if 𝒞G0 is solvable and G is solvable.

Similarly, if 𝒞 is integral, then so is 𝒞G0 but, as pointed out by the referee and contrary to the statement in [30, Proposition 4.1], the converse is not true in general; examples can be found among the non-degenerate fusion categories ℰ⁢(q,±) constructed in [14, Section 5].

Although it is not relevant to the contents of the present paper, we remark that if the support H of 𝒞G (which is a normal subgroup of G that coincides with G if 𝒞 is non-degenerate) has no nontrivial elementary abelian 2-group quotient, then the assumption that 𝒞G0 is integral implies that 𝒞 is integral as well. In fact, suppose that 𝒞G0 is integral, so that 𝒞G is weakly integral. It follows from [15, Theorem 3.10] that there exists a maximal integral fusion subcategory ℬ such that 𝒞G is an E-extension of ℬ, where E is an elementary abelian 2-group. In particular, 𝒞G0⊆ℬ. Since 𝒞G is an H-extension of 𝒞G0, then E is isomorphic to a quotient of H. By assumption, E must be trivial and thus 𝒞G, and therefore also 𝒞, are integral, as claimed.

Lemma 2.5

Let C be a braided fusion category. Then the subcategory Cad∩Cpt is symmetric.

Proof.

Suppose first that 𝒞 is non-degenerate. Then 𝒞ad=𝒞pt′, by [10, Corollary 3.27]. Therefore 𝒞ad∩𝒞pt=𝒞pt′∩𝒞pt is a symmetric subcategory.

Next, for an arbitrary braided fusion category 𝒞, let 𝒵⁢(𝒞) be the Drinfeld center of 𝒞. Since 𝒵⁢(𝒞) is non-degenerate, the category 𝒵⁢(𝒞)ad∩𝒵⁢(𝒞)pt is symmetric. The braiding of 𝒞 induces a canonical embedding of braided fusion categories 𝒞→𝒵⁢(𝒞). We may therefore identify 𝒞 with a fusion subcategory of 𝒵⁢(𝒞). Observe that 𝒞ad⊆𝒵⁢(𝒞)ad and 𝒞pt⊆𝒵⁢(𝒞)pt. Hence 𝒞ad∩𝒞pt⊆𝒵⁢(𝒞)ad∩𝒵⁢(𝒞)pt, and then 𝒞ad∩𝒞pt is symmetric, as claimed. ∎

Lemma 2.6

Let C be a braided fusion category such that Cad=C. Then Cpt′=C.

Proof.

Let ℬ⊆𝒞 be any fusion subcategory. By [10, Proposition 3.25] we have (ℬad)′=(ℬ′)co=𝒜, where 𝒜⊆𝒞 denotes the projective centralizer of ℬ. Letting ℬ=𝒞pt we find that 𝒞=(ℬad)′ equals the projective centralizer of 𝒞pt. By [10, Lemma 3.15], the projective centralizer of a fusion subcategory ℬ is a graded extension of the centralizer ℬ′. Since 𝒞=𝒞ad, this implies that 𝒞=𝒞pt′, as claimed. ∎

3 Grothendieck equivalence of fusion categories

Let 𝒞 and 𝒞~ be fusion categories. A Grothendieck equivalence between 𝒞 and 𝒞~ is a bijection f:Irr⁡𝒞→Irr⁡𝒞~ such that

(3.1)f⁢(𝟏)=𝟏,and Nf⁢(X),f⁢(Y)f⁢(Z)=NX,YZ,

for all X,Y,Z∈Irr⁡𝒞.

We say that 𝒞 and 𝒞~ are Grothendieck equivalent if there exists a Grothendieck equivalence between them.

Remark 3.1

Suppose f:Irr⁡𝒞→Irr⁡𝒞~ is a Grothendieck equivalence. Then the map f extends canonically to a ring isomorphism f:K0⁢(𝒞)→K0⁢(𝒞~).

In particular, f induces a bijection between the lattices of fusion subcategories of 𝒞 and 𝒞~. If 𝒟 is a fusion subcategory of 𝒞, we shall denote by f⁢(𝒟) the corresponding fusion subcategory of 𝒞~, that is, f⁢(𝒟) is the fusion subcategory whose simple objects are f⁢(X), X∈Irr⁡𝒟. Note that f restricts to a Grothendieck equivalence f:Irr⁡𝒟→Irr⁡f⁢(𝒟).

Proposition 3.2

Let C and C~ be fusion categories. Suppose that f:Irr⁡C→Irr⁡C~ is a Grothendieck equivalence. Then the following hold:

If X∈K0⁢(𝒞), then FPdim(f⁢(X)) = FPdim(X). Hence, if 𝒟 is a fusion subcategory of 𝒞, then FPdim(f⁢(𝒟))= FPdim(𝒟).
X∈Irr⁡𝒞 is invertible if and only if f⁢(X)∈Irr⁡𝒞~ is invertible.
If X∈Irr⁡𝒞, then f⁢(X*)=f⁢(X)*.
f⁢(𝒞(n))=𝒞~(n), for all n≥0. In particular, 𝒞 is nilpotent if and only if 𝒞~ is nilpotent.
f induces a group isomorphism f:U⁢(𝒞)→U⁢(𝒞~) such that f⁢(𝒞g)=𝒞~f⁢(g).

Proof.

(i) By Remark 3.1 we know that f extends to a ring isomorphism

f:K0⁢(𝒞)→K0⁢(𝒞~).

By [11, Lemma 8.3], FPdim:K0⁢(𝒞)→ℝ is the only ring homomorphism such that FPdim⁡(X)>0 for any 0≠X∈𝒞, so FPdim⁡(f⁢(X))=FPdim⁡(X), for all X∈Irr⁡𝒞.

(ii) This follows from (i), since the invertible objects of a fusion category are exactly those objects with Frobenius–Perron dimension 1.

(iii) Since f⁢(𝟏)=𝟏, we have

Nf⁢(X),f⁢(X*)𝟏=NX,X*𝟏=1.

Therefore f⁢(X*)=f⁢(X)*.

(iv) It follows from (iii) and the fact that f preserves fusion rules that

f⁢(𝒞ad)=𝒞~ad.

Then f induces by restriction a Grothendieck equivalence Irr⁡𝒞ad→Irr⁡𝒞~ad. An inductive argument implies that f⁢(𝒞(n))=𝒞~(n), for all n≥0.

(v) By definition, U⁢(𝒞)=U⁢(K0⁢(𝒞)) (see [15]) and K0⁢(𝒞) decomposes into a direct sum of indecomposable based K0⁢(𝒞)ad-bimodules

K0⁢(𝒞)=⊕g∈U⁢(𝒞)K0⁢(𝒞)g,

with K0⁢(𝒞)e=K0⁢(𝒞)ad. This decomposition is unique up to a permutation of U⁢(𝒞). By Remark 3.1, f extends to a ring isomorphism f:K0⁢(𝒞)→K0⁢(𝒞~) and by (iv) f restricts to a ring isomorphism

K0⁢(𝒞)ad=K0⁢(𝒞ad)≅K0⁢(𝒞~ad)=K0⁢(𝒞~)ad.

So for all g∈U⁢(𝒞), f⁢(K0⁢(𝒞)g)=K0⁢(𝒞~)g~, for a unique g~∈U⁢(𝒞~). Letting f⁢(g)=g~, we obtain a group isomorphism f:U⁢(𝒞)→U⁢(𝒞~) such that

f⁢(K0⁢(𝒞g))=K0⁢(𝒞~f⁢(g)).

This implies (v). ∎

Remark 3.3

Let G be a finite group. Observe that any G-grading on a fusion category 𝒞 with neutral component 𝒟 is uniquely determined by a G-grading on the Grothendieck ring K0⁢(𝒞) with neutral component K0⁢(𝒟). In particular, if 𝒞 and 𝒞~ are Grothendieck equivalent, then 𝒞 is G-graded with neutral component 𝒟 if and only if 𝒞~ is G-graded with neutral component 𝒟~, such that 𝒟~ and 𝒟 are Grothendieck equivalent.

Our first theorem concerns fusion categories with dihedral fusion rules.

Theorem 3.4

Let n be a natural number and let C be a fusion category. Suppose that C is Grothendieck equivalent to the category Rep⁡Dn, where Dn is the dihedral group of order 2⁢n. Then C is solvable.

Proof.

It follows from [25, Theorem 4.2] that a fusion category Grothendieck equivalent to the representation category of a dihedral group is group-theoretical. Then 𝒞 is group-theoretical, that is, it is Morita equivalent to a pointed fusion category 𝒞⁢(Γ,ω), where Γ is a group and ω is a 3-cocycle on Γ.

Suppose first that n is odd. Then the order of Γ is equal to 2⁢n and, since n is odd, Γ is solvable. Then 𝒞 is solvable too.

If n is even, then the center of Dn is of order 2 and Dn/Z⁢(Dn)≅Dn/2. Therefore, the category Rep⁡Dn is a ℤ2-extension of Rep⁡Dn/2; see [15, Example 3.2]. Since 𝒞 is Grothendieck equivalent to Rep⁡Dn, it is a ℤ2-extension of a fusion subcategory 𝒟1, where 𝒟1 is Grothendieck equivalent to Rep⁡Dn/2. Continuing this process, we find that the category 𝒞 is obtained by a sequence of ℤ2-extensions from a fusion subcategory 𝒟 such that 𝒟 is Grothendieck equivalent to Rep⁡Dm, with m an odd natural number. By the above, 𝒟 is solvable and therefore so is 𝒞. This finishes the proof of the theorem. ∎

The following consequence of Proposition 2.2 gives some restrictions that guarantee that the solvability of a fusion category is a Grothendieck invariant.

Proposition 3.5

Let p be a prime number. Suppose that C is a solvable fusion category such that G⁢(C)≅Zp and C has no simple objects of Frobenius–Perron dimension p. If C is Grothendieck equivalent to a fusion category C~, then C~ is solvable.

Proof.

By Proposition 2.2, 𝒞 is cyclically nilpotent. Therefore 𝒞~ is cyclically nilpotent, whence solvable. ∎

Remark 3.6

For all n≥2, the alternating group 𝔸n has no irreducible representation of degree 2.^[1] In addition, if n≥5 (Rep⁡𝕊4 is of type (1,2;2,1;3,2)), the symmetric group 𝕊n has no irreducible representation of degree 2 either. In fact, if V were such a representation, then the restriction V|𝔸n would not be irreducible. Hence, since 𝔸n has no nontrivial one-dimensional representations (because n≥5), it follows that V|𝔸n would be trivial. This is impossible, because the kernel of the restriction functor Rep⁡𝕊n→Rep⁡𝔸n is the pointed subcategory Rep⁡ℤ2⊆Rep⁡𝕊n.

Corollary 3.7

Let n≥5 be a natural number and let C be a fusion category. Suppose that C is Grothendieck equivalent to Rep⁡Sn. Then C is not solvable.

Proof.

The category Rep⁡𝕊n is not solvable. On the other hand, the group 𝕊n has two non-equivalent representations of degree one and no irreducible representation of degree two, in view of Remark 3.6. Hence G⁢(𝒞)≅ℤ2 and 𝒞 has no simple objects of Frobenius–Perron dimension 2. The result is thus obtained as a consequence of Proposition 3.5. ∎

Remark 3.8

Let G be a non-abelian finite simple group. If 𝒞 is a fusion category Grothendieck equivalent to Rep⁡G, then 𝒞pt=Vect and therefore 𝒞 is not solvable.

4 Examples of non-solvable fusion rules

4.1 Abelian extensions

Consider an abelian exact sequence of Hopf algebras

(4.1)k⟶kΓ⁢⟶𝑖⁢H⁢⟶𝜋⁢k⁢F⟶k,

where Γ and F are finite groups. Then (4.1) gives rise to actions by permutations

Γ⁢⟵⊲⁢Γ×F⁢⟶⊳⁢F

such that (Γ,F) is a matched pair of groups. Moreover, H≅kΓ#στkF is a bicrossed product with respect to normalized invertible 2-cocycles σ:F×F→kΓ, τ:Γ×Γ→kF, satisfying suitable compatibility conditions. See [18].

The multiplication and comultiplication of kΓ#στkF are determined in the basis {#⁡esx/s∈Γ,x∈F}, by the formulas

(4.2)(#⁡esx)⊗(#⁡ety)=#⁡δt,s⊲x⁢σs⁢(x,y)⁢esx⁢y,

(4.3)Δ⁢(#⁡esx)=∑g⁢h=s#⁡#⁡τx⁢(g,h)⁢eg(h⊳x)⊗ehx,

for all s,t∈Γ, x,y∈F, where σs⁢(x,y)=σ⁢(x,y)⁢(s) and τx⁢(s,t)=τ⁢(s,t)⁢(x). See [18]. The exact sequence (4.1) is split if σ and τ are the trivial 2-cocycles.

For all s∈Γ, the restriction of the map σs:F×F→k× to the stabilizer subgroup Fs=F∩s⁢F⁢s-1 is a 2-cocycle on Fs.

The irreducible representations of H≅kΓ#στkF are classified for pairs (s,Us), where s is a representative of the orbits of the action of F in Γ and Us is an irreducible representation of the twisted group algebra kσs⁢Fs, that is, a projective irreducible representation Fs with cocycle σs. Given a pair (s,Us), the corresponding irreducible representation is given by

(4.4)W(s,Us)=IndkΓ⊗k⁢FsH⁡s⊗Us.

Observe that dimW(s,Us)=[F:Fs]dimUs. See [21].

Remark 4.1

Recall that every matched pair (Γ,F) gives rise to a group structure, denoted F⋈Γ, on the product F×Γ in the form

(x,s)⁢(y,t)=(x⁢(s⊳y),(s⊲y)⁢t),

for x,y∈F, s,t∈Γ, where Γ⁢⟵⊲⁢Γ×F⁢⟶⊳⁢F are the associated compatible actions.

The group F⋈Γ has a canonical exact factorization into its subgroups F=F×{e} and Γ={e}×Γ; that is, F⋈Γ=F⁢Γ and F∩Γ={e}.

Conversely, every finite group G endowed with an exact factorization G=F⁢Γ into its subgroups F and Γ gives rise to canonical actions by permutations

Γ⁢⟵⊲⁢Γ×F⁢⟶⊳⁢F

making (Γ,F) into a matched pair of groups.

Suppose H≅kΓ#στkF is an abelian extension of kΓ by kF. It follows from [26, Theorem 1.3] that the category Rep⁡H is Morita equivalent to the pointed fusion category 𝒞(F⋈Γ,ω), where ω is a 3-cocycle on F⋈Γ arising from the pair (σ,τ) in an exact sequence due to G. I. Kac. In particular, there are equivalences of braided fusion categories

𝒵(RepH)≅RepD(H)≅RepDω(F⋈Γ),

where Dω(F⋈Γ) is the twisted Drinfeld double of F⋈Γ (see [7]). Note that Rep⁡H is solvable if and only if the group F⋈Γ is solvable.

Example 4.2

Let G be a finite group. Then the Drinfeld double D⁢(G) fits into a split cocentral abelian exact sequence

k⟶kG⟶D⁢(G)⟶k⁢G⟶k.

This exact sequence is associated to the adjoint action ⊲:G×G→G given by h⊲g=g-1⁢h⁢g, and to the trivial action ⊳:G×G→G.

The following lemma describes the group of invertible objects of the category Rep⁡H, when H is an abelian extension.

Lemma 4.3

Suppose H fits into an exact sequence (4.1). Then there is an exact sequence

1⟶F^⁢⟶π*⁢G⁢(Rep⁡H)⁢⟶i*⁢Γ0⟶1,

where F^ denotes the group of one-dimensional characters of F and Γ0 is the group Γ0={s∈ΓF:[σs]=1⁢ in ⁢H2⁢(Γ,(kF)×)}.

Proof.

The group G⁢(Rep⁡H) can be identified with the group G⁢(H*) of group-like elements in the dual Hopf algebra H*. In addition, H* fits into an abelian extension

(4.5)k⟶kF⁢⟶i*⁢H*⁢⟶π*⁢k⁢Γ⟶k.

The lemma follows from [28, Lemma 2.2]. ∎

Remark 4.4

Keep the notation in Lemma 4.3. Note that the dual exact sequence (4.5) is associated to the actions

F⁢⟵⊲′⁢F×Γ⁢⟶⊳′⁢Γ

defined in the form x⊲′s=(s-1⊳x-1)-1 and x⊳′s=(s-1⊲x-1)-1, for all x∈F, s∈Γ (see [18, Exercise 5.5]).

Hence the exact sequence of groups of Lemma 4.3 induces the transpose of the action ⊲′ of Γ0 on the abelian group F^.

Clearly, (4.5) is split if and only if (4.1) is split and, if this is the case, the exact sequence of groups in Lemma 4.3 is split as well.

Therefore, in the case where H is a split abelian extension, the group G⁢(Rep⁡H) is isomorphic to the semidirect product F^⋊Γ0 with respect to the action ⊲′ of Γ0 on F^.

Corollary 4.5

Let G be a finite group. Then the group of invertible objects of Rep⁡D⁢(G) is isomorphic to the direct product G/[G,G]×Z⁢(G).

Proof.

This is a consequence of Lemma 4.3, in view of Example 4.2 and Remark 4.4. In fact, we have G^≅G/[G,G] and the actions ⊳′:G×G→G and ⊲′:G×G→G in Remark 4.4 are given in this case by h⊳′g=h⁢g⁢h-1 and g⊲′h=g, for all g,h∈G. Then

G0={g∈G:h⊳′g=g⁢ for all ⁢h∈G}=Z⁢(G).

The corollary follows from the fact that the action ⊲′ is the trivial one. ∎

4.2 Examples associated to the symmetric group

Let n≥2 be a natural number. The symmetric group 𝕊n has an exact factorization 𝕊n=〈z〉⁢Γ, where Γ={σ∈𝕊n:σ⁢(n)=n}≅𝕊n-1 and z=(12⁢…⁢n), so that 〈z〉≅Cn. This exact factorization induces mutual actions by permutations

𝕊n-1⁢⟵⊲⁢𝕊n-1×Cn⁢⟶⊳⁢Cn

that make (𝕊n-1,Cn) into a matched pair of groups. The actions ⊲,⊳ are determined by the relations

(4.6)σ⁢c=(σ⊳c)⁢(σ⊲c),

for all σ∈Γ, c∈〈z〉.

Suppose n is odd, so that 〈z〉⊆𝔸n. Relations (4.6) imply that the subgroup Γ+=Γ∩𝔸n≅𝔸n-1 is stable under the action ⊲ of 〈z〉. Therefore the actions ⊲,⊳ induce by restriction a matched pair (𝔸n-1,Cn).

Remark 4.6

Let σ∈Γ. It follows from (4.6) that σ⁢z=zr⁢(σ⊲z), for some 0≤r≤n-1. Since σ⊲z∈Γ then (σ⊲z)⁢(n)=n, implying that r=b⁢(n), where b=σ⁢z.

Suppose that n≥4 is even and σ∈Γ∩𝔸n. Since z is an odd permutation and σ⊲z=z-r⁢σ⁢z, it follows that σ⊲z is even if and only if r is odd. Letting σ=(12⁢…⁢(n-1))∈Γ∩𝔸n, we find that r=b⁢(n)=σ⁢z⁢(n)=2; so that σ⊲z is an odd permutation. This shows that the subgroup Γ+=Γ∩𝔸n≅𝔸n-1 is not stable under the action ⊲ of 〈z〉 in this case.

Let us consider the associated Hopf algebras

Jn=#⁡k𝕊n-1k⁢Cn and Kn=#⁡k𝔸n-1k⁢Cn.

The categories Rep⁡Jn, Rep⁡Kn are Morita equivalent to the categories Rep⁡𝕊n and Rep⁡𝔸n, respectively; see Remark 4.1. In particular, Rep⁡Jn and Rep⁡Kn are not solvable, for all n≥5.

Observe that Jn* is a split abelian extension of kCn by k⁢𝕊n-1 associated to the actions ⊲′ and ⊳′ in Remark 4.4.

Remark 4.7

Suppose n≥5. It follows from [29, Theorem 5.2] that the Hopf algebras Jn, Kn, Jn*, Kn* admit no quasitriangular structure. In particular, the fusion categories Rep⁡Jn, Rep⁡Kn, Rep⁡Jn* and Rep⁡Kn* admit no braiding.

In addition, there are equivalences of braided fusion categories

(4.7)Rep⁡D⁢(Jn)≅Rep⁡D⁢(𝕊n),Rep⁡D⁢(Kn)≅Rep⁡D⁢(𝔸n).

It follows from Corollary 4.5 that there are group isomorphisms

G⁢(D⁢(𝕊n)*)≅𝕊n/[𝕊n,𝕊n]×Z⁢(𝕊n)≅ℤ2,

for all n≥3, and similarly, G⁢(D⁢(𝔸n)*)=1, for all n≥5.

Lemma 4.8

The pointed subcategory Rep⁡D⁢(Sn)pt≅Rep⁡Z2 is a Tannakian subcategory of Rep⁡D⁢(Sn).

Proof.

It follows from the description of the irreducible representations in (4.4), that the one-dimensional representations of D⁢(𝕊n) are parameterized by pairs (s,Us), where s∈𝕊n is a central element and Us is a one-dimensional representation of 𝕊n. Since Z⁢(𝕊n)={e} and 𝕊n has two non-isomorphic one-dimensional representations trivial one and the sign representation Sg, it follows that

Rep⁡D⁢(𝕊n)pt≅Rep⁡ℤ2.

Moreover, the unique nontrivial element of Rep⁡D⁢(𝕊n)pt corresponds to the pair (e,Sg). We have

s(e,Sg),(e,Sg)=|𝕊n||𝕊n|2⁢∑g∈𝕊nSg⁡(e)⁢Sg⁡(e)=|𝕊n||𝕊n|=1,

and

θ(e,Sg)=Sg⁡(e)deg⁢Sg=1,

where (sX,Y)X,Y∈Irr⁡(D⁢(𝕊n)) and θ denote the S-matrix and the ribbon structure of Rep⁡D⁢(𝕊n), respectively. See for instance [24, Section 3.1]. This shows that Rep⁡D⁢(𝕊n)pt is a Tannakian subcategory, as claimed. ∎

Lemma 4.9

Let n be an odd natural number. Then there is a central exact sequence of Hopf algebras

(4.8)k⟶kℤ2⟶Jn⁢⟶𝜋⁢Kn⟶k,

where the map π:Jn→Kn is induced by the inclusion An-1⊆Sn-1.

Proof.

The map π is defined in the form

π=j⊗id:Jn=#⁡k𝕊n-1k⁢Cn→Kn=#⁡k𝔸n-1k⁢Cn,

where j:k𝕊n-1→k𝔸n-1 is the canonical Hopf algebra map. Then π is a surjective Hopf algebra map.

Since the index of Kn in Jn is 2, it follows that Jnco⁡π≅k⁢ℤ2≅kℤ2 is a necessarily central Hopf subalgebra; see [27, Corollary 1.4.3]. ∎

Proposition 4.10

The following statements hold.

Rep⁡Jn is a ℤ2-extension of Rep⁡Kn, for all odd natural numbers n≥1.
Rep⁡Jn is a ℤ2-equivariantization of a fusion category 𝒟, for all even natural numbers n≥4.

Proof.

(i) This is an immediate consequence of Lemma 4.9. That is, since the sequence (4.8) is a central exact sequence of Hopf algebras, it follows that Rep⁡(Jn) is an ℤ2-graded fusion category with trivial component (Rep⁡(Jn))0=Rep⁡(Kn) (see [15, Theorem 3.8]). Therefore Rep⁡Jn is a ℤ2-extension of Rep⁡Kn.

(ii) Suppose that n≥4 is even. We first claim that Rep⁡Jn is not a ℤ2-extension of any fusion category. To see this, first note that it follows from [17, Lemma 3.4] that G⁢(Jn)=G⁢(k𝕊n-1)≅ℤ2, for all n≥2. Suppose that K is a central Hopf subalgebra of Jn such that K≅kℤ2; then K must necessarily coincide with k⁢G⁢(Jn). Observe that the action ⊲:𝕊n-1×Cn→𝕊n-1 gives rise to an action by algebra automorphism

⇀:Cn×k𝕊n-1→k𝕊n-1 such that x⇀eσ=eσ⊲x-1,

for all x∈Cn and σ∈𝕊n-1. If ϵ≠φ∈G⁢(Jn)=G⁢(k𝕊n-1), we have φ⁢(σ)=Sg⁡(σ), for all σ∈𝕊n-1 and φ=∑σ∈𝕊n-1Sg⁡(σ)⁡eσ.

Then G⁢(Jn)⊆Z⁢(Jn) if and only if z⇀φ=φ, if and only if 𝔸n-1 is stable under the action ⊲ of 〈z〉. This contradicts the observation in Remark 4.6. Hence Rep⁡Jn is not a ℤ2-extension of any fusion category, as claimed.

Let ℰ=Rep⁡D⁢(𝕊n)pt. By Lemma 4.8, ℰ≅Rep⁡ℤ2 is a Tannakian subcategory of Rep⁡D⁢(𝕊n). Since Rep⁡Jn is not a ℤ2-extension of any fusion category, it follows from [12, Propositions 2.9 and 2.10] that Rep⁡Jn must be a ℤ2-equivariantization of a fusion category 𝒟. We thus obtain (ii). ∎

Lemma 4.11

Let n≥5 be an odd natural number and let q be a prime number. Then the following hold:

The category Rep⁡Jn is not a ℤq-equivariantization of any fusion category.
Suppose that Rep⁡Jn is a ℤq-extension of a fusion category 𝒟~. Then q=2 and 𝒟~=Rep⁡Kn.
The category Rep⁡Kn is neither a ℤq-equivariantization nor a ℤq-extension of any fusion category.

Proof.

Let 𝒞 be one of the categories Rep⁡Jn or Rep⁡Kn. If 𝒞 is a ℤq-extension of a fusion category 𝒟~, then the Drinfeld center 𝒵⁢(𝒞) must contain a Tannakian subcategory equivalent to Rep⁡ℤq which maps to the trivial fusion subcategory Vect⊆𝒞 under the forgetful functor U:𝒵⁢(𝒞)→𝒞. In this case, the category 𝒟~ is canonically determined by the corresponding Tannakian subcategory. Dually, if 𝒞 is a ℤq-equivariantization, then 𝒵⁢(𝒞) must contain a Tannakian subcategory equivalent to Rep⁡ℤq which embeds into 𝒞 under the forgetful functor. See [12, Propositions 2.9 and 2.10].

Since n≥5, then the group of invertible objects of the Drinfeld center of 𝒞 coincides with ℤ2 if 𝒞=Rep⁡Jn, and it is trivial if 𝒞=Rep⁡Kn. Since Rep⁡ℤq is a pointed fusion category, we get (iii).

Suppose that 𝒞=Rep⁡Jn. Then 𝒞 is a ℤ2-extension of Rep⁡Kn. This implies that the pointed subcategory of 𝒵⁢(𝒞) is a Tannakian subcategory which maps to the trivial subcategory of 𝒞 under the forgetful functor. Hence, for every prime number q, 𝒞 is not a ℤq-equivariantization of any fusion category and we get (i). On the other hand, if it is ℤq-extension, then q=2 and the corresponding Tannakian subcategory of 𝒵⁢(𝒞) coincides with the pointed subcategory 𝒵⁢(𝒞)pt. Thus we obtain (ii). This finishes the proof of the lemma. ∎

Lemma 4.12

Let p be an odd prime number. Then the group G⁢(Jp*) is a semidirect product Cp^⋊〈σ〉, where σ∈Sp-1 is a (p-1)-cycle and the action of 〈σ〉 on Cp^ is induced by the action ⊲′:Cp×Sp-1→Cp. Moreover, the subgroup G⁢(Kp*)⊆G⁢(Jp*) is the semidirect product Cp^⋊〈σ2〉.

Proof.

The subgroup 𝕊p-1Cp of invariants of 𝕊p-1 under the action ⊳′ of Cp coincides with the subgroup of invariants under the action ⊲. It follows from [17, Corollary 5.2] that 𝕊p-1Cp is cyclic generated by a (p-1) cycle σ, i.e.

𝕊p-1Cp=〈σ〉

and therefore

G⁢(Jp*)≅Cp^⋊〈σ〉.

It follows from this that the invariant subgroup 𝔸p-1Cp is also cyclic generated by σ2. This implies the lemma, in view of Remark 4.4. ∎

Proposition 4.13

Let p be an odd prime number. Then the Hopf algebras Jp, Kp satisfy the following properties:

cd⁢(Jp)=cd⁢(Kp)={1,p}.
The groups G⁢(Jp*) and G⁢(Kp*) have trivial centers.

Proof.

Part (i) follows from the description of irreducible representations of crossed products in [21].

We next show (ii). Recall that Cp=〈z〉, where z=(12⁢…⁢p) and the actions ⊲, ⊳ are determined by the relation s⁢c=(s⊳c)⁢(s⊲c) in 𝕊p. So that the actions ⊲′, ⊳′ are determined by c⁢s=(c⊳′s)⁢(c⊲′s) in 𝕊p, for all s∈𝕊p-1 and c∈Cp.

It follows from the proof of [17, Lemma 3.2] that

z⊲′ai=zi,

for all i=1,…,p-1, where ai=(p-1,p-i). In addition, the stabilizer of z under the action ⊲′ coincides with the subgroup

Fz={a∈𝕊p-1:a⁢(p-1)=p-1}≅𝕊p-2.

These imply that, for all i=1,…,p-1, the stabilizer of zi coincides with the subgroup

Fzi={a∈𝕊p-1:a⁢(p-i)=p-i}.

In particular, Cp has no nontrivial fixed points under the action ⊲′.

On the other hand, the nontrivial powers of the (p-1)-cycle σ∈𝕊p-1 have no fixed point in {1,…,p-1}. Hence no nontrivial power of σ acts trivially on Cp under the action ⊲′.

By Lemma 4.12, G⁢(Jp*)=Cp^⋊〈σ〉 is a semidirect product with respect to the action induced by ⊲′, where σ is a (p-1)-cycle in 𝕊p-1. Then the center of G⁢(Jp*) consists of all pairs (e,x), where x∈〈σ〉 acts trivially on Cp under the action ⊲′.

Similarly, G⁢(Kp*)=Cp^⋊〈σ2〉 is a semidirect product with respect to the action induced by ⊲′ and the center of G⁢(Kp*) consists of all pairs (e,x), where x∈〈σ2〉 acts trivially on Cp under the action ⊲′.

Thus we obtain that the centers of the groups G⁢(Jp*) and G⁢(Kp*) are both trivial. ∎

Theorem 4.14

Let C~ be a fusion category. Suppose that C~ is Grothendieck equivalent to one of the categories Rep⁡Jp or Rep⁡Kp. Then C~ is not solvable.

Proof.

Suppose on the contrary that 𝒞~ is solvable and Grothendieck equivalent to 𝒞, where 𝒞=Rep⁡Jp or 𝒞=Rep⁡Kp. It follows from Proposition 3.2 that G⁢(𝒞~)≅G⁢(𝒞). By Proposition 4.13, the groups of invertible objects of Rep⁡Jp and Rep⁡Kp have trivial center. Then the center of G⁢(𝒞~) is trivial as well. It follows from Lemma 2.3 that, for every prime number q, the category 𝒞~ is not a ℤq-equivariantization of any fusion category. Therefore 𝒞~ must be a ℤq-extension of a fusion subcategory 𝒟~, and 𝒟~ is also a solvable fusion category. Hence 𝒞 is a ℤq-extension of a fusion subcategory 𝒟 such that 𝒟~ is Grothendieck equivalent to 𝒟. It follows from Proposition 4.10 and Lemma 4.11 that 𝒞=Rep⁡Jp, q=2 and 𝒟=Rep⁡Kp. Applying the same argument to the solvable fusion category 𝒟~ we get a contradiction. This shows that 𝒞~ cannot be solvable and finishes the proof of the theorem. ∎

4.3 Fusion rules of Rep⁡K5

In this subsection we determine explicitly the fusion rules of the category Rep⁡Kp in the case p=5. It follows from [21] that simple objects of the category Rep⁡K5 are parameterized by pairs (s,ρ), where s runs over a set of representatives of the orbits of C5 on 𝔸4 and ρ is an irreducible representation of the stabilizer Fs⊆C5. The dimension of the simple object Ss,ρ corresponding to the pair (s,ρ) is given by the formula dimSs,ρ=[C5:Fs].

The C5-action on 𝕊4 is explicitly determined in [17, Table 1]. We have in this case that there are ten fixed points and the remaining two orbits consist of five distinct elements each. Furthermore, there are exactly four distinct fixed points σ such that σ=σ-1 and both nontrivial orbits contain elements of order 2. In view of [17, Theorem 4.8], Rep⁡K5 has five invertible objects of order 2 and the five-dimensional simple objects are self-dual.

Let us denote by Y,Y′ the simple objects corresponding to the nontrivial orbits 𝒪,𝒪′, respectively. By [17, Table 1], we have

𝒪={(123),(243),(132),(13)⁢(24),(234)},

𝒪′={(124),(143),(134),(12)⁢(34),(142)}.

By Lemma 4.12, the group G⁢(Rep⁡K5) is isomorphic to the dihedral group D5 of order 10. The unique subgroup R of order 5 of G⁢(Rep⁡K5) coincides therefore with the stabilizer of Y and Y′ under left (or right) multiplication. Since every element s outside of R is of order 2, it follows that s⊗Y≅Y⊗s≅Y′. So that we have a decomposition

(4.9)Y⊗Y*≅Y⊗Y≅⊕r∈Rr⊕a⁢Y⊕b⁢Y′,

where a,b≥0 and a+b=4. Letting F:Rep⁡K5→𝒞⁢(𝔸4)=Rep⁡k𝔸4 denote the restriction functor, we obtain

F⁢(Y)=V(123)⊕V(243)⊕V(132)⊕V(13)⁢(24)⊕V(234),

F⁢(Y′)=V(124)⊕V(143)⊕V(134)⊕V(12)⁢(34)⊕V(142),

where, for each s∈𝔸4, Vs denotes the one-dimensional simple k𝔸4-module corresponding to s. Comparing these relations with (4.9), we find that a=b=2. Hence the fusion rules of Rep⁡K5 are determined by the condition

G=G⁢(Rep⁡K5)≅D5,g⊗Y=Y⊗g=Y′,

for every element g of order 2 of G, and

Y⊗Y≅⊕r∈Rr⊕2⁢Y⊕2⁢Y′≅Y′⊗Y′,

where R is the unique subgroup of order 5 of G.

4.4 The dual Hopf algebras Jn, Kn

Let n≥2 be a natural number and let Hn=Jn*. Recall that there is a split exact sequence of Hopf algebras

(4.10)k⟶kCn⟶Hn⟶k⁢𝕊n-1⟶k.

Suppose that n is odd. Let Ln=Kn*, so that there is a split exact sequence of Hopf algebras

(4.11)k⟶kCn⟶Ln⟶k⁢𝔸n-1⟶k.

Moreover, by Lemma 4.9 there is a cocentral exact sequence

(4.12)k⟶Ln⟶Hn⟶k⁢ℤ2⟶k.

Remark 4.15

Suppose n≥5. Since D⁢(Hn)≅D⁢(Jn), we have

G⁢(Rep⁡D⁢(Hn))≅ℤ2.

Let q be a prime number. If the category Rep⁡Hn is a ℤq-extension or a ℤq-equivariantization of a fusion category, then q=2.

Suppose n is odd. In view of [28, Proposition 3.5], Rep⁡Hn is a ℤ2-equivariantization of Rep⁡Ln. As in the proof of Lemma 4.11, we obtain that if n≥5, the category Rep⁡Hn is not a ℤq-extension of any fusion category. Similarly, Rep⁡Ln is not a ℤq-extension or a ℤq-equivariantization of any fusion category.

Lemma 4.16

Let n≥5 be a natural number. Then the following hold.

G⁢(Rep⁡Hn)≅ℤ2.
Rep⁡(H5) is of type (1,2;2,1;3,2;4,2;8,1) and Hn has no irreducible representation of dimension 2 , for all n>5.

Assume in addition that n is odd. Then

G⁢(Rep⁡Ln)=1, if n>5.
Rep⁡L5 is of type (1,3;3,1;4,3) and Ln has no irreducible representation of dimension 2.

Proof.

Consider the exact sequences (4.10) and (4.11). The respective invariant subgroups Cn𝕊n-1 and Cn𝔸n-1 are both trivial. Parts (i) and (iii) follow from Lemma 4.3.

Since Hn is a split abelian extension of kCn by k⁢𝕊n-1 and the action of 𝕊n-1 has two orbits {e} and {z,…,zn-1}, then the simple Hn-modules are classified by pairs (t,ρ), where either t=e and ρ is an irreducible representation of Fe=𝕊n-1, or t=z and ρ is an irreducible representation of

Fz={a∈𝕊n-1:a⁢(n-1)=n-1}≅𝕊n-2.

See [17, Lemma 3.2]. If St,ρ is the simple module corresponding to the pair (t,ρ), we have in addition dim⁡Se,ρ=dim⁡ρ, and

dimSz,ρ=[𝔸n-1:Fz]dimρ=(n-1)dimρ.

This implies the statement for H5 in part (ii).

Suppose that n>5. Then dim⁡Sz,ρ>2, for all ρ. As observed in Remark 3.6, the symmetric group 𝕊n-1 has no irreducible representation of degree 2. Therefore we also get that dim⁡Se,ρ≠2, for all ρ. In conclusion Hn has no irreducible representation of dimension 2, and we obtain (ii).

Suppose that n is odd. Similarly, Ln is a split abelian extension of kCn by k⁢𝔸n-1 and the action of 𝔸n-1 has two orbits {e} and {z,…,zn-1}. Hence the simple Ln-modules are classified by pairs (t,ρ), where either t=e and ρ is an irreducible representation of Fe∩𝔸n-1=𝔸n-1, or t=z and ρ is an irreducible representation of Fz∩𝔸n-1≅𝔸n-2. This implies that Rep⁡L5 is of the prescribed type. As before, dim⁡Sz,ρ>2, for all ρ, and since the alternating group 𝔸n-1 has no irreducible representation of degree 2, then also dim⁡Se,ρ≠2, for all ρ. So that Ln has no irreducible representation of dimension 2. This proves part (iv) and finishes the proof of the lemma. ∎

Lemma 4.17

Let C be a fusion category of type (1,3;3,1;4,3). Then C is not solvable.

Proof.

The assumption on the type of 𝒞 implies that the simple objects of dimensions 1 and 3 generate a fusion subcategory ℬ of 𝒞 of type (1,3;3,1) and moreover, every fusion subcategory of 𝒞 is contained in ℬ.

Suppose first that 𝒞 is a ℤq-extension of a fusion subcategory 𝒞0, for some prime number q. Then necessarily 𝒞0=ℬ and q=5. Hence we have a ℤ5-faithful grading 𝒞=𝒞0⊕𝒞1⊕…⊕𝒞4, with trivial component 𝒞0=ℬ. But 𝒞 has only three classes of simple objects outside of ℬ, entailing that such a decomposition is impossible.

Therefore 𝒞 must be a ℤq-equivariantization of a fusion category 𝒟, for some prime number q, where 𝒟 is also a solvable fusion category. Thus q=3 and 𝒞≅𝒟ℤ3. The description of simple objects of 𝒟ℤ3 together with the assumption on the type of 𝒞 imply that 𝒟 must be of type (1,4;4,1); cf. formula (2.2). Moreover, the action (by group automorphisms) of ℤ3 on the set of nontrivial invertible objects of 𝒟 must be transitive, hence G⁢(𝒟)≅ℤ2×ℤ2.

On the other hand, letting X be the unique noninvertible simple object of 𝒟, we must have

X⊗2≅⊕Y∈G⁢(𝒟)Y⊕3⁢X.

This means that the fusion category 𝒟 is a near-group fusion category of type (G,3), where G=G⁢(𝒟). Then it follows from [32, Theorem 1.2] that the group G⁢(𝒟) is cyclic, which leads to a contradiction. Therefore 𝒞 cannot be solvable. This finishes the proof of the lemma. ∎

Theorem 4.18

Let C be a fusion category and let n≥5 be a natural number. Then we have:

If 𝒞 is Grothendieck equivalent to Rep⁡Hn, then 𝒞 is not solvable.
Suppose that n is odd. If 𝒞 is Grothendieck equivalent to Rep⁡Ln, then 𝒞 is not solvable.

Proof.

To show (i), suppose first that n>5. By Lemma 4.16, G⁢(Rep⁡Hn)≅ℤ2 and Rep⁡Hn has no simple objects of dimension 2. The claim follows in this case from Proposition 3.5.

Consider the case n=5. Suppose on the contrary that 𝒞 is Grothendieck equivalent to Rep⁡H5 and 𝒞 is solvable. Then 𝒞 is of type (1,2;2,1;3,2;4,2;8,1) and, for any prime q, 𝒞 is not a ℤq-extension of any fusion subcategory, in view of Remark 4.15. Therefore 𝒞 must be a ℤ2-equivariantization of a solvable fusion category 𝒟 of dimension 60. The description of the simple objects of 𝒟ℤ2 together with the assumption on the type of 𝒞 imply that 𝒟 must be of type (1,3;3,1;4,3). Lemma 4.17 implies that 𝒟 is not solvable, which is a contradiction. Thus we get (i).

Let us show (ii). If n=5, the result follows from Lemma 4.17. Suppose next that n>5. Since a solvable fusion category contains nontrivial invertible objects, then part (ii) follows from Lemma 4.16 (iii). ∎

4.5 Further examples associated to the symmetric group

Let n≥2 be a natural number. Consider the matched pair (𝔸n,C2), where C2=〈(12)〉⊆𝕊n, the action ⊲:𝔸n×C2→𝔸n is given by conjugation in 𝕊n and the action ⊳:𝔸n×C2→C2 is trivial. The associated group 𝔸n⋈C2 is isomorphic to the symmetric group 𝕊n.

Let Bn=#⁡k𝔸nk⁢C2 be the split abelian extension associated to this matched pair. We have cd⁢(Bn)={1,2}. The fusion category Rep⁡Bn is Morita equivalent to 𝕊n and therefore it is not solvable if n≥5.

Remark 4.19

Since the action ⊳ is the trivial one and C2≅ℤ2, there is a cocentral exact sequence

(4.13)k⟶k𝔸n⟶Bn⟶k⁢ℤ2⟶k.

By [28, Proposition 3.5], Rep⁡Bn is a ℤ2-equivariantization of Rep⁡k𝔸n=𝒞⁢(𝔸n).

Moreover, since Rep⁡Bn is Morita equivalent to 𝕊n and the group of invertible objects of 𝒵⁢(Rep⁡𝕊n) is cyclic of order 2, it follows that for all prime number q, Rep⁡Bn is not a ℤq-extension of any fusion category and if it is a ℤq-equivariantization, then q=2 (compare with Proposition 4.10). In particular, not being a ℤq-extension, Bn has no nontrivial central group-like elements; that is, we have Z⁢(Bn)∩G⁢(Bn)={1}.

Our first statement concerns the dual Hopf algebra Bn*.

Theorem 4.20

Let C be a fusion category. Suppose that C is Grothendieck equivalent to Rep⁡Bn*, n≥5. Then C is not solvable.

Proof.

The dual Hopf algebra Bn* fits into a central exact sequence

(4.14)k⟶kℤ2⟶Bn*⟶k⁢𝔸n⟶k.

Therefore Rep⁡Bn* is a ℤ2-extension of Rep⁡𝔸n. Hence 𝒞 is a ℤ2-extension of a fusion category 𝒟, which is Grothendieck equivalent to Rep⁡𝔸n.

Suppose on the contrary that 𝒞 is solvable. Then so is 𝒟 and therefore we have 𝒟pt≠Vect. This implies that 𝔸n has nontrivial one-dimensional representations, which is a contradiction. Then 𝒞 cannot be solvable, as claimed. ∎

Lemma 4.21

The group G⁢(Rep⁡Bn) of invertible objects of the category Rep⁡Bn is isomorphic to the direct product C2^×CAn⁢(12), where CAn⁢(12) denotes the centralizer in An of the transposition (12).

Proof.

There is a group isomorphism G⁢(Rep⁡Bn)≅G⁢(Bn*). On the other hand, Bn* is a split abelian extension Bn*≅#⁡kC2k⁢𝔸n, associated to the adjoint action ⊳′:C2×𝔸n→𝔸n and the trivial action ⊲′:C2×𝔸n→C2. The result follows from Lemma 4.3. ∎

Theorem 4.22

Suppose n≥5. Let C~ be a fusion category Grothendieck equivalent to Rep⁡Bn. Then C~ is not solvable.

Proof.

Suppose first that n≥7. By Lemma 4.21, G⁢(Rep⁡Bn)≅C2^×C𝔸n⁢(12). Note that C𝔸n⁢(12) contains the subgroup

{σ∈𝔸n:σ⁢(1)=1,σ⁢(2)=2}≅𝔸n-2.

Since n≥7, the group 𝔸n-2 is not solvable. Then G⁢(𝒞~) is not solvable either and then 𝒞~ is not solvable.

It remains to consider the cases n=5 and 6. It follows from Lemma 4.21 that G⁢(Rep⁡B5)≅C2^×𝕊3 is non-abelian of order 12. Hence Rep⁡B5 is of type (1,12;2,27). Similarly, G⁢(Rep⁡B6)≅C2^×𝕊4 is non-abelian of order 48 and Rep⁡B6 is of type (1,48;2,168).

Suppose that there exist a solvable fusion category 𝒞~ which is Grothendieck equivalent to 𝒞, where 𝒞=Rep⁡B5 or Rep⁡B6. By Proposition 3.2, G⁢(𝒞~)≅G⁢(𝒞). Since, for every prime number q, 𝒞 is not a ℤq-extension of any fusion category, we have that 𝒞~ must be a ℤq-equivariantization of a fusion subcategory 𝒟, and 𝒟 is also a solvable fusion category. Moreover, q=2 because ℤq⊆Z⁢(G⁢(𝒞~)) and FPdim⁡𝒟=60 or FPdim⁡𝒟=360, respectively. Then there is an exact sequence of fusion categories

Rep⁡ℤ2⟶𝒞~⟶𝒟.

Since cd⁢(𝒞~)={1,2}, it follows that cd⁢(𝒟)={1,2}. The previous exact sequence induces an exact sequence of groups

1⟶ℤ^2⟶G⁢(𝒞~)⟶G0⁢(𝒟)⟶1,

where ℤ^2 denotes the group of invertible characters of ℤ2 and G0⁢(𝒟) is the subgroup of G⁢(𝒟) consisting of isomorphism classes of invertible objects which are ℤ2-equivariant. See [5, Remark 3.1]. As ℤ2 is a cyclic group, we have that G0⁢(𝒟) coincides with the subgroup of fixed points of the induced action of ℤ2 on the group of invertible objects of 𝒟.

Observe that since 𝒞 is also a ℤ2-equivariantization of 𝒞ℤ2≅𝒞⁢(𝔸5) or 𝒞⁢(𝔸6), respectively, the group G⁢(𝒞)≅G⁢(𝒞~) also fits into an exact sequence

1⟶ℤ^2⟶G⁢(𝒞)⟶G0⁢(𝒞ℤ2)⟶1.

In this case, the subgroup G0⁢(𝒞ℤ2) is isomorphic to C𝔸5⁢(12) or C𝔸6⁢(12), respectively. In addition, the group G⁢(𝒞) contains a unique normal subgroup of order 2. Therefore, G0⁢(𝒟)≅G0⁢(𝒞ℤ2) is a non-abelian subgroup of G⁢(𝒟).

Suppose first that n=5. In this case 𝒞=Rep⁡B5≅𝒞⁢(𝔸5)ℤ2. In particular, G0⁢(𝒟) is a subgroup of order 6 of G⁢(𝒟). A counting argument shows that G⁢(𝒟) can be of type (1,12;2,12) or else 𝒟 is pointed. Suppose that 𝒟 is of type (1,12;2,12). Then 𝒟ptℤ2⊆𝒞~ is a fusion subcategory of dimension 24 and type (1,12;2,3), containing the central subcategory Rep⁡ℤ2. Therefore 𝒞 has a fusion subcategory ℬ of type (1,12;2,3) containing the central subcategory Rep⁡ℤ2. (In fact, as observed before, G⁢(𝒞) contains a unique normal subgroup of order 2; see Lemma 2.3.)

Consider the de-equivariantization ℬℤ2⊆𝒞⁢(𝔸5). We have that dim⁡ℬℤ2=12 and thus ℬℤ2=𝒞⁢(H), where H is a (ℤ2-stable) subgroup of 𝔸5 of order 12. Since G⁢(Rep⁡B5)⊆ℬ, it follows that the subgroup H contains the invariant subgroup 𝔸5ℤ2=C𝔸5⁢(12)≅𝕊3. On the other hand, every subgroup of order 12 of 𝔸5 is isomorphic to 𝔸4, then H≅𝔸4. This leads to a contradiction, because 𝔸4 has no subgroup of order 6. This proves that 𝒟 cannot be of type (1,12;2,12).

Therefore 𝒟 must be a pointed fusion category. In this case 𝒟=𝒞⁢(Γ,ω), where ω:Γ×Γ×Γ→k* is a 3-cocyle and Γ is a solvable non-abelian subgroup of order 60. In addition ℤ2 acts on Γ by group automorphisms and Γℤ2≅𝕊3. Since Γ≠𝔸5, Γ can be isomorphic to 𝔸4×ℤ5, ℤ15⋊ℤ4 or ℤ15⋊(ℤ2×ℤ2).

If Γ≅𝔸4×ℤ5, the action of ℤ2 must fix 𝔸4 and ℤ5. Since |Γℤ2|=6, it follows that Γℤ2⊆𝔸4, and we reach a contradiction. Therefore Γ≅ℤ15⋊ℤ4 or ℤ15⋊(ℤ2×ℤ2). In this case Γ has a unique subgroup L of order 15, and then L is ℤ2-stable and 𝒞⁢(L)ℤ2 is a fusion subcategory of 𝒞~ of dimension 30. This implies that 𝒞 has a fusion subcategory of dimension 30. Such fusion subcategory must correspond to a quotient Hopf algebra of B5 of dimension 30, which is a contradiction, because Z⁢(B5)∩G⁢(B5)={1}. See [27, Corollary 1.4.3]. Thus 𝒟 cannot be pointed. This proves that if 𝒞~ is Grothendieck equivalent to Rep⁡B5 then 𝒞~ is not solvable.

Finally, let us consider the case n=6. In this case we have

𝒞:=Rep⁡B6≅𝒞⁢(𝔸6)ℤ2.

On the other hand, G0⁢(𝒟) is a subgroup of order 24 of G⁢(𝒟). As before, one can see that 𝒟 must be of type (1,24;2,84), (1,72;2,72), (1,120;2,60), or else 𝒟 is pointed.

Suppose that 𝒟 is of type (1,72;2,72) or (1,120;2,60). In these cases we have that 𝒟ptℤ2⊆𝒞~ is a fusion subcategory of dimension 144 or 240, respectively, containing the central subcategory Rep⁡ℤ2. Therefore 𝒞 has a fusion subcategory ℬ of dimension 144 or 240, respectively, containing the central subcategory Rep⁡ℤ2. The de-equivariantization ℬℤ2⊆𝒞⁢(𝔸6) is of dimension dim⁡ℬℤ2=72 or 120, respectively. Then ℬℤ2=𝒞⁢(H), where H is a ℤ2-stable subgroup of 𝔸6 of order 72 or 120, respectively. Since 𝔸6 has no subgroups of order 72 or 120, it follows that these types are not possible for 𝒟.

Suppose next that 𝒟 is of type (1,24;2,84). It follows from the description of the simple objects of 𝒟G and the fact that cd⁢(𝒟)=cd⁢(𝒞)={1,2}, that ℤ2 acts trivially on the set Irr⁡(𝒟); see (2.2). In particular,

G⁢(𝒟)=G0⁢(𝒟)≅C𝔸6⁢(12)≅𝕊4.

Since 𝒟 is solvable, then it is a ℤp-extension or a ℤp-equivariantization, where p is a prime number that divides the dimension of 𝒟, which is 360. If 𝒟 were a ℤp-equivariantization, then, by Lemma 2.3, ℤp⊆Z⁢(G⁢(𝒟)), which is a contradiction because G⁢(𝒟)≅𝕊4. Therefore 𝒟 must be a ℤp-extension of a fusion subcategory 𝒟e. The fusion subcategory 𝒟e is of dimension 72, 120 or 180. Furthermore, 𝒟e must be stable under the action of ℤ2, since this action is trivial on Irr⁡(𝒟). As before, this implies that 𝒞 contains a fusion subcategory ℬ of dimension 144, 240 or 360, respectively, containing the central subcategory Rep⁡ℤ2. Hence ℬℤ2=𝒞⁢(H), where H is a ℤ2-stable subgroup of 𝔸6 with order 72, 120 or 180, respectively. But 𝔸6 has no subgroups neither of order 72, 120 nor 180, therefore the type (1,24;2,84) is also impossible for 𝒟.

Suppose finally that 𝒟 is a solvable pointed fusion category. We have 𝒟=𝒞⁢(Γ,ω), where ω:Γ×Γ×Γ→k* is a 3-cocyle and Γ is a solvable non-abelian subgroup of order 360. In addition ℤ2 acts on Γ by group automorphisms and the subgroup Γ0 of fixed points of Γ under this ℤ2-action is of order 24. Let S be a Sylow 5-subgroup of Γ. Since Γ is solvable, there exist H, a Hall {2,5}-subgroup of Γ, and K, a Hall {3,5}-subgroup of Γ, such that S⊆H and S⊆K. A counting argument shows that S⊴H and S⊴K and so S⊴〈H,K〉=Γ. Hence S is the unique Sylow 5-subgroup of Γ and then S is ℤ2-stable. In this case 𝒞⁢(S,ω|S)ℤ2 is a fusion subcategory of 𝒞~ of dimension 10, containing the central subcategory Rep⁡ℤ2. Therefore 𝒞 has a fusion subcategory ℬ with dimension 10, containing the central subcategory Rep⁡ℤ2. The de-equivariantization ℬℤ2⊆𝒞⁢(𝔸6) is of dimension 5. Then ℬℤ2=𝒞⁢(T), where T is a ℤ2-stable subgroup of 𝔸6 of order 5. We have that

T={id,(a⁢b⁢c⁢d⁢e),(a⁢c⁢e⁢b⁢d),(a⁢d⁢b⁢e⁢c),(a⁢e⁢d⁢c⁢b)},

and without loss of generality we may assume a=1 and b=2. We thus reach a contradiction, since (12)⁢(12⁢c⁢d⁢e)⁢(12)=(21⁢c⁢d⁢e)≠(1⁢c⁢e⁢2⁢d). This proves that 𝒞~ cannot be solvable and finishes the proof of the theorem. ∎

5 Solvability and fusion rules of a braided fusion category

Let 𝒞 be a fusion category. Suppose that FPdim⁡𝒞 is an integer (which is always the case if 𝒞 is solvable). Then the adjoint subcategory 𝒞ad is integral [11, Proposition 8.27].

Assume that 𝒞 is braided. Recall that 𝒞 is solvable and integral; then either it is pointed or it contains a nontrivial Tannakian subcategory [30, Proposition 5.2].

Theorem 5.1

Let C, C~ be Grothendieck equivalent braided fusion categories. Suppose that C is solvable. Then the following hold:

𝒞~pt is a solvable fusion category and it is not trivial if 𝒞~ is not trivial.
If 𝒞~ is not pointed, then it contains a nontrivial Tannakian subcategory.

Proof.

We may assume that 𝒞 and 𝒞~ are not trivial. Since 𝒞 is solvable, it follows from [12, Proposition 4.5 (iv)] that 𝒞pt≠Vect. Therefore 𝒞~pt is not trivial.

Since 𝒞~ is braided, the pointed subcategory 𝒞~pt has abelian fusion rules, and therefore it is solvable. This shows part (i).

Suppose that 𝒞~ is not pointed, so that 𝒞 is not pointed either. Note that 𝒞ad is Grothendieck equivalent to 𝒞~ad. If 𝒞ad is a proper fusion subcategory, then an inductive argument implies that 𝒞~ad is solvable and therefore so is 𝒞~, because it is a U⁢(𝒞~)-extension of 𝒞~ad and the universal grading group U⁢(𝒞~) is abelian. Hence we may assume that 𝒞~ad=𝒞~ (in particular, the same is true for 𝒞). Since 𝒞 is solvable, its Frobenius–Perron dimension is an integer and therefore 𝒞 is in fact integral. Then 𝒞~ is also integral. To show part (ii) we may assume that 𝒞~ is not solvable, in view of [30, Proposition 5.2].

By part (i), 𝒞~pt is solvable and not trivial. Note that 𝒞~ cannot contain any nontrivial non-degenerate fusion subcategory. In fact, if 𝒞 were non-degenerate, then 𝒞~ad=𝒞~pt′⊊𝒞~, against the assumption. If, on the other hand, 𝒟~⊆𝒞~ were a proper non-degenerate subcategory, then 𝒞~≅𝒟~⊠𝒟~′, and both 𝒟~ and 𝒟~′ are Grothendieck equivalent to fusion subcategories of 𝒞. An inductive argument implies that 𝒟~ and 𝒟~′ are solvable and therefore so is 𝒞~.

Suppose that 𝒞~ contains no nontrivial Tannakian subcategory. It follows from [30, Lemma 7.1] that 𝒞~pt=𝒞~′≅sVect and G⁢[X~]=𝟏, for all simple objects X~ of 𝒞~. This implies that FPdim⁡𝒞pt=2 and G⁢[X]=𝟏, for all simple objects X of 𝒞.

On the other hand, since 𝒞 is solvable and 𝒞ad=𝒞, then 𝒞 must be a ℤp-equivariantization of a fusion category 𝒟 for some prime number p. In particular, 𝒞 contains a (pointed) fusion subcategory of dimension p, and therefore p=2. It follows from Lemma 2.1 that 𝒞 has a simple object X of Frobenius–Perron dimension 2. In addition, for every such simple object X, we have G⁢[X]=𝟏.

The Nichols–Richmond theorem implies that 𝒞 contains a fusion subcategory 𝒞¯ of type (1,2;2,1;3,2) or (1,3;3,1) or (1,1;3,2;4,1;5,1); see [31, Theorem 11] and [8, Theorem 3.4]. The first possibility cannot hold in this case, because the unique simple object of dimension 2 of 𝒞¯ is necessarily stable under the action of G⁢(𝒞¯)≅ℤ2. The second possibility contradicts the assumption that FPdim⁡𝒞pt is equal to 2. The third possibility is also discarded because 𝒞¯ must be a solvable fusion category, whence 𝒞¯pt≠Vect. This contradiction shows that 𝒞~ must contain a Tannakian subcategory, and hence (ii) holds. ∎

Proposition 5.2

Let C be a braided fusion category. Suppose that E⊆C is a Tannakian subcategory. Then C is solvable if and only if E′ is solvable.

Proof.

If 𝒞 is solvable, then every fusion subcategory of 𝒞 is solvable. In particular, ℰ′ is solvable, showing the ‘only if’ direction. Conversely, suppose that ℰ′ is solvable. Since ℰ is a Tannakian subcategory, it is symmetric, and therefore ℰ⊆ℰ′. Then ℰ is solvable. Let G be a finite group such that ℰ≅Rep⁡G as braided fusion categories. Then the group G is solvable, by [12, Proposition 4.5 (ii)].

Consider the G-crossed braided fusion category 𝒞G, so that 𝒞≅(𝒞G)G is an equivariantization. Furthermore, the category 𝒞G is a G-graded fusion category, and the neutral component 𝒞G0 of this grading satisfies (𝒞G0)G≅ℰ′ (see [10, Proposition 4.56 (i)]). Therefore 𝒞G0 is solvable. Since G is solvable, so is 𝒞G and also 𝒞≅(𝒞G)G. This proves the ‘if’ direction and finishes the proof of the proposition. ∎

Remark 5.3

Let 𝒞~ be a braided fusion category. Suppose that 𝒞~ is Grothendieck equivalent to a solvable braided fusion category 𝒞 and 𝒞~ is not solvable. Assume in addition that FPdim⁡𝒞~ is minimal with respect to these properties.

Then 𝒞~ must satisfy the following conditions:

𝒞~ad=𝒞~.
𝒞~pt≠Vect is a solvable fusion subcategory and (𝒞~pt)′=𝒞~.
𝒞~ contains a nontrivial Tannakian subcategory and for every Tannakian subcategory ℰ~, ℰ~′=𝒞~.
𝒞~ contains no proper non-degenerate fusion subcategories.

Indeed, (i) and (iv) can be shown as in the proof Theorem 5.1, (ii) follows from (i) and Lemma 2.6, and (iii) follow from Theorem 5.1 and Proposition 5.2.

6 The character table of a spherical fusion category

6.1 Spherical fusion categories

A spherical structure on a fusion category 𝒞 is a natural isomorphism of tensor functors ψ:id𝒞→()** such that

d+⁢(X)=d-⁢(X),

for all objects X of 𝒞, where d±⁢(X)=Tr±⁡(idX), and for every endomorphism f:X→X, Tr±⁡(f)∈k are defined as the compositions

Tr+⁡(f):𝟏⟶X⊗X*→ψX⁢f⊗idX**⊗X⟶𝟏

and

Tr-⁡(f):𝟏⟶X*⊗X**→id⊗f⁢ψX-1X*⊗X⟶𝟏.

Let 𝒞 be a spherical fusion category, that is, a fusion category endowed with a spherical structure. The quantum dimension of X∈𝒞 is denoted by

dX:=d+⁢(X)=d-⁢(X),

and the quantum dimension of 𝒞 is defined in the form dim⁡𝒞=∑X∈Irr⁡(𝒞)dX2. The quantum trace of an endomorphism f:X→X is denoted by

Tr⁡(f)=Tr+⁡(f)=Tr-⁡(f).

See [10, Section 2.4.3] and [11, Section 2.2].

Recall that a fusion category is called pseudo-unitary if its global dimension coincides with its Frobenius–Perron dimension. By [11, Proposition 8.24], every weakly integral fusion category is pseudo-unitary. It is shown in [11, Proposition 8.23] that every pseudo-unitary fusion category admits a canonical spherical structure with respect to which quantum dimensions of objects coincide with their Frobenius–Perron dimensions.

6.2 Modular categories and S-matrices

A premodular category is a braided fusion category equipped with a spherical structure. Equivalently, 𝒞 is a braided fusion category endowed with a ribbon structure, that is, a natural automorphism θ:id𝒞→id𝒞 satisfying

(6.1)θX⊗Y=(θX⊗θY)⁢cY,X⁢cX,Y,θX*=θX*,

for all objects X,Y of 𝒞 (see [2] and [10, Section 2.8.2]).

Let 𝒞 be a premodular category. The central charge of 𝒞 is the ratio

ξ⁢(𝒞)=τ+⁢(𝒞)dim⁡𝒞,

where dim⁡𝒞 is the positive square root and τ+⁢(𝒞)=∑X∈Irr⁡(𝒞)θX⁢dX2. See [10, Section 6.2].

The S-matrix of 𝒞 is defined in the form S=(SX⁢Y)X,Y∈Irr⁡(𝒞), where for all X,Y∈Irr⁡(𝒞),

SX,Y=Tr⁡(cY,X⁢cX,Y)∈k

is the quantum trace of the squared braiding cY,X⁢cX,Y:X⊗Y→X⊗Y.

A premodular category 𝒞 is called modular if the S-matrix is non-degenerate [33] or, equivalently, if it is non-degenerate [10, Proposition 3.7].

If 𝒞 is a spherical fusion category, its Drinfeld center 𝒵⁢(𝒞) is a modular category of global dimension dim⁡𝒵⁢(𝒞)=(dim⁡𝒞)2 and central charge ξ⁢(𝒵⁢(𝒞))=1 (see [22] and [10, Example 6.9]).

Suppose that 𝒞 is a modular category. Then for every X,Y,Z∈Irr⁡(𝒞), the multiplicity NX⁢YZ of Z in the tensor product X⊗Y is given by the Verlinde formula

(6.2)NX⁢YZ=1dim⁡𝒞⁢∑T∈Irr⁡(𝒞)SX⁢T⁢SY⁢T⁢SZ*⁢TdT,

where dT denotes the quantum dimension of the object T and dim⁡𝒞 is the quantum dimension of 𝒞. See [1, Theorem 3.1.14].

6.3 S-equivalence of spherical fusion categories

Definition 6.1

Let 𝒞 and 𝒟 be spherical fusion categories. We shall say that 𝒞 and 𝒟 are S-equivalent if there exists a bijection f:Irr⁡(𝒵⁢(𝒞))→Irr⁡(𝒵⁢(𝒟)) such that f⁢(𝟏)=𝟏 and Sf⁢(X),f⁢(Y)=SX,Y, for all X,Y∈Irr⁡(𝒞).

The following lemma summarizes some of the main properties of S-equivalence.

Lemma 6.2

Let C and D be spherical fusion categories. Furthermore, suppose that f:Irr⁡(Z⁢(C))→Irr⁡(Z⁢(D)) is an S-equivalence. Then the following hold:

df⁢(X)=dX, for all X∈Irr⁡(𝒞). In particular, dim⁡𝒵⁢(𝒞)=dim⁡𝒵⁢(𝒟).
f:Irr⁡(𝒵⁢(𝒞))→Irr⁡(𝒵⁢(𝒟)) is a Grothendieck equivalence.
For every fusion subcategory ℰ of 𝒵⁢(𝒞) we have f⁢(ℰ′)=f⁢(ℰ)′. In particular, f⁢(ℰ) is symmetric (respectively, non-degenerate) if and only if ℰ is.
For every fusion subcategory ℰ of 𝒵⁢(𝒞), f maps the projective centralizer of ℰ to the projective centralizer of f⁢(ℰ).

Proof.

For every simple object X of 𝒵⁢(𝒞) we have

dX=d𝟏⁢dX=S𝟏,X=S𝟏,f⁢(X)=d𝟏⁢df⁢(X)=df⁢(X),

and we get (i). Now part (ii) follows from (i) and the Verlinde formula (6.2). Part (iii) follows from the fact that two simple objects X and Y centralize each other if and only if SX,Y=dX⁢dY.

We now show part (iv). Let X and Y be simple objects of 𝒞. It follows from [10, Proposition 3.22] that X belongs to the projective centralizer of Y if and only if X belongs to the centralizer of Y⊗Y*. In view of (iii) this happens if and only if f⁢(X) centralizes f⁢(Z) for all Z∈Irr⁡(𝒞) such that NY⊗Y*Z≠0. Since, by (ii), f is a Grothendieck equivalence, we have f⁢(Y)*=f⁢(Y*) (Proposition 3.2 (iii)), and it follows that the last condition is equivalent to the condition that f⁢(X) centralizes f⁢(Y)⊗f⁢(Y)*, that is, f⁢(X) belongs to the projective centralizer of f⁢(Y). ∎

Theorem 6.3

Let C and D be S-equivalent spherical fusion categories. Then C is group-theoretical if and only if so is D.

Proof.

We have that 𝒞 is group-theoretical if and only if 𝒵⁢(𝒞) is group-theoretical. Suppose that this is the case. In particular 𝒵⁢(𝒞), and therefore also 𝒵⁢(𝒟), are integral. Since 𝒵⁢(𝒞) is a modular category, [9, Corollary 4.14] implies that it contains a symmetric subcategory ℰ such that ℰad′⊆ℰ. Since every S-equivalence preserves centralizers, symmetric subcategories and is a Grothendieck equivalence between 𝒵⁢(𝒞) and 𝒵⁢(𝒟), this implies that f⁢(ℰ) is a symmetric subcategory of 𝒵⁢(𝒟) and f⁢(ℰ)ad′=f⁢(ℰad′)⊆f⁢(ℰ) (see Proposition 3.2 (iv)). Hence 𝒵⁢(𝒟) and therefore also 𝒟 are group-theoretical. This implies the theorem. ∎

Lemma 6.4

Let G and Γ be finite groups and let ω:G×G×G→k* and ω′:Γ×Γ×Γ→k* be 3-cocycles on G and Γ, respectively. Suppose that the categories C⁢(G,ω) and C⁢(Γ,ω′) are S-equivalent. Then G is solvable if and only if so is Γ.

Proof.

It is enough to show the ‘if’ direction. Thus, let us assume that G is solvable. Let f:Irr⁡(𝒵⁢(𝒞⁢(Γ,ω′)))→Irr⁡(𝒵⁢(𝒞⁢(G,ω))) be an S-equivalence. The center of the category 𝒞⁢(Γ,ω′) contains a Tannakian subcategory ℰ equivalent to Rep⁡Γ as braided fusion categories. In view of Lemma 6.2, f⁢(ℰ) is a symmetric fusion subcategory of 𝒵⁢(𝒞⁢(G,ω)) which is Grothendieck equivalent to Rep⁡Γ. Being symmetric, f⁢(ℰ) is equivalent as a fusion category to the category Rep⁡F for some finite group F. Then F is solvable because 𝒵⁢(𝒞⁢(G,ω)) is solvable, by [12, Proposition 4.5]. Since the categories Rep⁡Γ and Rep⁡F are Grothendieck equivalent, the groups Γ and F have the same character table. This implies that Γ is solvable. Hence 𝒞⁢(Γ,ω′) is solvable, as claimed. ∎

Theorem 6.5

Let C and D be S-equivalent spherical fusion categories and suppose that C is group-theoretical. Then C is solvable if and only if D is solvable.

Proof.

Since 𝒞 is group-theoretical, 𝒵⁢(𝒞) is equivalent to the center of a pointed fusion category 𝒵⁢(𝒞⁢(G,ω)), for some finite group G and 3-cocycle ω on G. Hence 𝒵⁢(𝒞) contains a Tannakian subcategory ℰ equivalent to Rep⁡G as braided fusion categories, such that (dim⁡ℰ)2=dim⁡𝒵⁢(𝒞).

Being Grothendieck equivalent to 𝒵⁢(𝒞), 𝒵⁢(𝒟) is also group-theoretical, in view of Theorem 6.3. Thus 𝒵⁢(𝒞) is an integral modular category of dimension (dim⁡𝒟)2 and central charge 1. Note in addition that if f is an S-equivalence, then f⁢(ℰ) is a symmetric subcategory of 𝒵⁢(𝒟) such that dim⁡𝒵⁢(𝒟)=(dim⁡f⁢(ℰ))2. Theorem 4.8 of [9] implies that 𝒵⁢(𝒟) is equivalent to the center of a pointed fusion category, that is, we have 𝒵⁢(𝒟)≅𝒵⁢(𝒞⁢(Γ,ω′)), for some finite group Γ and 3-cocycle ω′ on Γ. Then the theorem follows from Lemma 6.4. ∎

Communicated by Robert M. Guralnick

Funding statement: This work was partially supported by CONICET and SeCYT–UNC.

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Received: 2016-1-16

Revised: 2016-2-17

Published Online: 2016-5-18

Published in Print: 2017-1-1

Articles in the same Issue

https://doi.org/10.1515/jgth-2016-0020

On fusion rules and solvability of a fusion category

Article

Abstract

1 Introduction

2 Preliminaries

2.1 Fusion categories

2.2 Nilpotent and solvable fusion categories

Proof.

Proof.

Proof.

2.3 Braided fusion categories

Proof.

Proof.

3 Grothendieck equivalence of fusion categories

Proof.

Proof.

Proof.

Proof.

4 Examples of non-solvable fusion rules

4.1 Abelian extensions

Proof.

Proof.

4.2 Examples associated to the symmetric group

Proof.

Proof.

Proof.

Proof.

Proof.

Proof.

Proof.

4.3 Fusion rules of Rep⁡K5

4.4 The dual Hopf algebras Jn*, Kn*

Proof.

Proof.

Proof.

4.5 Further examples associated to the symmetric group

Proof.

Proof.

Proof.

5 Solvability and fusion rules of a braided fusion category

Proof.

Proof.

6 The character table of a spherical fusion category

6.1 Spherical fusion categories

6.2 Modular categories and S-matrices

6.3 S-equivalence of spherical fusion categories

Proof.

Proof.

Proof.

Proof.

References

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

4.4 The dual Hopf algebras Jn, Kn