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arXiv:1208.1500v1 [math.QA] 7 Aug 2012

Near-group fusion categories and their doubles

David E. Evans

School of Mathematics, Cardiff University,

Senghennydd Road, Cardiff CF24 4AG, Wales, U.K.

e-mail: EvansDE@cf.ac.uk

Terry Gannon

Department of Mathematics, University of Alberta,

Edmonton, Alberta, Canada T6G 2G1

e-mail: tgannon@math.ualberta.ca

August 8, 2012

Abstract

A near-group fusion category is a fusion category C where all but 1 sim-

ple objects are invertible. Examples of these include the Tambara-Yamagami

categories and the even sectors of the D(1)

are infinitely many others. We classify the near-group fusion categories, and

compute their doubles and the modular data relevant to conformal field theory.

Among other things, we explicitly construct over 40 new finite depth subfac-

tors, with Jones index ranging from around 6.85 to around 14.93. We expect

all of these doubles to be realised by rational conformal field theories.

5

and E6 subfactors, though there

Contents

1 Introduction2

2 The near-group systems

2.1The numerical invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3First class: Near-group categories with n′= n − 1 . . . . . . . . . . . . . . . . . . . .

2.4The remaining class: n′a multiple of n . . . . . . . . . . . . . . . . . . . . . . . . . .

7

7

11

16

21

3Explicit classifications

3.1Which finite group module categories are near-group? . . . . . . . . . . . . . . . . .

3.2The type G + 0 classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3The near-group categories for the trivial group G . . . . . . . . . . . . . . . . . . . .

3.4 The type G + n − 1 C∗-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5At least as many T’s as S’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

25

26

26

26

28

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4Tube algebras and modular data

4.1The tube algebras of near-group systems . . . . . . . . . . . . . . . . . . . . . . . . .

4.2 The first class near-group C∗-categories: n′= n − 1 . . . . . . . . . . . . . . . . . . .

4.3The tube algebra in the second class . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4 The modular data for the double of G + n when n is odd

33

33

36

44

46. . . . . . . . . . . . . . .

References50

1Introduction

Considerable effort in recent years has been directed at the classification of subfactors

of small index. Subfactors of index ≤ 5 are now all known (see e.g. [23]). The

classification for index ≤ 4 was established some time ago. The Haagerup subfactor

[13] with Jones index (5+√13)/2 ≈ 4.30278, the Asaeda-Haagerup subfactor [1] with

index (5 +√17)/2 ≈ 4.56155, and the extended Haagerup subfactor [2] with index

≈ 4.37720, arose in Haagerup’s classification [13] of irreducible finite depth subfactors

of index between 4 and 3 +√3 ≈ 4.73205. A Goodman-de la Harpe-Jones subfactor

[11], coming from the even sectors of the subfactor corresponding to the A1,10⊂ C2,1

conformal embedding, has index 3 +√3. Then comes the Izumi-Xu subfactor 2221

[19] with index (5+√21)/2 ≈ 4.79 and principal graph in Figure 1, coming from the

G2,3⊂ E6,1conformal embedding.

The punchline is that, at least for small index, there are unexpectedly few sub-

factors. Does this continue with higher index? Are the aforementioned subfactors

exotic, or can we put them into sequences? In [19], Izumi realised the Haagerup and

Izumi-Xu subfactors using endomorphisms in Cuntz algebras, and suggested that his

construction may generalise. More precisely, to any abelian group G of odd order,

Izumi wrote down a nonlinear system of equations; any solution to them corresponds

to a subfactor of index (|G| + 2 +?|G|2+ 4)/2. He showed the Haagerup subfactor

solutions for the next several G, explained that the number of these depends on the

prime decomposition of |G|2+4, and argued that the Haagerup subfactor belongs to

an infinite sequence of subfactors and so should not be regarded as exotic.

Izumi in [19] also associated a second nonlinear system of equations to each finite

abelian group; to any solution of this system he constructs a subfactor of index

(|G| + 2 +?|G|2+ 4|G|)/2 and with principal graph 2|G|1, i.e. a star with one edge

an example). Izumi then found solutions for G = Zn(n ≤ 5) and Z2× Z2. G = Z1

and Z2correspond to the index < 4 subfactors A4and E6, respectively; his solution

for Z3 provides his construction for Izumi-Xu. An alternate construction of 2221,

involving the conformal embedding G2,3⊂ E6,1, is due to Feng Xu as described in

the appendix to [5] (see also [14]). As we touch on later in the paper, there may be

a relation between the series containing the Haagerup subfactor, and that containing

the Izumi-Xu subfactor.

corresponds to G = Z3, and that there also is a solution for G = Z5. In [10] we found

of length 1 and |G| edges of length 2 radiating from the central vertex (Figure 1 is

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Figure 1. The 231 principal graph

One of our tasks in this paper is to construct several more solutions to Izumi’s

second family of equations, strongly suggesting that this family also contains infinitely

many subfactors. But more important, in this paper we study a broad class of systems

of endomorphisms, the near-group fusion categories, including the Izumi-Xu series as

a special case. We obtain a system of equations, generalising those of Izumi, providing

necessary and sufficient conditions for their existence. We identify the complete list

of solutions to the first several of these systems, which permits us the construction

of over 40 new finite-depth subfactors of index < 15,

A fusion category C [9] is a C-linear semisimple rigid monoidal category with

finitely many simple objects and finite-dimensional spaces of morphisms, such that

the endomorphism algebra of the neutral object is C. The Grothendieck ring of a

fusion category is called a fusion ring. Perhaps the simplest examples are associated

to a finite group G: the objects are G-graded vector spaces ⊕gVg, with monoidal

product Vg⊗ V′

examples group categories. The category Mod(G) of finite-dimensional G-modules is

also a fusion category.

We’re actually interested in certain concrete realisations of fusion categories, which

we call fusion C∗-categories: the objects are endomorphisms (or rather sectors, i.e.

equivalence classes of endomorphisms under the adjoint action of unitaries) on some

infinite factor M, the spaces Hom(ρ,σ) of morphisms are intertwiners, and the prod-

uct is composition. Two fusion C∗-categories are equivalent iff they are equivalent as

fusion categories — all that matters for us is that the factor M exists, not which one

it is. Every finite-depth subfactor N ⊂ M gives rise to two of these, one correspond-

ing to the principal, or N-N, sectors and the other to the dual principal, or M-M,

ones. For example, given an outer action α of a finite group G on an infinite factor N,

we get a subfactor N ⊂ N×G = M coming from the crossed product construction:

the N-N system realises the group category for G, while the M-M system realises

Mod(G). Not all fusion categories can be realised as fusion C∗-categories (e.g. the

modular tensor categories associated to the so-called nonunitary Virasoro minimal

models are not fusion C∗-categories).

Perhaps the simplest nontrivial example of the extension of a fusion category is

when the category C has precisely 1 more simple object than the subcategory C0, and

the latter corresponds to a finite abelian group. More precisely, simple objects [g] in C0

correspond to group elements g ∈ G, with tensor product [g][h] = [gh] corresponding

to group multiplication. The simple objects of C consist of the [g], together with

h= (V ⊗ V′)gh. Its fusion ring is the group ring ZG. We call such

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some object we’ll denote [ρ]. Then [ρ] must be self-conjugate, [g][ρ] = [ρ] = [ρ][g], and

[ρ]2= n′[ρ]+?

of [1] must be 1).

We call these near-group categories of type G + n′. In this paper we restrict to

abelian G, and we reserve n always for the order of G. Examples of these have been

studied in the literature:

g∈G[g] (the multiplicities n′′

gin the second term must be independent of

g because of equivariance [g][ρ] = [ρ]; because [ρ] is its own conjugate, the multiplicity

• the Ising model and the module category of the dihedral group D4, which are

of type Z2+ 0 and Z2× Z2+ 0, respectively;

• more generally, the Tambara-Yamagami systems are by definition those of type

G + 0;

• the A4,E6and Izumi-Xu subfactors are of type G + n for G = Z1,Z2,Z3re-

spectively;

• more generally, Izumi’s second hypothetical family would be of type G + n;

• the D(1)

Z2+ 1, Z2+ 1, and Z3+ 2, respectively;

5

subfactor and the module category of groups S3and A4are of type

• more generally, the representation category of the affine group Aff1(Fq)∼=

Fq×F×

More precisely, Aff1(Fq) is the group of all affine maps x ?→ ax + b where a ∈ F∗

and b ∈ Fq. It has precisely q = pkconjugacy classes, with representatives (a,0)

and (1,1). It has precisely q − 1 1-dimensional representations, corresponding to the

characters of F∗

and is the nontrivial summand of the natural permutation representation of Aff1(Fq)

on Fqgiven by the affine maps: (a,b).x = ax + b.

In this paper we classify the near-group C∗-categories G+n′, in the sense that we

obtain polynomial equations in finitely many variables, whose solutions correspond

bijectively to equivalence classes of the near-group C∗-categories. Given any near-

group C∗-category C with n′> 0, we identify a natural subfactor ρ(M) ⊂ M whose

even systems are both identified with C. We also work out the principal graph of the

closely related subfactor ρ(M) ⊂ MG. By contrast, we can realise some but not all

C with n′= 0, as the even sectors of a subfactor.

There is a fundamental dichotomy here: n′either equals n−1, or is a multiple of

n, where as always n = |G|. When n′< n, we have a complete classification:

Fact. Let G be any abelian group of order n.

(a) There are precisely two C∗-categories of type G + 0.

(b) When n′is not a multiple of n = |G|, the only C∗-categories of type G + n′are

Mod(Aff1(Fn+1)), except for n = 1,2,3,7 which have 1,2,1,1 additional C∗-categories.

In all cases here, n + 1 is a prime power, n′= n − 1, and G = Zn+1.

qof a finite field Fqis of type Zq−1+ (q − 2).

q

q. The remaining irrep is thus of dimension?q(q − 1) − q − 1 = q−1,

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This is our Corollary 4 and Proposition 5 respectively, proven below. Type G+0

and type Zn+ n − 1 fusion categories were classified by Tambara-Yamagami [30]

and Etingof-Gelaki-Ostrik [8], respectively; we find that for these types, all fusion

categories can be realised as C∗-categories. Our proof of (a) is independent of and

much simpler than [30].

Conjecture 1.

inequivalent subfactors with principal graph 2n1whose principal even sectors satisfy

the near-group fusions of type G + n.

For every nontrivial cyclic group G = Zn, there are at least 2

We have verified this for n ≤ 13. For those n, the complete classification is given

in Table 2 below. In the process, we construct dozens of new finite depth subfactors of

small index with principal even sectors of near-group type. This classification for n =

3 yields a uniqueness proof (up to complex conjugation) for the principal even sectors

of the Izumi-Xu 2221 subfactor; this can be compared to Han’s uniqueness proof [14]

of the 2221 subfactor. Again, our proof is independent of and both considerably

shorter and simpler than the original one. We do not yet feel confident speculating

on systems with n′> n; the corresponding subfactors would have principal graph as

in Figure 3 below.

Two morals can be drawn from this paper together with our previous one [10]. One

is that there is surely a plethora of undiscovered finite-depth subfactors, of relatively

small index. This is in marked contrast to the observations of e.g. [24], who speak of

the ‘little desert’ in the interval 5 < [M : N] < 3+√5. The situation here is probably

very analogous to the classification of finite groups, which also is very tame for small

orders. The second moral is that, when the fusions are close to that of a group, a very

promising approach to the classification and construction of corresponding systems

of endomorphisms, equivalently C∗-categories, or the corresponding subfactors, is the

Cuntz algebra method developed in e.g. [19] and championed here. This approach

also makes the computation of the tube algebra and corresponding modular data

etc (to be discussed shortly) completely accessible. In contrast, the technique of

planar algebras is more robust, able to handle subfactors unrelated to groups, such

as Asaeda-Haagerup and the extended Haagerup. But planar algebra techniques

applied to e.g. the Haagerup fail to see that it (surely) lies in an infinite family. In

a few minutes the interested computer can construct several more subfactors of the

type described in Conjecture 1, using the Cuntz algebra method here, each of which

would be a serious challenge for the planar algebra method.

The underlying presence of groups here begs the question of K-theory realisations

of these fusion rings. For example, the fusion ring of the near-group C∗-categories

when n′= n − 1 can be expressed as KAff1(Fq)

in the other class, i.e. when n′∈ nZ?

An important class of fusion categories are the modular tensor categories [31],

which are among other things braided and carry a representation (called modular

data) of the modular group SL2(Z) of dimension equal to the rank of the category,

from which e.g. the fusion coefficients can be computed. These arise from braided

systems of endomorphisms on an infinite factor, from representations of completely

0

(1). Is there a K-theoretic expression

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rational conformal nets, or from the modules of a rational vertex operator algebra.

There is a standard construction, called the quantum or Drinfeld double, to go

from fusion categories (with mild additional properties) to modular tensor categories

[25]. We construct the doubles of our C∗-categories, following the tube algebra ap-

proach [18], and in particular explicitly compute its modular data. As with the

Haagerup series, our formulae are unexpectedly simple. This simplicity also chal-

lenges the perceived exoticness of these subfactors.

A natural question is, are these modular tensor categories realised by conformal

nets of factors, or by rational vertex operator algebras (VOAs)? Ostrik (see Appendix

A in [5]) shows that the double of Izumi-Xu 2221 has a VOA interpretation, in fact it

is the affine algebra VOA corresponding to G2,3⊕A2,1. No construction is known for

the large G. Curiously, this VOA conformally embeds into that of E6,1⊕A2,1(which

realises the fusions of the double of Z3), and this was where [10] suggests to look for

the VOA associated to the double of the Haagerup. Could there be a relation between

Izumi-Xu 2221 and the Haagerup? Other reasons suggest a relationship between 291

and the Haagerup. We discuss this latter possibility briefly in Subsections 3.5 and

4.4.

More generally we could consider quadratic extensions of a group category. More

precisely, let G be a finite group (not necessarily abelian) and suppose [ρ]∗= [ρ][gρ]

for some gρ∈ G. Let N be any subgroup of G: we require [g][ρ] = [ρ] iff g ∈ N.

Then [ρ][g] = [ρ] iff g ∈ gρNg−1

[g] for g ∈ G as well as [gi][ρ] for representatives giof cosets G/N. Let φ be any

isomorphism G/N → G/N′; we require [g][ρ] = [ρ][g′] iff g′∈ φ(gN). Then [ρ]2=

?

categories correspond to the choice N = G and gρ= 1; the Haagerup-Izumi series

[19, 10] corresponds to G = Z2n+1, N = 1, φ(g) = −g, n′

Haagerup subfactor at index (5 +√13)/2 corresponds to G = Z3. It would be very

interesting to extend the analysis in this paper to this larger class.

Here is a summary of our main results. Theorem 1 associates numerical invariants

to a near-group C∗-category, which according to Corollary 1 completely characterise

the category. Corollary 2 (and the end of Subsection 2.2) associate to each C∗-

category two subfactors and work out their principal graphs. Theorem 2 establishes

the fundamental dichotomy of near-group C∗-categories: either n′= n−1 or n′∈ nZ.

When n′= n− 1, Theorem 3 lists the identities necessarily obeyed by the numerical

invariants and shows they are also sufficient. Theorem 4 does the same when n|n′. In

Proposition 5 we find all near-group C-categories with n′= n−1; we see that almost

all of these are known. In Table 2 we list the first several with n′= n, and find that

almost none of these are known. In Theorem 5 and Corollary 6 we work out the tube

algebra and modular data for any near-group C∗-category with n′= n − 1. [19] had

found a very complicated expression for the modular data when n′= n; we notice in

Subsection 4.4 that it collapses to cosines.

ρ

=: N′. The simple objects in this category are

g∈N[g] +?

in′

i[gi][ρ]. We require φ to satisfy g−1

large class of examples should be accessible to a similar treatment. The near-group

ρφ(φ(g))gρ= g for all g ∈ G. This

i= 1; in particular, the

Note added in proof. After completing this manuscript, we received in July 2012

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[21] from Masaki Izumi, which overlaps somewhat the contents of our paper. In

particular, he also obtained necessary and sufficient conditions for the Cuntz algebra

construction to realise a near-group C∗-category of type G + n′. On the one hand,

unlike us, he does not require G to be abelian, and he allows the possibilities of

an H2-twist. On the other hand, unlike us, he does not address principal graphs

of associated subfactors, nor the tube algebra, nor the modular data (simplified or

otherwise) for the doubles, and he does not construct new solutions of the resulting

equations and hence does not construct new subfactors.

2 The near-group systems

2.1The numerical invariants

Let G be a finite abelian group (written additively) with order n = |G|, and as usual

write?G for its irreps. Let M be an infinite factor, ρ a self-conjugate irreducible

of G on M. Suppose the following fusion rules hold:

endomorphism on M with finite statistical dimension dρ< ∞, and α an outer action

[αgρ] = [ρ] = [ραg],

?

(2.1)

[ρ2] =

g

[αg] ⊕ n′[ρ], (2.2)

for some n′∈ Z≥0. Then the dρsatisfies d2

ρ= n′dρ+ n so

dρ=n′+√n′2+ 4n

2

=: δ.(2.3)

Let C(G,α,ρ) denote the fusion C∗-category generated by α,ρ. We call these, C∗-

categories of type G + n′.

Definition 1. By a pairing ?g,h? on G we mean a complex-valued function on G×G

such that for all fixed g ∈ G, both ?g,∗?,?∗,g? ∈?G. By a symmetric pairing we mean

for which the characters ?g,∗? are distinct for all g.

Note that a nondegenerate pairing is equivalent to a choice of group isomorphism

G →?G, g ?→ φg, by φg(h) = ?g,h?. The nondegenerate symmetric pairings for

Theorem 1. Let G,α,ρ be a C∗-category of type G + n′. Suppose in addition that

H2(G;T) = 1. Then there are n + n′isometries Sg,Tz(g ∈ G, z ∈ F) satisfying the

Cuntz relations, such that αgρ = ρ, ραg= Ad(Ug)ρ, for a unitary representation Ug

of G of the form

Ug=

?g,h?ShS∗

a pairing satisfying ?g,h? = ?h,g?. By a nondegenerate pairing we mean a pairing

G = Znare ?g,h? = e2πımgh/nfor some integer m coprime to n.

?

h

h+

?

z

uz,gTzT∗

gz,(2.4)

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where G permutes the z ∈ F and ?g,h? is a pairing on G. Moreover, αg(Sh) = Sg+h

and αg(Tz) = ? z(g)Tzfor some ? z ∈?G. Finally,

ρSg=

?

sδ−1?

?

h

?g,h?Sh+

?

h+

x,z

ux,gagx,zTxTz

?

U∗

g, (2.5)

ρ(Tz) =

?

h,x

? x(h)bz,xShT∗

x+

x,h

? x(h)b′

z,xTxShS∗

?

w,x,y

b′′

z;w,x,yTwTxT∗

y, (2.6)

for some sign s ∈ {±1} and complex parameters ay,z,bz,x,b′

F.

Proof. Our argument follows in part that of the first theorem of [20]. Because [αgρ] =

[ρ], there exists a unitary Wg∈ U(M) for each g ∈ G, satisfying αgρ = Ad(Wg)ρ.

But

Ad(Wg+h)ρ = αhαgρ = Ad(αh(Wg)Wh)ρ

z,x,b′′

z;w,x,y, for w,x,y,z ∈

(2.7)

for all g,h ∈ G, so αh(Wg)Wh = ξ(g,h)Wg+h for some 2-cocycle ξ ∈ Z2(G;T).

Because H2(G;T) = 1, we can require that ξ be identically 1, by tensoring Wgwith

the appropriate 1-coboundary. Since G is a finite group and α is outer, the α-cocycle

Wgis a coboundary, so there exists a unitary V ∈ U(M) so that Wg= αg(V∗)v for

all g ∈ G. This means Ad(αg(V ))αg(ρ) = Ad(V )ρ, i.e. αg(Ad(V )ρ) = Ad(V )ρ.

Thus if we replace ρ by Ad(V )ρ we obtain αgρ = ρ as endomorphisms, not just as

sectors.

This has exhausted most of the freedom in choosing ρ. The fusion [ραg] = [ρ]

means ραg = Ad(Ug)ρ for some unitaries Ug; because H2(G;T) = 1, we can in

addition insist that g ?→ Ugdefines a unitary representation of G. Note that we still

have a freedom in replacing Ugwith ψ(g)Ugfor any character ψ ∈?G.

ρ2(x) =

Sgαg(x)S∗

The fusion (2.2) means

?

g∈G

g+

?

z∈F

Tzρ(x)T∗

z, (2.8)

where Sgand {Tz}z∈Fare bases of isometries for the intertwiner spaces Hom(αg,ρ2)

and Hom(ρ,ρ2) respectively (so ρ2(x)Sg = Sgαg(x) etc). Then (2.8) implies Sg,Tz

obey the Cuntz relations. Since Ad(Ug)ρ2= ραgρ = ρ2, Uhmaps Hom(αg,ρ2) to

itself and Hom(ρ,ρ2) to itself, i.e. UhSg = µg(h)Sgand UhTz =?

and the matrices u define a unitary representation on Hom(ρ,ρ2). This gives us

?

wu(h)z,wTwfor

some µg(h),u(h)z,w∈ C. Since Uh+h′ = UhUh′, we have that µg∈?G for each g ∈ G,

Ug=µh(g)ShS∗

h

h+

?

z,y

u(g)z,yTzT∗

y.(2.9)

Define U′

of G. For this reason we may assume that µ0is identically 1.

g= µ0(g)Ug. Then ραg= Ad(U′

g)ρ and U′

gis still a unitary representation

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Similarly, αh maps Hom(αg,ρ2) to Hom(αg+h,ρ2) and Hom(ρ,ρ2) to itself, as

αhρ = ρ. This means αh(Sg) = ψg,hSg+hfor some nonzero ψg,h∈ C, and αhdefines

an n′-dimensional unitary G-representation on Hom(ρ,ρ2). Because H2(G;T) = 1

we can choose ψg,hto be identically 1. Because G is abelian, we can diagonalise the

n′-dimensional representation, i.e. choose our basis Tzso that αgTz= ? z(g)Tzfor some

Because ρ is self-conjugate and S0 ∈ Hom(id,ρ2), the isometry S0 will satisfy

S∗

For any T ∈ Hom(ρ,ρ2), define the right and left Frobenius maps R(T) =

√δT∗ρ(S0) and L(T) =

T∗ρ3(x)ρ(S0)=T∗ρ(ρ2(x)S0)=T∗ρ(S0x)

ρ(T∗ρ2(x))S0= L(T)ρ(x) so both L,R are conjugate-linear on the space Hom(ρ,ρ2).

R is surjective:

calculation shows that for any T,T′,T′′∈ Hom(ρ,ρ2), T∗ρ(T′)T′′∈ Hom(ρ,ρ2):

(T∗ρ(T′)T′′)ρ(x) = T∗ρ(T′ρ(x))T′′= ρ2(x)T∗ρ(T′)T′′.

Since 1 =?

ρ(S0) =SgS∗

TzT∗

? z ∈?G.

0ρ(S0) = s/δ for a sign s. Hence S∗

gρ(S0) = αg(S∗

0ρ(S0)) = s/δ.

√δρ(T∗)S0, as in Section 3.2 of [4]. Then ρ2(x)R(T) =

=

R(T)ρ(x) and ρ2(x)L(T)=

R(R(T)) = δρ(S0)∗Tρ(S0) = δρ(S∗

0ρS0)T = sT. A similar

gSgS∗

?

g+?

gρ(S0) +

zTzT∗

z, we find

?

zρ(S0) = sδ−1?

Sg+

?

TzR(Tz)

az,yTzTy

= sδ−1?

h

Sh+

?

z,y

(2.10)

for some complex numbers az,y, and covariance ρ(Sg) = ρ(αg(S0)) = Ad(Ug)ρ(S0)

forces (2.5).

We can identify the shape of ρ(T) similarly. Choose some Tz∈ Hom(ρ,ρ2); then

surjectivity of R implies there is some T′

find

?

=Sgαg(L(Tz)∗) +

w,g

?

αgρ(Tz) = ρ(Tz) forces bz(h,x) = bz,x? x(h) and b′

u(g) in (2.9) is a generalised permutation matrix. For each φ ∈?G let Tφ denote

⊕φTφ. Note that αgρ(Sh) = ρ(Sh) implies (among other things) the selection rule:

u(g)x,y?= 0 ⇒ ? y = ? xµgfor µg∈?G defined by µg(k) = µk(g). This means there is

isomorphism from Tφto Tµgφ(with inverse u(−g)).

Define H to be the set of all h ∈ G such that µh= 1; then H is a subgroup of G.

Choose a set O of orbit representatives of this G-action φ ?→ µgφ on?G, and a set C of

9

z∈ Hom(ρ,ρ2) such that Tz= R(T′

z). We

ρ(Tz) =

g

SgS∗

gρ(Tz) +

√

δ

?

TwT∗

w,g

TwT∗

wρ(T′∗

z)ρ2(S0)SgS∗

g+

?

w,x

TwT∗

wρ(Tz)TxT∗

x

?

g

?

wαg(L(Tz))αg(S0)S∗

?

z(x,h) = b′

g+

?

?

w,x

Tw(T∗

wρ(Tz)Tx)T∗

x

=

g,x

bz(g,x)SgTx+

g,x

b′

z(x,h)TxSgS∗

g+

w,x

Tw(T∗

wρ(Tz)Tx)T∗

x.

z,x? x(h), which gives (2.6).

All that remains is to show that a basis Tzof Hom(ρ,ρ2) can be found for which

the (possibly empty) subspace of Hom(ρ,ρ2) on which αgacts as φ, so Hom(ρ,ρ2) =

a pairing ?g,h? on G such that µh(g) = µg(h) = ?g,h?. Each u(g) defines a linear

Page 10

coset representatives for G/H. Note that u restricts to a unitary representation of H

on each space Tφ; for each representative φ ∈ O choose a basis Fφof Tφdiagonalising

this H-representation. For any ψ ∈?G there will be a unique choice of representatives

u(kψ) of the basis for Tφψ. Our basis F = {Tz} of the intertwiner space Hom(ρ,ρ2)

will be the union of these bases Fψfor the subspaces Tψ. For any basis vector Tx∈ F

and g ∈ G, write Tx = u(kx)Ty for some representative kx ∈ C and basis vector

Ty∈ Fφfor φ ∈ O, and g + kx= h + k′for k′∈ C and h ∈ H. Then because G

is abelian we have u(g)xx′ = u(g + kx)yx′ = u(h)yyδTx′,u(k′)Ty. Now write ux,g= uy,h

and define gx to label the basis vector Tgx= u(k′)Tz. Then gx defines a G-action on

F, and u(h) is diagonal ∀h ∈ H for this basis, and the matrix entries u(g)x,yequal

ux,gδy,gxas desired.QED to Theorem 1

φψ∈ O,kψ∈ C such that µkψφψ= ψ; let the basis Fψon Tψ be the image under

Corollary 1. Let G be a finite abelian group with H2(G;T) = 1 and choose any

C∗-category C of type G + n′. Then the sign s, pairing ?∗,∗?, and complex numbers

ux,g,ax,y,bz,x,b′

automorphism of G.

z,x,b′′

z;w,x,yform a complete invariant of C, up to gauge equivalence and

By gauge equivalence we mean equivalence up to a change of basis on Hom(ρ,ρ2)

(in contrast, relative rescaling of the Shwould wreck (2.10) so isn’t allowed). More

precisely, for each representative φ ∈ O and ψ ∈?H let Tφ,ψ denote the subspace

for h ∈ H. Then gauge equivalence amounts to a change-of-basis Pφ,ψ ∈ U(Tφ,ψ).

Let P ∈ U(Hom(ρ,ρ2)) be the direct sum of ?O? copies of each Pφ,ψ and define

Told

x

w

. Then anew= PTaoldP, bnew= P−1boldP, b′new= P−1b′oldP,

b′′new

permuting Sh?→ Sφh, Tx?→ Tφxwhere?

introduce more possibilities (e.g. Ugneed only be a projective representation). Recall

that for an abelian group, H2(G;T) = 1 iff G is cyclic. We will see shortly that either

n′= n − 1 or n′∈ nZ; when n′= n − 1 G must be cyclic and therefore in this case

the hypothesis H2(G;T) = 1 in Theorem 1 etc is redundant and can be dropped.

of Hom(ρ,ρ2) spanned by Tx ∈ F with ? x(g) = φ(g) for g ∈ G, and ux,h = ψ(h)

=?

φx = φ? x, ?g,h? ?→ ?φg,φh?, etc.

wPx,wTnew

z′,w′,x′,y′b′′old

z;w,x,y=?

The requirement that H2(G;T) = 1 is made for simplicity; dropping it would

z′;w′,x′,y′ Pz′,zPw′,wPx′,xPy′,y. An automorphism φ of G acts by

Corollary 2. Suppose C is a C∗-category of type G+n′for n′> 0. Then ρ(M) ⊂ M

is a subfactor with index d2

(dual) principal graph consisting of n vertices attached to a left vertex, n other vertices

attached to a right vertex, and the left and right vertices attached with n′edges (see

Figure 2 for an example). This has the intermediate subfactors ρ(M) ⊂ MG⊂ M

(where the G-action is given by αg) and ρ(M) ⊂ ρ(M)×G ⊂ M (using the G-action

ραgρ−1).

ρ, whose M-M and N-N systems are both type G+n′, with

10

Page 11

Figure 2. The principal graph for ρ3,2(M) ⊂ M

The subfactor ρ(M) ⊂ M is self-dual because ρ = ρ; the principal graphs for

ρ(M) ⊂ MGmatch those of ρ(M)×G ⊂ M because of the basic construction applied

to ρ(M) ⊂ ρ(M)×G ⊂ M. At the end of next subsection we compute the principal

graph of ρ(M)×G ⊂ M in all cases.

G acts on M through the αg; the relation αg· ρ = ρ says ρ(M) ⊂ MG. When

n′= 0, we see ρ(M) = MGby an index calculation. The index of ρ(M) ⊂ MGis

1 + n′n−1δ.

Recall the near-group C∗-categories listed in the Introduction. For example:

• The Tambara-Yamagami categories [30], which are of type G + 0, correspond

to δ =√n, µh(g) = ?g,h?, and s = ±1.

• The hypothetical Izumi-Xu family of subfactors (Section 5 of [19]) correspond

to the parameter choices s = 1, F =?G, n′= n, δ = (n +√n2+ 4n)/2,

b′

bers c,a(x),b(x) where a(z)2= ?z,z? and ?g,h? is a nondegenerate symmetric

pairing on G. We extend this pairing to?G through the group isomorphism

• The D(1)

δ = 2, s = 1, µh(g) = µh(g) = 1, ux,g= (−1)g, ax,x=

b′

µh(g) = ?g,h? = µh(g), ax,y =

√δ

−1δx,ya(x), ux,g = 1, bz,x = c√nδ

z;w,x,y= δy,wxa(x)b(zx)?z,y? for some complex num-

−1?z,x?,

z,x= a(z)c√n−1?z,x?, b′′

G →?G: ?xg,xh? := ?g,h?.

5

subfactor (Example 3.2 of [17]) has G = Z2, F =?G \ {1}, n′= 1,

x,x= a, b′′

√2

−1, bx,x= a√2

−1,

x;x,x,x= 0 for some complex number a satisfying a3= 1.

2.2Generalities

There is a fundamental bifurcation of the theory of near-group C∗-categories:

Theorem 2. Suppose C is a near-group C∗-category of type G + n′, and suppose

H2(G;T) = 1. Let ?∗,∗?,ux,s,a,b,b′,b′′be the parameters which Corollary 1 asso-

ciates to C.

(a) Either n′= n − 1, or n′= kn for some k ∈ Z≥0.

(b) Suppose n′= n − 1. Then δ = n, the pairing ?g,h? is identically 1, gx = x

∀x ∈ F and g ∈ G. The assignment x ?→ ? x bijectively identifies F with?G\{1} =:?G∗.

(c) Suppose n′= kn for k ∈ Z≥0. Then the pairing ?g,h? is nondegenerate, and for

any x ∈ F there is a unique gx∈ G such that ?

Proof. Let C = C(G,α,ρ) be of type G +n′, and let s,...,b′′be its numerical invari-

ants, and ?∗,∗? its pairing. Define subgroups H,H′of G by H = {h ∈ G|?h,g? =

There is a permutation σ of?G∗such that ux,g= (σ(x))(g) for all x and g. Finally,

ax,y=√δ

−1δy,x.

gxx = 1. Moreover, ux,gis identically

1, and ax,y?= 0 implies ? x? y = 1.

11

Page 12

1 ∀g} and H′= {h′∈ G|?g,h′? = 1 ∀g}. Let n′′= |H|. Write µg(h) = ?g,h? = µh(g)

as before. Note that the orders |H| and |H′| must be equal, since the row-rank of the

matrix ?g,h? will equal its column-rank.

Let us review some observations contained in the proof of Theorem 1. Recall

the coset representatives k ∈ C and orbit representatives φ ∈ O introduced in the

proof of Theorem 1. We saw there that the phases ux,hrestricted to h ∈ H forms

a representation of? H, which we’ll denote by ¨ x. Then we found there the formula

µ−k? x ∈ O and g +kx−k′∈ H. Recall the partition F = ∪Fφ,ψwhere φ ∈?G,ψ ∈? H;

φ ∈ O is the unique representative with φ|H= ? x|H.

phism of On,n′, ρ preserves the Cuntz relations. Firstly, ρ(Sg)∗ρ(Sh) = δg,hfor g ∈ H

is equivalent to

δg,0= nδ−2δg+H,H+

ux,g = ¨ x(g + kx− k′) valid for any g ∈ G and x ∈ F, where kx,k′∈ C satisfy

the G-action x ?→ gx on F, contained in Ug, bijectively relates F? x,¨ xto Fφ,¨ xwhere

Recall the Cuntz algebra On,n′ generated by Sgand the Tz. Being an endomor-

?

x,z

¨ x(g)ax,zagx,z. (2.11)

Putting g = 0 gives?

z|ax,z|2= δ−1for all x. Hitting both sides with ψ(g) for any

ψ ∈?H and summing over g ∈ H, (2.11) is equivalent to

1 − nn′′δ−2δH

where ¨ nψ denotes the number of x ∈ F with ¨ x = ψ. The TxS0S∗

coefficients of completeness 1 =?

δg,0= nδ−2δg+H′,H′ +

1,ψ= δ−1n′′¨ nψ, (2.12)

0Ty and S0S∗

g

gρSgρS∗

g+?

zρTzρT∗

?

zgive unitarity of the matrix

b′together with

z,x

|bz,x|2? x(g). (2.13)

Putting g = 0 into (2.13) tells us?

z|bz,x|2= δ−1, so (2.13) becomes

1 − nn′′δ−2δH

1,φ= δ−1n? nφ

(2.14)

for any φ ∈?G, where ? nφdenotes the number of x ∈ F with ? x = φ.

φ|H ?= 1, so (2.12),(2.14) say δ = n′′¨ nψ = n? nφ∈ Z. Then n = δ2− n′δ tells us δ

so ¨ n1= 0, so 1 − nn′′δ−2= 0 so n′′= n, i.e. H = G. Since |H′| = |H|, we know H′

would also equal G. This means ?g,h? is identically 1, so gx = x for all g,x. We’ve

just proved ? nφ= 1 − δφ,1= ¨ nφfor all φ ∈?G; in particular we can (and will) identify

σ of?G∗.

0. The element gxis then the unique one with µgx= ? x. We know from the proof

Now suppose H ?= 0. Then there exist ψ ∈? H and φ ∈?G such that ψ ?= 1 and

divides n, but δ = n? nφ≥ n, so δ = n. Hence n′= n − 1, so some ¨ nψmust vanish,

F with?G∗via x ?→ ? x, and then the assignment x ?→ uxcorresponds to a permutation

On the other hand, when H = 0, ?g,h? will be nondegenerate, and H′also equals

of Theorem 1 that the cardinalities ? nφand ? nµgφmust be equal for any φ ∈?G and

12

g ∈ G, and so in this case they all equal ? n1=: k ∈ Z≥0. Thus n′=?

φ∈? G? nφ= nk.

Page 13

Now, ux,gis uniquely defined by its values at g ∈ H, where it is a character, so in

this case it is identically 1 for all g ∈ G.

Return to the general case (i.e. arbitrary n′). The equivariance αgρ = ρ yields

the selection rule: ax,y ?= 0 implies ? x? y = 1. When n′= n − 1, this means ax,y =

QED to Theorem 2

a′(x)δy,xfor some a′(x) ∈ C. But?

For the convenience of the next two subsections, let us run through the identities

which must be satisfied by the numerical invariants s,a,b,b′,b′′, in order that they

define a near-group C∗-category of type n′. Let H = G respectively 0, and K = 0

respectively G, for n′= n − 1 and n|n′respectively. Then we know from Theorem 2

that ?∗,∗? is symmetric and nondegenerate on K. Write ux,g= ¨ x(g), where ¨ x ∈?G

of F (we already know this when n′= n−1, and will prove it in subsection 2.3 when

n divides n′).

Define endomorphisms ρ,αg,Ug on the Cuntz algebra On,n′, as in Theorem 1.

It is immediate that αgdefines a well-defined G-action on On,n′, and Ug a unitary

representation of G. In order for ρ to be a well-defined endomorphism on On,n′, we

need it to preserve the Cuntz relations S∗

and?

√

δ

z|ax,z|2= δ−1then implies |a′(x)|2= δ−1, so

a(x) := a′(x)√δ ∈ T.

equals 1 on K. Assume that ax,y=√δ

−1axδy,xfor some order-2 permutation x ?→ x

gSh= δg,h, S∗

gTz= 0 = T∗

zSg, T∗

zTz′ = δz,z′

gSgS∗

g+?

zTzT∗

z= 1. (ρSg)∗(ρSh) = δg,hreduces to (2.11). (ρSg)∗(ρTz) = 0

(or its adjoint) is equivalent to

− snδ? w,µg bz,w=

?

x

¨ x(g)agxb′′

z;x,gx,w,(2.15)

while the relation (ρTz)∗(ρTz′) = δzz′ (?ShS∗

δz,z′δy,y′ = nδ? y,?y′bz,ybz′,y′ +

and the unitarity of b′:?

h+?TxT∗

x) gives

?

w,x

b′′

z;w,x,yb′′

z′;w,x,y′

(2.16)

xb′z,xb′

z′,x= δz,z′. Finally, completeness 1 =?

?

¨ x(g)δgx,z¨x′(g)δgx′,z′ +

gρSgρS∗

g+

?

zρTzρT∗

zis equivalent to (2.13) (with H′= H), unitarity of b′, and

−δ−3/2az

g

¨ x(g)δgx,zµh(g) =

?

?

w,y

? y(h)bw,yb′′

b′′

w;x,z,y, (2.17)

δx,x′δz,z′ = δ−1azaz′

?

g

w,y

w;x,z,yb′′

w;x′,z′,y.(2.18)

To establish αgρ(x) = ρ(x) and ρ(αg(x)) = AdUg(ρ(x)) for all x, it suffices to

prove both for x = Shand x = Tz. The first follows from the bilinearity of ?g,h?,

that ?

The identity ρ(αgSh) = Ugρ(Sh)U∗

gx = µg? x, and that?x = ? x. The second follows from the factorisations of bz(g,x)

and b′

z(x,g) given in Theorem 1, and the selection rule b′′

z;w,x,y?= 0 ⇒ ? y = ? w? x.

gis built into (2.5), while ρ(αgTz) = Ugρ(Tz)U∗

gis

13

Page 14

implied by the covariances

bz,gx= ? z(g) ¨ x(g)bz,x, (2.19)

b′

z,gx= ? z(g) ¨ x(g)b′

z,x, (2.20)

b′′

z;gw,x,gy= ? z(g) ¨ w(g) ¨ y(g)b′′

z;w,x,y. (2.21)

All that remains is to consider are the fusion rules. We require Sgto be in the

intertwiner space Hom(αg,ρ2). We will follow as much as we can the proof of Lemma

5.1(a) in [19]. Thanks to Lemma 2.2 in [19], Sg∈ Hom(αg,ρ2) iff S∗

for all generators x. Hitting with α we see that this is true iff S∗

generators x. Because S∗

still goes through and S∗

have learned that Sg∈ Hom(αg,ρ2) iff both S∗

for all z. Those two identities are equivalent to

gρ2(x)Sg= αgx

0ρ2(x)S0= x for all

0ρ(Ug) = S∗

0ρ2(Sh)S0= Shwill follow once we know it for h = 0. So we

0ρ2(S0)S0= S0and S∗

g, the calculation in the middle of p.625 of [19]

0ρ2(Tz)S0= Tz

n′√δ

−1=

?

w,x

axbx,wb′

x,w, (2.22)

δz,w= snδ−1?

x

bz,xbx,w+ nδ−3/2?

?

x

b′

z,xbx,waw

+

x,y,x′,y′,z′

b′′

z;x′,x,ybx′,y′ b′′

x;y′,w,z′ by,z′ ,(2.23)

respectively, where we use (2.45) to simplify (2.23).

Clearly, if Tz is in the intertwiner space Hom(ρ,ρ2), then ρ(y∗ρ(x))Tz

ρ(y∗)Tzρ(x) for all generators x,y. Conversely, if ρ(y∗ρ(x))Tz = ρ(y∗)Tzρ(x) for

all generators x,y, then the calculation

?

=

ρ(Sg)ρ(S∗

=

ρ2(x)Tz=

g

ρ(SgS∗

g)ρ2(x)Tz+

?

w)Tzρ(x) = Tzρ(x)

w

ρ(TwT∗

w)ρ2(x)Tz

?

g

g)Tzρ(x) +

?

w

ρ(Tw)ρ(T∗

shows that T∗

imply Tz∈ Hom(ρ,ρ2). We compute

T∗

zρ2(x)Tz= ρ(x) for all generators x, and Lemma 2.2 of [19] then would

wρ(Ug) =

?

x,y,z

Vwxyz(g)TxT∗

yT∗

z+

?

h,y

Wwhy(g)ShS∗

hT∗

y+

?

h,z

Xwzh(g)TzS∗

h,

where

Vwxyz(g) = δ−1axay

?

k

?k,g? ¨ w(k) ¨ z(k)δkw,xδkz,y+

?

z′,y′

¨z′(g)b′′

z′;w,x,y′ b′′

gz′;z,y,y′,

Wwhy(g) = ? y(h) ? w(h)

?k,g − h?δkw,z¨ w(k) +

?

z

¨ z(g)b′

z,wb′gz,y,

Xwzh(g) = δ−3/2az

?

k

?

x,y

¨ x(g)b′′

x;w,z,y? y(h)bgx,y.

14

Page 15

The identity ρ(S∗

hρ(Sg))Tw= ρ(S∗

h)Twρ(Sg) gives

?g,h?Wwky(g) = sδ?g,k?

?

z

¨ z(g)ahw,zagz,hy, (2.24)

s?g,h?Vwxyz′(g) = δw,gz′ δy,gx¨ z(h)¨ hw(g) ¨ x(g) ¨ w(h)a(h−g)wahw, (2.25)

Xwzh(g) = 0. (2.26)

Using (2.26), the identity ρ(T∗

xρ(Sg))Tw= ρ(T∗

x)Twρ(Sg) gives

?

?

?

?

? w(k)?g,k − h?b′x,w= s

b′x,w? w(h)?h,g? ¨ z(g)δy,gz= s

s?g,h?b′′

√

δagx¨ x(g)

y

bgx,y? y(k)Wwhy(g),

agxbgx,x′?x′(h)Vwzyx′(g),

b′′

gx;w′,x′,yWwhy(g),

(2.27)

√

δ ¨ x(g)

x′

(2.28)

x;w,gx′,w′

¨

gx′(g)ax′ = ¨ x(g)agx

y

(2.29)

sδy,gy′¨y′(g)b′′

x;w,gx′,w′¨x′(g)ax′ = ¨ x(g)agx

z′

b′′

gx;w′,x′,z′ Vwy′yz′(g). (2.30)

ρ(S∗

gρ(Tz))Tx= ρ(S∗

g)Txρ(Tz) and ρ(T∗

xρ(Tz))Tw= ρ(T∗

√

δ

? w(g)bz,wb′w,x= ¨ x(g)agxb′

?x′(g)bz,x′ b′′

? x(k) ¨ w(k)δky,w=

b′

x)Twρ(Tz) now simplify to

?

w

z,−gx, (2.31)

√

δ

?

x′

x′;x,w,y= ¨ y(g) ¨ x(g)agxb′′

?

y′,zb′′

z;gx,gy,w, (2.32)

b′x,wbz,y− δ−3/2ayb′

z,x

?

k

w′,z′,x′

b′′

z;x,w′,z′ bw′,x′ b′′

z′;w,y,x′,(2.33)

?

y′,w′,x′

y′;x,w′,x′ b′

w′,yb′

x′,w= b′′

x;w,z,y,(2.34)

?

y′

b′′

x;w,y′,x′b′′

z;y′,y,z′ − δ−1ayaz′ b′

z,x

?

=

h

? x(h)¨x′(h) ¨ w(h)δhy,x′δhz′,w

b′′

?

w′,y′,w′′

z;x,w′,w′′ b′′

w′;x′,y,y′ b′′

w′′;w,z′,y′,(2.35)

where (2.33) was simplified using the selection rules akw,y ?= 0 ⇒ µk? w? y = 1 and

of b′and the selection rule for b′′.

Let M be the weak closure of the Cuntz algebra On,n′ in the GNS representation

of a KMS state (as in [17], Remark 4.8). Then the endomorphisms αgand ρ extend

to M and obey the same fusions as sectors. Note that the αgare outer because if

they were implemented by a unitary, it would have to commute with ρ since αgρ = ρ,

but as ρ is irreducible only the scalars can commute with it.

We have proved:

b′′

z′;w,y,x′ ?= 0 ⇒?x′= ? w? y, both obtained earlier. (2.34) was simplified using unitarity

15

Page 16

Proposition 1. Fix an abelian group G = H × K, where H = G or H = 0, and

a symmetric pairing ?g,h? nondegenerate on K. Let ax,y =

permutation x ?→ x of the index set F with x = x and?x = ? x, and suppose b′′obeys

the equations (2.11), unitarity of b′, (2.13), (2.15)-(2.35). Then αg,ρ defined as in

Theorem 1 yield a near-group C∗-category of type G + n′for n′= ?F?.

We can now identify the principal graph of the subfactor ρ(M)×G ⊂ M of index

d2

ι : ρ(M) ⊂ ρ(M)×G and j : ρ(M)×G → M. Then as M-M sectors the canonical

endomorphism jj is a subsector of the canonical endomorphism jιιj = ρ2, i.e. of

?

for [jj] is?

Could this 3-cocycle be related to the 3-cocycle appearing at the end of Subsection

4.2?

Now consider n′a multiple of n. When n′= 0 there is nothing to say: the

subfactor has index 1 so is trivial. When n′> 0, there is only one possibility for [jj],

namely [α + 0] + n′n−1[ρ]. We recover the graph as in Figure 3, i.e. the 2n1 graph

but with the n valence-2 vertices attached to the central vertex with n′n−1edges.

This generalises the paragraph concluding Section 5 of [19], which in the G + n case

considered there associates a subfactor M ⊃ ρ(M)×G with canonical endomorphism

[α0] ⊕ [ρ], index δ + 1 and principal graph 2n1.

√δ

−1axδy,x for some

the selection rule b′′

z;w,x,y?= 0 ⇒ ? y = ? w? x. Suppose the quantities s,a,b,b′,b′′satisfy

ρ/n = 1 + n′n−1dρ, introduced at the end of Section 2.1. Write the inclusions as

g[αg] + n′[ρ], which contains [α0].

Consider first n′= n−1; then dρ= n is the desired index, so the only possibility

g[αg]. We see that the principal graph matches that of the orbifold

MG⊂ M. This means this subfactor is isomorphic to MG⊂ M, up to a 3-cocycle.

Figure 3. Principal graph for the intermediate subfactor for type Z3+ 6

2.3First class: Near-group categories with n′= n − 1

Theorem 2 says that there are two classes of near-group C∗-categories: n′= n−1 and

n′∈ nZ. In this subsection we focus on the former, and identify a complete set of

relations satisfied by the numerical invariants s,...,b′′of Corollary 1. We know from

Corollary 2 that these systems are always realised by the even part of a subfactor.

Theorem 3(a) Let G be an abelian group of order n. Put δ = n and F =?G∗:=

σ(a) = σ−1a, σ3= id,and σ(σaσb) = σ2aσ(ba),

?G \ {1}. Let σ be a permutation of?G∗satisfying

(2.36)

16

Page 17

for all a ?= b ∈ G.

√n−1δy,σ xb(x), b′

tities a(x) ∈ {1,s}, b(x),b′′(x,y) ∈ T (provided xy ?= 1). Suppose these parameters

satisfy a(x) = a(σx) = sa(x), b(σx) = sb(x), b(x)b(σx)b(σ2x) = sa(x) and

Put ? x = x, ux,g = (σ x)(g), ax,y =

√n−1δy,xa(x), bx,y =

x,y= sδy,σ2xb(x)a(x), and b′′

z;w,x,y= δz,σw σyδy,wxb′′(w,x) for quan-

b′′(x,y) = sa(y)a(σ(xy)σx)b′′(xy,y)

b′′(x,y) = sa(x)b(σxσ(xy))b′′(x,xy)

∀xy ?= 1,

∀xy ?= 1,

∀xy ?= 1,

(2.37)

(2.38)

b′′(x,y) = sb(x)b(y)b(xy)b(σ2xσy)b′′(σ2x,σy)

b′′(σwσx,σxσ(xy))b′′(x,y)b′′(w,xw) = b′′(w,xyw)b′′(xw,y),

(2.39)

(2.40)

where the last equation requires w ?= xy, xy ?= 1, and w ?= x. Then αg, Ug, and ρ

defined as in Theorem 1 constitute a near-group C∗-category of type G + (n − 1).

3(b) Conversely, let C be a near-group C∗-category of type G + (n − 1), and assume

H2(G;T) = 1. Then C is C∗-tensor equivalent to one in part 3(a).

Proof. We’ll prove part (b) first. Theorem 2(b) tells us δ = n, we can identify F

with?G∗through x ?→ ? x, ax,y=√n−1a(x)δy,x, and there is a permutation σ of?G∗

Recall the Cuntz algebra On,n′ generated by the Sgand Tz. Select representatives

z ∈ R of each Z2-orbit {z,z} in F; then by rescaling the Tzappropriately we can fix

the values of az,zto be 1 for z ∈ R. Now, if Twis in the intertwiner space Hom(ρ,ρ2),

then ρ(y∗ρ(x))Tw = ρ(y∗)Twρ(x) for all generators x,y. In particular, the TySkS∗

coefficient of ρ(S∗

sδ−1?

Putting g = h = 0 in (2.53) gives axax= s, and hence az= s for all z ∈ R.

We get selection rules for b,b′,b′′through the equivariance αgρ = ρ and the identity

ρ(αgTz) = Ugρ(Tz)U∗

implies z σx = 1; and b′

√n−1δy,σ xb(x), b′

b(x),b′(x),b′′(w,x) ∈ C. Note that b′′(w,x) = 0 when wx = 1 because 1 is a forbidden

value for y = wx. (2.13) forces bx∈ T. g = h = 0 in (2.53) forces unitarity of the

matrix b′. The S0S0S∗

coming from Tw∈ Hom(ρ,ρ2)) gives b′(x) = sa(x)b(x). The TxTyT∗

1 =?

δx,x′δy,y′ = δy,xδy′,x′δx,x′ + b′′(x,y)b′′(x′,y′)

obtaining ux,gas (σx)(g).

k

hρ(Sg))Tw= ρ(S∗

h)Twρ(Sg) reads

z

¨ z(g)b′z,wb′

gz,y= ¨ w(h)¨ y(h)

?

z

¨ z(g)ahw,zagz,hy.(2.41)

g, namely: b′′

z,x?= 0 implies z = σ(x). Therefore we can write bx,y =

x,y= δy,σ xb′(x), and b′′

z;w,x,y?= 0 implies y = wx and z = σwσy; bz,x?= 0

z;w,x,y= δz,σw σyδy,wxb′′(w,x) for quantities

0coefficient of the identity ρ(T∗

xρ(S0))Tw= ρ(T∗

x)Twρ(S0) (again

y′T∗

x′ coefficient of

gρSgρS∗

g+?

wρTwρT∗

wcollapses now to

?

w,z

δw,σxσyδz,xyδw,σx′σy′δz,x′y′ . (2.42)

Choose any x,y ∈?G∗with xy ?= 1; we claim that the only solution x′,y′∈?G∗with

xy = x′y′and σxσy = σx′σy′is x = x′and y = y′: otherwise (2.42) would force

17

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b′′(x,y)b′′(x′,y′) = 0, which contradicts (2.42) with x = x′and y = y′. Thus each

b′′(x,y) ∈ T (provided xy ?= 1).

The S0S∗

gether with b′= sδab, gives σ2z = σz and b(z)b(z)b(σz) = a(z)a(σz)a(σ2z), re-

spectively σ(σaσb) = σ2aσ(ba) and (2.38). Taking σ of the complex conjugate of

σ(σaσb) = σ2aσ(ba), we obtain σ3= 1; iterating (2.38) twice gives b(y) = sb(σy).

The TyTz′S0S∗

TyS0S∗

(2.40).

Now that we know σ has order dividing 3, we know we can choose the Z2-orbit

representatives R so that azis constant on σ-orbits. Indeed, if σix = x for some i

and x, then σ−ix = σ−ix and s must equal 1, so there is nothing to do; when there

is no such i,x, there is no obstruction to putting all of x,σx,σ2x in R.

Conversely, suppose s,a(x),b(x),b′′(x,y), and σ are as in Theorem 3(a). We need

to verify the conditions of Proposition 1 are satisfied. For this purpose note that:

(i)?

(iii) that given a pair y,z ∈?G∗, there will be w,x ∈?G∗with b′′

The permutation in Proposition 1 is the usual complex-conjugation x of

characters. One easily computes Wwhy(g) = δw,y(σ2w)(g) and Vwxyz(g) =

δw,zδx,y(σ2w)(g)(σx)(g). In the last term of (2.23), x,y,y′,z′are determined from

x′,z,w, and is nonzero precisely when z = w ?= x′. That the second term in Xwzh(g)

vanishes, follows because σ2(w)σ(z) = wz implies wz = 1; this can be seen directly

from (2.36) but is trivial once we have Proposition 2 below. Both (2.57),(2.30) fol-

low from (2.37), (2.48) comes from (2.38), and both (2.33),(2.34) follow from (2.39).

(2.35) follows from (2.40) and (when y = x′) (2.37),(2.38). QED to Theorem 3

0respectively TyT∗

wcoefficients of ρ(S∗

gρ(Tz))Tx = ρ(S∗

g)Txρ(Tz), to-

0coefficient of ρ(T∗

0and TyTz′S0S∗

xρ(S0))Tw = ρ(T∗

xρ(Tz))Tw= ρ(T∗

x)Twρ(S0) recovers (2.37).

x)Twρ(Tz) give (2.39) and

The

0coefficients of ρ(T∗

xx(g) = nδg,0− 1 since x runs over?G∗; (ii) that given a pair w,x ∈?G∗, there

w = σ2((σy)z) and x = yw) iff z ?= σy.

will be y,z ∈?G∗with b′′

z;w,x,y?= 0 (namely y = wx and z = σwσy) iff x ?= w; and

z;w,x,y?= 0 (namely

When such a permutation σ exists, we get a solution by taking s = 1, a(x) =

b(x) = b′′(x,y) = 1. This solution corresponds to Mod(Aff1(Fq)), as we explain at

the end of Subsection 4.3. It is possible to classify all solutions σ to (2.36) — they

are essentially unique when they exist. The key observation in the following proof

(the relation to finite fields) is due to Siehler [29]. (Incidentally, an implicit unwritten

hypothesis throughout [29] is that G is abelian.)

Proposition 2. Let G be a finite abelian group which possesses a solution σ to (2.36).

Then G∼= Zq−1for some prime power q = pk. Moreover, if σ′is any other solution

to (2.36), then σ′= ασα−1for some group automorphism α ∈ Aut(G). Conversely,

any G = Zq−1for q = pkhas exactly |Aut(G)| = φ(q − 1) solutions σ to (2.36).

Proof. Suppose G has a solution σ. For convenience in the following proof, write

G multiplicatively. Then [29] explains how to give G ∪ {0} the structure of a field

F: the multiplicative structure of F is the multiplication in G, supplemented by

0x = x0 = 0; let −1 be the unique element in G of order gcd(2,1 + |G|) and write

−x = −1x; addition in F is defined by x + y = (σ(−x−1y))−1x when x,y ∈ G and

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x ?= −y, supplemented by 0 + x = x + 0 = x and x + (−x) = 0. This means G is

the multiplicative group of the finite field F, and thus is isomorphic to Zq−1for some

power q of a prime.

Call this field Fσ. Suppose there is a second solution σ′. Let α be the field

isomorphism Fσ→ Fσ′. Then α restricts to a group isomorphism from F×

F×

σ′ = G, i.e. α ∈ Aut(G). Conversely, given any solution σ to (2.36) and α ∈ Aut(G),

we get a new additive structure on Fσgiven by x +′y = ((ασα−1)(−x−1y))−1x etc,

corresponding to a solution ασα−1to (2.36). This is a bijectiion, since α can be

recovered from σ′= ασα−1.

Conversely, any G = Zq−1 with q a power of a prime, can be regarded as the

multiplicative group F×of a finite field with q elements (so 0 ∈ G corresponds to

1 ∈ F×). Then σ(x) = (1 − x)−1works.

Corollary 3. Consider any equivalence class of C∗-categories of type G + n − 1.

Fix any finite field Fn+1, identify G = F×

assignment of signs a′(x) ∈ {1,s} such that a′(x) = a′(σx) = sa′(x). There is a set L

of functions f :?G∗×?G∗→ Z, defined in the proof below, such that any C∗-category

• a(x) = a′(x) and σx = σ′x for all x;

• b(x) = sa(x) for all x ?= −1; in addition, b(−1) = sa(−1) unless n + 1 is a

power of 3 in which case b(−1) must be a third root of unity ω;

•

Conversely, any two C∗-categories with numerical invariants satisfying these con-

straints, and with identical b(x) and b′′(x,y), will be equivalent. Finally, s = 1 unless

n + 1 = q is a power of 2.

σ= G to

QED to Proposition 2

n+1and define σ′x = 1/(1 − x). Fix any

is equivalent to one with:

?

(x,y)b′′(x,y)f(x,y)= 1 for all f ∈ L.

Proof. Note that gauge equivalence by a diagonal matrix P with entries in T, permits

us to change a(x)new= PxPxa(x)old, bnew(x) = PxPσxbold(x), and b′′new(w,x) =

Pσw σ(wx)PwPxPwxb′′old(w,x). First note that, for any given x, we can change both

signs a(x) and a(x) (and leave all other a(y) unchanged) by taking Px = Px = i

and all other Py= 1. Now choose any x with σx ?= x. Without loss of generality

assume both x ?= −1 and σ2(x) ?= −1.

Px= sa(x)bold(x) = Px, Pσx= 1 = Pσx, Pσ2x= sa(x)bold(σx) = Pσ2xand all other

Py= 1. This gives bnew(x) = sa(x) = bnew(σx). Then b(x)b(σx)b(σ2x) = sa(x) forces

b(σ2x) = sa(x), and b(σy) = sb(y) forces b(σix) = a(x) = a(x).

−1 ∈?G∗precisely when G has even order, i.e. precisely when Fn+1 has odd

a(x) = 1, and by the previous paragraph we know b(x) = 1 unless σx = x. When

σx = x, the relation b(x)b(σx)b(σ2x) = sa(x) says b(x) will be a third root of 1.

Thanks to Proposition 2, we can find a finite field Fqwith G = F×

σx = 1/(1 − x). Suppose σx = x. Then x2− x + 1 = 0 in Fq, so x3= −1. If 3 does

not divide n, the only solution to x3= −1 is x = −1, but σ(−1) = −1 iff q is a power

Then x ?= x and σ2x ?= σ2x, so take

characteristic. In this case, a(−1) = sa(−1) = sa(−1) so s = +1. This forces all

q∼= Znfor which

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of 3, i.e. n ≡ 2 (mod 3). When 3 divides n, there are thus exactly 2 fixed-points

of σ. Let f be one of these. Then we calculate b′′(f,f) = b′′(f,f) from (2.37), and

b′′(f,f) = b(f)b′′(f,f) from (2.38), so b(f) = b(f) = 1.

By Corollary 1, two C∗-categories are equivalent, if they have identical numerical

invariants s,a,b,b′,b′′modulo gauge equivalence and automorphism of G. Fixing σ

fixes the automorphism of G. Suppose we also fix a and b. The remaining gauge

freedom are the quantities Pz= Pσz= Pz∈ T. Note that z = σiz iff σ−i(z) = −1

(the only order-2 element in F×

isomorphic to S3; the even-length orbits in?G∗are precisely those which don’t contain

gauge phase Pz∈ T is arbitrary. Associate to each pair (x,y) ∈?G∗×?G∗, xy ?= 1, a

Z-span of these ? v(x,y) is a lattice (of dimension ≤ ℓ), and thus will have a Z-basis; L

consists of those linear combinations?

We will show in Proposition 5 below that s = a = b = b′= b′′= 1 except for

n = 1,2,3,7. One can see from the data in Subsection 3.4 that for all n < 32, the

basis defining L consists of exactly ℓ of the ? v(x,y).

Let C(b,b′′) denote the equivalence class corresponding to the constraints of Corol-

lary 3 (we suppress the choice of field F and subset L, though these are implicit).

Note that there is an obvious product structure on the collection of equivalence

classes of type G + n − 1 C∗-categories: C(b1,b′′

(b1b2)(x) = b1(x)b2(x) and (b′′

a(x) = a1(x)a2(x) and b′(x) = b′

satisfy the conditions of Theorem 3(a) and Corollary 3, and thus uniquely deter-

mine an equivalence class of type G + n − 1 C∗-categories. Likewise, the identity

is s = a = b = b′= b′′= 1, and the inverse is complex-conjugation. We find this

abelian group structure very useful in Section 4.2 (though we will find in Section 3.4

that this group is usually trivial!).

In any case, this group of equivalence classes should be closely related to the set

H2((Zk

IX.8 (see also [15]). Those equivalence classes parametrise deformations of Kac al-

gebras (equivalently, depth-2 subfactors) possessing what we would call near-group

fusions of type Zpk−1+ pk− 2. We return to this briefly in Section 4.2.

Note that in all cases in Corollary 3, b(σx) = b(x) = sb(x). In the following we fix

a finite field Fq, identify the labels x ∈?G∗with the entries of Fq\ {0,1}, and choose

and σ(2) = −1. Note from the proof that σ will have exactly 0,1,2 fixed-points

respectively, for n ≡ 1,2,0 (mod 3).

By Corollary 2, the principal graph for ρ(M) ⊂ M when n = 2 is D(1)

McKay graph for binary S3. This suggests an alternate construction of the subfactor

ρ(M) ⊂ M, at least when b = b′′= 1. Construct a central extension BAff1(Fq) of

q). Together, σ and complex-conjugation form a group

−1. Let ℓ be the number of even-length orbits; the point is that for each of these, the

vector ? v(x,y) ∈ Zℓsuch that the gauge action is b′′(x,y) ?→ b′′(x,y)?

(x,y)f(x,y)? v(x,y), f(x,y) ∈ Z, corresponding

to such a basis.QED to Corollary 3

zP? v(x,y)z

z

. The

1) ∗ C(b2,b′′

1(x,y)b′′

2(x)). Then s,a,b,b′,b′′will obviously also

2) = C(b1b2,b′′

2(x,y). (Of course s = s1s2,

1b′′

2) where

1b′′

2)(x,y) = b′′

1(x)b′

p,Zpk−1);T)/∼ defined in Chapter III of [22], as suggested by their Theorem

σx = (1 −x)−1, so that (when Fqhas odd characteristic, i.e. n is even) σ(−1) = 1/2

5, the

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Page 21

Aff1(Fq) by Z2— it will have precisely 2 n-dimensional irreps (one of which, denoted

ρ′, is faithful) and 2n 1-dimensional irreps, and its McKay graph (which consists of

a node for each irrep and m edges connecting node i and j if k is the multiplicity of

irrep j in the tensor product of irrep i with ρ′) is the desired principal graph. The

irreps of BAff1(Fq) are separated into even and odd ones, depending on whether or

not the centre is in the kernel; the even vertices are precisely the irreps of Aff1(Fq).

To construct the subfactor, start with the index-n2subfactor

CI ⊗ Mn×n⊗ Mn×n⊗ ··· ⊂ Mn×n⊗ Mn×n⊗ Mn×n⊗ ··· ,

identifying BAff1(Fq) with its image K ⊂ Mn×nusing the faithful irrep ρ′, and then

take fixed-points:

(2.43)

CI ⊗ MK

n×n⊗ MK

n×n⊗ ··· ⊂ MK

n×n⊗ MK

n×n⊗ MK

n×n⊗ ··· .(2.44)

The principal graph is constructed as in [11]; for this purpose it is important that ρ′

is self-conjugate, as explained in [32].

2.4The remaining class: n′a multiple of n

In this section we identify a complete set of relations satisfied by the numerical

invariants s,...,b′′of Corollary 1, for the remaining class of near-group C∗-categories,

namely where the fusion coefficient multiplicity n′is a multiple of the order n of the

abelian group G. Corollary 2 tells us that when n′is a positive multiple of n, the

system will be realised by the even parts of a subfactor. There seems no reason to

expect all solutions with n′= 0 to arise from subfactors.

The main result in Theorem 4 is part (b). As usual, we identify the basis {Tz} = F

of Hom(ρ,ρ2) found in Theorem 1, with the set of labels {z}. For a given symmetric

pairing ?,? for G, fix a function ǫ?,?: G → T satisfying ǫ?,?(−g) = ǫ?,?(g) and ǫ?,?(g +

h) = ?g,h?ǫ?,?(g)ǫ?,?(h). An immediate consequence is that ǫ?,?(g)2= ?g,g?. For

example, if n = |G| is odd, the unique such function is ǫ?,?(g) = ?g,g?(n−1)/2, while if

G = Z2kand ?g,h? = exp(mπıgh/k) one of the two is ǫ?,?(g) = exp(−mπıg2/(2k)).

Theorem 4(a) Let G be a finite abelian group. Let ?k,k′? = ?k′,k? ∈ T be a

nondegenerate symmetric pairing on G. For each ψ ∈?G let Fψ

x = (? x, ˙ x, ˇ x) where ? x, ˙ x ∈?G, ˇ x ∈ F˙ x

Ugas in Theorem 1, where ux,g= 1 and gx = (µg? x,µ−g˙ x, ˇ x) for µg(k) = ?g,k?. Define

ax= sx˙ x(gx)ǫ?,?(gx), where gx∈ G is as in Theorem 2(c), for some signs sx∈ {1,s}

and some permutation x ?→ x of F as in Proposition 1. Then C(G,α,ρ) is a C∗-

category of type G + n′, provided the following equations are satisfied: sgx = sx,

1be a (possibly empty)

parameter set and define Fφψ

φ

= Fψ

1for all φ,ψ ∈?G; then F is the set of all triples

? x. Let n′= ?F? = n?

√δ

ψ?Fψ

1? and define δ by

(2.3). Let Sg,Tzbe standard Cuntz generators, for g ∈ G and z ∈ F. Define αgand

ρ(Sg) and ρ(Tz) by (2.5),(2.6) where ax,y =

−1axδy,x and b′

x,y= s√δaxbx,y, for

21

Page 22

sx= ssx,˙x = ˙ x, by,x= sbx,y,

bz,gx= ? z(g)bz,x;

?

(2.45)

? z(g)b′′

z;gw,x,gy= b′′

?

√

δ

z;w,x,y= ˙ w(g)ǫ?,?(g)b′′

bx,ybz,y= δx,z= s

−gz;gw,−gx,y;

b′

(2.46)

δ

y

y,w

x,yb′

y,wb′

w,z; (2.47)

δ−1b′′

z;w′,x′,y= s

−1aw′

?

w

bz,wb′′

w;w′,y,x′= δ−1ax′ azb′′

?

z;y,x′,w′

=

w,x

by,wbw′,xb′′

x′;x,z,w; (2.48)

n−1?

?

?

δ? y,?x′

y

b′′

z;y,x,y= −δ−1δz,xδ? x,1= δ? x,1δ? z,? x

z;w,x,yb′′

?

y

b′′

y;x,y,z; (2.49)

w,y

b′′

z′;w,x′,y= δz,z′δx,x′ − nδ−1δ? x,?x′δz,xδz′,x′ ;

w;y,z′,x′ = δx,x′δz,z′ − δ−1δgxx,gzzδgxx′,gzz′ ;

(2.50)

w,y

b′′

w;y,z,xb′′

(2.51)

?

y′

b′′

x;w,y′,x′b′′

z;y′,y,z′ − δ−1b′

?

z,xax′ δy,x′ awδz′,w

= δ? y,?x′

w′,y′,w′′

b′′

z;x,w′,w′′ b′′

w′;x′,y,y′ b′′

w′′;w,z′,y′; (2.52)

as well as the selection rules bx,y?= 0 ⇒ ˙ x˙ y = ? x? y, and b′′

H2(G;T) = 1. Then there exist quantities ?∗,∗?,s,a,b,b′,b′′satisfying the above

equations and relations, such that the corresponding C(G,α,ρ) is tensor-equivalent to

C.

Proof. We will prove part (b) first. Recall the Cuntz algebra On,n′ generated by the

isometries Sgand Tz. The desired selection rule for b′′follows from the equivariance

αgρ = ρ. The identity ρ(αgTz) = Ugρ(Tz)U∗

namely the left-sides of (2.45) and (2.46), along with b′

coefficient of the Cuntz relation δz,z′ = ρT∗

The TySkS∗

because Twlies in the intertwiner space Hom(ρ,ρ2)) reads

?g,h?sδ−1?

Putting g = h = 0 in (2.53) gives?

For each φ ∈?G, recall the subspace Fφ consisting of all T ∈ Hom(ρ,ρ2) with

22

z;w,x,y?= 0 ⇒ ? y = ? w ? x.

(b) Conversely, let C be a C∗-category of type G + n′for n′∈ nZ and suppose

gimplies the covariances for b,b′,b′′,

z,gx= ? z(g)b′

z,x.The S0S∗

0-

zρTz′ forces the unitarity of b′.

hρ(Sg))Tw= ρ(S∗

kcoefficient of the identity ρ(S∗

h)Twρ(Sg) (which holds

z

b′z,wb′

gz,y=

?

z

ahw,zagz,hy. (2.53)

yax,yay,z = sδ−1δx,z. Applying the triangle

y|ax,y|2= δ−1, yields ay,x= sax,y.inequality to this, and comparing with?

αg(T) = φ(g)T ∀g ∈ G. Using the gauge freedom discussed after Corollary 1, we

Page 23

can now simplify the form of a. We need this unitary change-of-basis P to commute

with each U(g), i.e. to satisfy P(Fφ) = Fφand Pkx,ky = Px,y for all k ∈ G. For

any g ∈ G define A(g)x,y= δ?

? y = µg? x. The latter tells us each A(g) is a map from Fφto Fµgφ. We verify that

representation of G. Moreover, covariance yields A(g)Uk= ?g,k?UkA(g). Hence the

set of unitary operators A(g)U−gfor all g ∈ G commute and so can be simultaneously

diagonalised; choose P so that all A(g)U−g are diagonal on F1, and P is defined

on arbitrary Fφby requiring Pkx,ky= Px,y. This means A(g)x,y= ǫx(g)δy,gxfor all

x,y ∈ F, for some ǫx(g) ∈ T. Hence the properties of A reduce to ǫkx(g) = ?g,k?ǫx(g)

and ǫx(g)ǫgx(h) = ǫx(g + h). We see that ψx := ǫxǫ?,? lies in?G.

agx,−gy= ǫx(g)ax,y. We need to refine this P further.

Define an equivalence relation on F by x ∼ x′iff there exists a sequence x =

x0,x1,...,xm = x′and y1,...,ymin F such that the entries axi−1,yiand axi,yiare

nonzero for all 1 ≤ i ≤ m. Let Xxdenote the equivalence class containing x and write

Tx for spanw∈Xx{Tw}. Then whenever ax,y ?= 0, a restricts to the indecomposable

blocks Tx→ Ty, where it is unitary. An induction argument (the base step of which

was done in the previous paragraph) verifies that any w ∈ Xxhas ? w = ? x. Moreover,

Choose any x ∈ F1, and suppose ax,y?= 0. Consider first the case where Xxand

Xyare disjoint, and fix some bijection π : Xy→ Xx. Define a unitary u on Tx+ Ty

to be the identity on Tx and to be a|Tx◦ π on Ty, and replace a on Tx+ Ty with

uTau. Otherwise we have the case Xx= Xy, so we can make use of some facts from

linear algebra (see section 4.4 of [16]) which say that: (i) when a complex matrix

B is both symmetric and normal, then there exists a real orthogonal matrix Q and

a diagonal matrix D such that B = QDQT; (ii) when a complex matrix B is both

skew-symmetric and normal, then there exists a real orthogonal matrix Q such that

QTBQ = 0 ⊕ ··· ⊕ 0 ⊕?

a|Tx) is in fact unitary, so both the zj and the diagonal entries of D lie in T and

we can adjust Q by the square-roots of those numbers and maintain unitarity. The

result is a matrix a in the form described in the statement of Theorem 4, where we

write πx = x and ˙ x(g) = ψx(g), and decompose each Fφinto ⊕ψFψ

x ∈ Fψx

g,x, ǫx(g) = ˙ x(g)ǫ?,?(g) and g(? x, ˙ x, ˇ x) = (µg? x,µ−g˙ x, ˇ x) as desired.

S∗

zax,gzay,z. Then (2.53) tells us for any g,k ∈ G the

covariance A(g)x,y= ?g,k?A(g)kx,kyand the selection rule A(g)x,y= 0 unless both

A(0) = I, A(g)A(k) = A(g + k) and A(−g) = A(g)∗so g ?→ A(g) defines a unitary

The triangle

inequality applied to A(g)x,gx, together with δ?

z|axz|2= 1, gives us the covariance

the invertibility of a says ax,y?= 0 implies the cardinalities ?Xx? = ?Xy? are equal.

j

?

0zj

0

−zj

?

for zj ∈ C×. Our matrix B here (namely

φwhere ˙ x = ψ for

? x; then for any

φ. This means we write x ∈ F as a triple (? x, ˙ x, ˇ x) where ˇ x ∈ F˙ x

Because Sh∈ Hom(αh,ρ2), we have

gρ2(S0)Sg= Sg= S∗

0ρ2(Sg)S0= S∗

0ρ(Ugρ(S0)U∗

g)S0= (ρ(Ug)∗S0)∗ρ2(S0)(ρ(Ug)∗S0).

But the intertwiner space Hom(αg,ρ2) is one-dimensional, and thus

β(g)S∗

g= S∗

0ρ(Ug)(2.54)

23

Page 24

for some scalars β(g) with |β(g)| = 1. Because ρ(Ug+h) = ρ(Ug)ρ(Uh), we have from

(2.54) and αh-covariance that β ∈?G. The S∗

δ2δµg,µh+

h-coefficient of (2.54) reads

?

β(g)δg,h=n

x,z

bz,xbgz,x? x(h). (2.55)

The triangle inequality applied to (2.55) with g = h forces β = 1 = µgµgon G (i.e.

the pairing ?∗,∗? is symmetric).

Putting g = h = 0 in (2.27) gives b′

the left-side of (2.47). ρ(S∗

its S0S∗

and the left-side of (2.48). Combining the left-side of (2.48) with ay,x= sax,yand

the unitarity of

?

Choose g ∈ G so that ? wµg= 1; replacing y with gx and applying?

the value for A(g) and our expression for b′into (2.53) with h = 0, and applying

the triangle inequality and δ?

Because Tw ∈ Hom(ρ,ρ2), we get ρ(T∗

and our formula for A(g), the TyTz′S0S∗

z,w= sδazbz,w. Hence the unitarity of b′implies

gρ(Tz))Tx= ρ(S∗

wcoefficients, together with b′= sδab, gives the right-side of (2.47)

√δb, gives

?

g)Txρ(Tz) holds because Tx∈ Hom(ρ,ρ2);

0and TyT∗

z′

bz,z′ b′′

z′;x,y,w= s

z′

bz′,zb′′

z′;x,y,w. (2.56)

xagxto (2.56), this

simplifies by (2.15) to?

z′bz,z′bz′,w= s?

x|bx,y|2= 1, yields bgw,gy= ǫw(g)ǫy(g)bw,y; comparing

with covariance (2.45) yields the selection rule for b given in the theorem.

xρ(Sg))Tw = ρ(T∗

0coefficient of this identity gives

z′bz′,zbz′,w, i.e. by,x= sbx,y. Substituting

x)Twρ(Sg). Using (2.53)

b′′

x;w,−gz′,yaz′ = agxb′′

gx;y,z′,−gw?g,g?ǫw(g),(2.57)

which simplifies to the second equality of (2.48) (for g = 0) and the right-side of

(2.46). The left-side of (2.49) comes from (2.15), while its right-side comes from the

T∗

using the left-side of (2.48) twice. (2.51) is a simplification of (2.18). Equations

(2.34),(2.35) simplify to the right-side of (2.48) (after applying the other parts of

(2.48)) and (2.52), respectively. (2.50) follows directly from (2.16) and (2.48). This

concludes the proof of part (b).

To prove part (a), we need to verify the conditions of Proposition 1, given the

equations listed in Theorem 4. Unitarity of b′follows from that of√δa and√δb. The

covariances (2.45),(2.46), agx = ǫ?,?(g)˙ x(g)ax and b′

(covariance for b′follows from that for a and b) are used repeatedly to simplify

the expressions of Section 2.2.For example, these covariances immediately give

Wwhy(g) = ?g,h?˙ y(g)ǫ?,?(g)δw,gy.

(2.15) follows from multiplying the left-side of (2.49) by?

selection rule b′

bx,w. (2.34) involves all three identities in (2.48). To obtain (2.35) from (2.52), use

three times the left-side of (2.46) with g = g′

wT∗

zcoefficient of (2.54). (2.50) follows by multiplying (2.16) by?

z,z′bz,ubz′,vand

gz,hw= ?g,h?? z(h)? w(g)ǫz(g)b′

z,w

zbz′,zand replacing x

with gx and y with gy. (2.17) involves the right-side of (2.49). (2.27) involves the

x,w?= 0 ⇒ ˙ x? x = ˙ w? w. Verifying (2.28) requires the selection rule for

xgy, as well as the selection rule for b′′.

24

Page 25

Equation (2.50) and the left-side of (2.49) tells us that the inverse of the matrix

b(w,y),(z,x):= b′′

another form of (2.50):

?

(2.16) arises from?

and use (2.48), (2.16), and both sides of (2.47). From (2.18) we compute Vwxyz(g) =

˙ z(g)ǫ?,?(g)δw,gzδgx,y.

z;w,x,yis b′

(w,y),(z,x):= b′′

z;w,x,y− ˙ w(gx)ǫ?,?(gx)δw,gxyδz,x. Hence we obtain

z,x

b′′

z;w,x,yb′′

z;w′,x,y′ = δw,w′δy,y′ − δ−1δgww′,gyy′δgww,gyy. (2.58)

z,z′by′,zbw′,z′ applied to (2.50), while (2.18) follows from (2.58).

(2.23) comes from (2.50) and the formula for δ. To see (2.33), hit it with?

QED to Theorem 4

w,yby′,wbz′′,y

3 Explicit classifications

Recall that G is the abelian group formed by the group-like simple objects, so n = |G|

is the number of S’s in the Cuntz algebra On,n′ of Section 2. The fusion coefficient

n′= Nρ

class’) or n′is a multiple of n (second class). In this subsection we explicitly solve our

equations for small n or n′. But first we address the question of the direct relation of

near-group systems to character rings of groups K, a question begged by the examples

in the Introduction.

ρρ= ?F? will be the number of T’s. We know that either n′= n − 1 (‘first

3.1 Which finite group module categories are near-group?

We have seen several examples of finite groups K whose module categories are near-

group. For example, the module categories Mod(D4) and Mod(Q8) for the order-8

dihedral and quaternion groups, are both of type Z2× Z2+ 0, and those exam-

ples apparently motivated Tambara-Yamagami to study their class of categories [30].

Similarly, the even sectors of the D(1)

5

subfactor satisfies the S3fusions and, more gen-

erally, the module categories for the affine groups Aff1(Fq) are of type Zq−1+q−2. The

complete list of groups whose representation rings possess the near-group property

has been rediscovered several times, but perhaps originated with Seitz:

Proposition 3. [28] The complete list of all finite groups K whose module category

Mod(K) is a near-group category of type G+n′, for some abelian group G and some

n′∈ Z≥0, is:

(a) |K| = 2kfor k odd, its centre is order 2, and G/Z∼= Z2× ···Z2. In this case,

G = K/Z(K), dρ= δ = 2(k−1)/2, and n′= 0.

(b) K∼= Aff1(Fq) for some finite field Fq. In this case, G∼= Zq−1, dρ= q − 1, and

n′= q − 2.

The groups in part (a) are called extraspecial 2-groups; there are precisely 2 of

them for each odd k > 1. We will see next subsection that most C∗-categories of type

G + 0 are not Mod(K) for some K. In contrast, Proposition 5 below says that all

but 5 C∗-categories of type G + n − 1 will be Mod(K) for K in part (b).

25

Page 26

3.2The type G + 0 classification

As a special case of Theorem 4, we recover the Tambara-Yamagami classification [30]:

Corollary 4. The equivalence classes of C∗-categories of type G + 0 are in one-

to-one correspondence with either choice of sign s and any choice of nondegenerate

symmetric pairing ?,? on G, up to automorphism of G.

Proof. Because n′= 0, the parameters a,b,b′,b′′must be dropped from all equations

in Proposition 1. All that remains is the sign s and the symmetric pairing ?∗,∗?,

which will be nondegenerate for G. QED to Corollary 4

The proof in [30] is independent and much longer, involving a detailed study of

the pentagon equations in the category. It is worth remarking that [30] prove that G

must be abelian (whereas we assume it).

These usually don’t seem to be realised by a subfactor. As explained after Corol-

lary 2, for both choices of signs the subfactors ρ±(M) ⊂ M are equivalent to the

MG⊂ M subfactor.

3.3 The near-group categories for the trivial group G

It is generally believed that there are a finite number of fusion categories of each

rank, so in particular one would expect that for each finite G, there are only finitely

many near-group C∗-categories of type G + n′for arbitrary n′. In fact, we are led

to expect that only finitely many cyclic groups G = Zn, when n + 1 is not a prime

power, will have C∗-categories of type G + n′for n′> 0.

Nevertheless, until we can bound n′given a G, it seems to be nontrivial to classify

all near-group C∗-categories whose group-like objects form a given group G. The

only example we can fully work out is G = {0} (although we expect the tube algebra

analysis for n′> n should yield classifications for other groups of small orders).

Proposition 4. Up to equivalence, there are precisely 3 near-group C∗-categories of

type {0} + n′: namely, two of type {0} + 0, and one of type {0} + 1.

The possibility n′= 0 is Tambara-Yamagami and so is covered by Corollary 4,

while n′= 1 is most easily handled using Corollary 5 below. The reason there can be

no examples with n′> 1 is that such a solution would yield a fusion category with

rank 2 and a fusion coefficient = n′≥ 2, and no such fusion category can exist [27].

There are precisely 4 rank 2 fusion categories (2 of type {0} + 0 and 2 of type

{0} + 1). The one which is not realised as a fusion C∗-category is known as the

Yang-Lee model, corresponding to one of the nonunitary Virasoro minimal models.

The nonunitary minimal models can never be realised as C∗-categories, so it is no

surprise that Yang-Lee is missing from Proposition 4.

3.4 The type G + n − 1 C∗-categories

By Corollary 3, the only parameters we need to identify are a sign s when n + 1 is

a power of 2, a third root of unity ω when n + 1 is a power of 3, and the b′′

x,ywhen

26

Page 27

xy ?= 1. The complete classification for n < 32 is collected in Table 1; the only value

for the entries n ?= 1,2,3,7 is the permutation σ. Recall n + 1 must be a power of

a prime and G must be cyclic. In Table 1 we identify F with the subset G \ {0}.

ω there is any third root of 1. In the σ column of the table we use cycle notation,

writing A for 10, B for 11, etc. For G = Z7, a(x) = 1,1,s,1,s,s for x = 1,2,...,6

respectively, and b′′is given by

It is elementary to verify from Corollary 3 that each entry in Table 1 yields an

inequivalent solution to the equations of Theorem 3.

b′′(i,j) =

1 s s 1 s ∗

1 1 s s ∗ s

s s 1 ∗ 1 1

s 1 ∗ 1 s s

s ∗ 1 s 1 1

∗ s 1 s 1 1

. (3.1)

G

Z1

Z2

Z3

Z4

Z6

Z7

Z8

Z10

Z12

Z15

Z16

Z18

Z22

Z24

Z26

Z28

Z30

Z31

#

2

3

2

1

1

2

1

1

1

1

1

1

1

1

1

1

1

1

σsa

∅

1

b

∅

ω

sa

1

1

b′′

∅

∅

(1)

±1

1

±1

1

1

±1 11s1ss sa (see above)

111

111

111

111

111

111

111

111

111

111

111

111

(1)

(1)

1,s

1

1

b′′

xx= ax

1

1

(123)

(234)

(142)(365)

(165)(273)

1

1

1

1

1

1

1

1

1

1

1

1

(159)(276)(843)

(16B)(378)(459)

(1B3)(276)(4EC)(8D9)

(1AD)(2E8)(36F)(4B9)(5C7)

(19H)(2EB)(4G7)(5CA)(6D8)

(1EI)(236)(48L)(5FD)(7H9)(ACB)(GJK)

(129)(3EJ)(5AL)(6CI)(7GD)(8HB)(FMN)

(139)(2LG)(4FK)(5OA)(6BM)(7EI)(8CJ)(HNP)

(1ER)(293)(4FN)(5DO)(6KG)(7HI)(8MC)(ABL)(JQP)

(1LN)(23A)(4JM)(6OF)(79T)(8BQ)(CHG)(DIE)(KRS)

(1HD)(23Q)(46L)(5ST)(7F9)(8CB)(APR)(EUI)(GOM)(JNK)

Table 1. The C∗-categories of type G + n′for |G| = n′+ 1 ≤ 31

Proposition 5. If a C∗-category is of type G + n′with n′?∈ nZ, then G = Zn,

n + 1 =: q is a prime power, and n′= n − 1. There is precisely one C∗-category of

type Zn+ n − 1, namely Mod(Aff1(Fq)), except for n = 1,2,3,7 which have precisely

1,2,1,1 additional C∗-categories, collected in Table 1.

Proof. We know n′= n − 1 from Theorem 2, and G∼= Zn where n = q − 1 for

some prime power q = pk, by Proposition 2. Corollary 7.4 of [8] tells us that the

27

Page 28

only fusion categories of type Zn+ n − 1 are Mod(Aff1(Fq)), except for n = 1,2,3,7

where there are precisely 1,2,1,1 additional fusion categories. There always is at least

one C∗-category of type Zn+ n − 1, namely the one corresponding to the solution

b = b′′= 1, so it suffices to find 1,2,1,1 additional solutions the the equations of

Theorem 3, when n = 1,2,3,7 respectively. These are collected in Table 1. QED

Combining [8] and Proposition 5, we find that each near-group fusion category

with n′= n − 1 and G = Znhas a C∗-category structure, i.e. a system of endomor-

phisms (unique up to equivalence). To our knowledge, this gives the first construction

of the extra C∗-categories for Z3and Z7. As pointed out in Section 2.3, the collection

of (equivalence classes of) type G + n − 1 C∗-categories for a given G will form an

abelian group. We find that this group is always trivial, except for n = 1,2,3,7 when

it is Z2,Z3,Z2,Z2respectively.

Recall the discussion of the deformation parameters H2((Zk

the end of Section 2.3. We expect the group H2((Zk

should be possible to verify this from the results of [22], at least for n ≤ 4 and n = 6.

Certainly it says there is a unique depth-2 subfactor with principal even fusions of

type Zn+n−1 for those n, namely MH⊂ M×Zn−1for H = Zk

analysis of these subfactors). It is easy to compute H2((Zk

the definition, for n = 1,2, and we find indeed that it is trivial. In particular, this

means that only the s = 1 solution at n = 1, and only ω = 1 at n = 2, are realised

by depth-2 subfactors. The triviality of this for n = 5 follows from uniqueness results

for subfactors of index 5.

p,Zpk−1);T)/ ∼ near

p,Zpk−1);T) is trivial for all n. It

p(see [3] for a complete

p,Zpk−1);T) directly from

3.5At least as many T’s as S’s

Consider now type G+n′, where n′≥ n (and therefore must be a multiple of n). We

don’t know of any examples where n′> n. A natural approach to bounding n′, given

n, would be carrying through the tube algebra analysis for n′> n. After all, we find

in Section 4.2 below that this strategy is effective in pruning the possibilities for type

G + n − 1. Likewise, Ostrik’s analysis [27], which eliminated n′> n for G = {0},

investigated the modular data for the double. We will pursue this thought in future

work.

Corollary 5. Any solution to the equations of Proposition 1, for arbitrary G and

n′= n, has s = 1, F =?G, H = 0, ux,g= 1, and is of the form ax,y=√δ

complex numbers c,a(x),b(x) satisfying

−1δx,ya(x),

bz,x= c√nδ

−1?z,x?, b′

z,x= a(z)c√n−1?z,x?, b′′

z;w,x,y= δy,wxa(x)b(zx)?z,y? for some

?

?x,y?b(y) =√ncb(x), a(x)b(x) = b(x),

a(1) = 1, a(x) = a(x), a(xy)?x,y? = a(x)a(y),

?

b(xy)b(x) = δy,1− δ−1,

x

a(x) =√nc−3, (3.2)

b(0) = −1/δ,

y

(3.3)

?

x

?

x

b(xy)b(xz)b(x) = ?y,z?b(y)b(z) −

c

δ√n.

(3.4)

28

Page 29

Conversely, any a(x),b(x),c satisfying (3.2)-(3.4) yields a solution to the equations

of Proposition 1 in this way. Two C∗-categories C1,C2of type G + n are equivalent

iff c1= c2and there is a φ ∈ Aut(G) such that ?g,h?2= ?φg,φh?1, a2(x) = a1(φx),

and b2(x) = b1(φx).

Proof. Through the pairing ?∗,∗? we may identify?G, and hence F, with the group

?

axg,xh=√δ

so we get s = +1 by looking at g = 0. We get b′

(2.46) says b′′

now reduces to

c

?g,h − k?b′′

G.

gx0(h) = µg(h) = ?g,h?. So the action of G on F corresponds to addition in G:

hxg= xh+g.

Equivariance (2.45) forces bxg,xh= c√δ

−1a(g)δg,−hfor some numbers a(g) with |a(g)| = 1. Then a(−g) = sa(g),

More precisely, let x0 denote the x ∈ F with ? x0 = 1; then xg = gx0 has

−1?g,h? for some c with |c| = 1. Write

xg,xh= c?g,h?a(g) and covariance

g,kfor some numbers b′′

xg;xh,xk,xl= δl,h+k?g,h?b′′

g,k. The left-side of (2.48)

?

g

g,l= a(k)?h,k?b′′

h,k+l

(3.5)

and hence

b′′

h,k= ca(k)?h,k?

?

g

?g,h − k?b′′

g,0= a(k)?h,k?b′′

h−k,0. (3.6)

Writing b′′

QED to Corollary 5

g,0= b(g), it is now easy to verify that all equations of [19] are recovered.

This Corollary says that the case n′= n reduces to the generalisation of E6

introduced in Section 5 of [19].

Proposition 6. There are (up to equivalence) precisely 1, 2, 2, 2, 3, 4, 2, 8, 2, 4,

4, 4, 4 systems for G = Zn, 1 ≤ n ≤ 13, respectively. There is 1 solution each for

G = Z2×Z2and G = Z3×Z3, 2 solutions for Z2×Z6, 4 solutions for Z2×Z4, and

no solutions for Z2× Z2× Z2.

29

Page 30

Gc

?,?

1

1

1

1

−1

1

−1

1

1

2

2

1

−1

1

−1

1

−1

1

1

−1

−1

3

3

−3

−3

1

1

1

−1

1

−1

−1

1

7

1

7

1

1

−1

−1

1

1

−1

−1

−1

1

1

1

2

2

aQ′

(j1,...,j⌊n/2⌋)

∅

(0.78539816)

(−0.78539816)

(−2.8484536)

(2.8484536)

(−0.60623837, −1.5707963)

(−1.39163653, 1.57079632)

(−2.356194490, 2.356194490, 0)

(−1.256637, 1.256637)

(−1.0071249, 0.3425266)

(−0.3425266, −2.263762)

(−2.9552611, −0.055354168, 0.78539816)

(−2.915033694, 1.5909100, −2.3561944)

(2.9150336, −1.590910, 2.3561944)

(2.9552611, 0.055354168, −0.78539816)

(−1.05169, 1.7936250, −0.3143315)

(1.0516925, −1.793625, 0.31433)

(−0.87227636, 2.7042615, −2.9767963, 3.1415926)

(−2.9767963, −1.1334651, −0.87227635, 3.1415926)

(0.87227636, −2.7042615, 2.9767963, 3.1415926)

(2.9767963, 1.1334651, 0.87227635, 3.1415926)

(2.4640490, −3.0755747, −0.49188699, 0)

(−0.49188700, 1.5047784, 2.4640490, 0)

(−2.4640490, 3.0755747, 0.49188699, 0)

(0.49188700, −1.5047784, −2.4640490, 0)

(0.7853981, 1.77783, −2.497219, 1.570796, −0.7853981)

(−0.785398, 0.9924406, 1.42977,−1.57079, 0.7853981)

(0.785398, 1.42977, −1.777838,−1.570796, −0.785398)

(−0.7853981, −2.497219, −0.9924406, 1.5707963, 0.7853981)

(−2.69568, 1.367012, 1.41882, −2.38374)

(2.695680, −1.3670127, −1.418824, 2.383744)

(2.9557793, −1.2330109, −2.2802084, 0)

(−2.3665026, −3.0894639, 3.077894, 0.00650245, 0.785398)

(1.7756309, −0.6115079, −1.030618, 2.8686859, −2.3561944)

(−3.077894, 2.519776, −1.424024, 3.089463, −0.78539816)

(−1.3447773, −2.868685, −1.7756309, 0.64512913, 2.3561944)

(1.9464713, 2.0140743, −1.7487929, 0.3352432, −0.1427077)

(0.53877136, −2.8317431, 0.2827610, 0.46457259, 2.5063157)

(−2.8884206, 2.3090448, 0.85395967, 2.1781685, −1.4920749)

(−1.4807206, 0.87167704, −1.1775942, 2.0488391, 2.1420869)

(3.0822445, −0.3494640, −3.0450322, −0.7241984, −0.38234715, 1.570796)

(−0.6247574, −3.044463,−2.3415376, 0.4718634, 0.99777419, 1.5707963)

(−3.0822445, 0.34946402, 3.0450322, 0.7241984, 0.3823471, −1.570796)

(0.6247574, 3.0444636, 2.3415376, −0.47186343, −0.99777419, −1.5707963)

(−2.35619, 0.0611997, 2.469129, 0.89833, −1.88433, 0.785398, 1.57079)

(2.356194, −0.0611997, −2.4691295, −0.8983332, 1.884331, −0.785398, −1.570796)

(−2.4521656, 1.9847836, 0.42579608, 1.4322079, −1.4550587, 1.1404478)

(1.4550587, 1.3924399, −1.9847836, −1.2761619, 0.44776608, −1.4322079)

(−2.4805730, 3.0305492, 0.28372451, −0.04125417, 0.44928247, 2.9410534)

(−0.44928247, 2.2170122, −3.0305492, −1.892166, −2.9638949, 0.041254182)

Z1

Z2

11

1

2

ξ17

24

ξ7

24

ξ−1

12

ξ12

ξ3

8

ξ−3

8

1

−1

ξ3

ξ−1

3

ξ−1

24

ξ−5

24

ξ5

24

ξ24

ı

−ı

ξ24

ξ24

ξ−1

24

ξ−1

24

ξ7

24

ξ7

24

ξ17

24

ξ17

24

ξ5

12

ξ−5

12

ξ−5

12

ξ5

12

ξ−1

3

ξ3

1

ξ−1

8

ξ3

8

ξ8

ξ−3

8

ξ7

12

ξ−1

12

ξ12

ξ5

12

ξ7

24

ξ−5

24

ξ−7

24

ξ5

24

ξ−1

3

ξ3

−1

−1

1

1

(−1,−5)

(1,5)

1

−1

(3,−3)

(−3,3)

−1

1

1

1

1

1

1

1

1

1

1

−1

−1

1

1

1

−1

1

−1

1

−1

1

−1

(1,1)

(−1,1)

(1,−1)

(−1,−1)

1

1

1

1

1

−1

−1

1

1

1

1

1

−1

1

−1

(1,1)

(−1,1)

1

1

1

1

Z3

Z4

Z2× Z2

Z5

(1,1)

2

−2

(−1,−19)

(3,13)

(−3,−13)

(1,19)

−2

2

(−5,−7)

(−5,17)

(5,7)

(5,−17)

(1,11)

(1,−13)

(−1,−11)

(−1,13)

Z6

Z7

Z8

Z2× Z4

Z9

−2

2

2

Z3× Z3

Z10

Z11

2

1

−1

−2

Z12

Z2× Z6

Z13

3

3

1

1

Table 2. The C∗-categories of type G + n for |G| ≤ 13

All systems for G = Zn (n ≤ 4) and Z2× Z2, and the first one for Z5 were

constructed in Section 5 and Appendix A of [19]; the rest are to our knowledge new.

The column Q′will be relevant to subsection 4.4, where it will be explained.

In the table, we write ξk for exp(2πı/k). If ?,? is a nondegenerate symmetric

30

Page 31

pairing for a cyclic group G = Zn, then it equals ?g,h? = exp(2πımgh/n) for some

integer m coprime to n. When G = Zn′ ×Zn′′ in Table 2, any pairing appearing there

is of the form ?(g′,g′′),(h′,h′′)? = exp(2πıg′h′/n′) exp(2πımg′′h′′/n′′) for some m ∈ Z

coprime to n′′. This m is the entry appearing in the ?,? column.

Note that if aiare both solutions of (3.2) for fixed group G and pairing ?,?, then

a2= ψ a1for some ψ ∈?G with ψ2= 1. Thus for G of odd order, the unique a is

given by m as above, then a(g) = sg

for some s1,s2 ∈ {±1}. These signs s1 or (s1,s2) grace the fourth column of the

Table.

For the group G = Zn, the quantities b(g) are recovered through Table 2 from

the formula b(g) = eıjg/√n, for 0 < g ≤ n/2, b(0) = −1/δ, and b(−g) =

a(g)b(g). For the noncyclic group Zn′ × Zn′′, these parameters j are taken in or-

der (1,0),...,(⌊n′/2⌋,0),(0,1),(1,1),...,(⌊n′/2⌋,⌊n′′/2⌋).

The table lists representatives of equivalence classes of systems. Using Corollary 5,

it is easy to determine when numerical invariants determine equivalent systems. Note

that taking the complex conjugate of numerical invariants (i.e. the conjugate of c,

and the negatives of the ?,? and jicolumns) will yield another (possibly equivalent)

solution to the equations of Corollary 5. For example, consider the first entry for

G = Z5: although the complex conjugate solution has different j’s, it is equivalent

as the j’s are permuted back to each other through the Z5automorphism −1 ∈ Z×

On the other hand, complex conjugation interchanges the second and third systems

for Z5; these two are inequivalent because they have different c’s.

Apart from G = Z2, the solutions in the Table turn out to be precisely the

solutions to the linear equations (3.3) together with |b(g)| = 1/√n for g ?= 1 (which

is a consequence of the equations of Corollary 5). These values for jg are floating

point; to improve their accuracy arbitrarily is trivial using mathematics packages like

Maple (where you would change ‘Digits’ to say 200, and use ‘fsolve’ with the provided

seed values). The b(g)’s are in fact all algebraic, but providing the exact algebraic

expressions (though possible) would not be very enlightening.

We illustrate our method of establishing Proposition 6 and Table 2, with the

hardest case, namely n = 13. Up to automorphism of Z13, there are two possible

pairings: ?g,h? = ξmgh

solutions for m = 2). By (3.2) there are 3 possible values of c; choose c = −1 for

now. To find a complete list of candidate solutions, first solve the linear equations

(3.3) (breaking each b(g) into its real and imaginary parts). This determines each of

the 26 variables Reb(g), Imb(g), up to 4 real parameters, which we can take to be the

real and imaginary parts of b(11) and b(12). The norms |b(1)|2= ··· = |b(4)|2= 1/n

yield four independent quadratic identities obeyed by those parameters; by Bezout’s

Theorem they can have at most 2k(complex) solutions, and as always here this upper

bound is realised (i.e. no zeros have multiplicities). We ‘solved’ these equations using

a(g) = ?g,g?(n−1)/2. For G = Znor G = Zn′ × Zn′′, for n,n′,n′′even, with pairing

1exp(−πmg2/n) or a(g′,g′′) = sg′

1sg′′

2 respectively,

5.

13

for m = 1 or 2; choose m = 1 (this will also yield the

31

Page 32

floating point: our approximate solutions are

(Reb(12), Imb(12), Reb(11), Imb(11)) ∈ {(−.0277709, .9996143, .7988155, −.601576), (−.022834, −.999739,.585511, −.810664),

(.1154793, .9933099, .1774122, .9841366), (−.2122594, .9772133, −.0413084, .9991464), (−.3498383, .9368100, .9107102, −.4130459),

(.623311, .7819739, .7557187, −.6548963), (−.7923374, .6100831, −.1096719, .9939678), (.8581882, .5133351, .8230236, −.5680071),

(−.9560999, .2930407, .00007002229, .9999999), (.976436, −.2158069, .6983522, −.7157542), (.9014165, −.4329529, .5249502, −.8511329),

(−.8964165, −.4432124, −.1923449, .9813274), (−.7716105, −.6360951, −.402263,.9155241), (.2614237, −.9652241, −.3345549, .9423762),

(.1595449, −.9871906, −.5115023, .8592819), (.08134136, −.9966863, .4166086, −.9090859)}.

Of these, four also satisfy the remaining norms |b(5)|2= |b(6)|2= 1, and indeed all

equations (3.4). Two of these are related to the other two by the automorphism −1 ∈

Z×

13; the resulting two inequivalent solutions are collected in Table 2. Incidentally,

had we chosen either of the two remaining possibilities for c, we would have had 3

parameters from the linear system, 8 solutions to 3 norm equations, but none of these

8 would be solutions to all of the remaining 3 independent norm equations; thus those

other values of c don’t yield solutions (floating point calculations suffice, thanks to

Bezout).

We still need to verify that each of these are indeed solutions, i.e. that they satisfy

(3.2)-(3.4) exactly. For this purpose, define√13b(1)(g) to be the roots of

P(X) = X12+13 +√13 −√17 −√221

?

4

?

2

X11+53 + 8√13 − 13√17 − 7√221

− 416√17 −253√221

2

?

2

X10+637 + 67√13 − 59√17 − 43√221

+15935√13

2934

+1057407√13

455377401

?

+1057407√13

455377401

?

2929

X3+53 + 8√13 − 13√17 − 7√221

2

2

X9

+

7531

+1905√13

4

+

t

628

?

57473

29

−16341√17

−15721329√17

−4531√221

−4359784√221

77401

34

??

??

?

??

??

X8+

169

8

−16809 − 2587√13 + 2249√17 + 1127√221 + t

3812952

4553

X7

+1

2

−25283 − 7179√13 + 6279√17 + 1701√221

−15721329√17

+15935√13

X6+

169

8

?

−16809 − 2587√13 + 2249√17 + 1127√221 − t

?

4

+637 + 67√13 − 59√17 − 43√221

?

cients) of P(X) is clearly K = Q[√13,√17,t], which has Galois group (over Q) D4

generated by t ?→ sı

splitting field F of P(X) is a quadratic extension of Q[ξ13,√17,t], with Galois group

Z12over K given by b(j) ?→ b(ℓj), ξ13?→ ξℓ2

to prove the products of roots of P(X) include 13th roots of unity, and hence that

13b(g)b(−g) = a(g); the first prime p > 17 with√13,√17,

p = 101, and modulo 101

?

3812952

4553

−4359784√221

77401

−4531√221

X5

+

7531

+1905√13

4

− 416√17 −253√221

2

−

t

628

57473

−16341√17

X2+13 +√13 −√17 −√221

3434

X4

22

X + 1

for t = ı

sarily have modulus 1. The base field (i.e. the one generated over Q by the coeffi-

75090 + 2√13. The roots√13b(g) are thus algebraic integers, and neces-

?

75090 + s′2√13 and

√17 ?→ s′′√17, for all signs s,s′,s′′. The

13, for any ℓ ∈ Z×

13∼= Z12. (Use resultants

?

−75090 − 2√13 ∈ Zpis

P(X) ≡ 19+74X+55X2+62X3+74X4+77X5+39X6+86X7+8X8+47X9+73X10+61X11+X12

is irreducible in Z101[X], hence Gal(F/K) contains an element of order 12; etc.)

The 7 nontrivial automorphisms in Gal(K/Q), lifted to F, map b(g) to the three

32

Page 33

other solutions for n = 13 in Table 2, together with 4 analogous ‘shadow’ solutions

corresponding to δ′= (13 −√221)/2, which can also be estimated numerically. To

show that b(g) (as well as the other 3 candidates in Table 2 for n = 13) indeed satisfy

the remaining identities in (3.2)-(3.4), it suffices to replace each b(g) with 13/b(g),

multiply by δ and an appropriate power of√n to guarantee that the equations are

manifestly algebraic integers, and then evaluate the equations numerically for all 8

choices of b (the 4 from Table 2 and the 4 shadows). We used 200 digits of accuracy —

far more than necessary but trivial using Maple — and found that the equations held

to accuracy 10−190or so. The errors will therefore be algebraic integers, and from the

above we know that all of their Galois associates will have modulus << 1. This means

the errors must vanish identically, and we are done. (Incidentally, the polynomial

P(X) was found working backwards: the numerical analysis of the previous paragraph

suggested its existence and basic properties.)

The 1 in Proposition 6 for G = Z1corresponds to (i.e. is also implied by) the

uniqueness of the A3 subfactor, and the 2 systems for G = Z2 correspond to the

two versions of the E6subfactor. Note that our classification for uniqueness (up to

complex conjugation) for G = Z3corresponds to the uniqueness for even sectors of

the Izumi-Xu 2221 subfactor. The uniqueness of the Izumi-Xu subfactor was first

shown in the thesis of Han [14]. His proof is independent of ours: it involved planar

algebras, and was quite complicated.

Note the numerology nδHn= δIXn2, where (n +√n2+ 4)/2 is the dimension of

the nongrouplike simple objects in the Haagerup system for Zn[19], and δIXn2 is the

dimension of ρ for near-group C∗-categories of type Zn2+n2. This suggests comparing

the subfactor ρHn(M) ⊂ M with the subfactor ρIXn2(M) ⊂ MZn2, as they have the

same index, namely δ2

Lemma 3.10 of [12], while the principal graph of the latter is the completely different

291, so the connection (if indeed there is one) is not simply this.

Hn. For n = 3, the principal graph of the former is Figure (4’) in

4Tube algebras and modular data

4.1The tube algebras of near-group systems

In this section we compute the tube algebras, for any solution to the equations of

Proposition 1. Our notation will be as in [18]. We can assume F ?= ∅, i.e. n′?= 0, as

the tube algebra for the Tambara-Yamagami systems was computed in Section 3 of

[19].

Let ∆ = {αg,ρ}g∈Gbe a finite system of endomorphisms, as in Section 2. The

tube algebra Tube ∆ is a finite-dimensional C∗-algebra, defined as a vector space by

Tube∆ = ⊕ξ,η,ζ∈∆Hom(ξ · ζ,ζ · η).(4.1)

Given an element X of Tube∆, we write (ξζ|X|ζη) for the restriction to

Hom(ξζ|X|ζη), since the same operator may belong to two distinct intertwiner spaces.

For readability we will often write g for αg. In our case the intertwiner spaces are

33

Page 34

Hom(αg+h,αgαh) = C1, Hom(ρ,αgρ) = C1, Hom(ρ,ραg) = CUg, Hom(αg,ρ2) =

CSg and Hom(ρ,ρ2) = spanz∈F{Tz}.

Bg,h = (gρ|Uh|ρh), Cg,z = (gρ|Tz|ρρ), Dg,z = (ρρ|UgT∗

E′

tube algebra is:

?

where Ag,g= CBg,g⊕ spank∈GAg,k, Ag,h= CBg,h(for g ?= h), Ag,ρ= spanz∈FCg,z,

Aρ,g= spanzDg,z, Aρ,ρ= spankEk⊕ spankE′

The C∗-algebra structure of Tube ∆ is as follows: multiplication is given by

?

where ρρ= ρ and ρg= αg, and: when ζ = g and ζ′= h, then the unique ν is g + h

and the unique T(ν,i) is 1; when ζ = g and ζ′= ρ, the unique ν is ρ and the unique

T(ν,i) is 1; when ζ = ρ and ζ′= g, the unique ν = ρ and the unique T(ν,i) is Ug;

and when ζ = ζ′= ρ, then ν runs over all g ∈ G, with T(g,i) = Sg, as well as ν = ρ,

with T(g,i) running over all Tz. Moreover, the adjoint is

Denote the elements Ag,h = (gh|1|hg),

z|ρg), Ek = (ρk|U∗

z|ρρ). Then the vector space structure of the

k|kρ),

k= (ρρ|SkS∗

k|ρρ), E′′

wz= (ρρ|TwT∗

Tube∆ =

g,h

Ag,h⊕

?

g

Ag,ρ⊕

?

g

Aρ,g⊕ Aρ,ρ, (4.2)

k⊕ spanw,zE′′

w,z.

(ξζ|X|ζη)(ξ′ζ′|Y |ζ′η′) = δη,ξ′

ν≺ζζ′

?

i

(ξν|T(ν,i)∗ρζ(Y )Xρξ(T(ν,i)|νη′)), (4.3)

(ξζ|X|ζη)∗= dζ(ηζ|ρζ(ρξ(R

∗

ζ)X∗)Rζ|ζξ),(4.4)

where dg= 1, dρ= δ, Rg= 1 = Rg, and Rρ= S0, Rρ= sS0.

Let σ be a finite sum of sectors in ∆. A half-braiding for σ is a choice of unitary op-

erator Eσ(ξ) ∈ Hom(σρξ,ρξσ) for each ξ ∈ ∆, such that for every X ∈ Hom(ρζ,ρξρη),

XEσ(ζ) = ρξ(Eσ(η))Eσ(ξ)σ(X).

For our systems this reduces to

(4.5)

Eσ(g + h) = αg(Eσ(h))Eσ(g),

Eσ(ρ) = αg(Eσ(ρ))Eσ(g),

UgEσ(ρ) = ρ(Eσ(g))Eσ(ρ)σ(Ug),

SgEσ(g) = ρ(Eσ(ρ))Eσ(ρ)σ(Sg),

TzEσ(ρ) = ρ(Eσ(ρ))Eσ(ρ)σ(Tz),

(4.6)

(4.7)

(4.8)

(4.9)

(4.10)

for all g,h ∈ G, z ∈ F. There may be more than 1 half-braiding associated to a

given σ; in that case we denote them by Ej

half-braiding is equivalent to knowing the matrix units of the corresponding simple

summand of Tube∆ (the matrix units ei,jof a matrix algebra isomorphic to Mk×k

are a basis satisfying ei,jem,l = δj,mei,l — e.g. the standard basis of Mk×k). If we

decompose σ =?

k =?kg+ kρ.

34

σ. As we will see shortly, knowing the

gkgαg+ kρρ into a sum of irreducibles, then each half-braiding Ej

will correspond to a distinct matrix subalgebra of Tube∆ isomorphic to Mk×k, where

σ

Page 35

The dual principal graph for the Longo-Rehren inclusion of ∆ can be read off from

the collection of half-braidings as follows. On the bottom are the simple sectors of ∆;

on the top row are the (inequivalent) half-braidings Ej

connect Ej

is called alpha-induction and plays a central role in much of the theory. See Figure 4

for an example, which follows from the tube algebra analysis of Subsection 4.2. (The

principal graph for the Longo-Rehren inclusion is much simpler: just ∆ × ∆ on the

bottom and ∆ on the top, with edge multiplicities given by fusion multiplicities.)

σ. If we write σ =?kgαg+kρρ,

σto αgwith kgedges, and to ρ with kρedges. This forgetful map Ej

σ?→ σ

Figure 4. The dual principal graph for the double of ∆(Z3+ 2)

The point is that the centre of the tube algebra is nondegenerately braided. A

nondegenerately braided system comes with modular data:

Definition 2. Unitary matrices S = (Sa,b)a,b∈Φ,T = (Ta,b)a,b∈Φare called modular

data if S is symmetric, T is diagonal and of finite order, the assignment?0

01

that S2

1

−1

0

??→ S,

?11

??→ T generates a representation of SL2(Z), and there is some index 0 ∈ Φ such

?

are nonnegative integers.

a,0> 0 ∀a ∈ Φ and the quantities

Na,b,c:=

d∈Φ

Sa,dSb,dSc,d

S0,d

(4.11)

The a ∈ Φ are called primaries; 0 ∈ Φ is called the identity; the Na,b,c are

called fusion coefficients and (4.11) is called Verlinde’s formula. S2will necessarily

be a permutation matrix, called charge-conjugation C. In the case of modular data

associated to nondegenerately braided fusion categories, the primaries Φ label the

simple objects, the quantities Nc

ring. Nondegenerately braided systems of endomorphisms always satisfy the stronger

inequality Sa,0> 0, so we will assume this for now. The quantum-dimension Sa,0/S0,0

in this case is the statistical dimension da=

the global dimension. When Sa,0= S0,0then a has an inverse in Φ and is called a

simple-current.

a,b:= Na,b,Ccare the structure constants of the fusion

?[M : a(M)] and 1/S0,0=??

ad2

ais

35

Page 36

In the case of the centre of Tube∆, the primaries are in one-to-one correspon-

dence with the simple summands of Tube∆, or equivalently with the half-braidings

defined above. These matrices can be computed once we know the matrix entries

Ej

simple objects (with multiplicities) in σ. In fact, the diagonal entries (η′,α′) = (η,α)

suffice to determine S,T. These matrix entries can be computed from either the

half-braidings, or from the diagonal matrix units e(σj)i,i, as follows. Let Wσ(η,α) be

an orthonormal basis of Hom(ρη,σ); then we have

σ(ξ)(η,α),(η′,α′)∈ Hom(ρη· ρξ,ρξ· ρη′) for each irreducible ξ ∈ ∆, as η,η′run over all

Ej

σ(ξ)(η,α),(η′,α′)= ρξ(Wσ(η′,α′)∗)Ej

e(σj)(η,α),(η′,α′)=

λ?dηdη′

σ(ξ)Wσ(η,α), (4.12)

dσ

?

ξ

dξ(ηξ|Ej

σ(ξ)(η,α),(η′,α′)|ξη′),(4.13)

where

λ = n + δ2= 2n + n′δ(4.14)

is the global dimension. The entries of the diagonal unitary matrix T and symmetric

unitary matrix S are determined from the matrix entries Ej

Tσj,σj = dξφξ(Ej

Sσj,σ′j′ =dσ

λ

(ξ,α)

σ(ξ)(η,α),(η,α)through:

σ(ξ)(ξ,α),(ξ,α)),(4.15)

?

dξφξ(Ej′

σ′(η)∗

(ξ,α),(ξ,α)Ej

σ(ξ)∗

(η,α′),(η,α′)),(4.16)

where φξ is the standard left inverse of the endomorphism ρξ, defined by φξ(x) =

R∗

all ξ (counting multiplicities) in σ′while η is any (fixed) irreducible in σ.

ρξρξ(x)Rρξ. In (4.15), ξ can be any irreducible in σ, and in (4.16) the sum is over

4.2The first class near-group C∗-categories: n′= n − 1

As before, we will sometimes write ax,bx,b′′

show that, as a C∗-algebra,

x,yfor a(x),b(x),b′′(x,y). We will first

Tube∆∼= C2n+1⊕ Mn×n⊕ (M2×2)n2−n,(4.17)

unless s = −1 and n = 7, in which case

Tube∆∼= C7⊕ M7×7⊕ (M2×2)44.(4.18)

It will prove useful to know the formulae

ρUg=

?

h

ShS∗

g+h+

?

w

σ2w(g)TwUgT∗

w,(4.19)

b′′(x,y) = axayaxyb(σ2xσy)b′′(y,x),

b′′(σ2(−σx),σ(−σx)) = b′′(x,σ2x).

(4.20)

(4.21)

36

Page 37

(4.19) is implicit in Section 2.3. (4.20) follows from the sequence (2.37),(2.39),(2.37).

(4.21) is trivial when n is odd (where −1 = 1), so it suffices to take s = a = b = 1;

then apply the sequence (2.37),(2.39),(2.38) to the left-side. Note also that we always

have a−x= axand b−x= bx(when both sides are defined).

Theorem 5. Consider any type G + n − 1 C∗-category, i.e. let s,a,b,b′′be any

solution of the equations of Theorem 3.

(a) There is precisely one half-braiding Eαgfor any g ∈ G: Eαg(h) = 1 and Eαg(ρ) =

(−1)n′gUg. We get the diagonal matrix entries Eg(h)g,g = 1, Eg(ρ)g,g = (−1)n′gUg.

These half-braidings correspond to central projections

πg= n−1(n + 1)−1?

(b) There is precisely one half-braiding E?

g:= (n + 1)−1?

The corresponding matrix entries are EΣ(g)h,h= 1, EΣ(ρ)g,g= n−1(−1)gn′+1Ug.

(c) There are precisely n − 1 half-braidings Ew

2 matrix algebra A(g,w) = Span{pg,w,Cg,w,Dg,σw,Dg,σwCg,w}. The corresponding

matrix entries are Ew

h

Ag,h+ (n + 1)−1(−1)gn′Bg,g.(4.22)

gαg. It corresponds to the n × n matrix

algebra A(?α), spanned by Bg,h(g ?= h) and

π′

h

Ag,h+ (n + 1)−1(−1)gn′+1Bg,g

∀g .(4.23)

ρ+αgfor any g ∈ G, one for each 2 ×

ρ+g(h)g,g= w(h), Ew

ρ+g(h)ρ,ρ= (−1)n′hw(g + h)U∗

?

σw,xσ2w σ2(wx)σ2(wx)(g)σ(wσ(wx))(g)TxT∗

h, and

Ew

ρ+g(ρ)ρ,ρ=

h

(−1)n′hw(h)ShS∗

h

+n−2awbw

?

x

b′′

σ2w σ2(wx),xb′′

−wxσ2(wx).

(d) When s = ω = 1 or n ≤ 3, there are precisely n + 1 half-braidings Eψ

rally parametrised by the characters ψ ∈?

Eψ

k

ρ, natu-

F+

n+1. The matrix entries are Eψ

ρ(g)ρ,ρ =

(−1)gn′U∗

gand

ρ(ρ)ρ,ρ= ζ1ψ(1)

?

(−1)kn′SkS∗

k+

?

x

ζxψ(σx)TxT∗

σ2x,(4.24)

where ζ1,ζx is any particular solution to the equations (4.41) (when b′′and b are

identically 1, take ζ identically 1).

(e) When s = −1 and n > 3, then n = 7 and there are precisely 2 half-braidings for

σ = 2ρ, with matrix entries Es1

Es1

g

2ρ(g)(ρ,s2),(ρ,s2)= U∗

?

gand

2ρ(ρ)(ρ,s2),(ρ,s2)= ıs1

SgS∗

g+ ıs2T1T∗

5+ s1s2ıT6T∗

4,(4.25)

where s1,s2∈ {±1}.

37

Page 38

(f) There are no C∗-categories of type G + n − 1 with ω ?= 1 and n > 2.

Proof. From (4.3) we obtain the formulae

Ag,hAk,l= δg,kAg,h+l, Ag,hBk,l= δg,kBk,l Bg,hBk,l= δh,kδg,l

?

m

Ag,m+ δh,kcghlBg,l,

Ag,hCk,z= δg,kz(h)Ck,z, Bg,hCk,z= 0, Cg,wDk,z= δg,kδσw,zazbz

?

(−1)n′hw(h)E′

h

w(h)Ag,h,

Dg,zCk,w= δg,kδz,σwn−1awbw

?

h

(−1)n′hw(h + g)Eh+ δg,kδz,σwb−2

w

?

h

h

+δg,kδz,σw

?

x

b′′

σ2w σ2(wx),xb′′

σw,xσ2w σ2(wx)σ2(wx)(g)σ(wσ(wx))(g)E′′

x,−wxσ2(wx),

Cg,zEk= z(k)(−1)n′kCg,z, Cg,zE′

Cg,zE′′

k= s(−1)n′kz(k − g)Cg,z,

w,x= Cg,zδx,σ(zw)x(g)σ2w(g)b′′(xσ2w,z)b′′(z,xσ2w),

where cghl:=?

e′′

zz(g)σz(h)σ2z(l). Using (−1)n′h= z(h)σz(h)σ2z(h), we get cghh=

chgh = chhg = (−1)n′h(nδg,h− 1). Write e =

x= E′′

?

h(−1)n′hEh, e′=

?

h(−1)n′hE′

h,

x,σ2x. From (4.4) we obtain

A∗

g,h= Ag,−h, B∗

2, e′′

z

g,h= Bh,g, C∗

∗= δz,2e′+ azbzb′′(σz,σ2(−σz))e′′

g,z= sz(g)azbzDg,σz,

e∗= e, e′∗= sδ−1,1e′+ se′′

σ2(−σz).

We use in these expressions that σ(2) = −1, where 2 denotes the element 1+1 in the

corresponding field Fn+1; when the characteristic of Fn+1is 2, then neither −1 = 1

nor 2 = 0 lie in F, so the corresponding terms should be ignored.

Let’s begin by solving the half-braiding equations for (a).

nience, we’ll solve this for any near-group C∗-category. Since Hom(αg+h,αg+h) and

Hom(αgρ,ραg) are both one-dimensional, we need to find numbers cg,h,c′

each g,h ∈ G, such that Eαg(h) = cg,hand Eαg(ρ) = c′

and c′

(4.10) becomes

−sδ−3/2?

¨ z(g)c′−1

g

c′−1

g

?g,g?¨ z(g)z(g) ¨ x(g)δw,gzδy,gx= Vwxyz(g),

where W,V are defined in the proof of Proposition 1. It is easy to see from Theorems

3 and 4 that (4.26) is automatically satisfied for both classes.

When n′= n−1 this condition becomes c′

z. Using the finite field expression σx = (1−x)−1of Proposition 2, z σz σ2z collapses

to −1 ∈ Fq, so this expression for c′

n is odd. In this case, (4.9) is automatically satisfied.

For later conve-

g∈ C for

gUg. (4.7) and (4.10) give cg,h

g, respectively. The remaining three equations are then automatically satisfied.

h

?g + k,h?

¨

(−hy)(h)ayδ−hy,z′ =

?

z,x

¨ z(g)x(k)bz,xb′′

gz,z′,y,x,(4.26)

z(g)?g,h − g?δx,gz= Wxhz(g),(4.27)

(4.28)

g= z(g)σz(g)σ2z(g) is independent of

gsimplifies to c′

g= (−1)gif n is even, or c′

g= 1 if

38

Page 39

Now turn to the proof of (b). We see from (4.3) or the products collected at

the beginning of this proof, that A(?α) is an n2-dimensional C∗-algebra, so is a

δg,hBh,k, so the projections π′

an n-dimensional space in A(?α) satisfying π′

To show the π′

of Tube∆, it suffices to show that π′

and all g ?= h. This is clear from the products listed earlier.

The C∗-algebras Ag,gare each isomorphic to Cn+1, with projections πzof Theorem

5(a), π′

pg,z= n−1?

Together with the n × n matrix algebra of Theorem 5(b) and the projections πg

of Theorem 5(a), these projections pg,z span all of?

Now turn to (c).Each A(g,w) is clearly a 4-dimensional C∗-algebra, and is

noncommutative since Cg,wDg,σw ∈

Therefore each A(g,w) is isomorphic as a C∗-algebra to the 2 × 2 matrix alge-

bra. Each A(g,w) is readily seen to be orthogonal to πg and A(?α). Note that

∪g,w{Cg,w,Dg,σw,Cg,wDg,σw,Dg,σwCg,w} ∪ {e,e′,e′′

maximal as a simple C∗-subalgebra of Tube∆, it suffices to verify that Cg,wDg′,w′ =

0 = Dg′,w′Cg,wunless g = g′and σw = w′, Cg,we = Cg,we′= Cg,we′′

orthogonalities are either trivial or follow from these). This is elementary.

The diagonal matrix units are pg,wand n−1awbwDg,σwCg,w. From this we obtain

the desired quantities.

Now, turn to (d). We compute

direct sum of matrix algebras. The π′

gobey π′

g

∗= π′

g, π′

gπ′

h= δghπ′

g, and π′

gBh,k=

gare the diagonal matrix units. For any g, there is

gx = x, namely the span of π′

gmust belong to an n × n block and hence A(?α) is Mn×n(C).

gπk= Bg,hπk= π′

g,Bg,h

(h ?= g), so π′

gare minimal, i.e. A(?α) is maximal as a simple C∗-subalgebra

gCk,z= Bg,hCk,z= 0 for all k,z

gof Theorem 5(b), and for all z ∈?G∗,

h

z(h)Ag,h. (4.29)

g,hAg,h. Note that pg,zπk =

pg,zπ′

k= 0 and p∗

g,z= pg,z.

?

hAg,h and Dg,σwCg,w ∈ Aρ,ρ are distinct.

Cg,wDg,σw is a scalar multiple of pg,w.A basis for

?

gAg,ρ+?

gAρ,g+ Aρ,ρ is

x}x∈? G∗.To verify that A(g,w) is

x= 0 (the other

e2= ne, ee′= ne′= e′e, ee′′

z= ne′′

z= e′′

ze, e′e′= n−1sδ−1,1e + a2b2e′′

1/2,

e′e′′

z= n−1δz,2e + bzb−σzb′′(σ(−σz),σ2(−σz))e′′

e′′

σ(−σz),

σ(−σz),

ze′= n−1δz,2e + b′′(σ2z,z)e′′

z,yb′′(σ2z,y)e′′

e′′

ze′′

w= n−1sbwazb′′

z,σ2zδσw,−σze + awδz,wbze′+ b′′

wequation x,x′,y are defined by σx = −σw σz, σx′= xσz, and

σy = σ2z σw. Manifestly, the span of e,e′and all e′′

algebra, which we’ll call A(ρ). It is immediate now that it is orthogonal to πg,A(?α)

bxb−σxb′′(σ(−σx),σ2(−σx)) = b′′(σ2x,x)

sbwazb′′

x,x′b′′

σ(−σw σz σ(wz)),

where in the e′′

ze′′

xis an n + 1-dimensional C∗-

and all A(g,w). Let’s identify when it is abelian. First note that

for x ?= 2,(4.30)

z,σ2z= sbzawb′′

w,σ2w

for σz = −σw ,

for z = w.

(4.31)

awbz= azbw

(4.32)

39

Page 40

Indeed, (4.30) follows by applying (4.20) twice and (4.21) once; any by appearing

in these expressions can be replaced with say. To see (4.31), use (4.21) and the

substitution −σz = σw; again, b = sa here. (4.32) is trivial.

Thus to conclude the argument that A(ρ) is commutative, it suffices to ver-

ify that b′′(x,x′)b′′(z,y′)b′′(σ2z,y′) is invariant under the switch z ↔ w.

(2.37),(2.40),(2.38) and (4.20), we obtain

Using

b′′

x,x′ = b′′(w,σzσ2w)axb2

xbyzaσ2w σzbσ2w σz

2a(σz σw)ab(σ2wσ(σ2wσz)), (4.33)

b′′

z,y= b′′(σ2z,σw)ab(z)bwbyzb(σ2wσz)b(σ2z σw), (4.34)

b′′(σ2z,y) = b′′(z,σ2z σw)bzbx′ bxb(σ2z σw), (4.35)

where we write ab(z) for a(z)b(z) etc. We likewise have

b′′(σ2w,σz) = b′′(σ2z,σw)sab(z)ab(w)ab(σ2z σw)b(σ2wσz)

2. (4.36)

Since x,x′,y depend on w,z, they will be affected by the switch w ↔ z: in particular

we find x becomes σ2x, y becomes σ2wσz, and x′is unchanged.

commutative iff

Thus A(ρ) is

ab(σ2wσz)ab(σ2wσ(σ2wσz)) =

ab(σ2z σw)ab(σ2z σ(σ2z σw))ab(σ2z σw)b(σ2wσz)

2. (4.37)

Consider first the generic case, where ω = 1. We find that, provided σw ?= −σz

and z ?= w, e′′

or n ≤ 3.

Much more subtle is when ω is a primitive third root of unity.

−1 = σ(−1) and s = 1; ax= bx= 1 except for b−1= ω. Then A(ρ) is commutative

iff

b(σ2wσz)b(σ2wσ(σ2wσz)) = b(σ2z σw)b(σ2z σ(σ2z σz))

ze′′

w= se′′

we′′

z. This means A(ρ) will be commutative provided s = ω = 1

In this case

(4.38)

for all w,z with σw ?= −σz and z ?= w. The only possible way this equation can be

violated is if at least one of those b’s doesn’t equal 1.

Suppose first σ2z σ(σ2z σw) = −1. Then hitting both sides with σ, we get −σz =

x. This implies σx′= −1, i.e. x′= −1, and hence σw = −σ2x. Therefore

− 1 = σ(−1) = σ(σwσ2x) = σ2wσ(σxw) = σ2wσ(−σz σww)

and thus σ2wσ(σ2w σz) = −1. Thus in (4.38), b(σ2wσ(σ2wσz)) and b(σ2z σ(σ2z σz))

are always equal.

Finally, suppose σ2z σw = −1. Then σ2wσz = x cannot equal −1. This means

that for any pair x,z with σ2z σw = −1 and σz ?= −σx and z ?= x, e′′

Consider now the case where A(ρ) is commutative, i.e. where either s = ω = 1,

or n < 7. Then the n + 1 minimal central projections in Aρ,ρare scalar multiples of

π(?ζ) = (n2+ n)−1e + (n + 1)−1ζ1e′+ (n + 1)−1?

40

(4.39)

ze′′

x= ωe′′

xe′′

z.

x

ζxe′′

x

(4.40)

Page 41

for some ζ1,ζx∈ C×. These must satisfy e′π(?ζ) = βe′π(?ζ) and e′′

scalars βe′,βx∈ C. This yields the equations

ζ2= ζ1, ζx= saxbxζ1ζx, ζσ(−σx)= bxb−σxζ1ζxb′′

ζσ(−σz σxσ(xz))= ζxζzb′′

xπ(?ζ) = βxπ(?ζ) for

x,σ2x

(z ?= x,σ2(−σx)),

(x ?= 2),

x,x′ b′′

z,yb′′

σ2z,y

(4.41)

as well as ζ2

First, note we can solve these equations in the special case that b′′and b are

identically 1. In this case, identify G with F×

for any of the n + 1 characters of the additive group F+

shows these n + 1 choices of ζ’s all work, and so must exhaust all solutions.

Recall that the C∗-categories with n′= n−1 form a group: C(b1,b′′

C(b1b2,b′′

ζ(1)xζ(2)x will be a solution for C(b1b2,b′′

particular solution?ζ for a given C(b,b′′), all other solutions for that category C(b,b′′)

are obtained by multiplying that particular solution by the solutions ψ(σx) for C(1,1).

Since the sum of the minimal projections of A(ρ) must equal the unit n−1e, we

know now that the π(?ζ) given in (4.40) are indeed the minimal projections (i.e. the

coefficient λ−1for e is correct).

When s = −1, both e,e′are central elements.

dimensional. Then there is some Z =?

0,1,x,x. Then e′′

that of e′′

that A(ρ) will indeed be a sum of two matrix subalgebras. Note that βe + γe′=

(βe + γe′)2= β2ne + 2βγne′− γ2n−1e forces β = (2n)−1and γ = ±ı/2. Thus the

identities in the two matrix subalgebras are 1±:= (2n)−1e±ıe′/2, and the two matrix

subalgebras will be the images e±(A(ρ)) =: A(ρ)±, and are of equal dimension.

Let’s finish off the proof of (e). The C∗-algebra A(ρ)+ is spanned by 1+ and

z+:= (γze′′

choose these square-roots so that γz= ıγz, so z+= z+). We compute z2

each z+is invertible. Provided w ?∈ {z,z}, we have

1= s when n + 1 is even.

n+1and take ζ1= ψ(1) and ζx= ψ(σx)

n+1∼= (Zp)k: a little effort

2)∗C(b2,b′′

2) =

1b′′

2). Let π(?ζ(i)) be solutions of (4.41) for b′′

irespectively; then ζx =

2). This implies that if you have any

1b′′

Suppose the centre is not 2-

x?= 0 which commutes with all e′′

xcxe′′

w.

Suppose cx?= 0; then as long as n > 3 it will be possible to choose a z not equal to

zZ and Ze′′

σ(σ2xσ2z)σ(xz)), which will be nonzero because cxis. This contradiction means

zwill differ by a sign in at least one coefficient (namely

z− ıγze′′

z)/2 where γz=?b′′(σz,z) (since b′′(σz,z) = −b′′(σ2z,z), we can

+= 1+, so

z+w+= βz,w

?wz + 1

w + z

?

+

(4.42)

for some βz,w∈ C, where we use addition in the finite field Fn+1to simplify notation.

But provided w ?∈ {z,z}, we know e′′

choose some x ?∈ {z,z,w,w,(wz + 1)/(w + z),(w + z)/(wz + 1)} — this is possible

as long as n > 7. Then

ze′′

w= −e′′

we′′

zand hence z+w+= −w+z+. Now

− (z+w+)x+= x+(z+w+) = −z+x+w+= z+w+x+,

so z+x+w+ = 0, which contradicts invertibility of z+,w+,x+. This contradiction

shows that s = −1 requires n ≤ 7.

(4.43)

41

Page 42

When ω ?= 1, say ω = e±2πı/3, we see that every z ?= −1 will have precisely one x

such that σ2z σx = −1. This means e′′

using finite field notation for addition as usual. Note that e′′

e′′

in the s = −1 argument, we see that the centre cannot be more than 3-dimensional.

Therefore A(ρ) here must be a sum of 3 matrix subalgebras. We compute the corre-

sponding identities as before, obtaining 1ζ= (3n)−1e+e∓2πı/9ζe′/3+e∓2πı/9ζ2e′′

each third root ζ of unity. Thus A(ρ) will be a sum of the matrix subalgebras A(ρ)ζ:=

1ζA(ρ) = span{1ζ,1ζe′′

see that 1ζe′′

commutative. Since they are also simple, each must be C, and we have that n+1 = 3.

This concludes the proof of (f).QED to Theorem 5

ze′′

x= e′′

xe′′

zunless x ∈ {−1/(1+z),−(1+1/z)},

−1/(1+z)∈ Ce′e′′

zand

−(1+1/z)∈ Ce′′

−1ez. The centre of A(ρ) manifestly contains e,e′,e′′

−1, and exactly as

−1/3 for

z}. Now, 1ζe′′

xwill always commute, and thus each subalgebra A(ρ)ζwill be

x∈ C1ζe′′

ziff x ∈ {z,−1/(1+z),−(1+1/z)}, so we

zand 1ζe′′

We already knew (from Proposition 5) that s = 1 for n ?= 1,3,7 and ω = 1 for

n ?= 2, but we wanted to derive it directly to demonstrate the effectiveness of the

structure of the tube algebra in constraining the sets of solutions. Let us now give

particular solutions for ζ in the known cases where a,b,b′′are not all identically 1.

When n = 1 and s = 1, there are no parameters ζ. When n = 2 and ω = e±2πı/3,

take ζ1= e∓2πı/9= ζ−1. When n = 3 and s = −1, take ζ1= ζω2 = ı and ζω= 1,

where we identify F4with Z2[ω] for ω3= 1 and identify G with F×

Corollary 6. Fix any finite field F = Fqfor q = n+1, and identify G with F×. Here

is the modular data of the double of any system covered by Theorem 3. The global

dimension is λ = n2+n. The primaries come in 4 families, parametrised as follows:

4.

• g ∈ G;

• the symbol Σ;

• w + h for w ∈?G∗and h ∈ G;

s = −1 and n = 7).

Then the T and S matrices are given in block form by

• either ρψfor ψ ∈?

Fn+1(when s = ω = 1 or n ≤ 3), or ρs1for s1∈ {±} (when

T = diag

?

1;1;w(h);ζ1ψ(1)

?

, (4.44)

S =

1

n + 1

n−1

1

1

n

0

w(g)(n + 1)n−1

0

(n + 1)n−1w′(h)w(h′)

0

1

−1

0w(g)(n + 1)n−1

1

−1

?

x∈F×ζx

2ψ(σx)ψ′((σx)−1)

. (4.45)

except for n = 7 when s = −1, when

T = diag

?

1;1;w(h);±ı

1

7

0

−2

?

,(4.46)

S =1

8

7−1

1

7w(g)

2

8

7w(g)

0

2

−2

0

8

8

7w′(h)w(h′)

0

−4s1s′

1

.(4.47)

42

Page 43

The primary labelled g corresponds to the half-braiding with σ = αgin Theorem

5(a); Σ corresponds to σ =?αgin 5(b); w+g corresponds to the half-braiding Ew

The proof of Corollary 6 is an elementary calculation based on the matrix entries

listed in Theorem 5, as well as the formulae of Section 4.1. The most difficult is the

bottom-right block in the S matrix. Consider n + 1 odd (so s = 1). Then

?

g

?

h

?

z

?

w

which simplifies down to the given expression.

Note that when s = ω = 1, this recovers the modular data for the double of

the affine group Aff1(Fn+1) (see e.g. [7, 6] for the general theory of finite group

modular data and its twists by cocycles in H3(G;T)). Recall that the primaries of the

(untwisted) double of a finite group are pairs (g,ψ) of a conjugacy class representative,

and an irrep of the centraliser of g in the full group. The primaries of the double

of C(1,1) and the double of Aff1(Fn+1)∼= Fq×F×

what we call g ∈ G corresponds to the pair (e,ψ) where e is the identity and ψ is

a 1-dimensional representation of Aff1(Fn+1); Σ corresponds to (e,ρ) where ρ is the

n-dimensional irrep; w + h corresponds to conjugacy class (w,0) and irrep g ∈?

Note that in each case the modular data is inequivalent for the two systems at

n = 1, the three at n = 2, the two at n = 3 and at n = 7. This then verifies that for

n = 1,2,3,7, the solution b = b′′= 1 corresponds to Mod(Aff1(Fq)) (for the other n,

this is clear by the uniqueness in Proposition 5. This inequivalence of the modular

data also means that those systems are not Morita equivalent, i.e. there cannot exist

a subfactor N ⊂ M for which the principal even sectors form say the s = +1 system

at n = 7 and the dual principal sectors form the s = −1 system at n = 7.

The modular data for the 3 systems with G = Z2was also computed in Section

4 of [19], where it was remarked that the modular data corresponds to that of the

double of S3∼= Aff1(Z2) and its twists by order-3 cocycles in H3(S3;T)∼= Z6. The

order-3 twist arises because all that the twist is allowed to affect is the primary

corresponding to conjugacy class (1,1) and (projective) irreps ψ of its centraliser Z3.

In other words the cocycle must be nontrivial on Z3< S3, and coboundary on the

Z2subgroups.

ρ+g

of 5(c); ρj or ρ,ρ′or ρ,ρ′,ρ′′correspond to the half-braidings with σ = ρ in 5(d).

Sρψ,ρψ′ =n2

λS∗

0

ζ1ψ(1)

?

(−1)gρ(Sg)ρ(Sg)∗+

?

z

ζzψ(σz)ρ(Tσ2z)ρ(Tz)∗

?

?

?

×

ζ1ψ′(1)

?

(−1)hρ(Sh)ρ(Sh)∗+

?

w

ζwψ′(σw)ρ(Tσ2w)S0ρ(Tw)∗

= n−1

ζ1ψ(1)T∗

2T∗

1/2+

?

ζzφ(σz)bzb′′(σ2(−σz),σ(−σz))T∗

σ(−σz)T∗

σ2(−σz)

×

ζ1ψ′(1)T1/2T2+

?

ζwφ′(σw)bwb′′(σ2(−σw),σ(−σw))Tσ2(−σw)Tσ(−σw)

?

,

qmatch up quite nicely as follows:

F×

q;

ρψcorresponds to conjugacy class (1,1) and irrep ψ of the centraliser F+

q.

43

Page 44

It appears that s = −1 for n = 1,3,7 likewise corresponds to twisted modular

data. This is clear for n = 1: H3(Z2;T)∼= Z2and s = ±1 corresponds to ±1-twisted

data for Z2∼= Aff1(F2). Aff1(F4) is isomorphic to the alternating group A4, and the

natural restriction of H3(A4;T) to the subgroup Z2

data appears to agree with the twist by some cocycle in that Z3

yet fleshed out the details. For n = 7, something analogous will hold, except now we

want a twist which, when restricted to Z3

(because the number of primaries for s = −1, n = 7 is less than for s = 1, n = 7).

Such cocycles do indeed exist here.

2< A4is Z3

2. The s = −1 modular

2, although we haven’t

2< Z3

2×Z7, is not ‘CT’ in the sense of [6]

4.3The tube algebra in the second class

Consider now the near-group systems of type G + n′where n′∈ nZ. It is certainly

expected that only finitely many n′will work for a given n; in fact to our knowledge

all known near-group categories for abelian G have n′∈ {0,n−1,n}. Last subsection,

we used the existence and properties of the tube algebra for n′= n−1 to prove that

s = ω = 1 except for n = 1,2,3,7. Likewise, we expect that the existence of the tube

algebra should constrain the possible values n′∈ nZ.

As a preliminary step towards working out the tube algebra structure here for

n′> n, consider the half-braidings for σ = αg, when n divides n′. We find (following

the proof of Theorem 5(a)) that Eαgexists here iff, whenever ? z = ? w = 1, we have

c′

˙ z(g) = ˙ w(g). Indeed, when n|n′, equations (4.27) and (4.28) are both satisfied iff

g= ǫ?,?(g) ˙ z(g)? z(g) (4.48)

is independent of z. In this case, the only condition from (4.9) is c′

˙ gz = µ−g˙ z while ?

c′

g

This condition (independence of z) is automatically true when n = n′(the case

considered in [19]) because then ? z uniquely determines z. At this time, we don’t

When the half-braiding Eαgexists, it is unique and defined by Eαg(αh) = ?g,h?

and Eαg(ρ) = ǫ?,?(g)˙ z(g)? z(g)Ug, for any choice of z. This then allows us to compute

braidings Eαgexist), we have

g

2= ?g,g?. Because

gz = µg? z, it suffices to consider z with trivial ? z. Moreover, because

˙ z =˙z, if ˙ z(g)? z(g) is independent of z then it must be real and hence in ±1. Thus

2= ǫ?,?(g)2= ?g,g? and (4.9) is automatic.

know whether it is also true when n′> n — for all we know, n′> n is never realised.

the corresponding parts of the S and T matrices. In particular (assuming all half-

Tαg,αg= Eαg(αg) = ?g,g?, (4.49)

Sαg,αh= λ−1Eαg(αh)∗Eαh(αg)∗= λ−1?g,h?

2, (4.50)

where λ is given in (4.14).

In the remainder of this subsection we turn to n′= 0 and n′= n. When n′= 0,

(4.10) no longer applies and we have two half-braidings for αggiven by choosing either

sign in c′

g. The remaining half-braidings for n′= 0 are for σ = ρ, with precisely 2n

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half-braidings, and precisely one each for σ = αg+ αh for each g ?= h. In this

Tambara-Yamagami case, as analysed in Section 3 of [19], Tube∆ is isomorphic as a

C∗-algebra to C4n⊕(M2×2)n(n−1)/2, and elementary expressions for the modular data

fall out directly.

Thanks to Proposition 6, the case n = n′reduces to that studied in [19], and so

its tube algebra is fully analysed in Section 6 of [19]. We find there is a unique half-

braiding and simple summand C in Tube∆ for each σ = αg, while each σ = ρ + αg

corresponds to a unique summand M2×2and half-braiding. σ = ρ + αg+ αh(g ?= h)

gives a unique braiding and summand M3×3. Finally, there are exactly n(n + 3)/2

half-braidings with σ = ρ, and each contributes a C to Tube∆. Thus

Tube∆n′=n∼= Cn(n+5)/2⊕ (M2×2)n⊕ (M3×3)n(n−1)/2.

The modular data for n′= n is described in [19] as follows. First, find all functions

ξ : G → T and ω ∈ T, τ ∈ G such that

(4.51)

?

g

ξ(g) =√nω2a(τ)c3− nδ−1,(4.52)

c

?

k

b(g + k)ξ(k) = ω2c3a(τ)ξ(g + τ) −√nδ−1, (4.53)

ξ(τ − g) = ω c4a(g)a(τ − g)ξ(g), (4.54)

?

k

ξ(k)b(k − g)b(k − h) = c−2b(g + h − τ)ξ(g)ξ(h)a(g − h) − c2δ−1.(4.55)

There will be a total of n(n + 3)/2 such triples (ωj,τj,ξj).

The n(n + 3) primaries fall into four classes:

1. n primaries, denoted ag, g ∈ G;

2. n primaries, denoted bhfor h ∈ G;

3. n(n − 1)/2 primaries, denoted ck,l= cl,kfor k,l ∈ G, k ?= l;

4. n(n + 3)/2 primaries, denoted dj, corresponding to the triples (ωj,τj,ξj).

We can write the S and T matrices in block form as

T = diag(?g,g?;?h,h?;?k,l?;ωj),

(δ + 1)?g,h′?−2

?h,h′?−2

(δ + 2)?k + l,h′?

−δ?τj,h′?

(4.56)

S =1

λ

?g,g′?−2

(δ + 1)?h,g′?−2

(δ + 2)?k + l,g′?

δ?τj,g′?

(δ + 2)?g,k′+ l′?

(δ + 2)?h,k′+ l′?

(δ + 2)(?k,k′??l,l′? + ?k,l′??l,k′?)

0

δ?g, τj′?

−δ?h,τj′?

0

Sj,j′

, (4.57)

where

Sj,j′ = ωjωj′

?

g

?τj+τj′+g,g?+δωjωj′c6a(τj)a(τj′)n−1?

g,h

ξj(g)ξj′(h)?τj− τj′ + h − g,h − g?.

(4.58)

This is all perfectly simple, except for the n(n + 3)/2 × n(n + 3)/2 block Sj,j′.

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Page 46

4.4The modular data for the double of G+n when n is odd

The point of this subsection is to compute the mysterious part (4.58). We will show

that, rather unexpectedly, Sj,j′ is built up from a quadratic form on an abelian group

of order n + 4.

Definition 3. Let G be any finite abelian group. By a nondegenerate quadratic

form Q on G we mean a map Q : G → Q/Z such that Q(−g) ≡ Q(g) (mod 1) for

all g ∈ G, and ?,?Q : G × G → T defined by ?g,h?Q = e2πı(Q(g+h)−Q(g)−Q(h))is a

nondegenerate symmetric pairing in the sense of Definition 1.

For example, when G = Znfor n odd, these are precisely Q(g) = mg2/n for any

integer m coprime to n. More generally, for |G| odd, the nondegenerate quadratic

forms and nondegenerate symmetric pairings are in natural bijection.

case, we can always write G as Zn1× ··· × Znkwhere Q restricted to each Zniis

nondegenerate and ?Zni,Znj? = 1 for i ?= j. When |G| is even, things are more

complicated but G will have precisely |G/2G| nondegenerate quadratic forms for

each nondegenerate symmetric pairing ?,?. The map a : G → T of Corollary 5 is the

exponential of a nondegenerate quadratic form.

Given a nondegenerate quadratic form Q and a ∈ Z, define the Gauss sum

1

?|G|

For example, note from (3.2) that the quantity c3of Corollary 5 is a Gauss sum.

Provided aQ is nondegenerate, αQ(a) will be a root of unity (this is a consequence of

Proposition 7(a) below). All Gauss sums needed in this paper can be computed from

the classical Gauss sums, corresponding to G = Znand Q(g) = mg2/n, which equal

where

b

is the Jacobi symbol. For n even, these classical Gauss sums are not

modulus 1, because mg2/n is degenerate in Zn.

In such a

αQ(a) =

?

k∈G

exp(2πıaQ(k)).

?am

n

?

for n ≡41

for n ≡43

for n ≡42

ı?am

n

?

am

0

(1 ± ı)?n

?

when n ≡40 and a ≡4±m

,

?a

?

Proposition 7(a) Let Q be a nondegenerate quadratic form on any abelian group G.

Define matrices

SQ

g,h=

α

?|G|?g,h?Q,TQ

g,h= β δg,hexp(2πıQ(g)),(4.59)

for any α,β ∈ C. Then SQ,TQdefine modular data iff α = ±1 and β3= ααQ(1). In

this case, the identity is a0.

(b) Let G,G′be abelian groups of odd order n and n + 4 respectively. Choose any

nondegenerate quadratic forms Q and Q′on them, and write ?g,h? = ?g,h?(n+1)/2

Q

and

46

Page 47

?β,γ?′= ?β,γ?(n+5)/2

Let Φ consist of the following n(n+3) = n+n+n(n−1)/2+n(n+3)/2 elements: ag

∀g ∈ G; bh∀h ∈ G; ck,l= cl,k∀k,l ∈ G with k ?= l; and dm,γ= dm,−γ∀m ∈ G,γ ∈ G′,

γ ?= 0. Define

TQ,Q′= diag(?g,g?;?h,h?;?k,l?;?m,m??γ,γ?′),

where λ is given in (4.14). Then these define modular data iff αQ(1)αQ′(1) = −1.

The identity is a0.

Q′

for all g,h ∈ G, β,γ ∈ G′, so Q(g) = ?g,g? and Q′(γ) = ?γ,γ?′.

SQ,Q′=1

λ

?g,g′?2

(δ + 1)?g,h′?2

?h,h′?2

(δ + 2)?k + l,h′?

−δ?m, h′?2

(δ + 2)?g, k′+ l′?

(δ + 2)?h,k′+ l′?

?k,k′??l,l′? + ?k,l′??l,k′?

0

δ?g, m′?2

−δ?h,m′?2

0

?

(δ + 1)?h,g′?2

(δ + 2)?k + l,g′?

δ?m, g′?

(δ + 2)

??

2

−δ?m, m′??γ,γ′?′+ ?γ,γ′?′?

The straightforward proof is by direct calculation: S2= C, S∗= CS, ST∗S =

TS∗T, and Verlinde’s formula (4.11). In (a), α2= 1 arises from the requirement that

S2be a permutation matrix. The conditions α3β3= αQ(−1) and αQ(−1)αQ′(−1) =

−1 for (a) and (b) respectively both come from ST∗S = TS∗T. We find that the

fusion coefficients of part (a) are Nk

and charge-conjugation C acts by −1. In (b), charge-conjugation sends ag?→ a−g,

bh?→ b−h, ck,l?→ c−k,−l, dm,γ?→ d−m,γ. The nonzero fusion coefficients there are

Nag,ah,ak=Nag,bh,bk= Nbg,bh,bk= Nbg,bh,dk,γ= δ(g + h + k);

Nag,ch,k,ch′,k′=δ(g + h + h′)δ(g + k + k′) + δ(g + h + k′)δ(g + h′+ k) ∈ {0,1};

Nag,dh,β,dk,γ=δ(g + h + k)δβ,γ;

Nbg,bh,ck,k′=Nbg,ck,k′,dh,γ= δ(2g + 2h + k + k′);

Nbg,ch,k,ch′,k′=δ(2g + h + k + h′+ k′) + δ(g + h + h′)δ(g + k + k′)

+ δ(g + h + k′)δ(g + h′+ k) ∈ {0,1,2};

Nbg,dh,β,dk,γ=δ(g + h + k)(1 − δβ,γ);

Ncg,h,cg′,h′,dk,γ=δ(g + h + g′+ h′+ 2k);

Ncg,h,dk,β,dk′,γ=δ(g + h + 2k + 2k′);

Ncg,h,cg′,h′,cg′′,h′′=δ(g + h + g′+ h′+ g′′+ h′′)(1 + δ(g + g′+ g′′) + δ(g + h′+ g′′)

+ δ(g + g′+ h′′) + δ(g + h′+ h′′)) ∈ {0,1,2};

Ndg,γ,dg′,γ′,dg′′,γ′′=δ(g + g′+ g′′)(1 − δ(γ + γ′+ γ′′) − δ(γ − γ′+ γ′′) − δ(γ + γ′− γ′′)

− δ(γ − γ′− γ′′)) ∈ {0,1},

where we write δ(g) = 1 or 0 depending on whether or not g = 0.

We’ll let MDG,G′(Q,Q′) denote the modular data of (b). Of course, associating

SL2(Z) representations to quadratic forms is an old story. See for instance [26],

who study these in similar generality (though their G are p-groups, and they require

β = 1), and call these Weil representations. To our knowledge, Proposition 7(b) is

completely new, but what is more important is its relation to near-group doubles:

g,h= δk,g+h, every primary is a simple-current,

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Conjecture 2. When |G| = n is odd, the modular data for G+n is MDG,G′(Q,Q′),

where Q is the nondegenerate quadratic form on G corresponding to ?,?, and Q′is a

nondegenerate quadratic form on some abelian group G′of order n + 4.

This is true for all groups G of odd order ≤ 13 (recall in Table 2). The easi-

est way to verify modular datum are equivalent is to first identify their T matrices

(straightforward since they must have finite order) and then compare the floating

point values of the S matrices — see Section 4.1 of [10] for details. In all cases in

Table 2, G′ ∼= Zn+4, except for the first entry for G = Z5when G′= Z3× Z3. The

quadratic form Q′is then identified in Table 2 by the integer m′in the Q′column.

For the first entry of G = Z5, Q′(γ1,γ2) = (γ2

Even for n = 3, this is vastly simpler than the modular data as it appears in

Example A.1 of [19]. In fact we have no direct proof that they are equal for n = 3 —

our proof that they are the same is that they both yield nonnegative integer fusions,

they have the same T matrices, and their S matrices are numerically close. The

simplicity of this modular data MDQ,Q′(G,G′) supports our claim that the doubles

of these near-group categories G + n should not be regarded as exotic. We would

expect that these doubles are realised by rational conformal nets of factors, and by

rational vertex operator algebras.

The quantum-dimensions Sx,0/S0,0are 1,δ+1,δ+2,δ for primaries of type a,b,c,d

respectively. We see from the above that the agare simple-currents, and obey the

fusions ag∗ ah= ag+h, ag∗ bh= bg+h, ag∗ ch,k= cg+h,g+k, and ag∗ dh,γ = dg+h,γ.

They form a group isomorphic to G, and act without fixed-points. They supply the

ultimate explanation for the G-action of Proposition 6.7 of [19]. The phases ϕg(x)

defined by Sagx,y= ϕg(y)Sx,yare ?g,h? for ah,bh,dh,γ, and ?g,k + l? for ck,l.

The Galois symmetry is useful in understanding the modular invariants. For

ℓ ∈ Z×

equals 1. Similarly, bℓ

1+ γ2

2)/3.

n(n+4), aℓ

g= aℓgor bℓgdepending on whether or not the Jacobi symbol

g= bℓgor aℓgrespectively. Finally, cℓ

All parities ǫℓ(x) = +1 except for ǫℓ(dg,γ) =

Galois symmetry is what led us to the simplified modular data given above.

A modular invariant is a matrix Z with nonnegative integer entries (often formally

written as a generator function Z =?

There are exactly 3 type I modular invariants when both n and n+4 are prime (e.g.

for n = 3):

?

Z2=

g

g,h

?

ℓ

n(n+4)

?

g,h= cℓg,ℓhand dℓ

. The requirement of a coherent

g,γ= dℓg,ℓγ.

?

ℓ

n(n+4)

?

a,bZa,bchachb), with Z0,0= 1, which commutes

with the modular data S,T. It is called type I if Z can be written as a sum of squares.

Z1=

g

|ag|2+

?

g

|bg|2+

?

?

|cg,h|2,

g,h

|cg,h|2+

?

Z3= |a0+ b0+

g,γ

|dg,γ|2,

?

|ag+ bg|2+ 2

?

g?=0

cg,0|2.

The most important modular invariants are the monomial ones, of form Z =

|ΣZa,0cha|2, as explained in Section 1.3 of [10], as they give a canonical endomor-

phism θ as a sum of sectors, and can be used to recover the original system from

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its double. This is M¨ uger’s forgetful functor [25]. For example, there are exactly 3

monomial modular invariants for the modular data of the double of the Haagerup

subfactor; these should correspond bijectively to the three systems found in [12] which

are Morita equivalent to the principal even Haagerup system (see their Theorem 1.1),

as each of those must correspond to a monomial modular invariant.

We see however that for the MDG,G′(Q,Q′) modular data there is only one generic

monomial modular invariant, namely Z3. This suggests that Grossman-Snyder per-

haps isn’t as interesting here as it was for the Haagerup (at least not for general n).

On the other hand, recall our comments earlier that the type G + n systems with

n = n′= ν2may be related to the Haagerup-Izumi system for groups of order ν.

Consider first G = Znfor n = ν2a perfect square and write H = νG∼= Zν; then

MDG,G′(Q,Q′) has at least one other monomial invariant, namely

?

Alternatively, when G = H1× H2where each Hi∼= Zν, and ?H1,H2?Q= 1 (which

can always be arranged), another monomial invariant is

Z4= |

h∈H

ah+

?

h∈H

bh+ 2

?

h<h′∈H

ch,h′|2. (4.60)

Z′

4= |a0+ b0+

?

h∈H1,h′∈H2

c(h,0),(0,h′)|2.(4.61)

We would expect that systems of type Zν2 +ν2or Zν×Zν+ν2should have nontrivial

quantum subgroups in the sense of [12].

It isn’t difficult to see why n+4 arises here, i.e. why it can’t be replaced by some

other positive integer n′. In particular, after some work, the nonzero fusions of the

form Nb,b,bare found to be 4/(n′− n);,and the ST∗S = TS∗T calculation requires

the product of Gauss sums for G and G′to be −1, which forces 4|(n′− n).

When G has even order, the situation is similar but (as always with n even)

somewhat messier; we will provide its modular data elsewhere. Again we have n

simple-currents (the ag), but for each g ∈ G of order 2, agnow has n/2 fixed-points,

which complicates things. The T entries for the first several even G are provided by

the pairs (m′,m′′) in the Q′column of Table 2, and from this the S matrix follows

quickly from the equations of the last subsection. In particular,

?

n

n(n+4)

Tdg,γ,dg,γ=

ξ?g,g?

n

ξm′γ2

n+4

ξm′′(1+nγ)2

if γ + n/2 is odd

if γ + n/2 is evenξ?m,m−1?

where τγ= 0,1 for γ +n/2 odd respectively even. Here 1 ≤ γ ≤ (n+4)/2 and g ∈ G

except for γ = (n + 4)/2 when g ∈ G/2.

Recall our observation in Section 3 of [10] that the modular data of the double of

the Haagerup-Izumi series at G = Znresembles that of the affine algebra so(n2+4)(1)

at level 2. The analogous statement here is that the modular datum of the double of

type Zn+n near-group systems resemble that of the affine algebra so(n+4)(1)at level

2. In particular, for an appropriate choice of Q′(corresponding to m′= (n+3)/2), this

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Page 50

recovers Td0,γ,d0,γand Sd0,γ,d0,γ′. This could hint at ways to construct the corresponding

vertex operator algebra.

Acknowledgement.

The authors thank the Erwin-Schr¨ odinger-Institute, Cardiff School of Mathematics, and

Swansea University Dept of Computer Science, for generous hospitality while researching this paper.

They thank Masaki Izumi for sharing with us an early draft of [20] and informing us of [8], and

Eric Rowell for informing us of [29]. Their research was supported in part by EPSRC, EU-NCG

Research Training Network: MRTN-CT-2006 031962, and NSERC.

References

[1] Asaeda, M., Haagerup, U.: Exotic subfactors of finite depth with Jones indices (5 +√13)/2 and (5 +√17)/2.

Commun. Math. Phys. 202, 1–63 (1999).

[2] Bigelow, S., Morrison, S., Peters, E., Snyder, N.: Constructing the extended Haagerup planar algebra. arXiv:

math.OA0909.4099.

[3] Bisch, D., Haagerup, U.: Composition of subfactors: new examples of infinite depth subfactors. Ann. Sci.´Ecole

Norm. Sup. 29, 329–383 (1996).

[4] B¨ ockenhauer, J., Evans, D.E., Kawahigashi, Y.: On α-induction, chiral generators and modular invariants for

subfactors. Commun. Math. Phys. 208, 429–487 (1999).

[5] Calegari, F., Morrison, S., Snyder, N.: Cyclotomic integers, fusion categories, and subfactors. Commun. Math.

Phys. 303, 845–896 (2011); arXiv: 1004.0665.

[6] Coste, A., Gannon, T., Ruelle, P.: Finite group modular data. Nucl. Phys. B581, 679–717 (2000).

[7] Dijkgraaf, R., Vafa, C., Verlinde, E., Verlinde, H.: The operator algebra of orbifold models. Commun. Math.

Phys. 123, 485–526 (1989).

[8] Etingof, P., Gelaki, S., Ostrik, V.: Classification of fusion categories of dimension pq. Int. Math. Res. Notices

2004, no. 57, 3041–3056 (2004).

[9] Etingof, P.,Nikshych, D., Ostrik,V.:On fusion categories. Ann. Math. 162,

arXiv:math.QA/0203060.

[10] Evans, D. E., Gannon, T.: The exoticness and realisability of twisted Haagerup-Izumi modular data. Commun.

Math. Phys. 307, 463–512 (2011); arXiv:1006.1326.

[11] Goodman, F.M., de la Harpe, P., Jones, V.F.R.: Coxeter graphs and towers of algebras. vol. 14, Mathematical

Sciences Research Institute Publications (Springer-Verlag, New York 1989).

[12] Grossman, P., Snyder, N.: Quantum subgroups of the Haagerup fusion categories. Commun. Math. Phys. 311,

617–643 (2012); arXiv:1102.2631v2.

[13] Haagerup, U.: Principal graphs of subfactors in the index range 4 < [M : N] < 3 +√3. In: Subfactors. H.

Araki et al (eds.). World Scientific, 1994, pp.1–38.

[14] Han, R.: A Construction of the “2221” Planar Algebra. Ph.D. Thesis (University of California, Riverside)

(2010); arXiv:1102.2052v1.

[15] Hong, J.H., Szymanski, W.: Composition of subfactors and twisted bicrossed products. J. Operator Theory

37, 281–302 (1997).

[16] Horn, R.A., Johnson, C.R.: Matrix Analysis. (Cambridge University Press, 1985).

[17] Izumi, M.: Subalgebras of infinite C*-algebras with finite Watatani indices, I. Cuntz Algebras. Commun. Math.

Phys. 155, 157–182 (1993).

[18] Izumi, M.: The structure of sectors associated with Longo-Rehren inclusions, I. General theory. Commun.

Math. Phys. 213, 127–179 (2000).

[19] Izumi, M.: The structure of sectors associated with Longo-Rehren inclusions, II. Examples. Rev. Math. Phys.

13, 603–674 (2001).

[20] Izumi, M.: (notes on the Haagerup series). January 2012.

[21] Izumi, M.: (notes on near-group categories).

[22] Izumi, M., Kosaki, H.: Kac algebras arising from composition of subfactors: general theory and classification.

Mem. Amer. Math. Soc. 158 no. 750, (2002).

[23] Morrison, S., Snyder, N.: Subfactors of index less than 5, part 1: the principal graph odometer. Commun.

Math. Phys. 312, 1–35 (2012); arXiv:1007.1730.

[24] Morrison, S., Peters, E.: The little desert? Some subfactors with index in the interval (5,3 +√5); arXiv:

1205.2742.

[25] M¨ uger, M.: From subfactors to categories and topology II. The quantum double of tensor categories and

subfactors. J. Pure Appl. Alg. 180, 159–219 (2003).

581–642 (2005);

50