arXiv:2002.03220v2 [math.QA] 30 May 2020

AUTO-EQUIVALENCES OF THE MODULAR TENSOR CATEGORIES OF

TYPE A, B, C AND G

CAIN EDIE-MICHELL

Abstract. We compute the monoidal and braided auto-equivalences of the modular tensorcategories C(slr+1, k), C(so2r+1, k), C(sp2r, k), and C(g2, k). Along with the expected simplecurrent auto-equivalences, we show the existence of the charge conjugation auto-equivalenceof C(slr+1, k), and exceptional auto-equivalences of C(so2r+1,2), C(sp2r, r), C(g2,4). We endthe paper with a section discussing potential applications of these computations, including therelationship of these computations to the program to classify quantum subgroups of the simpleLie algebras. Included is an appendix by Terry Gannon, which discusses the group structure ofthe auto-equivalences of C(slr+1, k).

1. Introduction

Modular tensor categories have increasingly become ubiquitous in many areas of modernmathematics. Several important appearances include the superselection sectors of a rationalconformal field theory [21], the even parts of certain subfactor standard invariant [4], and therepresentation category of a quantum group [38] (after an appropriate semisimplification hasbeen applied). With these far ranging applications, structure results for modular categories arein high demand.

Given a modular tensor category C, a particularly useful invariant one can compute is thegroup of auto-equivalences of C; either braided (which we denote EqBr(C)), or plain monoidal(which we denote Eq(C)). The usefulness of these two invariants is mainly due to the variousconstructions one can perform on C once these are computed. For example, given a finite groupG, and a homomorphism G→ EqBr(C) one can (if certain obstructions vanish) construct a newmodular tensor category called the gauging of C [7]. Further, for every braided auto-equivalenceof C, one can construct a “quantum subgroup” (in the sense of [42]) of C [8, Corollary 3.8].

Roughly speaking, all known modular tensor categories come from two sources. The firstis from Drinfeld centres of spherical fusion categories. The braided auto-equivalences of theseexamples have been extensively studied, with particular focus on the Drinfeld centres of fusioncategories coming from exotic subfactors [28, 29, 27]. The second source comes from quantumgroups. Let g be a simple Lie algebra, and g the corresponding affine Lie algebra. Then, forevery k ≥ 1, the category of level k integrable representations of g has the structure of a modulartensor category [38]. These categories are usually denoted C(g, k). Unlike the Drinfeld centreexamples, very little is known about the auto-equivalences of these modular tensor categories.To the authors best knowledge, the only case known is when g = sl2 [13]. Hence it is a naturalquestion to compute these auto-equivalence groups for all the higher rank g.

It has been a long standing open problem to classify the quantum subgroups of the simpleLie algebras, which was initially laid out by Ocneanu [42] and Gannon [23]. These quantumsubgroups have important applications to the various areas of mathematics that modular tensorcategories intertwine. For example, such quantum subgroups classify defect sectors in Chern-Simons topological quantum field theories, and can be used to construct highly non-trivialsubfactors of the hyperfinite type II1 factor. These vast applications has meant that an enormousamount of effort has been spent on trying to classify these quantum subgroups. However untilrecently, only a small amount of progress has been made, with only the sl2 and sl3 cases being

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2 CAIN EDIE-MICHELL

completed [23]. One of the big stumbling blocks for this program is that for each Lie algebra,there are a-priori an infinite number of levels where a quantum subgroup may occur, which makessearching and classifying them very difficult. Recent breakthrough results of Schopieray [47] andGannon [22] solve this issue, giving effective bounds on the levels for which exceptional quantumsubgroups can occur. With these recent results, we will now very have an list exhaustive list ofpotential quantum subgroups of the low rank simple Lie algebras. The problem then shifts toconstructing these quantum subgroups. The braided auto-equivalences classified in this paperwill allow the construction of a whole new plethora of quantum subgroups of the simple Liealgebras, completing a significant portion of the program laid out by to classify all quantumsubgroups of the simple Lie algebras.

In this paper we are able to compute both Eq(C) and EqBr(C), where C ≃ C(g, k) with g asimple Lie algebra of type Ar, Br, Cr, or G2 for all ranks and levels. Our main theorem is asfollows.

Theorem 1.1. Let r and k positive integers. We have:

C Eq(C) EqBr(C)C(slr+1, k) r = 1 and k = 2 {e} {e}

2 exactly divides gcd(r + 1, k) Zc2 ×Z×n′ ×Z2 ×Zn′′2

Zc+p+t2

otherwise Zc2 ×Z×n′ ×Zn′′ Zc+p+t2

C(so2r+1, k) k = 1 {e} {e}k = 2 and every prime p dividing Zω(2r+1)+1

2 Zω(2r+1)−12

2r + 1 satisfies p ≡ 1 (mod 4)k = 2 and there exists a prime p dividing Zω(2r+1)

2 Zω(2r+1)−12

2r + 1 such that p ≢ 1 (mod 4)k ≥ 3 and k ≡ 0 (mod 2) Z2 {e}k ≥ 3 and k ≡ 1 (mod 2) Z2 Z2

C(sp2r, k) r = 2 and k = 1 {e} {e}r = k and r ≡ 0 (mod 2) Z2 ×Z2 {e}r = k and r ≡ 1 (mod 2) Z2 {e}r ≠ k and rk ≡ 1 (mod 2) {e} {e}r ≠ k and rk ≡ 0 (mod 4) Z2 {e}r ≠ k and rk ≡ 2 (mod 4) Z2 Z2

C(g2, k) k = 4 Z2 Z2

k ≠ 4 {e} {e}.

Where

● n′′ = gcd(r + 1, k∞),● n′ = n

n′′ ,

● c ∶=⎧⎪⎪⎨⎪⎪⎩

0, if k ≤ 2 or r = 1

1, if k ≥ 3 and r ≠ 1,

● t ∶=⎧⎪⎪⎨⎪⎪⎩

0, if r is even, or if r is odd and k ≡ 0 (mod 4), or if k is odd and r ≡ 1 (mod 4),1, otherwise,

● p is the number of distinct odd primes that divide r + 1 but not k, and● ω(n) is the number of distinct prime divisors of n.

Remark 1.2. We remark that the result for g = g2 in the above theorem is entirely reliant onresults of Ostrik and Snyder that has yet to appear in print.

AUTO-EQUIVALENCES OF THE MODULAR TENSOR CATEGORIES OF TYPE A, B, C AND G 3

The auto-equivalences we classify in this Theorem can be broken into three classes. The firstare the simple current auto-equivalences. These occur whenever there is an invertible object inmodular category satisfying some mild conditions on its self-braid eigenvalue. These are the vastmajority of the auto-equivalences in the classification. They should not be considered exotic, oreven exciting. All constructions one can perform with simple current auto-equivalences resultin essentially trivial information.

The second class of auto-equivalences are the charge-conjugation auto-equivalences, whichoccur for C(slr+1, k) at all levels for r ≥ 2. These are the braided auto-equivalences which mapX ↦ X∗. These can be considered semi-exceptional, as they give rise to interesting gaugedcategories, and new quantum subgroups. However, they exist for all r and k, and their existenceessentially follows from classical Lie theory.

The final class are the truly exceptional auto-equivalences, which occur sporadically, and canonly be shown to exist through a variety of ad-hoc methods. The existence and classification ofthese exceptional auto-equivalences is the true substance of this paper. We find such exceptionalauto-equivalences for the following categories:

C(so2r+1,2) for all r ≥ 2,

C(sp2r, r) for all r ≥ 2, and

C(g2,4).The author is still exploring the exciting consequences of the existence of these exceptionalauto-equivalences. For now we wish to draw attention to the quantum subgroups that can beconstructed from the exceptional auto-equivalences which are braided. These are the Z2 worth of

auto-equivalences of C(g2,4), and the Zω(2r+1)−12 worth of auto-equivalences of C(so2r+1,2). For

the former case, we explicitly work out the structure of the quantum subgroup of g2 in the finalsection of this paper. We find that it is a quantum subgroup whose existence was hypothesised in[19]. For the latter case, the existence of these exceptional braided auto-equivalences gives a newmassive family of quantum subgroups of so2r+1. The general structure of this massive family isnot immediately clear to the author, though preliminary analysis of the low rank cases suggeststhat the quantum subgroups have a very rich structure. As delving further into describing thesequantum subgroups is beyond the scope of this (already large) paper, we leave it for a futurepublication.

Let us briefly try to outline our methods used to prove Theorem 1.1. As the techniquesrequired to deal with each of the cases of Lie type A, B, C, and G are vastly different, it ishard to give a detailed overview in this introduction. Each of these cases is dealt with in itsown separate section of the paper, and a detailed outline of the method used for that case canbe found at the beginning of each section.

A very rough outline for the general method is as follows. First we show that the group ofgauge auto-equivalences (auto-equivalences which are trivial on objects) for each of the examplesis trivial. To show this we argue that such a gauge auto-equivalence would imply the existenceof a automorphism of a certain planar algebra associated to each category. We can then directlycompute the planar algebra automorphisms for the relevant examples, and deduce the desiredresult.

Once we have shown that the gauge auto-equivalence group of the category is trivial, we knowthat the auto-equivalence groups are subgroups of the groups of fusion ring automorphisms ofthe categories. Fortunately for this paper (in fact, one of the main incentives to begin thispaper) is the results of [24], which compute the fusion ring automorphisms of all the categoriesC(g, k). Hence our task now reduces to showing which of these automorphisms are realised aseither braided, or monoidal auto-equivalences. For this task, a variety of ad-hoc techniques aredeveloped. For the details of these techniques, see the beginning introduction of each sectionwhere we outline the construction of the auto-equivalences. Of particular note, we find that not

4 CAIN EDIE-MICHELL

every fusion ring automorphism of C(g, k) lifts to an auto-equivalence, a fact we found surprising.Examples of this phenomenon can be seen in the C(so2r+1,2) and C(g2,3) cases where we havefusion ring automorphisms which are not realised as auto-equivalences of the category.

Ideally the goal is to generalise the main theorem to classify the auto-equivalences for allcategories C(g, k), i.e. also deduce the cases where g ∈ {so2r, e6, e7, e8, f4}. There are severalproblems the author is currently unable to deal with to solve this problem. The first is thatthere is no planar algebra we can study for the remaining exceptionals in order to leverageinformation about the gauge auto-equivalences groups. Without knowledge of the gauge auto-equivalence group, it is impossible to determine the auto-equivalences. Hence, either planaralgebras for the exceptionals have to be deduced, or new techniques for computing gauge auto-equivalences have to be developed. The issue of realisability is also a problem. For the categoriesC(so2r,2) and C(f4,4) there exist exceptional auto-equivalences. The author was unable to findad-hoc methods for constructing these. We expect that the techniques for constructing theexceptional auto-equivalences of C(so2r+1,2) will generalise in some sense to the C(so2r,2) case.However, there are fundamental differences between the two that need to be worked out. Theexceptional auto-equivalence of C(f4,4) remains completely mysterious.

We end the paper with a section discussing potential applications of Theorem 1.1. In our mainTheorem, we show the type A categories have the charge conjugation braided auto-equivalence.As this auto-equivalence has order 2, we can potentially gauge this Z2 action to obtain a new2-parameter family of modular tensor categories. A quick obstruction argument shows that thisgauging is always possible when r is even, and we suspect that gauging is also possible when ris odd. We leave determining the structure of these new modular tensor categories as a futureapplication, as it goes far beyond the scope of this paper. In type B for certain ranks at level2 we show the existence of an exceptional auto-equivalence with the peculiar property that itis reverse braided. We observe that the categories with these peculiar auto-equivalences areprecisely the categories used in the hypothetical “modular grafting” procedure to construct theIzumi-Haagerup doubles. We present the evidence for a connection between reverse braided auto-equivalences, and modular grafting, and end with an open question regarding the relationshipbetween these two concepts. In type C, we found the existence of the exceptional level-rankduality auto-equivalence when r = k. We identify a new planar algebra which is constructedas fixed points of this auto-equivalence. We identify a potential generator, and give some basicrelations. We leave it as an open question to determine a full presentation of this new planaralgebra. Finally, for type G, we use the exceptional braided auto-equivalence at level 4 toconstruct a new quantum subgroup. The existence of this quantum subgroup was hypothesisedin [19].

Finally, in an appendix written by Terry Gannon, the explicit group structure of the auto-equivalences of C(slr+1, k) is determined.

Acknowledgements

We are grateful to many people for their assistance throughout this project. We wish to thankNoah Snyder for his help regarding several technical aspects of this paper, and Scott Morrisonfor many helpful conversations, along with his insight into the modular data of the centre ofZ(EH). We also thank Rob Culling, Cesar Galindo, Terry Gannon, and Andrew Schopierayfor various conversations that helped in writing this paper. Finally we thank Pinhas Grossmanfor comments on an early draft of this paper.

This material is based work supported by the National Science Foundation under Grant No.DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institutein Berkeley, California, during the Spring 2020 semester. In addition, the author gratefullyacknowledges support from the AMS-Simons travel grant


2. Preliminaries

We direct the reader to [15] for the basics on fusion categories. For the purpose of this paperwe will work with fusion categories over the field C.

Modular tensor categories.A braided fusion category is a fusion category, along with a collection of natural isomorphisms

cX,Y =

X Y

XY

∶X ⊗ Y → Y ⊗X

satisfying the topologically implied coherence axioms.A pivotal structure on a fusion category C is a monoidal functor ψ ∶ ∗∗ → IdC . We refer

to a fusion category along with a pivotal structure as a pivotal fusion category. When usinggraphical calculus for pivotal fusion categories, we will often suppress ψ, as it makes for muchcleaner pictures, and it is always clear where the component of ψ should belong in a givenpicture.

Given a pivotal fusion category C we can define traces on the endomorphism algebras of C.Let f ∈ Hom(X,X), we have

Tr+(f) ∶= f

X

X

= evX ○(f ⊗ idX∗) ○ (ψX ⊗ idX∗) ○ coevX∗ ,

Tr−(f) ∶= f

X

X

= evX∗ ○(idX∗ ⊗f) ○ (idX∗ ⊗ψ−1X ) ○ coevX .

We say the pivotal fusion category C is spherical if Tr+(f) = Tr−(f) for all f . For a sphericalfusion category, we define the categorical dimension of an object X as

dim(X) = Tr+(idX) = Tr−(idX).Given a spherical braided fusion category we can construct two numerical invariants. These

are the S and T matrices, defined by

SX,Y ∶= Tr−⊗Tr+(cY,X ○ cX,Y ) = Y X

TX,X ∶= Tr+(cX,X)Tr+(idX) =

X

X

.

6 CAIN EDIE-MICHELL

If the S matrix of C is invertible, we say that C is a modular tensor category. The main objectsof study in this paper will be a certain class of modular tensor categories that we will introduceshortly.

Modular tensor categories from Lie theory.Here we give a brief recap on the modular categories C(g, k), the main objects of study in

this paper. Surprising little knowledge of these categories is required for this paper, so we keepthe details to a minimum. For additional details we point the reader to [48] for an expositoryexplanation of the modular categories C(g, k), and to [1, 37] for a detailed exposition. We mainlyrestrict our attention to the case when g is one of slr+1, so2r+1, so2r+1, or g2, as these are theLie algebras we deal with in this paper.

Let g a simple Lie algebra, g its corresponding affine Lie algebra, and k a positive integer.The modular category C(g, k) consists of the level k integrable highest weight modules of g. Letr be the rank of g, then the simple objects of C(g, k) are parametrised by the set:

P+ ∶=⎧⎪⎪⎨⎪⎪⎩

r

∑j=1

λjΛj ∣ λj ∈ N,r

∑j=1

λja∨j ≤ k

⎫⎪⎪⎬⎪⎪⎭.

Here a∨j are the co-labels of g, given in Table 1 for the relevant Lie algebras in this paper.

g a∨j

slr+1 1

so2r+1

⎧⎪⎪⎨⎪⎪⎩

1 if j ∈ {1, r}2 otherwise

sp2r+1 1

g2

⎧⎪⎪⎨⎪⎪⎩

1 if j = 1

2 if j = 2

Table 1. Co-labels of the Lie algebras slr+1, so2r+1, sp2r, and g2

The fusion rules for the categories C(g, k) can be computed using the Weyl chamber of g. Thedetails of this procedure are complicated, so we direct the reader to [48, Section 5] for the details.We will recall relevant fusion rules of the categories C(g, k) as needed throughout the paper.While in theory it is possible to compute the associator for the categories C(g, k), for this paperwe avoid using the associators almost completely.

The categories C(g, k) have several choices for pivotal structures, indexed by characters ofthe universal grading group of the category C(g, k). For this paper we use the standard Hopfalgebra pivotal structure on the categories C(so2r+1, k) and C(g2, k), and Tureav’s unimodalpivotal structure on the categories C(slr+1, k) and C(sp2r, k). Importantly for us, this ensuresthat the object Λ1 satisfies certain conditions which allow us to use powerful planar algebramachinery. With these choices of pivotal structures we have that the quantum twists of thesimple objects in the categories C(so2r+1, k), C(sp2r, k) and C(g2, k) are given by the formulas:

C(slr+1, k) ∶ T∑rj=1 λjΛj= ((−1)∑rj=1 jλj) e

2πi2(1+k+r) (λ1,λ2,⋯,λr).K.(λ1+2,λ2+2,⋯,λr+2)T

C(so2r+1, k) ∶ T∑rj=1 λjΛj= e

2πi4(2r−1+k) (λ1,λ2,⋯,λr).K.(λ1+2,λ2+2,⋯,λr+2)T

C(sp2r, k) ∶ T∑rj=1 λjΛj= ((−1)∑rj=1,3,⋯ λj) e

2πi4(r+k+1) (λ1,λ2,⋯,λr).K.(λ1+2,λ2+2,⋯,λr+2)T


C(g2, k) ∶ Tλ1Λ1+λ2Λ2 = e2πi

6(4+k) (λ1,λ2).K.(λ1+2,λ2+2)T,

where the matrices K are Killing matrices of g, given in Table 2 for the relevant Lie algebrasfor this paper.

g K

slr+1 [ (r+1)min(i,j)−ijr+1 ]

i,j

so2r+1 [ 2 min(i,j)

2(δi,r+δj,r)]i,j

sp2r [min(i, j)]i,jg2 [2 3

3 6]

Table 2. The Killing matrices for the Lie algberas slr+1, so2r+1, sp2r, and g2.

Monoidal Auto-equivalences.Let C be a fusion category. We say an auto-equivalence F ∶ C → C is a monoidal auto-

equivalence of C if there exists a family of natural isomorphisms

τX,Y ∶ F(X)⊗F(Y )→ F(X ⊗ Y )satisfying a straightforward coherence condition. We call the collection τ a tensorator for C.

If C is a braided fusion category, then we say (F , τ) is a braided auto-equivalence of C if it isa monoidal auto-equivalence, and the following diagram commutes for all X,Y ∈ C.

F(X)⊗F(Y )cF(X),F(Y )ÐÐÐÐÐÐ→ F(Y )⊗F(X)

τX,Y×××Ö

×××ÖτY,X

F(X ⊗ Y )F(cX,Y )ÐÐÐÐÐ→ F(Y ⊗X)

If C is a pivotal fusion category, then we say (F , τ) is a pivotal auto-equivalence if

δX∗ ○F(ψX) = δ∗X ○ ψF(X),where

δX ∶= ((F(evX) ○ τX∗,X)⊗ idF(X)∗) ○ (idF(X∗)⊗ coevF(X)).Given a fusion category C we write Eq(C) for the monoidal category whose objects are

monoidal auto-equivalences of C, and whose morphisms are monoidal natural transformations.We can truncate this monoidal category to get Eq(C), the group of monoidal auto-equivalences ofC. In a similar fashion if C is a braided fusion category we can define EqBr(C) to be the monoidal

category of braided auto-equivalences of C, and EqBr(C), the group of braided auto-equivalencesof C.

Let K(C) be the fusion ring of C, i.e. the based ring whose objects are isomorphism classesof objects in C. A fusion ring automorphism of C is a based ring automorphism of K(C). Wewrite FusEq(C) for the group of fusion ring automorphisms of C.

Given any monoidal auto-equivalence of C we can forget the structure of the tensorator to geta fusion ring automorphism of C, hence we get a map

Eq(C)→ FusEq(C).In general this map is neither injective or surjective. Let us write FusEq(C) for the image ofthis homomorphism, i.e. the group of fusion ring automorphisms of C that are realised by a

8 CAIN EDIE-MICHELL

monoidal auto-equivalence. We call elements of FusEq(C) realisable fusion ring automorphisms

of C. As the map Eq(C)→ FusEq(C) is surjective we get a short exact sequence

0→ Gauge(C)→ Eq(C)→ FusEq(C)→ 0.

Here Gauge(C) is the kernel of the map Eq(C) → FusEq(C). This kernel corresponds to thegroup of monoidal auto-equivalences of C whose underlying functor is the identity i.e the trivialfunctor equipped with possibly interesting tensorators. We call such auto-equivalences of C,gauge auto-equivalences.

With the above short exact sequence in mind, we see that the process of computing Eq(C)for a given C splits into three parts:

(1) Compute the fusion ring automorphisms of C.(2) Determine which fusion ring automorphisms of C are realisable.(3) Compute the gauge auto-equivalences of C.

Step (1) is purely combinatorial, and is simply an exercise in ring theory given that the fusionring of C is known. Step (2) is more categorical, as it asks when one can equip a fusion ringautomorphism of C with a tensorator. There are many invariants one can check to show that afusion ring automorphism can not be equipped with a tensorator, however there is no generalpurpose solution for the converse problem. Our tools in this paper are a ad-hoc mix of directtensorator computations, Hopf algebra methods, and planar algebra arguments. Step (3) ispurely categorical, and one can think of the group Gauge(C) as telling us how many distincttensorators exist for a fusion ring automorphism, given one already exists to begin with. Thestandard approach to computing Gauge(C) is to use planar algebra techniques. Key to applyingthese planar algebra techniques will be the following Lemma.

Lemma 2.1. Let C a pivotal category, and (IdC , τ) a gauge auto-equivalence of C, then (IdC , τ)is a pivotal functor.

Proof. From the definition of a pivotal functor, the statement of this Lemma is equivalent toshowing δX∗ = δ∗X for all simple X. We compute

(2.1) δ∗X =

X∗∗

X∗∗

τX∗;X =

X∗∗

X∗∗

τX∗∗;X∗

τX∗∗X∗;X

τ−1

X∗∗;X∗X

=

X∗∗

X∗∗

τX∗∗;X∗

τX∗∗X∗;X


Let us separately consider the koru in the last diagram. As X is simple, the hom spaceHom(X∗∗ ⊗X∗ → 1) is 1-dimensional. Thus there exists a scaler α such that

τX∗∗X∗;X

X∗∗ X∗

= αX

∗∗X

∗

.

Appending the morphismX∗∗ X∗

−1

X

to both sides gives α ⋅ dim(X) on the right, and dim(X) on the left (after an application of thenaturality of τ). Hence, α = 1.

Returning to Equation (2.1) we thus have that

X∗∗

X∗∗

τX∗∗;X∗

τX∗∗X∗;X =

X∗∗

X∗∗

τX∗∗;X∗ = δX∗ ,

completing the proof. �

This Lemma shows us that every gauge auto-equivalence must appear as a planar algebraautomorphism of some planar algebra corresponding to C. We discuss this more in the planaralgebra section of these preliminaries.

Another useful Lemma regarding auto-equivalences is as follows.

Lemma 2.2. Suppose C is a spherical braided fusion category, and (F , τ) is a braided auto-equivalence of C that preserves categorical dimensions. Then

TX,X = TF(X),F(X).

Proof. Let X ∈ C be a simple object. We have

F(evX) ○ τX∗,X ○ (δ−1X ⊗ idF(X)) ∈ Hom(F(X)∗ ⊗F(X)→ 1) ∋ evF(X),

(idF(X)∗ ⊗δ∗−1X ) ○ (δX ⊗ δX∗) ○ τ−1

X∗,X∗∗ ○F(coevX∗) ∈ Hom(1→ F(X)∗ ⊗F(X)∗∗) ∋ coevF(X)∗ ,

F(ψX) ○ δ−1X∗ ○ δ∗X ∈ Hom(F(X)→ F(X)∗∗) ∋ ψF(X).

As X is simple, the Hom spaces above are 1-dimensional. Thus there exist scalers α,β, γ suchthat

F(evX) ○ τX∗,X ○ (δ−1X ⊗ idF(X)) = α evF(X),

10 CAIN EDIE-MICHELL

(idF(X)∗ ⊗δ∗−1X ) ○ (δX ⊗ δX∗) ○ τ−1

X∗,X∗∗ ○F(coevX∗) = β coevF(X)∗ ,

F(ψX) ○ δ−1X∗ ○ δ∗X = γψF(X).

As the functor (F , τ) preserves categorical dimensions, we have in particular that F(dim(X)) =dim(F(X)). Expanding out the term F(dim(X)) reveals that the product αβγ is equal to 1.

Now consider the term F(TX,X). Expanding this gives

F(TX,X) = αβγTF(X),F(X) = TF(X),F(X),

proving the statement of the Lemma. �

As the categorical dimensions of the objects in the categories C(g, k) agree with their Frobenius-Perron dimensions, any auto-equivalence will automatically preserve categorical dimensions.Hence we get the following Corollary.

Corollary 2.3. Let (F , τ) a braided auto-equivalence of C(g, k), then (F , τ) preserves the twistsof the simple objects.

This gives a useful obstruction for the existence of a braided auto-equivalence of C(g, k). Infact, for the examples we consider in this paper, we find that a fusion ring automorphism ofC(g, k) lifts to a braided auto-equivalence precisely when it preserves the twists of the simpleobjects. This phenomenon is not true for arbitrary braided fusion categories, and we only showthis to be true for our examples as a corollary of our main classification theorem.

A useful construction of monoidal auto-equivalences is through the process of simple currents.This process takes an invertible object in a modular tensor category C, and constructs a monoidalauto-equivalence. This process was first investigated for fusion rings in the language of conformalfield theory in [46]. For modular tensor categories, the process was studied in [11, 14]. Theconstruction given in [14] is somewhat awkward to use for the examples considered in thispaper. Instead we present the following construction, which can be proved in much the samefashion as [14].

Lemma 2.4. Let C be a modular tensor category, and g an invertible object of order M . Set qequal to the unique integer (modulo 2M) such that

σg,g = e2πi q2M idg⊗g,

and choose a ∈ {0,1,⋯,M − 1} such that

1 + aq is coprime to M.

Then there exists a monoidal auto-equivalence Fg,a of C defined on objects by

Fg,a(X) = g−an ⊗X,

where n is the unique integer (modulo M) such that σX,gσg,X = e2πi nM idg⊗X . The monoidal

auto-equivalence Fg,a is braided if and only if

a + a2q

2≡ 0 (mod M).

Proof. Nearly identical to the proof of [14]. �


Hopf algebras.A useful tool for constructing auto-equivalences of tensor categories is by studying the au-

tomorphisms of an associated Hopf algebra. Here we define Hopf algebras, and explain howsymmetries of certain Hopf algebras give rise to symmetries of certain tensor categories. Our mo-tivation for introducing these techniques is to construct the charge conjugation auto-equivalencesof C(sln, k).

A Hopf algebra H is an associative algebra, along with a coassociative colagebra structure∆ ∶ H → H ⊗H, a counit map ε ∶ H → C, and an antipode map S ∶ H → H, all of which mustsatisfy certain conditions. Further, we say that the Hopf algebra H is quasi-triangular if thereexists an element R ∈H ⊗H satisfying the Yang-Baxtor equation. Full definitions can be foundin [40]. We are intentionally light on the definition of a Hopf algebra, as we are interested inspecific concrete examples coming from deformed Lie algebras. These are the so called quantumgroups in the sense of Drinfeld [9].

Definition 2.5. [34] Let g be a simple Lie algebra with simple roots ∆, and let A the Cartanmatrix of g. The quasi-triangular Hopf algebra Uq(g) is defined by the generators:

Xi, Yi,Hi for i ∈ ∆

These generators satisfy the relations:

[Hi,Hj] = 0

[Xi,Xj] = δi,jsinh(qdiHi/2)

sinh(qdi/2)[Hi,Xj] = Ai,jXj

[Xi, Yj] = −Ai,jYj1−Ai,j

∑k=0

(−1)k [ 1−Ai,jk

]hiXki XjX

1−Ai,j−ki = 0 for i ≠ j

1−Ai,j

∑k=0

(−1)k [ 1−Ai,jk

]hiY ki YjY

1−Ai,j−ki = 0 for i ≠ j

The comultiplication ∆ ∶ Uq(g)→ Uq(g)⊗Uq(g) is defined by

∆(Hi) =Hi ⊗ 1 + 1⊗Hi,

∆(Xi) =Xi ⊗ eqdiHi/4 + e−qdiHi/4 ⊗Xi,

∆(Yi) = Yi ⊗ eqdiHi/4 + e−qdiHi/4 ⊗ Yi.

The conunit ε ∶ Uq(g)→ C is defined by

ε(Hi) = ε(Xi) = ε(Yi) = 0.

The antipode S ∶ Uq(g)→ Uq(g) is defined by

S(Hi) = −Hi S(Xi) = −ehdi/2Xi S(Yi) = −ehdi/2Yi.This Hopf algebra has an R-matrix, however we neglect to write it down.

The representation theory of Uq(g) gives a braided tensor category Rep(Uq(g)). We will beinterested in the special case of when q is a root of unity. In this case the category Rep(Uq(g)) isnot semisimple. However, we can take the semi-simplification (as in [17]) to obtain the braided

fusion category Rep(Uq(g)).We now have two ways of constructing a braided fusion category from a simple Lie algebra

g. We can either form C(g, k) or Rep(Uq(g)). A result of Finkelberg [20] shows that these


two constructions are essentially the same, by providing the following braided equivalence ofcategories:

C(g, k) ≃ Rep(Ue

πim(k+h)

(g))

where h is the dual coxeter number of g, and m = 1 for the Lie algebras ADE, m = 2 for the Liealgebras BCF , and m = 3 for the Lie algebra G. We remark that there are four exceptions to theabove equivalence, which are (g, k) = (E6,1), (E7,1), (E8,1), or (E8,2). These four exceptionswill not feature in the results or proofs of this paper.

The above equivalence will be useful to us, as it will allow us to perform certain computationsregarding the categories C(g, k) in the Hopf algebra Uq(g). In particular we have that Hopfalgebra automorphisms (see [40]) will induce monoidal and braided auto-equivalences of the

category Rep(Uq(g)), and hence also of Rep(Uq(g)). To make this precise, we have the followingnaturality result of Hopf algebras.

Theorem 2.6. Let H be a quasi-triangular Hopf algebra, and let Aut(H) be the group of Hopfalgebra automorphisms of H that preserve the R-matrix. Then there is an injection

Aut(H)→ EqBr(Rep(H)).

Applying this Theorem to the quasi-triangular Hopf algebras Uq(g) gives the following Corol-lary.

Corollary 2.7. Let g be a simple Lie algebra, with Dynkin diagram Γg. Then there is aninjection

Aut(Γg)→ EqBr(Rep(Uq(g))).

Proof. Let ∆ be the set of simple roots of g. A symmetry of Γg induces a symmetry σ of ∆which preserves the Cartan matrix of g, i.e. Ai,j = Aσ(i),σ(j). Using the generators and relationspresentation of Uq(g) we can see that σ induces a Hopf algebra automorphism φσ ∶ Uq(g)→ Uq(g)defined by

φσ(Hi) ∶=Hσ(i)

φσ(X±i ) ∶=X±

σ(i).

We still have to show that φσ preserves the R-matrix of Uq(g). To do this we will use theresults of [3]. For this next portion of the proof, we will freely use the notation and conventionsof this cited paper.

Following [3] we can write the R-matrix of Uq(g) in the form R ⋅K, where R ∈ Tq(b+⊗b−), and

K = q∑i,j∈∆A−1i,j(Hi⊗Hj). Clearly φσ preserves K, and φσ(R) is again an element of Tq(b+⊗b−) (as

φσ preserves the signs of the generators). Hence φσ(R) is of the form R′ ⋅K for R′ ∈ Tq(b+⊗ b−).As φσ preserves the comultiplication of Uq(g), we can apply [3, Theorem 7.1] to see that R andR′ only differ by a scalar. From the standard properties of the R-matrix, we can further deducethat this scalar is the identity. Hence φσ(R) = R. Hence we get an injection

Aut(Γg)→ EqBr(Rep(Uq(g))).Finally, any auto-equivalence of Rep(Uq(g)) will preserve the negligible ideal. Thus the above

map descends to an injection

Aut(Γg)→ EqBr(Rep(Uq(g))).�


The main purpose of this Corollary is that it gives the construction of the charge conjugationauto-equivalences of C(sln, k). These auto-equivalences occur due to the Z2-symmetry in theDynkin diagram for sln.

Planar algebras.Roughly speaking a planar algebra P is a collection of vector spaces {Pn ∶ n ∈ N}, along with a

multi-linear action of planar tangles. The full definition is fairly involved, so we point the readertowards [33] for the full definition. Planar algebras have appeared in many different contexts,and under many different names. Slightly different axiomitisations for the general idea of aplanar algebra include monoidal algebras [51], spiders [36], and towers of algebras [26].

It is well known that there is a one to one correspondence between planar algebras, and pivotalfusion categories with a distinguished ⊗-generating symmetrically self-dual object. Recently itwas shown in that this correspondence is functorial. This functorial relationship is very usefulfor this paper, as it will allow us to construct and classify certain auto-equivalences of pivotalfusion categories by studying planar algebra automorphisms of the corresponding planar algebra.

Let us describe this functorial relationship between planar algebras and pivotal categories.On one hand we have the category of based pivotal (braided) fusion categories. The objects of

this category are pairs (C,X), where C is a pivotal (braided) category, and X is a ⊗-generatingsymmetrically self-dual generating object. The morphisms (C1,X1) → (C2,X2) in this categoryare pivotal (braided) functors F ∶ C1 to C2 such that F(X1) ≅X2.

On the other hand we have the category of (braided) planar algebras. The objects of thiscategory are (braided) planar algebras with P0 ≅ C, and the morphisms are (braided) planar al-gebra homomorphisms, considered up to planar algebra natural isomorphism, see [12, AppendixA].

Given a based pivotal (braided) category (C,X) we can construct a (braided) planar algebraPA(C,X) by

PA(C,X)n ∶= HomC(X⊗n → 1).

Furthermore, given a pivotal (braided) functor (F , φ) ∶ (C1,X1) → (C2,X2) we define a planaralgebra morphism

φX1,X2φX⊗21 ,X1

⋯φX⊗n−11 ,X1

F ∶ PA(C1,X1)n → PA(C2,X2)n.

Conversely, given a planar algebra P we can define the based pivotal category CP of idempo-tents in P , and given a planar algebra homomorphism P1 → P2, we can define a based pivotal(braided) functor CP1 → CP2 . We are light on details of these converse constructions as theydon’t feature much in this paper. Additional details can be found in [30].

A powerful theorem of Henrqiques, Penneys, and Tener [30, Theorem A] shows that the aboveconstructions give equivalences between the category of based (braided) pivotal categories, andthe category of (braided) planar algebras. In particular this equivalence gives the following keyLemma for this paper.

Lemma 2.8. Let C be a pivotal fusion category, and X a symmetrically self-dual ⊗-generatingobject. Then there is an injection

Gauge(C)→ Aut(PA(C,X)).

Proof. By Lemma 2.1, every gauge auto-equivalence of C is pivotal. The result then follows byapplying [30, Theorem A]. �


Every planar algebra P has the (possible trivial) planar ideal consisting of negligible elements.We say an element f ∈ P is negligible if

f g = 0 for all g ∈ P.

It is a straightforward exercise to show the set of all negligible elements of P form a planar ideal.

Definition 2.9. We write P for the planar quotient of P by the negligible ideal.

The planar algebra P should be thought of as the “semisimplification” of P , as this con-struction corresponds to taking the semisimplification (in the sense of [17]) of the idempotentcategory of P . Giving an explicit description of the planar ideal of P can be difficult in practice.In the bulk of this paper we aim to construct planar algebra automorphisms of several examplesof semisimplified planar algebras. As we won’t have explicit descriptions of the negligible idealsfor these examples, we instead apply the following result, which shows that every planar algebraautomorphism descends to the semisimplification.

Proposition 2.10. Let P be a planar algebra, and φ a planar algebra automorphism of P . Thenφ preserves the negligible ideal of P .

Proof. For each f in the negligible planar ideal of P , we have to show that φ(f) is also in thenegligible planar ideal. Fix such a f , and let g ∈ P . As φ is a planar algebra automorphism,there exists some h ∈ P such that φ(h) = g. Therefore

φ(f ) g = φ(f ) φ(h) = φ

⎛⎜⎜⎜⎜⎜⎝

f h

⎞⎟⎟⎟⎟⎟⎠

= φ(0) = 0.

Thus φ(f) is in the negligible ideal of P . �

We now introduce the PH , BMW andG2 braided planar algebras, and explain their connectionto the modular categories C(slr+1, k), C(so2r+1, k), C(sp2r, k), and C(g2, k).

The planar algebra PH(δ, γ).Here we define the planar algebra PH(δ, γ), and explain its connection to certain endomor-

phism algebras in C(slr+1, k).

Definition 2.11. Let δ and γ non-zero complex numbers. The braided planar algebra PH(δ, γ)

has two generators ,S

∈ PH(δ, γ)3, along with the relations:

(i)S

= S ;

(ii) S = = 0;

(iii) S = 0;

(iv) = δ2−2δ ;


(v) SS = γ δ3−2δδ2−1

;

(vi) = δ2−3δ ;

(vii) S = −1δ S

;

(viii)SS

= (γ − 1)S

− γ δδ2−1

;

(ix) S

SS

= δ4γ+δ2(−2γ2+γ−2)+2(γ−1)2

δ(δ2−1) S+ γ(γ − 1) δ2

δ2−1.

There exists a braiding element in PH(δ, γ)4 which we neglect to write down, as it is compli-cated, and not important for this paper.

Combining several theorems gives the following isomorphism of planar algebras

(2.2) PA(C(slr+1, k), (Λ1 +Λr)) ≅ PH(δ, γ), when r ≥ 2 and k ≥ 3

where

δ = e2iπ(r+1)2(r+1+k) − e

−2iπ(r+1)2(r+1+k)

e2iπ

2(r+1+k) − e−2iπ

2(r+1+k)and γ = e

4iπ2(r+1+k) − e

(12+4(r+1))iπ2(r+1+k) + e

(8+8(r+1))iπ2(r+1+k) − e

4iπ(r+1)2(r+1+k)

e8iπ

2(r+1+k) − e(12+4(r+1))iπ

2(r+1+k) + e(4+8(r+1))iπ

2(r+1+k) − e4iπ(r+1)2(r+1+k)

Let us quickly explain how we obtain this isomorphism. From [35] we know that the Heckealgebras maps on to the endomorphism algebra of the representation Λ1 of Uq(slr+1) with q =e

2iπ(r+1)2(r+1+k) . The results of [6] compute the idempotents of the semisimplified Hecke algebra. From

these idempotents, we see the above map descends to an isomorphism between the semisimplified

Hecke algebras, and the endomorphism algebras of Λ1 ∈ Rep(Uq(slr+1)). Using the braiding and

duality maps of Rep(Uq(slr+1)), we can then get an isomorphism between the semisimplifiedHecke algebras, and the morphism spaces

HomRep(Uq(slr+1))

(Λ1 ⊗Λr)⊗n → 1).

The object Λ1⊗Λr contains the sub-object Λ1+Λr. Hence we can cut down the semi-simplifiedHecke algebra by the idempotent projecting onto Λ1 +Λr to obtain an isomorphism to

HomRep(Uq(slr+1))

((Λ1 +Λr)⊗n → 1).

The idempotent cut-down of the semisimplified Hecke algebras is, by the definition in [31,

Lemma 3.2], the box spaces of the planar algebra PH(δ, γ) with δ and γ as above. Hence, aftertranslating to planar algebra language, we obtain a planar algebra isomorphism

PA(Rep(Uq(slr+1)), (Λ1 +Λr)) ≅ PH(δ, γ).

Finally, applying the equivalences of braided categories from [20] gives the desired result.


The planar algebra BMW(q, r).Here we define the planar algebra BMW(q, r), and explain its connection to certain endomor-

phism algebras in C(so2r+1, k) and C(sp2r, k).

Definition 2.12. Let q ≠ ±1 and r non-zero complex numbers. The braided planar algebra

BMW(q, r) has a single generator ∈ BMW(q, r)4, satisfying Reidemiester moves 2 and 3,

along with the additional relations:

(i) = r

(ii) = r−1

(iii) − = (q − q−1)⎛⎜⎝

−⎞⎟⎠

The braiding in this planar algebra is simply the element .

An amalgamation of several theorems ([51, Section 7.7,b] and [20]) gives the following iso-morphisms of braided planar algebras (after some translating of languages):

(2.3) PA(C(so2r+1, k) ⊠Rep(Z2),Λ1 ⊠Csgn) ≅ BMW(q2, q4r) with q = e2πi

4(2r−1+k) .

(2.4) PA(C(sp2r, k),Λ1) ≅ BMW(q,−q2r+1) with q = e2πi

4(r+k+1) .

In the latter case, the object Λ1 ∈ C(sp2r, k) tensor generates, and thus the idempotent com-

pletion of the BMW(q,−q2r+1) braided planar algebra recovers the modular category C(sp2r, k).We have to be more careful in the former case, as the object Λ1⊠Csgn only tensor generates thesubcategory

(2.5) Ad(C(so2r+1, k)) ⊠Rep(Z2),

where

Ad(C(so2r+1, k)) ∶= ⟨X ⊗X∗ ∶X ∈ C(so2r+1, k)⟩.

Hence the idempotent completion of the braided planar algebra BMW(q,−q2r+1) gives the spher-ical braided tensor category from Equation 2.5.

The planar algebra G2(q).Here we define the planar algebra G2(q), and explain its connection to certain endomorphism

algebras in C(g2, k).

Definition 2.13. Let q a non-zero complex number. The braided planar algebra G2(q) has a

single generator ∈ G2(q)3, satisfying the relations:

(i) = q10 + q8 + q2 + 1 + q−2 + q−8 + q−10


(ii) = 0

(iii) = − (q6 + q4 + q2 + q−2 + q−4 + q−6)

(iv) = (q4 + 1 + q−4)

(v) = − (q2 + q−2)⎛⎜⎝

+⎞⎟⎠+ (q2 + 1 + q−2)

⎛⎜⎝

+⎞⎟⎠

(vi) = + + + +

− − − − − .

The braiding in this planar algebra is given by the element

1

1 + q2+ 1

1 + q−2+ 1

q2 + q4+ 1

q−2 + q−4.

In forthcoming work of Ostrik and Snyder, the following isomorphism of planar algebras isproven:

(2.6) PA(C(g2, k),Λ1) = G2(q) with q = e2πi

6(4+k) .

In this case, the object Λ1 ∈ C(g2, k) ⊗-generates the entire C(g2, k), and thus the idempotent

completion of the planar algebra G2(q) recovers the category C(g2, k). Additionally we have thefollowing planar algebra isomorphisms, which will prove useful later in this paper.

Lemma 2.14. There exist isomorphisms of planar algebras

G2(q) ≅ G2(−q) ≅ G2(q−1) ≅ G2(−q−1).Furthermore, the isomorphism

G2(q) ≅ G2(−q)is an isomorphism of braided planar algebras.

Proof. The isomorphism in each case simply sends the generator ↦ . It is routine to

check that all the relations are preserved by these isomorphisms. �

Tambara-Yamagami categories.A key tool for the computations in this paper are the Tambara-Yamagami categories [50].

We will see later on that we can use auto-equivalences of the Tambara-Yamagami categories toconstruct auto-equivalences of the categories C(so2r+1,2). The Tambara-Yamagami categoriesare particularly nice to work with as it is possible to explicitly write down associators for thesecategories.


Definition 2.15. Let G a finite abelian group, χ a symmetric bicharacter G × G → C×, andτ ∈ {±}. The Tambara-Yamagami category T Y(G,χ, τ) is the fusion category with simpleobjects are G ∪ {m}, and with fusion rules given by

i⊗ j ∶= i + j i⊗m ∶=m =∶m⊗ i and m⊗m ∶= ⊕Gg.The associator morphisms are trivial, except for

αi,m,j ∶= χ(i, j) idm

(αm,i,m)j,j ∶= χ(i, j) idj

(αm,m,m)i,j ∶=τ√∣G∣

χ(i, j)−1.

With such an explicit presentation of the Tambara-Yamagami categories we are able to com-pute the group of monoidal auto-equivalences. This result is certainly well-known to the experts,however we were unable to find a proof in the literature. Hence we give a quick proof.

Lemma 2.16. Let Aut(G,χ) be the group of automorphisms of G preserving χ. Then we have

Eq(T Y(G,χ, τ)) ≅ Aut(G,χ).

Proof. We begin by classifying the gauge auto-equivalences of T Y(G,χ, τ). Let (Id, τ) be such agauge auto-equivalence. Naively we have that τ is determined by ∣G∣2+2∣G∣+ ∣G∣ complex scalers.However solving for the hexagon commuting diagram shows that τ is completely determined bya function µ ∶ G→ C×, with

τi,j ∶=µ(i)µ(j)µ(ij) idi+j

τi,m ∶= µ(i) idm ∶= τm,i(τm,m)i = µ(i)−1 idi .

Furthermore, each µ ∈ Hom(G→ C×) gives a gauge auto-equivalence (Id, τµ) of T Y(G,χ, τ).Let µ1, µ2 ∈ Hom(G→ C×). We define a natural isomorphism η ∶ (Id, τµ1)→ (Id, τµ2) by

ηi ∶=µ1(i)µ2(i)

idi and ηm ∶= idm .

A direct computation shows that η is monoidal. Thus the group of gauge auto-equivalences ofEq(T Y(G,χ, τ)) is trivial.

As the fusion ring automorphisms of T Y(G,χ, τ) correspond to automorphisms of G, wehave that Eq(T Y(G,χ, τ)) ⊆ Aut(G). Explicitly for ψ ∈ Aut(G), the corresponding fusion ringautomorphism fixes m, and sends i ↦ ψ(i). A direct computation shows that there exists atensor structure for this fusion ring automorphism if and only if ψ preserves the symmetricbicharacter χ. �

Remark 2.17. An explicit isomorphism

Aut(G,χ)→ Eq(T Y(G,χ, τ))is given by

ψ ↦ (i↦ ψ(i)m↦m

,ψi,j = idi+j ψm,mi = idi

ψi,m = idm ψm,i = idm) .


Functorality of equivariantization.A key tool for constructing auto-equivalences in this paper is through the functorality of

equivariantization. In [10, Theorem 4.4] a 2-equivalence between the 2-category of G-crossedbraided fusion categories, and the 2-category of braided fusion categories over Rep(G) is given.The author was unable to understand the full details of this 2-equivalence. In this subsectionwe present our interpretation of the functorality of equivariantization, for both the braided andmonoidal cases.

We begin with the braided situation. Let G be a finite group, then we can form two categories.On one hand we have the category of G-crossed braided fusion categories, whose objects aretriples (C, ρ, σ), where C is G-graded fusion category, ρ is a monoidal functor G → Eq(C), and

ω is a family of natural isomorphisms

ωXg ,Y ∶Xg ⊗ Y → ρg(Y )⊗Xg,

satisfying various conditions. A morphism (C1, ρ1, σ1)→ (C2, ρ2, σ2) is a pair (F , η), where F isa monoidal functor C1 → C2, and η is a family of monoidal natural isomorphisms

ηg ∶ ρ2(g) ○F → F ○ ρ1(g),such that the following diagram commutes:(2.7)

F(Xg)⊗F(Y ) F(Xg ⊗ Y ) F(ρ2g(Y )⊗Xg)

(ρ2 ○F)(Y )⊗F(Xg) (F ○ ρ1g)(Y )⊗F(Xg)

τXg,Y

σ2F(Xg),F(Y )

F(σ1Xg,Y

)

ηgY ⊗idF(Xg)

τρ1g(Y ),Xg

On the other hand we have the category of braided fusion categories over Rep(G), whoseobjects are pairs (B,H), where B is a braided fusion category, and H is a fully faithful braidedfunctor Rep(G)→ B. A morphism (B1,H)→ (B2,H) is just a braided functor B1 → B2.

We now describe a functor from the category of G-crossed braided fusion categories, to thecategory of braided fusion categories over Rep(G).

Remark 2.18. We wish to point out that this category of braided fusion categories over Rep(G)is different from the category of braided fusion categories over Rep(G) in [10]. This differenceis the reason why we are only able to give a functor between category of G-crossed braided fusioncategories, to the category of braided fusion categories over Rep(G), and not an equivalence aswas done in [10].

Given (C, ρ, ω), then G-equivariantization produces a braided fusion category CG along witha faithful braided functor H ∶ Rep(G)→ CG defined by:

(V,π)↦ (1⊗ V,{π(g)}).

Given a morphism (F , η) ∶ (C1, ρ1, ω1)→ (C2, ρ2, ω2), we define a braided functor FG ∶ (C1)G →(C2)G by:

FG((X,{µg ∶X → ρ1g(X)})) ∶= (F(X),{(ηg)−1F(µg) ∶ F(X)→ ρ2

g(F(X))}).The tensor structure morphisms are exactly the tensor structure morphisms of F , which areG-equivariant morphisms from the monoidality of η. The functor FG is braided due to thecommutativity of (2.7).

In the monoidal situation we can also construct a functor. Let G a finite group, then weagain define two categories. On one hand we the category of fusion categories with G-action in a


similar way to the definition of the category of G-crossed braided fusion categories, but withoutany G-crossed braiding structure or conditions.

On the other hand we have the category of fusion categories over Rep(G), whose objects arepairs (C,H), where C is a fusion category, and H is a fully faithful braided functor Rep(G) →Z(C). A morphism (C1,H1)→ (C2,H2) is just a monoidal functor C1 → C2.

A similar construction as in the braided case gives a functor from category of fusion categorieswith G-action, to the category of fusion categories over Rep(G).

3. Auto-equivalences for Lie type A

In this section we compute the monoidal, and braided auto-equivalences of the categoriesC(slr+1, k). This case is in some sense easier than the others, as there are no exceptional auto-equivalences to construct. However, the categories C(slr+1, k) have many invertible objects,which give rise to a vast number of simple current auto-equivalences. Also, the existence of theZ2-symmetry of the Ar Dynkin diagram induces an order 2 braided auto-equivalence. Hence thecategories C(slr+1, k) still have a large amount of auto-equivalences that we have to construct,making this case non-trivial, even without the existence of exceptional auto-equivalences.

The outline of our computations is as follows.We begin by showing that the category C(slr+1, k) has no non-trivial gauge auto-equivalences.

To do this, we first study the gauge auto-equivalences of the category Ad(C(slr+1, k)). Weknow that (Λ1 + Λr) is a generating object of Ad(C(slr+1, k)), and further, the planar algebra

generated by (Λ1+Λr) is isomorphic to PH(δ, γ) for a specific choice of δ and γ. Thus, any gauge

auto-equivalence of Ad(C(slr+1, k)) will appear as a planar algebra automorphism of PH(δ, γ).Using the generators and relations presentation of PH(δ, γ) we compute the automorphism

group, finding several non-trivial automorphisms. In theory each of these non-trivial automor-phisms could lift to a gauge auto-equivalences of the category Ad(C(slr+1, k)). However, we areable to show that each of these non-trivial automorphisms lifts to a non-gauge auto-equivalence.

The proper way to do this would be to compute the idempotents of the planar algebra PH(δ, γ),and to show that each non-trivial automorphism must exchange non-isomorphic idempotents.

However, computing the idempotents of PH(δ, γ) proved to be too computationally intensive tocomplete. Instead we identify known pivotal auto-equivalences of the category Ad(C(slr+1, k))coming from charge-conjugation and level-rank duality, that fix the object (Λ1 + Λr). These

auto-equivalences must appear as automorphisms of the planar algebra PH(δ, γ), hence we are

left with no room in the automorphism group of PH(δ, γ) for a gauge auto-equivalence.With the fact that the gauge auto-equivalence group of Ad(C(slr+1, k)) is trivial, we can then

leverage the fact that C(slr+1, k) is a Zr+1-graded extension of Ad(C(slr+1, k)) to show that thegauge auto-equivalence group of C(slr+1, k) is also trivial.

As the category C(slr+1, k) has no non-trivial gauge auto-equivalences, the group of monoidalauto-equivalences of C(slr+1, k) are a subgroup of the group of fusion ring automorphisms. Theresults of [24] compute the order of the group of fusion ring automorphisms of C(slr+1, k), givingupper bounds on the size of the monoidal auto-equivalence group of C(slr+1, k). We find outthat these bounds are sharp, by constructing enough monoidal auto-equivalences of C(slr+1, k)to realise this upper bound. We construct these auto-equivalences using the charge-conjugationsymmetry of the Ar Dynkin diagram, and simple current auto-equivalences which we get fromthe large number of invertible objects in C(slr+1, k).

With the high level argument explained, let us begin with the finer details of the computation.


Lemma 3.1. Suppose δ ≠ ±√

2. The planar algebra PH(δ, γ) has planar algebra automorphismgroup (up to planar algebra natural isomorphism) as follows

Aut(PH(δ, γ)) =⎧⎪⎪⎨⎪⎪⎩

Z2, if γ ≠ 1,

Z2 ×Z2 if γ = 1.

Proof. As the planar algebra PH(δ, γ) is generated by the two elements ,S

∈ PH(δ, γ)3,

any planar algebra automorphism of PH(δ, γ) is completely determined by its value on these

two elements. Explicitly this means that φ ∈ Aut(PH(δ, γ)) is completely determined by fourscalers c1, c2, c3, c4 ∈ C, with

φ( ) = c1 + c2S

, and φ(S

) = c3 + c4S

.

Applying φ to relations (iii) through (vii) of PH(δ, γ) yields the following system of equations

δ − 2

δ= c2

1 (δ −2

δ) +

c22δ (δ2 − 2)γδ2 − 1

0 = c1c3 (δ −2

δ) +

c2c4δ (δ2 − 2)γδ2 − 1

δ (δ2 − 2)γδ2 − 1

= c23 (δ −

2

δ) +

c24δ (δ2 − 2)γδ2 − 1

c1 (δ2 − 3)δ

=c3

1 (δ2 − 3)δ

− 3c1c22δγ

δ2 − 1+ c

32δ

2(γ − 1)γδ2 − 1

c2 (δ2 − 3)δ

= −3c21c2

δ+ 3c1c

22(γ − 1) +

c32 (δ4γ + δ2 (−2γ2 + γ − 2) + 2(γ − 1)2)

δ (δ2 − 1)

−c3

δ=

c21c3 (δ2 − 3)

δ−δγ (2c1c2c4 + c2

2c3)δ2 − 1

+ c22c4δ

2(γ − 1)γδ2 − 1

−c4

δ= −c

21c4 + 2c1c2c3

δ+ (γ − 1) (2c1c2c4 + c2

2c3) +c2

2c4 (δ4γ + δ2 (−2γ2 + γ − 2) + 2(γ − 1)2)δ (δ2 − 1)

With an additional assumption that δ ≠√

2, a computer algebra program1 can solve theseequations to find the following solution sets, where in each case ε1, ε2 ∈ {+,−}:

● The solutions

φ1ε1,ε2 ∶ {c1 = ε1, c2 = 0, c3 = 0, c4 = ε2},

which exist for all parameter values δ and γ.● The solutions

φ2ε1,ε2 ∶ {c1 = ε1

√δ2 − 1(γ − 1)√

δ4γ + δ2γ2 − 2δ2γ + δ2 − γ2 + 2γ − 1, c2 =

c1δ

γ − 1, c3 = ε2

1 − c21

c2, c4 =

c1c2c3

c21 − 1

},

which exist for all δ when γ ≠ 1.● The solutions

φ3ε1,ε2 ∶ {c1 = 0, c2 = ε1

√δ2 − 1

δ, c3 = ε2

1

c2, c4 = 0},

which exist for all δ when γ = 1.

1Reduce from the Mathematica library


We can apply these solutions to the remaining relations of PH(δ, γ) to see that

φ1+,+, φ1

−.−, φ2+,+, and φ2

−.−

are planar algebra automorphisms of PH(δ, γ) when γ ≠ 1, and

φ1+,+, φ1

+,−, φ1−,+, φ1

−.−, φ3+,+, φ3

+,−, φ3−,+, and φ3

−.−

are planar algebra automorphisms of PH(δ, γ) when γ = 1.Finally, there exist planar algebra natural isomorphisms

φ○ε1,ε2 → φ○−ε1,−ε2 .

Hence we have two planar algebra automorphisms, up to planar algebra natural isomorphism,

of PH(δ, γ) when γ ≠ 1, and four when γ = 1. A direct computation shows that these planaralgebra automorphisms compose as in the statement of the Lemma. �

As a corollary, we are able to use this planar algebra computation to determine the gaugeauto-equivalences of Ad(C(slr+1, k)).

Corollary 3.2. We have

Gauge(Ad(C(slr+1, k))) =⎧⎪⎪⎨⎪⎪⎩

Z2, if r = 2 and k = 3,

{e} otherwise.

Proof. We prove this statement in several cases.

Case k = 1:In the k = 1 case we have that Ad(C(slr+1,2)) ≃ Vec which clearly has no non-trivial

gauge auto-equivalences.

Case k = 2:When k = 2 we have that Ad(C(slr+1,2)) is monoidaly equivalent to Ad(C(sl2, r + 1))

via [43, Theorem 5.1]. The category Ad(C(sl2, r + 1)) is shown to have trivial gauge auto-equivalence group in [12, Lemma 4.2].

Case k ≥ 3 and (r, k) ≠ (2,3):Recall that the object (Λ1+Λr) is a symmetrically self-dual ⊗-generator of Ad(C(slr+1, k)),

and that the planar algebra generated by (Λ1 +Λr) is isomorphic to PH(δ, γ) with

δ = e2iπ(r+1)2(r+1+k) − e

−2iπ(r+1)2(r+1+k)

e2iπ

2(r+1+k) − e−2iπ

2(r+1+k)and γ = e

4iπ2(r+1+k) − e

(12+4(r+1))iπ2(r+1+k) + e

(8+8(r+1))iπ2(r+1+k) − e

4iπ(r+1)2(r+1+k)

e8iπ

2(r+1+k) − e(12+4(r+1))iπ

2(r+1+k) + e(4+8(r+1))iπ

2(r+1+k) − e4iπ(r+1)2(r+1+k)

.

From [30, Theorem A] we know that the group of planar algebra automorphisms of

PH(δ, γ) corresponds to the group of pivotal auto-equivalences of Ad(C(slr+1, k)) that fixthe object (Λ1 + Λr). As the category Ad(C(slr+1, k)) has a unique pivotal structure, wehave that every monoidal auto-equivalence of Ad(C(slr+1, k)) is pivotal. Hence the group

of planar algebra automorphisms of PH(δ, γ) also corresponds to the group of monoidalauto-equivalences of Ad(C(slr+1, k)) that fix the object (Λ1 + Λr). Recalling the previousLemma, and translating the parameters, we find the categories Ad(C(slr+1, k)) for k ≠ r + 1have exactly two monoidal auto-equivalences that fix the object (Λ1+Λr), and the categoriesAd(C(slr+1, r+1)) have exactly four monoidal auto-equivalences that fix the object (Λ1+Λr).We now attempt to identify these auto-equivalences.


From Corollary 2.7 we know that the category C(slr+1, k) has the order 2 charge-conjugationauto-equivalence, which maps

r

∑i=1

λiΛi ↦r

∑i=1

λr−i+1Λi.

In particular, this charge-conjugation auto-equivalence restricts to a monoidal auto-equivalenceof Ad(C(slr+1, k)) fixes the object (Λ1+Λr). Further, the charge-conjugation auto-equivalencemoves the object (Λ2 + 2Λr) ∈ Ad(C(slr+1, k)). Therefore the charge-conjugation auto-equivalence restricts to a non-gauge auto-equivalence of Ad(C(slr+1, k)) that fixes (Λ1+Λr).

From [43, Theorem 5.1] there exists a level-rank duality auto-equivalence of Ad(C(slr+1, r+1)) which has order 2. The details of this auto-equivalence are complicated, and can befound in the cited paper. For us it suffices to know that this auto-equivalence fixes theobject (Λ1 + Λr), and the auto-equivalence is anti-braided, so it will only fix an object ifits twist is equal to ±1. Consider the object (Λ2 + Λr−1) ∈ Ad(C(slr+1, r + 1)). This object

has twist equal to e2πi 4r

4(r+1) when r ≥ 3. Hence the twist of this object is not equal to ±1when r ≥ 3, and so this object must be moved by the level-rank duality auto-equivalence.In particular, this auto-equivalence is not gauge if r ≥ 3.

We are now in place to compute the gauge auto-equivalence group of Ad(C(slr+1, k)).When k ≠ r+1 there are exactly two monoidal auto-equivalences of Ad(C(slr+1, k)) that fixthe object (Λ1 +Λr). One of these is the identity, and the other is charge-conjugation auto-equivalence. Thus there is no room for a non-trivial gauge auto-equivalence of Ad(C(slr+1, k))when k ≠ r + 1.

When k = r+1 there are exactly four monoidal auto-equivalences of Ad(C(slr+1, r+1)) thatfix the object (Λ1+Λr). One of these is the identity, another is the charge-conjugation auto-equivalence, and a third is the level-rank duality auto-equivalence. When r ≥ 3 the level-rankauto-equivalence moves the object (Λ2+Λr−1), yet the charge-conjugation auto-equivalencefixes this object. Thus the charge-conjugation and level-rank duality auto-equivalences aredistinct. Further these two auto-equivalences have order 2, thus their composition givesa fourth distinct auto-equivalence that moves the object (Λ2 + Λr−1), and fixes the object(Λ1+Λr). Thus we have constructed four non-gauge auto-equivalences of Ad(C(slr+1, r+1))that fix the object (Λ1+Λr), hence there is no room for a non-trivial gauge auto-equivalenceof Ad(C(slr+1, r + 1)) when r ≥ 3.

Case (r, k) = (2,3):A direct computation shows that there are only two fusion ring automorphisms of Ad(C(sl3,3)).

The single non-trivial fusion ring automorphism fixes the objects 1 and (Λ1 + Λ2), andexchanges the objects (3Λ1) and (3Λ2). We can realise this non-trivial fusion ring auto-morphism as the charge-conjugation auto-equivalence. However there are four monoidalauto-equivalences of Ad(C(sl3,3)) that fix (Λ1 +Λ2). Therefore two of these four monoidalauto-equivalences must be gauge.

�

With the knowledge that Ad(C(slr+1, k)) has no gauge auto-equivalences except for a single ex-ception, we can leverage this to show that C(slr+1, k) has no non-trivial gauge auto-equivalences,with no exceptions.

Theorem 3.3. The category C(slr+1, k) has trivial gauge auto-equivalence group.

Proof. We prove this statement in several cases

Case r = 1:


If r = 1, then the statement of this Theorem is exactly [12, Lemma 4.2] paired withLemma 2.8.

Case (r, k) = (2,3):If r = 2 and k = 3, then the object Λ1 generates an orientated A2 planar algebra with q =

e2iπ12 [36]. This planar algebra has two generators, and the automorphisms are parametrised

by a single non-zero complex number. Further, every element of this 1-parameter family ofplanar algebra automorphisms is naturally isomorphic to the identity. Hence, the orientatedversion of [30, Theorem A] (which has not appeared in print, but is known as folklore) impliesthat the group of pivotal auto-equivalences of C(sl3,3) that fix the object Λ1 is trivial. Asevery gauge auto-equivalence of C(sl3,3) is pivotal by Lemma 2.1, we have the result.

Case r ≥ 2 and (r, k) ≠ (2,3):For this general case we will use [11, Theorem 9.1]. This Theorem allows us to leverage

the fact that C(slr+1, k) is a Zr+1-graded extension of Ad(C(slr+1, k)) in order to computethe gauge auto-equivalences of C(slr+1, k). To apply this Theorem we must show threerequirements are satisfied. They are

(1) the category Ad(C(slr+1, k)) must have no non-trivial gauge auto-equivalences,(2) if z is an invertible object of Ad(C(slr+1, k)) such that there exists a non-trivial graded

component Cg of C(slr+1, k) such that for all Xg ∈ Cg we have z ⊗Xg ≅ Xg, then z ≅ 1,and

(3) the categories Ad(Ad(C(slr+1, k))) and Ad(C(slr+1, k)) must be monoidally equivalent.

Let us verify each of these requirements in turn.

1) The first requirement follows from Corollary 3.2.

2) The category C(slr+1, k) has the r non-trivial invertible objects {(kΛi) ∶ 1 ≤ i ≤ r}which may, or may not live in Ad(C(slr+1, k)) depending on k. Let Cg be the g-gradedcomponent of C(slr+1, k), where 1 ≤ g ≤ r+1. If the invertible object (kΛi) was a fixed pointfor all objects in Cg, then in particular it would fix the object (Λg), so we would have

(kΛi)⊗ (Λg) ≅ (Λg).Frobenius reciprocity implies (kΛi) is a sub-object of (Λg) ⊗ (Λg)∗. However we can de-compose this tensor product into simples to see

(kΛi) ⊂ (Λg)⊗ (Λg)∗ ≅min(g,r+1−g)

⊕j=0

(Λj +Λr+1−j).

Hence the only possibility for (kΛi) to fix Λg is when k = 2 and i = g = r+12 . Thus if k ≠ 2

we are done.If k = 2 then we need to find an object in C r+1

2that is not fixed by (2Λ r+1

2). Consider

the object (Λ1 +Λ r−12). We can directly compute that (2Λ r+1

2)⊗(Λ1 +Λ r−1

2) ≅ (Λ r+3

2+Λr).

Hence (Λ1 +Λ r−12) is not fixed by (2Λ r+1

2) and we are done. Thus the second requirement

is satisfied.

3) For the third requirement, it suffices to note that (Λ1+Λr) ⊗-generates Ad(C(slr+1, k)),and is contained in Ad(Ad(C(slr+1, k))) as (Λ1 +Λr) ⊂ (Λ1 +Λr)⊗ (Λ1 +Λr)∗.

As our three requirements are satisfied, we can apply [11, Theorem 9.1] to see

Gauge(C(slr+1, k)) =H2(Zr+1,C×) = {e}.


�

Let us move on to realising the fusion ring automorphisms of C(slr+1, k) as monoidal auto-equivalences. From the results of [24] we know that

∣FusEq(C(sl2,2))∣ = 1

∣FusEq(C(sl2, k))∣ = ∣{0 ≤ a ≤ 1 ∶ 1 + a is coprime to 2}∣ for k ≠ 2

∣FusEq(C(slr+1,1))∣ = ∣{0 ≤ a ≤ r ∶ 1 + a is coprime to r + 1}∣∣FusEq(C(slr+1,2))∣ = ∣{0 ≤ a ≤ r ∶ 1 + 2a is coprime to r + 1}∣∣FusEq(C(slr+1, k))∣ = 2∣{0 ≤ a ≤ r ∶ 1 + ka is coprime to r + 1}∣ for k ≥ 3 and r ≥ 2.

As we have shown that C(slr+1, k) has no non-trivial gauge auto-equivalences, we know that∣FusEq(C(slr+1, k))∣ gives an upper bound on the order of Eq(C(slr+1, k)). We will soon seethat this upper bound is sharp. Let us move on to constructing these ∣FusEq(C(slr+1, k))∣auto-equivalences.

Lemma 3.4. For every r and k we have that

Eq(C(slr, k))is a group of order

∣FusEq(C(slr+1, k))∣.

Proof.

Case r = 1:The case of r = 1 is exactly [12, Lemma 4.9].

Case r ≥ 2:For r ≥ 1 we first note that the charge-conjugation auto-equivalence of C(slr+1, k) con-

structed in Corollary 2.7 gives a non-trivial auto-equivalence of C(slr+1, k) for all r ≥ 2. Thischarge-conjugation auto-equivalence is characterised by the fact that it maps

Λ1 ↦ Λr.

Let us now fix an 0 ≤ a ≤ r such that 1+ ka is coprime to r + 1, and use this to constructan auto-equivalence of C(slr+1, k). We pick out the invertible object kΛ1 ∈ C(slr+1, k). Thisinvertible object has order

M = r + 1,

and self-braiding eigenvalue

eπi rk

2(r+1) .

As 1 + ka is coprime to r + 1, we apply the simple current construction from Lemma 2.4to get monoidal auto-equivalences FkΛ1,a of C(slr+1, k). We have that

σΛ1,kΛ1 ○ σkΛ1,Λ1 = e2πi rr+1 .

Thus

FkΛ1,a(Λ1) = (kΛ1)−ar ⊗Λ1 = (kΛa)⊗Λ1 =⎧⎪⎪⎨⎪⎪⎩

(k − 1)Λa +Λa+1, if a ≠ r(k − 1)Λa, if a = r.

Hence the monoidal auto-equivalences Fg,a are pairwise distinct for all a. Thus these simplecurrent auto-equivalences give

∣{a ∶ 1 + ka is coprime to r + 1}∣distinct auto-equivalences of C(slr+1, k).


There is the possibility that these simple current auto-equivalences overlap with thecharge conjugation auto-equivalences. To investigate when this occurs, we compute

Fg,a(Λ1) = Λr

if and only if k = 2 and a = r, or if k = 1 and a = r − 1. Thus, when paired with thecharge-conjugation auto-equivalence of C(slr+1, k) we have constructed

∣{a ∶ 1 + ka is coprime to r + 1}∣distinct auto-equivalences of C(slr+1, k) when k ∈ {1,2}, and

2∣{a ∶ 1 + ka is coprime to r + 1}∣distinct auto-equivalences of C(slr+1, k) when k ≥ 3. Thus we have realised the upper boundon Eq(C(slr+1, k)) given by ∣FusEq(C(slr, k))∣.

�

Let us explicitly describe these monoidal auto-equivalences of C(slr+1, k), by describing howthey act on the object Λ1. It follows from [35] that this completely describes the auto-equivalence.

Remark 3.5. For an a ∈ {0 ≤ a ≤ r ∶ 1 + ka is coprime to r + 1} and choice of sign ε ∈ {+,−},we get a monoidal auto-equivalence of C(slr+1, k) that sends

Λ1 ↦

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

(k − 1)Λa +Λa+1 if a ≠ r and ε = +,(k − 1)Λa if a = r, and ε = +,(k − 1)Λr+1−a +Λr−a if a ≠ r and ε = −,(k − 1)Λr+1−a if a = r, and ε = −.

The composition of the monoidal auto-equivalences of C(slr+1, k) is determined in Appendix A.Here it is found the group structure is as given in Theorem 1.1.

Finally, we investigate which of the monoidal auto-equivalences of C(slr+1, k) are braided.

Lemma 3.6. For all r and k we have

EqBr(C(slr+1, k)) = Zc+p+t2

where

c =⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

−1, if r = 1 and k = 2

0, if k ≤ 2

1, if k ≥ 3,

p is the number of distinct odd primes dividing r + 1, and

t =⎧⎪⎪⎨⎪⎪⎩

0, if either r is even, or r is odd and k ≡ 0 (mod 4), or k is odd and r ≡ 1 (mod 4),1, otherwise.

Proof. The simple current auto-equivalence Fg,a is braided if and only if

a2 − rka2

≡ 0 (mod r + 1).

From the analysis of [24], we have that there are 2p+t such choices for a, where p and t are inthe statement of the Lemma. It is shown in [24] that each of these braided Fg,a has order two.

Further, the charge-conjugation auto-equivalence is braided, giving us an additional order twobraided auto-equivalence when r ≥ 2, giving the c term in the Lemma. The special case of c = −1when r = 1 and k = 2 is to uncount the simple current auto-equivalence of C(sl2,2) coming froma = 1 that is naturally isomorphic to the identity.

�


4. Auto-equivalences for Lie type B

In this section we compute the monoidal, and braided auto-equivalences of the categoriesC(so2r+1, k). The outline of our computations is as follows.

We begin with the gauge auto-equivalences of C(so2r+1, k). We know that the planar algebra

generated by the object Λ1 ⊠Csgn ∈ C(so2r+1, k) ⊠Rep(Z2) is isomorphic to BMW(q2, q4r) with

q = e2πi

4(2r−1+k) . However, the object Λ1 ⊠Csgn only generates the subcategory

Ad(C(so2r+1, k)) ⊠Rep(Z2) ⊂ C(so2r+1, k) ⊠Rep(Z2).

Thus the planar algebra automorphism group of BMW(q2, q4r) will only give us informationabout the gauge auto-equivalences of Ad(C(so2r+1, k)) ⊠Rep(Z2).

A straightforward computation shows that the planar algebra BMW(q2, q4r) has trivial auto-morphism group, except in the special case where k = 2r + 1. A further computation shows thatthis non-trivial planar algebra automorphism corresponds to an auto-equivalence of

Ad(C(so2r+1, k)) ⊠Rep(Z2)which exchanges simple objects. Thus, the category Ad(C(so2r+1, k)) ⊠ Rep(Z2) has no non-trivial gauge auto-equivalences. From this fact, it immediately follows that Ad(C(so2r+1, k))also has no non-trivial gauge auto-equivalences. As C(so2r+1, k) is a Z2-graded extension ofAd(C(so2r+1, k)), we can now apply [11, Theorem 9.1] to see that C(so2r+1, k) also has no non-trivial gauge auto-equivalences.

The results of [24] compute the fusion ring automorphisms of C(so2r+1, k) as:

{e} if k = 1,

Z2 × (Z×2r+1/{±1}) if k = 2, and

Z2 if k > 2.

Thus to complete the computation of the auto-equivalences of C(so2r+1, k) we have to deter-mine which of the above fusion ring automorphisms are realised as either monoidal, or braidedauto-equivalences. The k = 1 case is trivial, as there are no non-trivial auto-equivalences. Thek > 2 case is also rather easy, as the single non-trivial fusion ring automorphism is alwaysrealised as a simple current automorphism. The difficult case is k = 2. Here we proceed byusing the modular data of C(so2r+1,2) to bound the size of the braided auto-equivalence groupof C(so2r+1,2), which turns out to be significantly smaller than Z2 × (Z×2r+1/{±1}). We thenperform a trick to leverage this bound to also determine a bound for the size of the monoidalauto-equivalence group of C(so2r+1,2). To realise these upper bounds we observe that thede-equivariantization of C(so2r+1,2) is a Tambara-Yamagami category, which we can explic-itly identify. Using Lemma 2.16 we can compute the auto-equivalence group of this Tambara-Yamagami category. We can then apply the functorality of equivariantization to extend theseauto-equivalences of the Tambara-Yamagami category, up to auto-equivalences of C(so2r+1,2).These auto-equivalences realise the upper bound we computed earlier, hence the computation iscomplete.

With the high level argument in mind, let us proceed with the details. We begin by computing

the planar algebra automorphisms of BMW(q, r). This computation will also be useful later on,for our computations in the Lie type C section.

Lemma 4.1. The planar algebra BMW(q, r) has no non-trivial automorphisms unless r = ±i.If r = ±i, then there exists a single non-trivial automorphism sending

↦ − .


This automorphism is not braided, and exchanges non-isomorphic simple objects in the idempo-tent completion.

Proof. As the planar algebra BMW(q, r) is generated by the single element

∈ BMW(q, r)4,

any planar algebra automorphism of BMW(q, r) is completely determined by its value on this

element. The box space BMW(q, r)4 is 3-dimensional, so φ ∈ Aut(BMW(q, r)) is determined bythree scalers α,β, γ ∈ C, with

φ⎛⎜⎝

⎞⎟⎠

= α + β + γ .

Solving for the relation

− = (q − q−1)⎛⎜⎝

−⎞⎟⎠

yields the equation

β − α = (γ − 1)(q − q−1).Solving for the Reidemeister 2 relation yields the additional three equations

αβ + γ2 − αγ(q − q−1) = 1,

α2 + β2 + αβ (r − r−1

q − q−1+ 1) + αγ

r+ γβr + γα(q − q−1) = 0,

γ(α + β) = 0.

Solving for all four of these above equations yields two solutions:

(α,β, γ) = (0,0,1) or (q − q−1,−(q − q−1),−1).The first of these solutions is just the identity, and the second we can write as

φ⎛⎜⎝

⎞⎟⎠

= − .

Solving for the relation

= r

reveals that the second is an automorphism only if r = −r−1, i.e. only if r = ±i.Clearly the possible non-trivial φ satisfies Reidiemester 3, thus φ is a planar algebra automor-

phism of BMW(q,±i), and hence of BMW(q,±i) by Proposition 2.10.

As the braiding in BMW(q,±i) is the crossing generator, the non-trivial automorphism φ isclearly not braided.

The idempotent completion of BMW(q,±i) contains the two distinct simple objects

± iq

(q ± i)2∓ iq

(q ∓ i)(q ± i)2∓ q2

(q ∓ i)(q ± i)2


and±iq

(q ± i)2± q2

(q ∓ i)(q ± i)2± iq

(q ∓ i)(q ± i)2.

The non-trivial automorphism φ exchanges these two idempotents, and thus exchanges thecorresponding non-isomorphic simple objects in the idempotent completion. �

Specialising this Lemma to the C(so2r+1, k) case gives the following Corollary.

Corollary 4.2. The category C(so2r+1, k) has no non-trivial gauge auto-equivalences.

Proof. Let us begin with the special case of k = 1, which requires special treatment. Thecategories C(so2r+1,1) have Ising fusion rules, hence these categories have no non-trivial gaugeauto-equivalences by [13].

In the general case of k ≥ 2, we will use [11, Theorem 9.1]. This Theorem allows us to leveragethe fact that C(so2r+1, k) is a Z2-graded extension of Ad(C(so2r+1, k)), in order to compute thegauge auto-equivalences of C(so2r+1, k). To apply this Theorem we must show three requirementsare satisfied. They are

(1) The category Ad(C(so2r+1, k)) must have no non-trivial gauge auto-equivalences,(2) if z is an invertible object of Ad(C(so2r+1, k)) such that for every object X in the non-

trivially graded component of C(so2r+1, k) we have z ⊗Xg ≅Xg, then z ≅ 1, and(3) the categories Ad(Ad(C(so2r+1, k))) and Ad(C(so2r+1, k)) must be monoidally equiva-

lent.

Let us verify each of these requirements in turn.

1) From Lemma 2.8, every gauge auto-equivalence of Ad(C(so2r+1, k))⊠Rep(Z2) will appearas a planar algebra automorphism of the planar algebra generated by X, where X is any ⊗-generator of Ad(C(so2r+1, k))⊠Rep(Z2). The object Λ1⊠Csgn is a particularly nice ⊗-generatorof Ad(C(so2r+1, k)) ⊠ Rep(Z2), as we have the following isomorphism of planar algebras fromEquation (2.3)

PA(Ad(C(so2r+1, k)) ⊠Rep(Z2),Λ1 ⊠Csgn) ≅ BMW(q2, q4r) with q = e2πi

4(2r−1+k) .

From Lemma 4.1, the planar algebra BMW(q2, q4r) only has a single non-trivial automorphismwhen k = 2r + 1, which corresponds to an auto-equivalence of Ad(C(so2r+1, k) ⊠ Rep(Z2) thatexchanges distinct simple objects. Thus, the category Ad(C(so2r+1, k)) ⊠Rep(Z2) has no non-trivial gauge auto-equivalences. As a non-trivial gauge auto-equivalence of Ad(C(so2r+1, k))would canonically extend to a non-trivial gauge auto-equivalence of Ad(C(so2r+1, k))⊠Rep(Z2),we see that Ad(C(so2r+1, k)) has no non-trivial gauge auto-equivalences.

2) The only non-trivial invertible object of Ad(C(so2r+1, k)) is the object kΛ1. The non-trivialgraded component of C(so2r+1, k) contains the object Λr, and we have the fusion

(kΛ1)⊗ (Λr) ≅ ((k − 1)Λ1 +Λr).Thus, when k > 1 (a case we have already covered), the invertible object kΛ1 is not a fixed pointfor the non-trivially graded component of C(so2r+1, k).

3) To show that Ad(Ad(C(so2r+1, k))) ≃ Ad(C(so2r+1, k)), we have to break up into two casesdepending on the parity of k.

If k is even, then consider the object kΛr ∈ Ad(C(so2r+1, k)). We have the fusion

(kΛr)⊗ (kΛr)∗ ⊃ Λ1,

therefore Λ1 ∈ Ad(Ad(C(so2r+1, k))). As Λ1 ⊗-generates Ad(C(so2r+1, k)), we thus have

Ad(Ad(C(so2r+1, k))) ≃ Ad(C(so2r+1, k)).


If k is odd and not equal to 1, then consider the object (k−1)Λr ∈ Ad(C(so2r+1, k)). We havethe fusion

((k − 1)Λr)⊗ ((k − 1)Λr)∗ ⊃ Λ1,

therefore Λ1 ∈ Ad(Ad(C(so2r+1, k))). As Λ1 ⊗-generates Ad(C(so2r+1, k)), we thus have

Ad(Ad(C(so2r+1, k))) ≃ Ad(C(so2r+1, k)).

As all three conditions are satisfied, we can apply [11, Theorem 9.1] to see that the gaugeauto-equivalence group of C(so2r+1, k) is isomorphic to H2(Z2,C×) = {e}. �

Now that we have shown the categories C(so2r+1, k) have no non-trivial gauge auto-equivalences,we next have to determine which fusion ring automorphisms of C(so2r+1, k) are realised as eithermonoidal or braided auto-equivalences. There are three cases to work through, k = 1, k = 2, andk ≥ 3.

Case: k = 1.When k = 1, there are no non-trivial fusion ring automorphisms of C(so2r+1, k), thus

EqBr(C(so2r+1,1)) = Eq(C(so2r+1,1)) = Gauge(C(so2r+1,1)) = {e},as in the statement of Theorem 1.1.

Case: k ≥ 3.When k ≥ 3, there is a single non-trivial fusion ring automorphism of C(so2r+1, k), charac-

terised by the fact it maps

(Λr)↔ ((k − 1)Λ1 +Λr).We will show this automorphism is always realised by a monoidal auto-equivalence of C(so2r+1, k),

which we construct as a simple current auto-equivalence.Consider the order two invertible object (kΛ1) ∈ C(so2r+1, k). This invertible object has self

braiding eigenvalue 1 is k is even, and −1 if k is odd. Thus we can use Lemma 2.4 to get amonoidal auto-equivalence of C(so2r+1, k) when k is even, and a braided auto-equivalence whenk is odd. We compute that this auto-equivalence maps

(Λr)↦ (Λr)⊗ (kΛ1) ≅ ((k − 1)Λ1 +Λr).Thus we have, for k ≥ 3,

Eq(C(so2r+1, k)) = Z2 and EqBr(C(so2r+1, k)) =⎧⎪⎪⎨⎪⎪⎩

{e}, if k is even,

Z2, if k is odd.

As in the statement of Theorem 1.1.

Case: k = 2.By far and away the most difficult case are the categories C(so2r+1,2), due to the existence of

many exotic fusion ring automorphisms. To simplify notation we give these categories explicitpresentations. We label the simple objects of C(so2r+1,2) as:

1, Z,X1,X2, Yi

with 1 ≤ i ≤ r. The objects 1 and Z have dimension 1, the objects X1, and X2 have dimension√2r + 1, and the objects Yi all have dimension 2. The commutative fusion rules are given by

Z ⊗Z ≅ 1

Z ⊗X1 ≅X2

Z ⊗X2 ≅X1

Z ⊗ Yi ≅ Yi


X1 ⊗X1 ≅X2 ⊗X2 ≅ 1⊕⊕i

Yi

X1 ⊗X2 ≅ Z ⊕⊕i

Yi

Yi ⊗ Yj ≅ Ymin(i+j,2r+1−i−j) ⊕ Y∣i−j∣ i ≠ jYi ⊗ Yi ≅ 1 +Z + Ymin(2i,4r+2−2i).

The twists of the simple objects 1 and Z are 1, the twists of the objects X1 and X2 have 8 foldsymmetry with respect to r, which we present in Table 3. The twists of the simple objects Yjare given by the formula

tYj = e2iπ j2r2r+1 .

r mod 8 tX1 tX2

0 1 -1

1 e2iπ 18 e2iπ 5

8

2 e2iπ 14 e2iπ 3

4

3 e2iπ 38 e2iπ 7

8

4 -1 1

5 e2iπ 58 e2iπ 1

8

6 e2iπ 34 e2iπ 1

4

7 e2iπ 78 e2iπ 3

8

Table 3. Twists of the simple objects X1 and X2 in C(so2r+1,2)

The group of automorphisms of the C(so2r+1,2) fusion ring is isomorphic to Z2 × Z×2r+1/(±).The automorphism corresponding to the Z2 factor simply sends

X1 ↔X2,

and fixes all other objects. The automorphism corresponding to an element m ∈ Z×2r+1/(±) sends

Yi ↦ Ymin(mi (mod 2r+1)),−mi (mod 2r+1)),

and fixes all other objects.As there are no non-trivial gauge auto-equivalences of C(so2r+1,2), we have that

Eq(C(so2r+1,2)) ⊆ Z2 ×Z×2r+1/(±).It turns out that this upper bound for Eq(C(so2r+1,2)) is not sharp, as there are fusion ringautomorphisms of C(so2r+1,2) that are not realisable as monoidal auto-equivalences. We beginour computations by giving improved bounds. While not a-priori, these new bounds will besharp.

Lemma 4.3. We have

EqBr(C(so2r+1,2)) ⊆ {n ∈ Z×2r+1 ∶ n2 = 1}/(±).Proof. To determine an upper bound for the group of braided auto-equivalences of C(so2r+1,2)we calculate the group of fusion ring automorphisms that preserve the twists of the simpleobjects.

As X1 and X2 always have differing twists, the fusion ring automorphism X1 ↔X2 does notpreserve twists.

Let n ∈ Z×2r+1/(±), and consider the fusion ring automorphism

Yi ↦ Ymin(ni (mod 2r+1)),−ni (mod 2r+1)).


A calculation shows that this fusion ring automorphism preserves the twists of all the simpleobjects if and only if n2 ≡ 1 (mod 2r + 1). �

Using similar techniques, we can prove the following Lemma, giving a necessary conditionfor there to be a braided equivalence between C(so2r+1,2) and its braided opposite. While thisLemma may seem irrelevant, we will soon find an important use for it when trying to improvethe bound on Eq(C(so2r+1,2)).

Lemma 4.4. There exists a braided equivalence C(so2r+1,2) → C(so2r+1,2)rev only if Z×2r+1

contains an element n such that n2 = −1.

Proof. Let F ∶ C(so2r+1,2) → C(so2r+1,2)rev a braided equivalence. As the underlying monoidalcategories of C(so2r+1,2) and C(so2r+1,2)rev are the same, we have that

F(Yi) = Y revmin(ni (mod 2r+1)),−ni (mod 2r+1))

for some n ∈ Z×2r+1. In particular we have that Y1 = Y revmin(n (mod 2r+1)),−n (mod 2r+1)). As F

preserves twists we get the equation

e2iπ r2r+1 = e2iπ −n

2r2r+1 ,

which implies that n2 = −1 (mod 2r + 1). �

We now return to improving the upper bound on Eq(C(so2r+1,2)). We achieve this bound bya combinatorial argument, counting the size of the group EqBr(Z(C(so2r+1,2))) in two ways.On one hand we give an upper bound for EqBr(Z(C(so2r+1,2))) by considering fusion rules andtwists of the Drinfeld center. On the other hand we count the size of EqBr(Z(C(so2r+1,2)))using the following equation

∣EqBr(Z(C(so2r+1,2)))∣ = ∣Eq(C(so2r+1,2))∣ ⋅ ∣EqBr(C(so2r+1,2))∣.This equation follows from the fact that the data for an invertible bimodule over C(so2r+1,2)consists of an invertible C(so2r+1,2) module, and an outer auto-equivalence of C(so2r+1,2) [28].As the category C(so2r+1,2) is braided and has no non-trivial gauge auto-equivalences, we havethat Out(C(so2r+1,2)) = Eq(C(so2r+1,2)). Hence we can combine the two ways to count the sizeof EqBr(Z(C(so2r+1,2))) to give an upper bound for Eq(C(so2r+1,2)). Working through thisapproach gives the following Lemma.

Lemma 4.5. We have

∣Eq(C(so2r+1,2))∣ ≤⎧⎪⎪⎨⎪⎪⎩

4∣EqBr(C(so2r+1,2))∣ if Z×2r+1 contains an element n such that n2 = −1

2∣EqBr(C(so2r+1,2))∣ otherwise.

Proof. Let us begin by showing that the category Z(C(so2r+1,2)) has no non-trivial braidedgauge auto-equivalences. This fact will allow us to apply combinatorial techniques for theremainder of the proof.

Aiming for a contradiction, suppose Z(C(so2r+1,2)) had a non-trivial braided gauge auto-equivalence F . Then [16, Theorem 1.1] would imply the existence of a non-trivial invertiblebimodule M over C(so2r+1,2). As F is gauge, [32, Proposition 3.1] tells us that the rank of Mis equal to the rank of C(so2r+1,2). The modular category C(so2r+1,2) has no non-trivial gaugeauto-equivalences via Corollary 4.2, and so in particular it has no non-trivial braided gauge auto-equivalences. Thus [16, Theorem 5.2] combined with [5] show that the only invertible modulecategory over C(so2r+1,2) with maximal rank is the trivial module. Hence M is equivalent toC(so2r+1,2) as a left C(so2r+1,2)-module, which then implies

M ≃ C(so2r+1,2)G


as an C(so2r+1,2)-bimodule, with G an outer auto-equivalence of C(so2r+1,2). As M is non-trivial, the auto-equivalence G must be non-trivial, and in particular must be non-gauge byCorollary 4.2. However, we can now use [39, Section 3] to see that G ⊠ Gop and F are natu-rally isomorphic, up to an inner auto-equivalence of Z(C(so2r+1,2)). Due to the braiding onZ(C(so2r+1,2)), any inner auto-equivalence must be gauge, and so the functors G ⊠ Gop and Fmust act the same on objects. As G is non-gauge, we thus have F is also non-gauge, giving thecontradiction.

As the category Z(C(so2r+1,2)) has no non-trivial braided gauge auto-equivalences, we canbound the size of EqBr(Z(C(so2r+1,2))) by considering fusion rules and twists, as the groupEqBr(Z(C(so2r+1,2))) is a subgroup of the group of fusion ring automorphisms of Z(C(so2r+1,2))that preserve twists. We will refer to such fusion ring automorphisms as braided fusion ring au-tomorphisms for the remainder of the proof.

Consider the objects X1 ⊠ 1rev and 1 ⊠Xrev1 in

Z(C(so2r+1,2)) = C(so2r+1,2) ⊠ C(so2r+1,2)rev.

These objects ⊗-generate the factors C(so2r+1,2) and C(so2r+1,2)rev respectively. Thus anybraided fusion ring automorphism of Z(C(so2r+1,2)) that fixes these objects must be of theform F ⊠ G for F a braided fusion ring automorphism of C(so2r+1,2), and G a braided fusionring automorphism of C(so2r+1,2)rev.

We now consider F a F a braided fusion ring automorphism of Z(C(so2r+1,2)) that moveseither of the objects X1 ⊠ 1rev and 1 ⊠Xrev

1 . Simply considering dimensions shows that thereare eight possible objects that each of these two objects can be sent to, they are

X1 ⊠ 1rev,X1 ⊠Zrev,X2 ⊠ 1rev,X2 ⊠Zrev,1 ⊠Xrev1 , Z ⊠Xrev

1 ,1 ⊠Xrev2 , and Z ⊠Xrev

2 .

By considering dimensions and twists, we can determine that F(X1 ⊠ Xrev1 ) must be either

X1 ⊠ Xrev1 or X2 ⊠ Xrev

2 . With this information, we are able to compute Table 4, showingpossibilities for F(1 ⊠Xrev

1 ), given F(X1 ⊠ 1rev). In particular the twists of the objects for thetwo possibilities of F(1⊠Xrev

1 ) always differ by −1. This shows that F is completely determinedby where it sends the object X1 ⊠ 1rev. This gives us a naive bound

∣EqBr(Z(C(so2r+1,2)))∣ ≤ 8∣EqBr(C(so2r+1,2))∣2.

To improve this bound we now split the proof up in to cases.

F(X1 ⊠ 1rev) F(1 ⊠Xrev1 )

X1 ⊠ 1rev Ô⇒ 1 ⊠Xrev1 or Z ⊠Xrev

2

X1 ⊠Zrev Ô⇒ 1 ⊠Xrev2 or Z ⊠Xrev

1

X2 ⊠ 1rev Ô⇒ 1 ⊠Xrev2 or Z ⊠Xrev

1

X2 ⊠Zrev Ô⇒ 1 ⊠Xrev1 or Z ⊠Xrev

2

1 ⊠Xrev1 Ô⇒ X1 ⊠ 1rev or X2 ⊠Zrev

Z ⊠Xrev1 Ô⇒ X2 ⊠ 1rev or X1 ⊠Zrev

1 ⊠Xrev2 Ô⇒ X2 ⊠ 1rev or X1 ⊠Zrev

Z ⊠Xrev2 Ô⇒ X1 ⊠ 1rev or X2 ⊠Zrev

Table 4. Possibilities for F(1 ⊠Xrev1 ).

Case: r ≡ 0 (mod 4) and Z×2r+1 contains an element n such that n2 = −1By considering dimensions and twists there are four possible objects in the image ofX1 ⊠ 1rev under F . They are

X1 ⊠ 1rev,X1 ⊠Zrev,1 ⊠Xrev1 , Z ⊠Xrev

1 .


Case: r ≡ 2 (mod 4) and Z×2r+1 contains an element n such that n2 = −1By considering dimensions and twists there are four possible objects in the image ofX1 ⊠ 1rev under F . They are


2 .

Case: r ≡ 1 (mod 2) and Z×2r+1 contains an element n such that n2 = −1The first supplement to quadratic reciprocity shows that this is a vacuous case.

Case: r ≡ 1 (mod 2) and Z×2r+1 does not contain an element n such that n2 = −1By considering dimensions and twists there are two possible objects in the image of X1 ⊠1rev under F . They are

X1 ⊠ 1rev and X1 ⊠Zrev.

Case: r ≡ 0 (mod 4) and Z×2r+1 does not contain an element n such that n2 = −1By considering dimensions and twists there are four possible objects in the image ofX1 ⊠ 1rev under F . They are


1 .

Aiming towards a contradiction, suppose that F(X1 ⊠ 1rev) = 1 ⊠Xrev1 . Then F induces

a braided equivalence C(so2r+1,2) → C(so2r+1,2)rev. However this is a contradiction byLemma 4.4, thus

F(X1 ⊠ 1rev) ≇ 1 ⊠Xrev1 .

Aiming towards another contradiction, suppose that there exist braided fusion ring au-tomorphisms F1,F2 such that F1(X1 ⊠ 1rev) = X1 ⊠ Zrev and F2(X1 ⊠ 1rev) = Z ⊠Xrev

1 .From Table 4 we see that F1(1⊠Xrev

1 ) = Z ⊠Xrev1 . Therefore F−1

1 ○F2 is a braided fusionring automorphism sending X1 ⊠ 1rev to 1 ⊠Xrev

1 , which induces the same contradictionas before.Together, we see there are at most two objects in the image of X1⊠1rev under F a braidedfusion ring automorphism of C(so2r+1,2).

Case: r ≡ 2 (mod 4) and Z×2r+1 does not contain an element n such that n2 = −1By considering dimensions and twists there are four possible objects that a braided fusionring automorphism can send X1 ⊠ 1rev to. They are


2 .

The same arguments as in the previous case show that at most two of these objects canbe the image of X1 ⊠ 1rev under F .

This case by case analysis gives us the following bound

∣EqBr(Z(C(so2r+1,2)))∣ ≤⎧⎪⎪⎨⎪⎪⎩

4∣EqBr(C(so2r+1,2))∣2 if Z×2r+1 contains an element n such that n2 = −1

2∣EqBr(C(so2r+1,2))∣2 otherwise..

Combining this with the equality

∣EqBr(Z(C(so2r+1,2)))∣ = ∣Eq(C(so2r+1,2))∣ ⋅ ∣EqBr(C(so2r+1,2))∣,gives the statement of the Lemma. �

We move on to showing that the bounds on Eq(C(so2r+1,2)) from Lemma 4.5, and onEqBr(C(so2r+1,2)) from Lemma 4.3 are sharp. Our main tool to construct monoidal andbraided auto-equivalences of the category C(so2r+1,2) is to identify the de-equivariantization ofC(so2r+1,2) by the Rep(Z2) category ⟨1, Z⟩. It turns out that computing the auto-equivalences


of this de-equivariantization is much easier. We then use the functorality of equivariantizationto obtain auto-equivalences of the category C(so2r+1,2).

There are two structures in play here, depending on if we think of C(so2r+1,2) as a braidedcategory, or just a plain monoidal category. We begin by considering the braided case, which isslightly easier.

It is clear from the fusion rules and T-matrix of C(so2r+1,2) that there is unique fully faithfulbraided functor Rep(Z2)→ C(so2r+1,2), defined by

triv↦ 1 and sgn↦ Z.

By de-equivariantizating C(so2r+1,2) we get a triple of data (D, ρ, σ), where D is a Z2-gradedfusion category, ρ a monoidal functor ρ ∶ Z2 → Eq(D), and σ is a family of Z2-crossed braidings

X ⊗ Y1 → ρ(1)[Y1]⊗X,such that the category DZ2 is braided equivalent to C(so2r+1,2). Our goal is to determine asmuch information about this triple (D, ρ, σ) as possible.

From the results of [2] we know that the category D is a Tambara-Yamagami categoryT Y(Z2r+1, χ, τ), where χ is a symmetric bicharacter Z2r+1 × Z2r+1 → C×, and τ ∈ ±. Fur-thermore they show that at the level of objects ρ(1) fixes m, and sends i ↔ −i. Using upa degree of freedom, we can thus arrange Z2r+1 so that the forgetful functor C(so2r+1,2) →C(so2r+1,2)//Rep(Z2) ≃ T Y(Z2r+1, χ, τ) sends

Yi ↦ i⊕ −i.The Z2-crossed braided category T Y(Z2r+1, χ, τ) has a braided subcategory with Z2r+1 fusion

rules. By [45] such categories are completely classified by q a 2r+1-th root of unity. The associ-ator on (Z2r+1, q) is trivial, and the braiding of two objects i and j is given by qij . To determineq we note that the adjoint subcategory of C(so2r+1,2) is braided equivalent to (Z2r+1, q)Z2 .Forgetting gives a braided functor that sends

Yi ↦ i⊕ −i.In particular this implies that the twist of 1 ∈ (Z2r+1, q) is equal to the twist of Y1. Thus we

have that q = e2iπ r2r+1 .

As T Y(Z2r+1, χ, τ) is a Z2-crossed braided extension of (Z2r+1, q), we have that the single non-invertible simple object of T Y(Z2r+1, χ, τ) forms a module category over the braided category(Z2r+1, q). The unique rank one module over (Z2r+1, q), is Vec with trivial structure morphisms.Using the theory of G-crossed braided extensions [16], we can see that the associator morphism

(i⊗m)⊗ j → i⊗ (m⊗ j)in T Y(Z2r+1, χ, τ) is given by the braiding of i and j in (Z2r+1, q). This completely determinesthe bicharacter χ, giving us

χ(i, j) = qij = e2iπ ijr2r+1 .

We neglect to determine τ .

Lemma 4.6. Let n ∈ {n ∈ Z×2r+1 ∶ n2 = 1}, then there exists a monoidal auto-equivalence Fn ofT Y(Z2r+1, χ, τ) defined by

Fn(m) =m and Fn(i) = ni (mod 2r + 1).The structure constants for Fn(m) are trivial. The map n↦ Fn(m) gives an isomorphism

{n ∈ Z×2r+1 ∶ n2 = 1}→ Eq(T Y(Z2r+1, χ, τ)).

Proof. This is a direct application of Lemma 2.16 and Remark 2.17. It is routine to check thatn ∈ Z×2r+1 preserves χ if and only if n2 = 1. �


Now we determine ρ ∶ Z2 → Eq(T Y(Z2r+1, χ, τ)). Trivially we have that ρ(0) is the iden-

tity on T Y(Z2r+1, χ, τ). As mentioned before, ρ(1) restricts to the (Z2r+1, q) to give theauto-equivalence i ↦ −i. From Lemma 4.6 there is a unique monoidal auto-equivalence ofT Y(Z2r+1, χ, τ) that restricts to i ↦ −i on the Z2r+1 subcategory. Thus ρ(1) ≅ F2r. We neglectto determine the tensor structure of ρ.

Finally we determine the Z2-crossed braiding σ on T Y(Z2r+1, χ, τ). Solving [10, Equations(82) and (83)] using a slight alteration of the techniques used in [49] gives that there are fourpossibilities for σ, depending on a choice of square root of both χ(1,1) and τ .

σi,j ∶= χ(i, j) idi+j

σi,m ∶= χ(1,1)i2

2 idm

(σm,m)i ∶= σi,mχ(i, i)

¿ÁÁÁÀ τ√

2r + 1

2r

∑j=0

σj,m idi .

Solving for equation [10, Equation (81)] shows that χ(1,1) 12 = χ(1,1)2r2

, uniquely determiningthe choice of square root of χ(1,1). This allows us to write

σi,m = χ(1,1)2i2r2

idm .

Without τ we are unable to determine its choice of square root. Thus we have two possible Z2

crossed braidings. Thankfully the following key Lemma is independent from both the choice ofτ , and the choice of square root of τ .

Lemma 4.7. Let n ∈ {n ∈ Z×2r+1 ∶ n2 = 1}, then there exist monoidal natural isomorphisms

η0 ∶ ρ(0) ○Fn → Fn ○ ρ(0)and

η1 ∶ ρ(1) ○Fn → Fn ○ ρ(1)defined by η0

X = idFn(X) and η1X = idF2rn(X).

Furthermore, the pair (Fn, η) is a automorphism of (T Y(Z2r+1, χ,±), ρ, σ) in the category ofZ2-crossed braided fusion categories.

Proof. As the tensor structure morphisms for the monoidal functor Fn are trivial, the naturalisomorphisms η0 and η1 are easily checked to be monoidal.

To verify that (Fn, η) is a automorphism of (T Y(Z2r+1, χ,±), ρ, σ) in the category of Z2-crossed braided fusion categories, we have to check the commutativity of the diagram (2.7).Working through all possibilities for all objects in T Y(Z2r+1, χ,±) shows that commutativity isequivalent to the following equations

Fn(σi,j) = σni,nj for all i, j ∈ Z2r+1,

Fn(σi,m) = σni,m for all i ∈ Z2r+1,

Fn(σm,i) = σm,ni for all i ∈ Z2r+1,

Fn(σm,m) = σm,m.Expanding these gives the equations:

χ(i, j) = χ(ni, nj),

χ(1,1)2i2r2 = χ(1,1)2n2i2r2

,

χ(1,1)2i2r2

χ(i, i)

¿ÁÁÁÀ τ√

2r + 1

2r

∑j=0

χ(1,1)2j2r2 = χ(1,1)2n2i2r2

χ(ni, ni)

¿ÁÁÁÀ τ√

2r + 1

2r

∑j=0

χ(1,1)2j2r2 ,


for all i, j ∈ Z2r+1.

Each of these follows from the fact that χ(1,1) is a 2r + 1-th root of unity, and

n2 ≡ 1 (mod 2r + 1).�

With the above Lemma in hand we can now apply the functorality of equivariantization toget a homomorphism

n↦ FZ2n ∶ {n ∈ Z×2r+1 ∶ n2 = 1}→ EqBr(C(so2r+1,2))

We explicitly compute that FZ2n fixes the objects 1, Z,X1,X2, and sends the object Yi ↦

Ymin(ni (mod 2r+1)),−ni (mod 2r+1)). This homomorphism is surjective (although not a-priori), butnot injective.

Lemma 4.8. The kernel of the homomorphism n↦ FZ2n is {1,−1}.

Proof. Let n ∈ Z×2r+1 such that n2 = 1 such that FZ2n ≅ IdC(so2r+1,2). Then Y1 = FZ2

n (Y1) =Ymin(n (mod 2r+1)),−n (mod 2r+1)), giving us that n = ±1.

On the other hand the auto-equivalence FZ22r is a gauge auto-equivalence of C(so2r+1,2), as it

fixes all the simple objects. Thus by Corollary 4.2 FZ22r is trivial. �

Corollary 4.9. There is an injection

{n ∈ Z×2r+1 ∶ n2 = 1}/(±)→ EqBr(C(so2r+1,2)).This injection realises the upper bound on EqBr(C(so2r+1,2)) from Lemma 4.3, hence we

haveEqBr(C(so2r+1,2)) ≅ {n ∈ Z×2r+1 ∶ n2 = 1}/(±).

Let ω(2r + 1) be the number of distinct prime divisors of 2r + 1. We have that

Zω(2r+1)2 ≅ {n ∈ Z×2r+1 ∶ n2 = 1},

with an explicit isomorphism sending

{p±1 , p±2⋯, p±k}↦k

∑i=1

±2r + 1

pi(mod 2r + 1).

This allows us to writeEqBr(C(so2r+1,2)) ≅ Zω(2r+1)−1

2

as in the statement of Theorem 1.1.

We now consider C(so2r+1,2) as just a monoidal category, forgetting the braided structure.The monoidal category C(so2r+1,2) has two structures as a category over Rep(Z2), given by thetwo central functors ⟨Z⟩→ Z(C(so2r+1,2)) defined by:

F+ ∶ Z ↦ Z ⊠ 1rev and F− ∶ Z ↦ 1 ⊠Zrev.

Thus de-equivariantizing C(so2r+1,2) gives us two fusion categories with Z2 action, corre-sponding to the two lifts F + and F −. We call these two fusion categories with Z2 action

(D+, ρ+) and (D−, ρ−)respectively, where D± are fusion categories, and ρ± ∶ Z2 → Eq(D±) are monoidal functors,

such that DZ2± = C(so2r+1,2).

It follows from the braided case that D+ = T Y(Z2r+1, χ, τ), with

χ(i, j) = e2iπ ijr2r+1 .


As DZ2± = C(so2r+1,2) we can count the simple objects of both categories to see that ρ(1) can only

fix the simple objects 0 and m in T Y(Z2r+1, χ, τ). From Lemma 4.6 there is a unique monoidalauto-equivalence of T Y(Z2r+1, χ, τ) with this property. Hence we have that ρ+(1) = F2r.

As F− factors through C(so2r+1,2)rev, we have that D− = (D+)op = T Y(Z2r+1, χ−1, τ). The

same argument as in the ρ+ case shows that ρ− = F2r.

Lemma 4.10. Let n ∈ Z×2r+1 such that n2 = −1, then there exists a monoidal equivalence Fn ∶T Y(Z2r+1, χ,±)→ T Y(Z2r+1, χ

−1,±) that fixes m, and sends i↦ ni. The structure constants forthese monoidal equivalences are trivial. Furthermore there exist monoidal natural isomorphisms

η0 ∶ ρ−(0) ○Fn → Fn ○ ρ+(0)

and

η1 ∶ ρ−(1) ○Fn → Fn ○ ρ+(1)

defined by η0X = idFn(X) and η1

X = idF2rn(X).Further, the pair (Fn, η) gives a morphism

(C(so2r+1,2), F +)→ (C(so2r+1,2), F −)

in the category of fusion categories with Z2-action.

Proof. This proof is a slight altercation of the proofs of Lemmas 4.6 and 4.7. �

Using the functorality of equivariantization, we get for each n ∈ Z×2r+1 such that n2 = −1, amorphism in the category of fusion categories over Rep(Z2):

FZ2n ∶ (C(so2r+1,2), F +)→ (C(so2r+1,2), F −).

Forgetting the central structures gives monoidal auto-equivalences of C(so2r+1,2) for each n ∈Z×2r+1 such that n2 = 1. Combining these new monoidal auto-equivalences, with the braidedauto-equivalences we have already constructed, we obtain a homomorphism

n↦ FZ2n ∶ {n ∈ Z×2r+1 ∶ n2 = ±1}→ Eq(C(so2r+1,2)).

Explicitly we compute that FZ2n fixes the objects 1, Z,X1,X2, and sends Yi ↦ Ymin(ni (mod 2r+1)),−ni (mod 2r+1)).

The same as in the braided case, we can show that the kernel of the homomorphism n↦ FZ2n

is {1,−1}, and thus we get an injection

{n ∈ Z×2r+1 ∶ n2 = ±1}/{±}→ Eq(C(so2r+1,2)).

We can use the boson Z to construct a simple current monoidal auto-equivalence of C(so2r+1,2)sending X1 ↔X2, and fixing all other simple objects. This gives us an injection

Z2 × {n ∈ Z×2r+1 ∶ n2 = ±1}/{±}→ Eq(C(so2r+1,2)),

realising the upper bound on Eq(C(so2r+1,2)) from Lemma 4.5.Via the first supplement to quadratic reciprocity, we have that Z2r+1 has a solution to n2 = −1

if and only if every prime p dividing 2r + 1 satisfies p ≡ 1 (mod 4). This allows us to write

Eq(C(so2r+1,2)) =⎧⎪⎪⎨⎪⎪⎩

Z2 ×Z2 ×EqBr(C(so2r+1,2)) if every prime p dividing 2r + 1 satisfies p ≡ 1 (mod 4)Z2 ×EqBr(C(so2r+1,2)) otherwise.

As in the statement of Theorem 1.1.


5. Auto-equivalences for Lie type C

In this section we compute the monoidal, and braided auto-equivalences of the categoriesC(sp2r, k). The outline of our computations is as follows.

We begin by showing that there are no non-trivial gauge auto-equivalences of the categoriesC(sp2r, k). Recall that the object Λ1 ⊗-generates C(sp2r, k). Thus any non-trivial gauge auto-equivalence of C(sp2r, k) will appear as a non-trivial planar algebra automorphism of the planaralgebra generated by Λ1. We know that this planar algebra is isomorphic to

BMW(q,−q2r+1) with q = e2πi

4(r+k+1) .

We can reuse a computation from the Lie type B section to show that this planar algebra hasno non-trivial automorphisms that give rise to gauge auto-equivalences of C(sp2r, k). Hence thecategory C(sp2r, k) has no non-trivial gauge auto-equivalences.

The results of [24] compute the fusion ring automorphisms of C(sp2r, k) as:

{e} if r = 2 and k = 1 ,

{e} if r ≠ k and rk is odd ,

Z2 if r ≠ k and rk is even ,

Z2 if r = k and r is odd ,

Z2 ×Z2 if r = k and r is even .

We find that all of these fusion ring automorphisms are realised as monoidal auto-equivalences,though only some of them are braided. The first four cases above are easily dealt with byconstructing the auto-equivalences as simple current auto-equivalences. The latter two casesare more interesting, as there exists an exotic fusion ring automorphism. We show this exotic

automorphism is realised by lifting the interesting planar algebra automorphism of BMW(q, i)seen in Lemma 4.1.

Let us being our computations.

Lemma 5.1. The category C(sp2r, k) has no non-trivial gauge auto-equivalences

Proof. Consider the object Λ1 ∈ C(sp2r, k). Recall from Equation 2.4 that the planar algebragenerated by Λ1 is isomorphic to

BMW(q,−q2r+1) with q = e2πi

4(r+k+1) .

As the object Λ1 ⊗-generates C(sp2r, k), we can apply Lemma 2.8 to see that every non-trivialgauge auto-equivalence of C(sp2r, k) appears as a non-trivial planar algebra automorphism

of BMW(q,−q2r+1). From Lemma 4.1, the planar algebra BMW(q,−q2r+1) only has a singlenon-trivial automorphism when k = r, which corresponds to an auto-equivalence of C(sp2r, k)that exchanges distinct objects. Thus the category C(sp2r, k) has no non-trivial gauge auto-equivalences. �

Next up we have to show that all fusion ring automorphisms of C(sp2r, k) are realisable. Forthis we break up into cases.

Case: r = 2 and k = 1 or r ≠ k and rk is odd.In this case we have no non-trivial fusion ring automorphisms of C(sp2r, k), thus

EqBr(C(sp2r, k)) = Eq(C(sp2r, k)) = Gauge(C(sp2r, k)) = {e},



Case: r ≠ k and rk is even.In this case the upper bound of the auto-equivalence group is Z2. The modular category

C(sp2r, k) has an order 2 invertible object kΛr which we can use to construct a simple currentauto-equivalence. This invertible object has self-braiding eigenvalue 1 when rk ≡ 0 (mod 4), and−1 when rk ≡ 2 (mod 4). Thus we can use Lemma 2.4 to construct a (possibly trivial) monoidalauto-equivalence of C(sp2r, k). This auto-equivalence sends

Λ1 ↦ Λ1 ⊗ (kΛr) ≅ (Λr−1 + (k − 1)Λr),

hence this auto-equivalence is non-trivial apart from the case r = 2 and k = 1, which we havealready dealt with. This auto-equivalence is braided if and only if (kΛr) has self-braidingeigenvalue −1, i.e if and only if rk ≡ 2 (mod 4). Thus we have

Eq(C(sp2r, k)) = Z2 and EqBr(C(sp2r, k)) =⎧⎪⎪⎨⎪⎪⎩

{e}, if rk ≡ 0 (mod 4),Z2, if rk ≡ 2 (mod 4).


Case: r = k. When r = k, the category C(sp2r, r) has an exotic order 2 fusion ring automorphismdefined as follows. For an object ∑ri=1 λiΛi consider the Young diagram whose j-th row consists

of r−∑ki=k−j λi boxes. The transpose of this Young diagram corresponds to another simple object

of C(sp2r, r), under the same combinatorial interpretation. The image of the object ∑ri=1 λiΛi isexactly the simple object corresponding to this transposed Young diagram. As a quick example,we work out where Λ2 gets sent to under this exotic automorphism.

(Λ2) ∼TrÐ→ ∼ (2Λ1).

This exotic automorphism of the C(sp2r, r) fusion ring is completely characterised by thefact that it fixes the object Λ1, while not being gauge. The fact this automorphism fixesthe object Λ1 is particularly convenient for us, as if the automorphism is realisable, then itwill appear as a non-trivial planar algebra automorphism of the planar algebra generated byΛ1. From Equation 2.4 we know that the planar algebra generated by Λ1 is isomorphic to

BMW(e2πi

4(2r+1) ,−i). Further, from Lemma 4.1, there exists a planar algebra automorphism of

BMW(e2πi

4(2r+1) ,−i) that corresponds to a non-braided monoidal auto-equivalence of C(sp2r, r)that exchanges distinct simple objects. By the unique characterisation of the above exoticfusion ring automorphism, we must have that this non-trivial planar algebra automorphism of

BMW(e2πi

4(2r+1) ,−i) corresponds to a non-braided monoidal auto-equivalence of C(sp2r, r) thatrealises the exotic fusion ring automorphism. We denote this non-trivial auto-equivalence as thelevel-rank duality auto-equivalence of C(sp2r, r).

When r is odd, there is only a single non-trivial automorphism of the C(sp2r, r) fusion ring.The non-trivial fusion ring automorphism is realised by the level-rank duality auto-equivalence.Thus when r is odd, we have

Eq(C(sp2r, r)) = Z2 and EqBr(C(sp2r, k)) = {e}

as in the statement of Theorem 1.1.When r is even, there are three non-trivial automorphisms of the C(sp2r, r) fusion ring. The

invertible object kΛr ∈ C(sp2r, r) is a has self-braiding eigenvalue 1, as r2 ≡ 0 (mod 4), so wecan apply the same arguments as in the rk ≡ 1 (mod 2) and r ≠ k case to construct an order 2,


non-braided auto-equivalence that sends

Λ1 ↦ (Λr−1 + (k − 1)Λr).

Furthermore the level-rank duality auto-equivalence gives us a non-braided order 2 auto-equivalencethat fixes Λ1. These two auto-equivalences are clearly not inverse to each other, thus their com-position gives us a third non-trivial monoidal auto-equivalence. This composition is not braided,as it maps Λ1 ↦ (Λr−1 + (k − 1)Λr), which have differing twists. Thus when r is even, we have

Eq(C(sp2r, r)) = Z2 ×Z2 and EqBr(C(sp2r, k)) = {e}


6. Auto-equivalences for Lie type G

In this section we compute the monoidal, and braided auto-equivalences of the categoriesC(g2, k). We remark that the results of this section are dependant on work that has yet toappear in print by Ostrik and Snyder.

The outline of our computations is as follows.

We begin with a standard planar algebra computation, showing that the G2(q) planar algebraonly has no non-trivial automorphisms, up to planar algebra natural isomorphism. Hence thecategory C(g2, k) has no non-trivial gauge auto-equivalences.

The results of [24] compute the fusion ring automorphisms of C(g2, k) as

{e} if k ≠ {3,4},Z3 if k = 3,

Z2 if k = 4.

When k = 3 or k = 4, we see the category C(g2,3) has exotic fusion ring automorphisms.For k = 3 we find that these exotic automorphisms do not lift to the category C(g2, k). Ourargument here is to consider the planar algebra generated by the object Λ1, and the planaralgebra generated by the image of Λ1 under the automorphism. We show that these two planaralgebras are non-isomorphic, thus the automorphism can not lift to a monoidal auto-equivalence.For k = 4 we show the exotic fusion ring automorphism does lift to a braided auto-equivalence.Our argument is similar to the k = 3 case, where we consider the planar algebra generated bythe object Λ1, and the planar algebra generated by the image of Λ1 under the automorphism.However in this case we show that planar algebras are isomorphic as braided planar algebras,and hence there is a braided auto-equivalence of C(g2, k) realising the fusion ring automorphism.

Let us begin our computations for this section.

Lemma 6.1. The planar algebra G2(q) has no non-trivial automorphisms, up to planar algebranatural isomorphism.

Proof. Let φ be a planar algebra automorphism of G2(q), then φ is completely determined by

how it acts on the generator . As the 3-box space of G2(q) is 1-dimensional, we must

have that

φ⎛⎜⎝

⎞⎟⎠

= α

for some α ∈ C.


As φ must preserve the relations of G2(q), we can see from relation (iii) of the G2(q) planaralgebra that α = ±1. It is routine to check that the automorphisms

φ⎛⎜⎝

⎞⎟⎠

= ±

are consistent with the other relations, and the automorphisms preserve the negligible ideal by

Proposition 2.10. Thus we have two planar algebra automorphisms of G2(q). However thesetwo automorphisms are isomorphic to each other via the natural isomorphism

µ ∶= − .

�

The object Λ1 ⊗-generates C(g2, k), and we have the isomorphism of planar algebras

PA(C(g2, k),Λ1) = G2(q) with q = e2πi

6(4+k) ,

from Equation (2.6). Hence we can apply Lemma 2.8 to get the following Corollary.

Corollary 6.2. The categories C(g2, k) have no non-trivial gauge auto-equivalences.

Now that we have shown that the categories C(g2, k) have no non-trivial gauge auto-equivalences,we move on to determining which of the fusion ring automorphisms of C(g2, k) are realisable.We break into cases.

Case: k ≠ {3,4}.In this case there are no non-trivial fusion ring automorphisms. Thus

EqBr(C(g2, k)) = Eq(C(g2, k)) = Gauge(C(g2, k)) = {e},as in the statement of Theorem 1.1.

Case: k = 3.When k = 3, there are two non-trivial fusion ring automorphisms, which are

Λ1 ↦ Λ2 ↦ 3Λ1 ↦ Λ1,

and its inverse.

Lemma 6.3. The fusion ring automorphism Λ1 ↦ Λ2 ↦ 3Λ1 ↦ Λ1 does not lift to a monoidalauto-equivalence of the category C(g2,3).

Proof. Using Frobenius-Schur indicators, computed from the modular data of C(g2,3), we can seethat the object Λ2 is symmetrically self-dual. Thus we can form the planar algebra PA(C(g2,3),Λ2).We begin the proof by identifying this planar algebra.

As dimHom(1 → Λ⊗n2 ) is equal to 1,0,1,1,4,10 for 0 ≤ n ≤ 5, we have from [41, Theorem B]

that there is a planar algebra injection G2(q) → PA(C(g2,3),Λ2) for q some primitive root ofunity.

As the categorical dimension of Λ2 is equal to

e10iπ21 + e

8iπ21 + e

2iπ21 + 1 + e

−2iπ21 + e

−8iπ21 e

−10iπ21 ,

we must have that q = eNiπ21 for some N ∈ {1,4,5,16,17,20,22,25,26,37,38,41}. The quantum

twist of Λ2 is e8iπ7 , which implies that N has to be either 5 or 26. By Lemma 2.14 we know

that the planar algebras G2(e5iπ21 ) and G2(e

26iπ21 ) are isomorphic, thus we have a planar algebra

injection

G2(e5iπ21 )→ PA(C(g2,3),Λ2).


Aiming towards a contradiction of the statement of the Lemma, suppose there was a monoidalauto-equivalence Γ of C(g2,3) that sends Λ1 ↦ Λ2. Then Γ induces a monoidal equivalence ofbased categories (C(g2,3),Λ1) → (C(g2,3),Λ2), and hence by [30, Theorem A] an isomorphismof planar algebras

PA(C(g2,3),Λ1)→ PA(C(g2,3),Λ2).

We know from Equation (2.6) that PA(C(g2,3),Λ1) ≅ G2 (eiπ21 ), so we have an injection of

planar algebras

G2(e5iπ21 )→ G2(e

iπ21 ).

As the 3-box space of both these planar algebras is 1-dimensional, we must have that the trivalent

vertex of G2(e5iπ21 ) is mapped to a scaler multiple of the trivalent vertex of G2(e

iπ21 ). However,

the bubble and triangle popping values of the trivalent vertex in each of these planar algebrasare different, and there exists no rescaling of the trivalent vertex to arrange that these two valuesare the same for both planar algebras. Hence we have a contradiction, and thus the fusion ringautomorphism Λ1 ↦ Λ2 ↦ 3Λ2 ↦ Λ1 does not lift to a monoidal auto-equivalence of C(g2,3). �

As the non-trivial fusion ring automorphism of C(g2,3) does not lift, we see

EqBr(C(g2,3)) = Eq(C(g2,3)) = Gauge(C(g2,3)) = {e},as in the statement of Theorem 1.1.

Case: k = 4.When k = 4 there is a single non-trivial fusion ring automorphism, which sends

Λ1 ↔ 2Λ2 and Λ2 ↔ 4Λ1,

and fixes the other simple objects.

Lemma 6.4. This non-trivial fusion ring automorphism of C(g2,4) lifts to a braided auto-equivalence.

Proof. Using the modular data of C(g2,4), we can use Frobenius-Schur indicators to verify thatthe object 2Λ2 is symmetrically self-dual. Thus we can form the planar algebra PA(C(g2,4),2Λ2).Using the same argument as in the proof of Lemma 6.3 we can show that there is an injectionof planar algebras

G2(q)→ PA(C(g2,4),2Λ2)for q some primitive root of unity.

The categorical dimension of 2Λ2 is equal to

e10iπ24 + e

8iπ24 + e

2iπ24 + 1 + e

−2iπ24 + e

−8iπ24 e

−10iπ24 ,

so we must have that q = eNiπ24 for some N ∈ {1,5,19,23,25,29,43,47}. The quantum twist

of 2Λ2 is i, which allows us to further deduce that N ∈ {1,5,25,29}. The Hopf link of 4Λ2 is

3√

2+√

6 (2√

2 + 3)+2, which shows that N ∈ {1,25}. By Lemma 2.14 we know that the planar

algebras G2 (eiπ24 ) and G2 (e

25iπ24 ) are isomorphic as braided planar algebras, hence there is an

injection of braided planar algebras

G2 (eiπ24 )→ PA(C(g2,4),2Λ2).

As the map exchanging Λ1 ↔ 2Λ2 and Λ2 ↔ 4Λ1 is a fusion ring homomorphism, we have thedimensions of the spaces

Hom(1→ (4Λ2)n) and Hom(1→ (Λ1)n)


agree. Hence the planar algebras G2 (eiπ24 ) and PA(C(g2,4),2Λ2) have box spaces of the same

dimensions, and so the injection

G2 (eiπ24 )→ PA(C(g2,4),2Λ2)

is in fact an isomorphism of braided planar algebras.By [30, Theorem A], the above isomorphism of planar algebras lifts to a braided equivalence

of based categories

(C(g2,4),Λ1)→ (C(g2,4),2Λ2).Forgetting the basing gives a braided auto-equivalence of C(g2,4) that sends Λ1 to 2Λ2. Asthere is a unique non-trivial fusion ring automorphism of C(g2,4), we must have that thisautomorphism lifts. �

As the non-trivial fusion ring automorphism of C(g2,4) lifts, we thus have

Eq(C(g2,4)) = EqBr(C(g2,4)) = Z2


7. Future Directions and Applications

A new family of modular tensor categories from the charge-conjugation auto-equivalenceof C(slr+1, k).

For the modular tensor categories C(slr+1, k) we show the existence (for r ≥ 2) of the semi-exceptional charge conjugation braided auto-equivalence. This auto-equivalence has order 2,hence we get a group action

Z2 → EqBr(C(slr+1, k)).The technique of gauging [7] allows us to take a group action on a modular tensor category,

and construct a new modular tensor category, given that two certain cohomological obstructionsvanish. For the example of C(slr+1, k), the first of these cohomological obstructions will be anelement of

H3(Z2, Inv(C(slr+1, k))) =H3(Z2,Zr+1),with Z2 acting on Zr+1 via the inverse map, and the second will be an element of

H4(Z2,C×) = {e}.As latter obstruction group is trivial, the corresponding obstruction vanishes for free. The

former obstruction group is isomorphic to

Zr+1/(2Zr+1) =⎧⎪⎪⎨⎪⎪⎩

{e} if r is even,

Z2 if r is odd.

Thus for r even, we can gauge C(slr+1, k) by the charge conjugation auto-equivalence to obtaina new family of modular tensor categories. While we have the abstract existence of thesecategories, almost all other information about them remains mysterious.

Question 7.1. For even r, determine the structure of the modular tensor categories obtainedby gauging C(slr+1, k) by the charge conjugation auto-equivalence.

Even for the simplest non-trivial case of C(sl3,3) it is hard to say anything about the resultingmodular tensor category. Hence, it appears new techniques will be required to answer thisquestion.

In another direction, it would be interesting to study the obstruction living in H3(Z2,Zr+1)when r is odd.


Reverse-braided auto-equivalences of C(so2r+1, k), and connections with modulargrafting.

When k = 2 and every prime p dividing 2r + 1 is congruent to 1 mod 4, we show the existenceof a distinguished exceptional monoidal auto-equivalence of C(so2r+1,2). This monoidal auto-equivalence has the peculiar property that it is reverse braided in the sense that

F⎛⎜⎝

⎞⎟⎠

= .

The existence of such auto-equivalences is rare for general modular tensor categories, as it impliesrestrictive conditions on the central charge of the category, and on the twists of objects whichare fixed points.

The first few examples of categories C(so2r+1,2) with such reverse braided auto-equivalencesare

C(so5,2), C(so5,13), C(so17,2), C(so25,2), and C(so29,2).

Playing numerology, we notice that that (apart from 2r + 1 = 25), these categories all appearas ingredients of the hypothetical “modular grafting” construction [18]. It is conjectured thatthese categories can be grafted with certain dihedral groups to obtain the Drinfeld centres ofthe Haagerup-Izumi categories. The difficulty in this conjecture is that there is no idea ofwhat grafting should look like at the level of modular tensor categories. The appearance ofthese reverse braided auto-equivalences for these examples leads one to suspect that such auto-equivalences may play important role in making the modular grafting construction rigorous.

While this is all wild speculation, we do have some evidence for a connection. We firstnotice that the objects from C(so2r+1,2) that appear in the modular grafting are exactly thenon-fixed simple objects of the reverse braided auto-equivalence. By this we mean precisely,that if {Xi ∶ 1 ≤ i ≤ m} are the non-fixed simple objects of C(so2r+1,2), then there exists anm×m block in the S-matrix of the Drinfeld centre of the hypothetically corresponding Haagerup-Izumi category that exactly corresponds to the m×m sub-matrix of the S-matrix for C(so2r+1,2)obtained by restricting to the objects {Xi ∶ 1 ≤ i ≤m}. The same fact holds true for the T-matrixas well.

We also notice that the only other quantum group modular tensor category that could possiblyhave a reverse braided auto-equivalence is C(f4,4). This auto-equivalence has 12 non-fixed simpleobjects, and we find that the 12×12 block of restricted modular data for C(f4,4) exactly appearsin the modular data for the centre of extended Haagerup [25]. Further, we can identify Z4⋉Z5 asthe group component of the modular grafting, and construct a fairly natural looking formula forthe modular data of centre of extended Haagerup, starting from the modular data of C(f4,4) andthe Drinfeld centre of Z4⋊Z5. This strongly suggests that the centre of extended Haagerup shouldfit into the hypothetical modular grafting procedure, and that reverse braided auto-equivalenceswill play a key role in making this procedure precise.

While of course this could all just be coincidence, we believe there is sufficient evidence toinvestigate the following question.

Question 7.2. Can reverse braided auto-equivalences be used to give a rigorous definition ofmodular grafting.

A new family of planar algebras from the level-rank duality auto-equivalence ofC(sp2r, r).

Recall that C(sp2r, r) has exceptional order two level-rank duality auto-equivalence, whichwe constructed from an exceptional planar algebra automorphism φ of the planar algebra


BMW(q, i). This exceptional planar algebra automorphism was defined by

φ⎛⎜⎝

⎞⎟⎠

= − .

By taking fixed points of the planar algebra BMW(q, i) under this Z2-action, we can construct

a new planar algebra BMW(q, i)Z2.

While this planar algebra is easily described as the fixed points of φ, is always desirable tohave a presentation of a planar algebra by generators and relations. For the planar algebra

BMW(q, i)Z2there is a canonical 6-box generator given by

S ∶= + + + + +

+2(q − 1)(q + 1) (2iq2 + q − i) (−2 + q(q − i))

q(1 + q(q(2 + q(q + 4i)) − 4i))⎛⎜⎝

+ +⎞⎟⎠

+ −2i (q2 − 1)2 (−3 + q(3q − 4i))q(1 + q(q(2 + q(q + 4i)) − 4i))

⎛⎜⎝

+⎞⎟⎠.

which is uniquely characterised (up to scaler) by the fact that it is uncappable in every position,and invariant under rotation by π degrees. Finding more complicated relations that this gen-erator satisfies however proves to be an exhausting task. However, finding such relations couldreveal new ideas for skein theories involving 6-boxes, and hopefully pave the way for more exoticplanar algebras based on similar skein theories.

Question 7.3. Find a generators and relations presentation for the sub-planar algebra of

BMW(q, i)Z2generated by S.

A non-commutative algebra object in C(g2,4).For any modular tensor category C, there exists an isomorphism from the group of braided

auto-equivalences of C, to the group of invertible modules over C [16]. The goal of this Sectionwill be to determine the invertible module over C(g2,4) corresponding to the exceptional braidedauto-equivalence

Λ1 ↔ 2Λ2 and Λ2 ↔ 4Λ1.

We begin by describing how to take a braided auto-equivalence of a modular tensor categoryC, and construct an invertible module over C.Construction 7.4. Let F ∈ EqBr(C), then the object

AF ∶= ⊕X∈Irr(C)

X ⊗F−1(X∗)

has the structure of an algebra object in C. Except in the trivial case of C = Vec, the algebraAF is not simple. Let

AF = ⊕Aibe a decomposition of AF into simple algebra objects. The invertible module corresponding toF is ModC(Ai) for any choice of algebra Ai. As the simple algebra objects {Ai} are pairwiseMorita equivalent, the choice of Ai does not affect the resulting module category.

For a given braided auto-equivalence F , it is easy to determine the algebra object AF . In thecase of the exceptional braided auto-equivalence of C(g2,4) we have that

A(Λ1↔2Λ2 and Λ2↔4Λ1) =


(0)⊕5 ⊕ (Λ1)⊕4 ⊕ (2Λ1)⊕8 ⊕ (3Λ1)⊕9 ⊕ (4Λ1)⊕5 ⊕ (Λ2)⊕5 ⊕ (Λ1 +Λ2)⊕12 ⊕ (2Λ1 +Λ2)⊕10 ⊕ (2Λ2)⊕4.

However it is difficult in practice to determine how the algebra AF decomposes into simplealgebras. To determine how the algebra A(Λ1↔2Λ2 and Λ2↔4Λ1) decomposes into simple algebraobjects {Ai}, we apply the techniques of [28] to compute an exhaustive list of possible simplealgebra objects in C(g2,4). Via [5], we know that the rank of ModC(Ai) is equal to 5 (thenumber of fixed points of the exceptional braided auto-equivalence). Thus we can restrict ourattention to simple algebras in C(g2,4) whose corresponding module category has rank 5. Thisreduction significantly reduces the complexity of the computation.

Lemma 7.5. Let A ∈ C(g2,4) be an algebra whose corresponding module category has rank 5.Then A is contained in the list

A1 ∶= (0)⊕ (3Λ1)⊕ (4Λ1)⊕ (Λ2)A2 ∶= (0)⊕ (2Λ1)⊕ (3Λ1)⊕ (4Λ1)⊕ (Λ2)⊕ (Λ1 +Λ2)⊕2 ⊕ (2Λ1 +Λ2)A3 ∶= (0)⊕ (2Λ1)⊕2 ⊕ (3Λ1)⊕ (4Λ1)⊕ (Λ2)⊕ (Λ1 +Λ2)⊕2 ⊕ (2Λ1 +Λ2)⊕2

A4 ∶= (0)⊕ (Λ1)⊕ (2Λ1)⊕ (3Λ1)⊕ (4Λ1)⊕ (Λ2)⊕ (Λ1 +Λ2)⊕2 ⊕ (2Λ1 +Λ2)⊕2 ⊕ (2Λ2)A5 ∶= (0)⊕ (Λ1)⊕ (2Λ1)⊕2 ⊕ (3Λ1)⊕3 ⊕ (4Λ1)⊕ (Λ2)⊕ (Λ1 +Λ2)⊕3 ⊕ (2Λ1 +Λ2)⊕2 ⊕ (2Λ2)A6 ∶= (0)⊕ (Λ1)⊕ (2Λ1)⊕3 ⊕ (3Λ1)⊕3 ⊕ (4Λ1)⊕2 ⊕ (Λ2)⊕2 ⊕ (Λ1 +Λ2)⊕4 ⊕ (2Λ1 +Λ2)⊕3 ⊕ (2Λ2)A7 ∶= (0)⊕ (Λ1)⊕2 ⊕ (2Λ1)⊕4 ⊕ (3Λ1)⊕5 ⊕ (4Λ1)⊕3 ⊕ (Λ2)⊕3 ⊕ (Λ1 +Λ2)⊕4 ⊕ (2Λ1 +Λ2)⊕4 ⊕ (2Λ2)⊕2

A8 ∶= (0)⊕ (Λ1)⊕2 ⊕ (2Λ1)⊕4 ⊕ (3Λ1)⊕5 ⊕ (4Λ1)⊕3 ⊕ (Λ2)⊕3 ⊕ (Λ1 +Λ2)⊕5 ⊕ (2Λ1 +Λ2)⊕4 ⊕ (2Λ2)⊕2

A9 ∶= (0)⊕ (Λ1)⊕2 ⊕ (2Λ1)⊕5 ⊕ (3Λ1)⊕5 ⊕ (4Λ1)⊕3 ⊕ (Λ2)⊕3 ⊕ (Λ1 +Λ2)⊕6 ⊕ (2Λ1 +Λ2)⊕5 ⊕ (2Λ2)⊕2.

Proof. We use the techniques of [28] to find all potential simple algebras in C(g2,4). We thenthrow out all potential simple algebras whose corresponding potential module category does nothave rank 5. �

With this finite list of possible simple algebra objects, it now becomes a combinatorial problemto determine how A(Λ1↔2Λ2 and Λ2↔4Λ1) decomposes. While a-priori there should be no reasonto expect a unique decomposition, in this case we are lucky, and can exactly determine thedecomposition.

Lemma 7.6. The algebra A(Λ1↔2Λ2 and Λ2↔4Λ1) ∈ C(g2,4) decomposes into simple algebras as

A3 ⊕A4 ⊕A4 ⊕A5 ⊕A5.

Proof. We brute force check all possible 1287 different ways of choosing 5 simple algebras fromthe list in Lemma 7.5, and see that A3 ⊕ A4 ⊕ A4 ⊕ A5 ⊕ A5 is the only one that is equal toA(Λ1↔2Λ2 and Λ2↔4Λ1). �

Corollary 7.7. The objects A3, A4, and A5 in C(g2,4) have the structure of simple algebraobjects.

The module fusion graph of ModC(A3) is


The category C(g2,4) is unitary, hence this graph is the principal graph of a subfactor viaPopa’s embedding theorem [44]. The existence of this subfactor was hypothesised in [19].

Appendix A. The group structure of Eq(C(slr+1, k)).By Terry Gannon

In this appendix we explicitly determine the group structure of Eq(C(slr+1, k)). The goal willbe to prove the following Theorem.

Theorem A.1. Let r, k ≥ 1. Write n = r + 1 and d = gcd(n, k). Then we have

Eq(C(slr+1, k)) ≅ Z2 × {b ∈ Z×dn ∶ b ≡ 1 (mod d)}

where c ∶=⎧⎪⎪⎨⎪⎪⎩

0 if k ≤ 2 or r = 1

1 if k ≥ 3 and r ≥ 2.

More explicitly, this subgroup of Z×dn is isomorphic to

⎧⎪⎪⎨⎪⎪⎩

Z×n′ ×Z2 ×Zn′′2

if 2 exactly divides d

Z×n′ ×Zn′′ otherwise,

where we write n = n′n′′ for n′′ = gcd(n, k∞), so n′ is coprime to k.

We have seen that auto-equvialences of C(slr+1, k) come in two forms, the charge conjugationauto-equivalences, and the simple current auto-equivalences. The charge conjugation auto-equivalences of C(slr+1, k) are already well understood. They exist and are distinct from thesimple current auto-equivalences when both k ≥ 3 and r ≥ 2. Each of these charge conjugationauto-equivalences has order 2, and commutes with all other auto-equivalences. More complicatedis the composition of the simple current auto-equivalences. Let us study these in detail.

Recall that the simple current auto-equivalences of C(slr+1, k) are parameterised by the setG ∶= {a ∈ Zn ∶ 1 + ka is coprime to r + 1} via the map a ↦ FkΛ1,a. We begin by determiningthe group structure on this set which corresponds to the composition of simple current auto-equivalences.

Let a, b two elements in the set G. Let X ∈ C(slr+1, k), and n the unique integer (modulor + 1) such that

σX,kΛ1 ○ σkΛ1,X = e2πi nr+1 id(kΛ1)⊗X .

Then by the definition of the simple current auto-equivalence we have FkΛ1,b(X) = (kΛ−bn)⊗X,and so FkΛ1,a(FkΛ1,b(X)) = (kΛ−am) ⊗ (kΛ−bn) ⊗ X, where m is the unique integer (modulor + 1) such that

σ(kΛ−bn)⊗X,kΛ1○ σkΛ1,(kΛ−bn)⊗X = e2πi m

r+1 id(kΛ1)⊗(kΛ−bn)⊗X .

A direct computation gives that m = n + nbk. Hence

FkΛ1,a(FkΛ1,b(X)) = (kΛ−an(1+bk))⊗ (kΛ−bn)⊗X = (kΛn(−a−b−kab))⊗X.


Thus the auto-equivalence FkΛ1,a ○FkΛ1,b behaves the same on objects as the auto-equivalenceFkΛ1,a+b+kab. As the category C(slr+1, k) has no non-trivial gauge auto-equivalences, this impliesthat

FkΛ1,a ○FkΛ1,b ≅ FkΛ1,a+b+kab.

Hence we can endow the set G with the group structure

a ⋅ a′ ∶= a + a′ + aa′kto capture the composition of the simple current auto-equivalences.

Lemma A.2. The group G is isomorphic to the subgroup G(n, d) ∶= {b ∈ Z×dn ∶ b ≡ 1 (mod d)}of Z×dn.

Proof. There exists integers `,m such that `k +mn = d. Since `kd ≡ 1 (mod nd ), we have ` is

coprime to nd . In fact, we can choose ` so that it is coprime to n: choose an integer L so that

L ≡⎧⎪⎪⎨⎪⎪⎩

1 (mod p) if p ∣ gcd(`, n)0 (mod p) if p ∣ n and p /∣ `

then `′ ∶= ` +Lnd and m′ ∶=m −Lkd also satisfy d = `′k +m′n, though `′ is coprime to n.Since ` is coprime to n, the map b ↦ `b on Zn is a bijection. Because 1 + db ≡ 1 + k`n

(mod n), it restricts to a bijection G(n, d) → G as sets. Comparing the product in G(n, d), i.e(1 + db)(1 + db′) = 1 + d(b + b′) + d2bb′, with that of G, i.e. (`b) ⋅ (`b′) = `b + `b′ + d`bb′ (mod n),we see then that this map defines a group isomorphism. �

All that remains is to determine G(n, d) as a product of cyclic groups. Let us write νp for themultiplicity of the prime p in n, and ηp for the multiplicity of the prime p in d. Then Z×n ≅∏pZ×pνpand G(n, d) = ∏pG(pνp , pηp), and it suffices to work locally. Trivially, G(pν ,1) ≅ Z×pν , and so

G(n, d) ≅ Zn′ ×G(n′′, d).

Lemma A.3. Assume 0 < η < ν and p is prime. As groups,

G(pν , pη) ≅⎧⎪⎪⎨⎪⎪⎩

Z2 ×Z2ν−1 if p = 2 and η = 1

Zpν otherwise.

Proof. First note that for any p, b↦ 1+pηb ∶ Zpν → G(pν , pη) is a bijection of sets, so ∣G(pν , pη)∣ =pν .

If p is odd, then it is a classical group theory result that Z×pν+η ≅ Zp−1×Zpν+η−1 . Then G(pν , pη)will be a subgroup of the cyclic subgroup Zpν+η−1 , hence is itself cyclic with G(pν , pη) ≅ Zpν asgroups.

When p = 2 we have that Z×2ν+η ≅ Z2×Z2ν+η−2 , where the Z2 factor can be taken to be generatedby −1 ∈ Z×2ν+η , and Z2ν+η−2 taken to be all c ∈ Zν+η with c ≡ 1 (mod 4). When η ≥ 2, G(2ν ,2η)must then be a subgroup of the cyclic subgroup Z2ν+η−2 , and so must be isomorphic to Z2ν . Whenη = 1, the condition c ≡ 1 (mod 2)η in G(2ν ,2η) is automatically satisfied, so G(2ν ,2) = Z×ν+1

and we’re done. �

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