SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3
TOBIAS J. HAGGE AND SEUNG-MOON HONG
Abstract. We classify all fusion categories for a given set of fusion rules with
three simple object types. If a conjecture of Ostrik is true, our classification
completes the classification of fusion categories with three simple object types.
To facilitate the discussion we describe a convenient, concrete and useful varia-
tion of graphical calculus for fusion categories, discuss pivotality and sphericity
in this framework, and give a short and elementary re-proof of the fact that
the quadruple dual functor is naturally isomorphic to the identity.
1. Introduction
Let k be an algebraically closed field. A fusion category C over k is a k-linear
semi-simple rigid monoidal category with finitely many (isomorphism classes of)
simple objects, finite dimensional morphism spaces, and End(1) ∼= k. See [7] or [4]
for definitions, and [2] for many of the known results about fusion categories.
The rank r of C is the number of isomorphism classes of simple objects in C. Let
{xi}1≤i≤r be a set of simple object representatives. The fusion rules for C are a
set of r × r N-valued matrices N = {Ni}1≤i≤r, with (Ni)j,k denoted Nkij or, when
convenient, Nxkxixj , such that xi⊗xj ∼=
1≤k≤r N
ijxk. In the sequel, assume k = C.
Fusion categories appear in representation theory, operator algebras, conformal
field theory, and in constructions of invariants of links, braids, and higher dimen-
sional manifolds. There is currently no general classification of them. Classifications
of fusion categories for various families of fusion rules have been given in work by
Kerler ([3]), Tambara and Yamagami ([12]), Kazhdan and Wenzl ([5]), and Wenzl
and Tuba ([13]).
For a given set of fusion rules, there are only finitely many monoidal natural
equivalence classes of fusion categories. This property is called Ocneanu rigidity (see
[2]). It is not known whether or not the number of fusion categories of a given rank
is finite. If one assumes a modular structure, the possibilities up to rank four have
been classified by Belinschi, Rowell, Stong and Wang in [11]. Ostrik has classified
fusion categories up to rank two in [10], and constructed a finite list of realizable
fusion rules for braided categories up to rank three in [9], in which the number of
categories for each set of fusion rules is known. The rank two classification relies in
an essential way on the theory of modular tensor categories; Ostrik shows that the
quantum double of a rank two category must be modular, and uses the theory of
modular tensor categories to eliminate most of the possibilities. The classification
of modular tensor categories is of independent interest; in many contexts one must
assume modularity.
We consider the only set of rank three fusion rules which is known to be realizable
as a fusion category but which has no braided realizations. Ostrik conjectured in
http://arxiv.org/abs/0704.0208v2
2 TOBIAS J. HAGGE AND SEUNG-MOON HONG
[9] that a classification for this rule set completes the classification of rank three
fusion categories.
The axioms for fusion categories over C reduce to a system of polynomial equa-
tions over C. In this context, Ocneanu rigidity, roughly translated, says that nor-
malization of some of the variables in the equations gives a finite solution set. In
this case, one can compute a Gröbner basis for the system and obtain the solutions
(see [1]). However, normalization becomes complicated when there are i, j, k such
that Nkij > 1. The fusion rules we consider are the smallest realizable set with this
property.
2. Main theorem and outline
Theorem 1 (Main Theorem). Consider the set of fusion rules with three simple
object types, x, y and 1. Let 1 be the trivial object, and let x⊗ x ∼= x⊕ x⊕ y⊕ 1,
x⊗ y ∼= y ⊗ x ∼= x and y ⊗ y ∼= 1. Then the following hold:
(1) Up to monoidal natural equivalence, there are four semisimple tensor cat-
egories with the above fusion rules. A set of associativity matrices for one
of these categories is given in Appendix A. Applying a nontrivial Galois
automorphism to all of the coefficients gives a set of matrices for any one
of the other three categories.
(2) The categories in part 1 are fusion categories.
(3) The categories in part 1 do not admit braidings.
(4) The categories in part 1 are spherical.
The structure of the remainder of the paper is as follows:
Section 3 describes the notation and categorical preliminaries used in later parts
of the paper. It constructs a canonical representative for each monoidal natural
equivalence class of fusion categories. This construction is really just two well
known constructions, skeletization and strictification, applied in sequence. These
constructions, taken together, form a bridge between the category theoretic lan-
guage in the statements of the theorems and the algebra appearing in the proofs.
For some of the calculations in this paper the translation between the category the-
ory and the algebra is already widely known, but there are some subtleties when
discussing pivotal structure that justify the treatment. Section 3 concludes by de-
scribing the algebraic equations corresponding to the axioms for a fusion category,
using the language of strictified skeletons.
Section 4 proves part 1 of Theorem 1. The proof amounts to solving the variety
of polynomial equations defined in the previous section, performing normalizations
along the way in order to simplify calculations. The section ends by arguing that
the nature of the normalizations guarantees that the solutions obtained really are
monoidally inequivalent. Section 5 proves part 2 of Theorem 1 by explicitly com-
puting rigidity structures.
Part 3 of Theorem 1 follows from Ostrik’s classification of rank three braided
fusion categories in [9]. Section 6 gives a direct proof by showing that there are no
solutions to the hexagon equations.
Section 7 defines pivotal and spherical structures and discusses their properties.
The focus is on the question of whether every fusion category is pivotal and spher-
ical. A novel and elementary proof that the quadruple dual functor is naturally
isomorphic to the identity functor is given. This proof makes use of the strictified
skeleton construction developed in Section 3. The section concludes by describing
SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 3
what a pivotal category which does not admit a spherical structure would look like.
In particular, it must have at least five simple objects.
Section 8 proves part 4 of Theorem 1 by computing explicit pivotal structures
for the four categories given in Section 4, and invoking a lemma from Section 7 for
sphericity.
3. Preliminaries and notational conventions
This paper uses the “composition of morphisms” convention for functions as
well as morphisms, and left to right matrix multiplication. For calculations of the
fusion rules, our treatment is similar to [12], but the notation differs superficially
for typographic reasons. The notation captures algebraic data sufficient to classify
a fusion category up to monoidal natural equivalence, and is reviewed later in this
section.
We assume that the reader is familiar with the notions of a monoidal category,
a monoidal functor and a monoidal equivalence; for precise definitions, see [7].
Recall that a monoidal category is equipped with an associative bifunctor ⊗ and
a distinguished object 1. Reassociation of tensor factors in a monoidal category is
described by a natural isomorphism of trifunctors α : (−⊗−)⊗− → −⊗ (−⊗−).
Tensor products with 1 have natural isomorphisms ρ : −⊗1 → − and λ : 1⊗− → −.
These isomorphisms are subject to a coherency condition, namely that for any pair
of multifunctors there is at most one natural isomorphism between them which may
be constructed from λ, ρ, α and their inverses, along with Id and ⊗. This coherency
condition is well known to be equivalent to the statement that the category satisfies
the pentagon and triangle axioms (see [7] for a proof).
The triangle equations are the equations ρx⊗y = αx,1,y ◦ (x⊗λy) for all ordered
pairs (x, y) of objects. Here, and in the sequel when the context is unambiguous,
the name of an object is used as a shorthand for the identity morphism on that
object. The pentagon equations, defined for all tuples of objects (w, x, y, z), are as
follows (see also Figure 1):
(αx,y,z ⊗ w) ◦ αx,y⊗z,w ◦ (x⊗ αy,z,w) = αx⊗y,z,w ◦ αx,y,z⊗w,
When studying fusion categories up to monoidal equivalence, one may choose
categories within an equivalence class which have desirable attributes. Since the
categorical properties considered in this paper (fusion rule structure, monoidality,
pivotality, sphericity, presence of braidings) are all well known to be preserved un-
der monoidal equivalence, the desirable attributes may be assumed without loss. In
particular, one may construct, given an arbitrary fusion category, a canonical rep-
resentative for that category’s equivalence class in which one may replace instances
of the words “is isomorphic to” with “equals”.
3.1. Skeletization. The skeleton CSKEL of an arbitrary category C is any full
subcategory of C containing exactly one object from each isomorphism class in
C. If C is semi-simple, every object in C is isomorphic to a direct sum of simple
objects in CSKEL. One may then assume without loss that the objects of CSKEL
consist of simple object representatives and direct sums of such.
It is a well known fact that CSKEL may be given a monoidal structure such
that CSKEL and C are monoidally equivalent. The proof is a straightforward but
tedious extension of Maclane’s proof of the natural equivalence an ordinary category
4 TOBIAS J. HAGGE AND SEUNG-MOON HONG
and its skeleton (see section IV.4 in [7]). In that proof, one defines a family of
isomorphisms ix from objects x to their isomorphic representatives in CSKEL and
uses it to construct a pair of functors F and G which give a natural equivalence.
For the extension, the ix are used to define the tensor product functor on CSKEL,
as well as α, λ, ρ and the monoidal structures for F and G. One then writes out
all of the relevant commutative diagrams and removes any compositions ix ◦ i−1x .
The result in each case is a commutative diagram in C.
3.2. Strictification. Given a monoidal category C, one may construct a strict
monoidal category CST R equivalent to C. In a strict monoidal category, α, λ and ρ
are the identity. It is common practice to assume that a monoidal category is strict
without explicit reference to the construction. However, by using the construction
explicitly we will be able to pick a canonical representative for an equivalence class of
monoidal categories and provide a natural interpretation of the graphical calculus.
Strictification of a monoidal category is analogous to the construction of a tensor
algebra; it gives an equivalent strict category CSTR by replacing the tensor product
with a strictly associative formal tensor product. The objects of CSTR are finite
sequences of objects in C. Morphism spaces of the form
Mor((a1, a2, . . . , am−1, am), (b1, b2, . . . , bn−1, bn))
are given by
Mor(a1 ⊗ (a2 ⊗ . . . (am−1 ⊗ am) . . . ), b1 ⊗ (b2 ⊗ . . . (bn−1 ⊗ bn) . . . ))
in C. The tensor product on objects is just concatenation of sequences, for mor-
phisms it is the tensor product in C pre and post-composed with appropriate asso-
ciativity morphisms. Monoidal equivalence of C with CST R is proven in section XI.3
of [7].
It is not usually possible to make a fusion category strict and skeletal at the
same time. However, the category (CSKEL)ST R, while not a skeleton, is still unique
up to strict natural equivalence. Also, it is a categorical realization of a graphical
calculus, as will shortly become clear. The next subsection describes what strictified
skeleta of fusion categories look like, up to strict equivalence.
3.3. Strictified skeletal fusion categories. A strictified skeletal fusion category
C is as follows: Let N be a set of fusion rules for a set of objects S. Then the objects
in C are multisets of finite sequences of elements of S. C has a tensor product ⊗,
which is defined on objects by pairwise concatenation of sequences, distributed over
elements of multisets. Direct sum of objects is given by multiset disjoint union.
A strand is an object which is a sequence of length one. Strands correspond
to simple object types. If x, y and z are strands, define Mor(x ⊗ y, z) to be a k
vector space isomorphic to kN
xy . For brevity, V yx will denote Mor(x, y), and tensor
products will be omitted when the context is clear. A morphism is (n,m)-stranded
if its source and target are sequences of length n and m, respectively. A morphism
is (n)-stranded if it is (m,n−m)-stranded for some 0 ≤ m ≤ n.
Semi-simplicity of C means that for all objects w, x, y and z there are vector space
isomorphisms
v∈S V
xy ⊗ V zwv ∼= V zwxy ∼=
v∈S V
wx ⊗ V zvy . The first isomorphism
is given by f ⊗ g → (Idw ⊗ f) ◦ g, the inverse of the second by h⊗ l → (h⊗ Idy) ◦ l.
The composition of the two isomorphisms is denoted αzw,x,y. Additionally, each
morphism space V zxy has an algebraically dual space V
z , in the sense that there
are bases {vi}i ⊂ V zxy and {wi}i ⊂ V xyz such that wi ◦ vj = δijIdz.
SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 5
In a strictified skeleton, the trivial object 1 is the zero length sequence. There
is a strand which is isomorphic to the trivial object, but not equal. This strand
shall be taken to be 1 in the sequel. This choice makes the category non-strict,
since λ and ρ are no longer the identity, but it is convenient for graphical calculus
purposes.
(Right) rigidity in a strictified skeleton means that there is a set involution ∗ on
the strands, and each strand x has morphisms bx : 1 → x⊗ x∗ and dx : x∗ ⊗ x → 1
such that (Idx∗ ⊗ bx) ◦ (dx ⊗ Idx∗) = Idx∗ and (bx ⊗ Idx) ◦ (Idx ⊗ dx) = Idx.
This implies that Nzxy = N
z∗x. Left rigidity is similar, and in the sequel, the right
rigidity morphism for x∗ will be defined to be the left rigidity morphism for x.
Define ∗, b, and d on concatenations of strands such that dx⊗y = (Idy∗ ⊗ dx ⊗
Idy) ◦ dy and extend to direct sums. Then there is a contravariant (right) dual
functor ∗ which sends f ∈ V yx to f∗ = (Idy∗ ⊗ bx)◦ (Idy∗ ⊗f ⊗ Idx∗)◦ (dy ⊗ Idx∗) ∈
y∗ . The definition of a left dual functor is similar, and the two duals are inverse
functors by rigidity.
Monoidality for a strictified skeletal fusion category implies that the α are iden-
tity morphisms. For this to be true, it is necessary and sufficient that the following
equation holds for all objects x, y, z, w, and u. Each instance will be referred to as
Pux,y,z,w in the sequel.
αty,z,wV
V syzα
x,s,w ◦
αtx,y,zV
V szwV
V syzV
V syzV
V sxyV
is equal to
V szwα
x,y,s ◦ τ ◦
V sxyα
s,z,w :
V szwV
V szwV
V sxyV
V sxyV
Here ⊗ for vector spaces and morphisms are omitted, and τ is the isomorphism
interchanging the first and the second factors of vector space tensor products (see
Figure 1).
3.4. Remarks.
(1) Every fusion category is monoidally naturally equivalent to a strictified
skeleton. Also, two naturally equivalent strictified skeleta have an invert-
ible equivalence functor that takes strands to strands. This implies that
equivalences are given by permutations of strands along with changes of
basis on the (2, 0) and (2, 1)-stranded morphism spaces.
(2) The functor ∗∗ fixes objects. The isomorphisms Jx,y : x∗∗⊗y∗∗ → (x⊗y)∗∗
associated with ∗∗ in the definition of a monoidal functor (see [7]) may be
taken to be trivial. There is an invertible scalar worth of freedom in the
choice of each bx, dx pair.
(3) Semi-simplicity allows every morphism to be built up from (3)-stranded
morphisms. Choosing bases for the (3)-stranded morphisms allows mor-
phisms in C to be characterized as undirected trivalent graphs with labeled
6 TOBIAS J. HAGGE AND SEUNG-MOON HONG
((x⊗ y)⊗ z)⊗ w
αx,y,zw
uukkk
αxy,z,w
(x⊗ (y ⊗ z))⊗ w
αx,yz,w
(a) (x ⊗ y)⊗ (z ⊗ w)
αx,y,zw
x⊗ ((y ⊗ z)⊗ w)
xαy,z,w // x⊗ (y ⊗ (z ⊗ w))
s,t V
s,t V
V sxyα
s,z,woo
s,t V
αtx,y,zV
s,t V
s,t V
V syzα
x,s,w
s,t V
y,z,wV
V szwα
x,y,s
Figure 1. (a) Pentagon equality and (b) corresponding equality
edges and vertices, subject to associativity relations given by the penta-
gon equations. The labels for the edges are isomorphism types of simple
objects; the labels for the vertices are basis vectors for the corresponding
morphism spaces. This gives a categorically precise interpretation of an
arrowless graphical calculus for C.
(4) If C is pivotal (see Section 7 for the definition), a well known construction
allows one to add a second copy of each object and get a strict pivotal
category. This construction gives a graphical calculus with arrows on the
strands.
(5) Strictified skeleta give any categorical structure preserved under natural
equivalence (and any functorial property preserved under natural isomor-
phism) a purely algebraic description.
4. Proof of Theorem 1 part 1:possible tensor category structures
In this section we classify, up to monoidal equivalence, all C-linear semisimple
tensor categories with fusion rules given in Theorem 1. This amounts to solving
the matrix equations described in the previous section. The simplest equations
(those involving 1× 1 matrices) are solved first, and normalizations are performed
as necessary in order to simplify the equations.
4.1. Setting up the pentagon equations. The fusion rules are given by x⊗x ∼=
1⊕y⊕x⊕x, x⊗y ∼= y⊗x ∼= x, and y⊗y ∼= 1. The non-trivial vector spaces are V 111,
V x1x ,V
x1, V
1y , V
y1, V
xy, V
yx, V
yy, V
xx, V
xx,and V
xx, and they are all 1-dimensional
except the last space which is 2-dimensional.
Let’s choose basis vectors in each space. If we fix any non-zero vector v111 ∈ V 111,
then there are unique vectors vx1x ∈ V x1x, vxx1 ∈ V xx1,v
1y ∈ V
1y, and v
y1 ∈ V
y1 such
that the triangle equality holds. For the other spaces, choose any non-zero vectors
SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 7
in each space and denote them by vxxy ∈ V xxy, vxyx ∈ V xyx, v1yy ∈ V 1yy, v1xx ∈ V 1xx,
vyxx ∈ V yxx, v1 and v2 ∈ V xxx where the two vectors v1 and v2 are linearly independent.
There are 30 associativities. It is a well known fact that if at least one of
the bottom objects is 1 then the associativity is trivial. That is, with the above
basis choices the matrix for αzu,v,w is trivial if at least one of the u, v and w is 1.
Now we have ten non-trivial 1-dimensional associativities, αyy,y,y,α
x,y,y,α
y,y,x,α
x,y,x,
αyx,y,x,α
y,x,y,α
x,x,y,α
x,x,y, α
y,x,x, and α
y,x,x, five non-trivial 2-dimensional ones,
αxx,y,x,α
x,x,y,α
y,x,x,α
x,x,x, and α
x,x,x, and one 6-dimensional one, α
x,x,x.
4.2. Normalizations. With the above basis choices we obtain a basis for each
tensor product of vector spaces in a canonical way and can parameterize each
associativity and pentagon equation. However, at this point our basis elements
have not been uniquely specified, and we should expect to obtain solutions with
free parameters. As the calculation progresses it will be convenient to simplify the
pentagon equations by requiring certain coefficients of certain associativity matrices
to be 1 or 0. These normalizations should be thought of as restrictions on the basis
choices made above. Normalizations simplify the equations and have an additional
advantage: once the set of possible bases is sufficiently restricted, Ocneanu rigidity
[2] guarantees a finite set of possibilities for the associativity matrices of fusion
categories, which can be found algorithmically by computing a Gröbner basis.
4.3. Associativity matrices. The following are the 1-dimensional associativities:
αyy,y,y : v
y1 7→ ayy,y,yv1yyv
αxx,y,y : v
x1 7→ axx,y,yvxxyvxxy
αxy,y,x : v
yx 7→ axy,y,xv1yyvx1x
α1x,y,x : v
xx 7→ a1x,y,xvxxyv1xx
αyx,y,x : v
xx 7→ ayx,y,xvxxyvyxx
αxy,x,y : v
yx 7→ axy,x,yvxyxvxxy
α1x,x,y : v
xx 7→ a1x,x,yvyxxv1yy
αyx,x,y : v
xx 7→ ayx,x,yv1xxv
α1y,x,x : v
yy 7→ a1y,x,xvxyxv1xx
αyy,x,x : v
y1 7→ ayy,x,xvxyxvyxx
where associativity coefficients are all non-zero.
For 2-dimensional and 6-dimensional associativities we need to fix the ordering
of basis elements in each Hom vector space. The orderings are as follows:
{vxyxv1, vxyxv2} for V xx(yx), {vxxyv1, vxxyv2} for V x(xy)x,
{vxxyv1, vxxyv2} for V xx(xy), {v1v
xy, v2v
xy} for V x(xx)y,
{v1vxyx, v2vxyx} for V xy(xx), {v
yxv1, v
yxv2} for V x(yx)x,
{v1v1xx, v2v1xx} for V 1x(xx), {v1v
xx, v2v
xx} for V 1(xx)x,
{v1vyxx, v2vyxx} for V
x(xx)
, {v1vyxx, v2vyxx} for V
(xx)x
{v1xxvxx1, vyxxvxxy, v1v1, v1v2, v2v1, v2v2} for V xx(xx),
and {v1xxvx1x, vyxxvxyx, v1v1, v1v2, v2v1, v2v2} for V x(xx)x.
With these ordered bases, each associativity has a matrix form (recall that we are
using the right multiplication convention). That is, αxx,y,x is given by the invertible
2 × 2 matrix axx,y,x, and αxx,x,y is given by the invertible 2 × 2 matrix axx,x,y,etc.,
and finally αxx,x,x is given by the invertible 6× 6 matrix axx,x,x.
8 TOBIAS J. HAGGE AND SEUNG-MOON HONG
4.4. Pentagon equations with 1× 1 matrices. Considering only nontrivial as-
sociativities, there are 17 1-dimensional pentagon equations, 14 2-dimensional pen-
tagon equations, 6 6-dimensional ones, and 1 16-dimensional one. Without redun-
dancy, the following are the 1-dimensional equations:
P xx,y,y,y : a
y,y,y a
x,y,y = a
x,y,y.
P 1x,x,y,y : a
x,y,ya
x,x,y a
x,x,y = 1 ,
P 1x,y,x,y : a
y,x,y a
x,y,x = a
x,y,x ,
P yx,y,x,y : :a
y,x,y a
x,y,x = a
x,y,x ,
P 1x,y,y,x : a
y,y,x a
x,y,y = (a
x,y,x)
P yx,y,y,x : a
y,y,x a
x,y,y = (a
x,y,x)
P xy,x,y,y :(a
y,x,y)
2 = 1 ,
P 1y,y,x,x : a
y,x,x a
y,x,x a
y,y,x = 1 ,
P 1y,x,x,y : a
x,x,y a
y,x,x = a
y,x,x a
x,x,y
If we normalize the basis we may assume axy,y,x, a
x,y,x and a
x,x,y to be 1 (for
normalization see [12] or [6]), and we can solve the above 1-dimensional equations.
Here is the solution:
ayy,y,y = a
x,y,y = a
x,x,y = 1, a
y,x,y = a
x,y,x = ±1, a1y,x,x = ayy,x,x = ±1.
Let’s say g := axy,x,y = a
x,y,x and h := a
y,x,x = a
y,x,x in the sequel. Also let
A := axx,y,x, B := a
x,x,y, D := a
x,x,x, E := a
x,x,x, F := a
y,x,x and Φ := a
x,x,x for
brevity.
4.5. Pentagon equations with 2× 2 or 6× 6 matrices. Now, the following are
the 2-dimensional pentagon equations using the above 1-dimensional solutions:
P xy,y,x,x : F
2 = Id2
P xy,x,y,x : gAF = FA
P xx,y,y,x : A
2 = Id2
P xy,x,x,y : gBF = FB
P xx,y,x,y : gBA = AB
P xx,x,y,y : B
2 = Id2
P 1y,x,x,x : EF = D
P yy,x,x,x : DF = E
P 1x,y,x,x : FDA = D
P yx,y,x,x : FEA = gE
P 1x,x,y,x : ADB = D
P yx,x,y,x : AEB = gE
P 1x,x,x,y : BE = D
P yx,x,x,y : BD = E
It should be noted that for this particular category the large number of one
dimensional morphism spaces gives us q-commutativity relations and matrices with
±1 eigenvalues, which are of great help when simplifying the pentagon equations
by hand.
To analyze 2-dimensional and 6-dimensional pentagon equations, at first let’s
look at the isomorphism τ interchanging the first and the second factors of tensor
products. This change of basis is necessary for 6-dimensional pentagon equations
because the image basis of the matrix for αux,y,zw and the domain basis of the matrix
for αuxy,z,w may not be the same. For P
x,y,x,x, τ is an isomorphism from the space
V 1xxV
x1⊕V yxxV xxyV xxy⊕V xxxV xxyV xxx to V xxyV 1xxV xx1⊕V xxyV yxxV xxy⊕V xxyV xxxV xxx, both of
SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 9
which correspond to Hom((x⊗ y)⊗ (x⊗x), x). With the canonically ordered basis
{v1xxvxxyvxx1, vyxxvxxyvxxy, vivxxyvj} and {vxxyv1xxvxx1, vxxyvyxxvxxy, vxxyvivj}, respectively,
τ turns out to be I6. For P
y,x,x,x, P
x,x,y,x and P
x,x,x,y, τ is also I6. But for P
x,x,x,x,
it is τ1, and for P
x,x,x,x, it is τ2, defined as follows:
τ1 :=
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 0 1 0
0 0 0 1 0 0
0 0 0 0 0 1
, τ2 :=
0 1 0 0 0 0
1 0 0 0 0 0
0 0 1 0 0 0
0 0 0 0 1 0
0 0 0 1 0 0
0 0 0 0 0 1
Here are the 6 6-dimensional pentagon equations:
P xy,x,x,x ; Φ(I2 ⊕ I2 ⊗ F )
⊕ F ⊗ I2
⊕ I2 ⊗ F
P xx,y,x,x ;
⊕ F ⊗ I2
⊕A⊗ I2
= (I2 ⊕ I2 ⊗A)Φ
P xx,x,y,x ;
⊕A⊗ I2
⊕B ⊗ I2
= Φ(I2 ⊕ I2 ⊗A)
P xx,x,x,y ;
⊕B ⊗ I2
(I2 ⊕ I2 ⊗ B)Φ = Φ
⊕ I2 ⊗B
P 1x,x,x,x ; (I2 ⊕ I2 ⊗D)Φ = (I2 ⊕ I2 ⊗D)τ1
⊕ I2 ⊗D
P yx,x,x,x ; Φ
⊕ I2 ⊗ E
Φ = (I2 ⊕ I2 ⊗ E)τ2
⊕ I2 ⊗ E
If we normalize the basis {v1, v2} of V xxx, we may assume A is of the form
and then get g = −1 and h = 1 using the above equations. Following is the
computation for this:
At first we may assume that matrix A is of the Jordan canonical form, then
A = ±I2 or
from P xx,y,y,x. We eliminate the possibility A = ±I2 from
P xy,x,y,x, P
x,y,x,x and P
y,x,x,x which imply respectively that g = 1, F = ±I2 and
then det(Φ) = 0, since the first two columns of Φ are scalar multiples of each other.
So we conclude A =
. Now we eliminate the possibility g = 1 using P xy,x,y,x,
P xy,y,x,x and P
y,x,x,x, which imply F is a diagonal matrix, with entries ±1 and then
det(Φ) = 0, respectively. For the case A =
and g = −1, F is of the form
[ 0 f
1/f 0
from P xy,x,y,x and P
y,y,x,x, and B is of the form
1/b 0
from P xx,y,x,y and
P xx,x,y,y. If h = −1, the first column of Φ has to be zero by comparing the first and
the second columns of P xy,x,x,x, P
x,y,x,x, P
x,x,y,x, and P
x,x,x,y.
At this point we have fixed all 1-dimensional associativity matrices.
From the above equations, we get
[ 0 f
1/f 0
for F and
1/b 0
for B with the
relation f2 + b2 = 0 from P xy,x,x,y. We note that the diagonalization of A defines
each basis element v1 and v2 only up to choice of a nonzero scalar. By using up
one of these degrees of freedom, we may assume f = 1. Then from the above
6-dimensional equations, we get the following:
D = d
E = d
φ φ −wb w w −wb
φ −φ −wb w −w wb
x x −yb z y −zb
x x −zb y z −yb
x −x −yb z −y zb
−x x zb −y z −yb
10 TOBIAS J. HAGGE AND SEUNG-MOON HONG
4.6. The pentagon equation with 16 × 16 matrices. Now we analyze the 16-
dimensional pentagon equation P xx,x,x,x. It is convenient to express each Hom vector
space in two different ways and put basis permutation matrices into the pentagon
equation. The following are two expressions with ordered direct sum.
Hom(x(x(xx)), x) :
V xxxV
x1 ⊕ V xxxV yxxV xxy ⊕ V 1xxV xx1V xxx ⊕ V yxxV xxyV xxx ⊕ V xxxV xxxV xxx, and
V 1xxV
xx ⊕ V yxxV xxyV xxx ⊕ V xxxV 1xxV xx1 ⊕ V xxxV yxxV xxy ⊕ V xxxV xxxV xxx
Hom(x((xx)x)), x) :
V xxxV
x1 ⊕ V xxxV yxxV xxy ⊕ V 1xxV x1xV xxx ⊕ V yxxV xyxV xxx ⊕ V xxxV xxxV xxx, and
V 1xxV
xx ⊕ V yxxV xyxV xxx ⊕ V xxxV 1xxV xx1 ⊕ V xxxV yxxV xxy ⊕ V xxxV xxxV xxx
Hom((x(xx))x), x) :
V 1xxV
xx ⊕ V yxxV xxyV xxx ⊕ V xxxV 1xxV x1x ⊕ V xxxV yxxV xyx ⊕ V xxxV xxxV xxx, and
V xxxV
1x ⊕ V xxxV yxxV xyx ⊕ V 1xxV xx1V xxx ⊕ V yxxV xxyV xxx ⊕ V xxxV xxxV xxx
Hom((((xx)x)x), x) :
V xxxV
1x ⊕ V xxxV yxxV xyx ⊕ V 1xxV x1xV xxx ⊕ V yxxV xyxV xxx ⊕ V xxxV xxxV xxx, and
V 1xxV
xx ⊕ V yxxV xyxV xxx ⊕ V xxxV 1xxV x1x ⊕ V xxxV yxxV xyx ⊕ V xxxV xxxV xxx
Hom((xx)(xx), x) :
V 1xxV
x1 ⊕ V yxxV xxxV xxy ⊕ V xxxV 1xxV x1x ⊕ V xxxV yxxV xyx ⊕ V xxxV xxxV xxx, and
V 1xxV
1x ⊕ V yxxV xxxV xyx ⊕ V xxxV 1xxV xx1 ⊕ V xxxV yxxV xxy ⊕ V xxxV xxxV xxx
where each direct summand space has canonical ordered basis. For exam-
ple V xxxV
xx has basis {vivjvk} where (i, j, k) range from 1 to 2 in the order
(1, 1, 1), (1, 1, 2), (1, 2, 1), etc., and V xxxV
1x has {v1v1xxvx1x, v2v1xxvx1x}.
τ3 :=
⊕ I8 and
τ4 :=
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
Then the pentagon equation P xx,x,x,x is of the form:
(D ⊕ E ⊕ Φ⊗ I2)τ3(I2 ⊕A⊕ Φ̃)τ3(D ⊕ E ⊕ Φ⊗ I2)τ3
= τ3(I2 ⊕B ⊕ Φ̃)τ4(I2 ⊕ F ⊕ Φ̃)
where
φ 0 φ 0 −wb w w −wb 0 0 0 0
0 φ 0 φ 0 0 0 0 −wb w w −wb
φ 0 −φ 0 −wb w −w wb 0 0 0 0
0 φ 0 −φ 0 0 0 0 −wb w −w wb
x 0 x 0 −yb z y −zb 0 0 0 0
x 0 x 0 −zb y z −yb 0 0 0 0
x 0 −x 0 −yb z −y zb 0 0 0 0
−x 0 x 0 zb −y z −yb 0 0 0 0
0 x 0 x 0 0 0 0 −yb z y −zb
0 x 0 x 0 0 0 0 −zb y z −yb
0 x 0 −x 0 0 0 0 −yb z −y zb
0 −x 0 x 0 0 0 0 zb −y z −yb
SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 11
4.7. Solutions. We may assume x = 1 once we normalize basis vector vxxy. Then
from the equations, we get four explicit solution sets for the parameters b, φ, d, w, y,
and z. We list one solution here. All of its values lie in the field Q(
3, i); the
other solutions are obtained by applying Galois automorphisms. The full set of
associativity matrices for this solution is given in Appendix A.
b = i, φ =
, d =
e7πi/12, w =
e2πi/3,
(e−πi/3 + i), z =
e5πi/6.
4.8. Inequivalence of the solutions. To see that these solutions are monoidally
inequivalent, recall from the previous section that for strictified skeletons a natural
equivalence between two solutions to the pentagon equations is limited to change of
basis on the (2, 0) and (2, 1)-stranded morphism spaces, along with permutation of
the strands. In our case permutation of strands does not preserve the fusion rules.
Therefore, we must show that it is not possible to replicate the effect of a nontrivial
Galois automorphism by change of basis choices for the (2, 0) and (2, 1)-stranded
morphism spaces.
The Galois automorphism that fixes
3 and sends i to −i changes the eigenvalues
of the matrix a1x,x,x. However, a
x,x,x is determined by the basis choices v1 and v2,
and its rows and columns are indexed by v1 and v2. Thus changes to v1 and v2
conjugate a1x,x,x by a change of basis matrix, which doesn’t affect its eigenvalues.
Therefore this automorphism does not correspond to a change of basis.
The other two Galois automorphisms send
3 to −
3 and thus change the
value of the (1, 1) entry of axx,x,x. But this entry is invariant under change of basis.
Therefore no Galois automorphism corresponds to a change in basis, and the four
solutions given above are mutually monoidally inequivalent.
5. Proof of Theorem 1 part 2: rigidity structures
This section explicitly computes rigidity structures for the categories given in
the previous section. Rigidity implies that these categories are fusion categories.
Given v1xx ∈ V 1xx, choose a vector vxx1 ∈ V xx1 such that vxx1 ◦v1x,x = id1(see Figure
2). Now we define right death and birth, dx := v
xx : x ⊗ x → 1, bx := 1φv
1 : 1 →
x⊗ x (see Figure 3).
With these definitions, right rigidity is an easy consequence by direct computa-
tion. The following is a graphical version of it:
= = = idx,
= = = idx
where the first and the third equalities are from the definitions above, and the
second equalities are the associativity αxxxx and (α
−1, respectively.
12 TOBIAS J. HAGGE AND SEUNG-MOON HONG
, = 1
Figure 2. Graphical notation of v1xx and v
1 and property
:= =: , := 1
= = 1
Figure 3. Definitions of bx and dx, and elementary properties
(yx)z
αy,x,z // y(xz)
ycx,z
(xy)z
cx,yz
;;wwwwwwwww
αx,y,z
y(zx)
x(yz)
cx,yz // (yz)x
αy,z,x
;;wwwwwwwww
(yx)z
αy,x,z // y(xz)
yc−1z,x
(xy)z
c−1y,xz
;;wwwwwwwww
αx,y,z
y(zx)
x(yz)
c−1yz,x // (yz)x
αy,z,x
;;wwwwwwwww
Figure 4. Hexagon equalities
The same morphisms give a left rigidity structure when treated as left birth and
left death. Treat the objects y and 1 analogously by replacing φ with 1.
6. Proof of Theorem 1 part 3: the absence of braidings
The categories under consideration are known not to be braided (see [9]). How-
ever, once associativity matrices are known it is in principle not difficult to classify
braidings by direct computation. In this section we perform this computation and
show that no braidings are possible.
A braiding consists of a natural family of isomorphisms {cx,y : x ⊗ y → y ⊗ x}
such that two hexagon equalities hold:
(cx,y ⊗ z) ◦ αy,x,z ◦ (y ⊗ cx,z) = αx,y,z ◦ cx,yz ◦ αy,z,x and
((cy,x)
−1 ⊗ z) ◦ αy,x,z ◦ (y ⊗ (cz,x)−1) = αx,y,z ◦ (cyz,x)−1 ◦ αy,z,x.
SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 13
= rzx,y
= r̄zx,y
Figure 5. Isomorphisms Rzx,y and R̄
We define isomorphisms Rzx,y : V
yx → V zxy by f 7→ cx,y ◦ f and R̄zx,y : V zyx → V zxy
by f 7→ (cx,y)−1 ◦ f for any f ∈ V zyx. Figure 5 shows the 1-dimensional case
where rzx,y is nonzero and r̄
x,y = (r
−1. For higher dimensional spaces it can
be expressed as an invertible matrix, also denoted rzx,y on the canonically ordered
basis as before.
With this linear isomorphism, the hexagon equations are equivalent to the equa-
tions
⊕sRsx,zV tys ◦ αty,x,z ◦ ⊕sRsx,yV tsz = αty,z,x ◦ ⊕sV syzRtx,s ◦ αtx,y,z, and
⊕sR̄sx,zV tys ◦ αty,x,z ◦ ⊕sR̄sx,yV tsz = αty,z,x ◦ ⊕sV syzR̄tx,s ◦ αtx,y,z, which we still call
hexagon equations, referred to as Htx,y,z and H̄
x,y,z, respectively. These are illus-
trated graphically in Figure 6).
We show the absence of a braiding by assuming the existence and deriving a
contradiction.
We need five 2-dimensional hexagon equations as follows:
Hxy,x,x : R
y,x ⊗ I2 ◦ αxx,y,x ◦Rxy,x ⊗ I2 = αxx,x,y ◦ I2 ⊗Rxy,x ◦ αxy,x,x
H̄xy,x,x : R̄
y,x ⊗ I2 ◦ αxx,y,x ◦ R̄xy,x ⊗ I2 = αxx,x,y ◦ I2 ⊗ R̄xy,x ◦ αxy,x,x
Hxx,y,x : R
x,x ⊗ 1 ◦ αxy,x,x ◦Rxx,y ⊗ I2 = αxy,x,x ◦ 1⊗Rxx,x ◦ αxx,y,x
H̄xx,y,x : R̄
x,x ⊗ 1 ◦ αxy,x,x ◦ R̄xx,y ⊗ I2 = αxy,x,x ◦ 1⊗ R̄xx,x ◦ αxx,y,x
H1x,x,x : R
x,x ⊗ 1 ◦ α1x,x,x ◦Rxx,x ⊗ 1 = α1x,x,x ◦ I2 ⊗R1x,x ◦ α1x,x,x
These are of the following forms, respectively:
(rxy,x)
= rxy,x
1/b 0
(rxx,y)
= (rxx,y)
1/b 0
rxx,y
(rxy,x)
]−1 [
]−1 [
= d2r1x,x
where
represents the matrix rxx,x.
From the first four equations, we get rxy,x = b, r
x,y = 1/b, −n = rxx,yk, m = rxx,yl,
−n = rxy,xk, which imply k = n = 0 since rxx,y 6= rxy,x as above. Now from the final
one we get l2 = dr1x,x(b+1) and −blm = dr1x,x(1+ b), and the later equality means
l2 = −dr1x,x(1 + b) by substituting m = rxx,yl. We get easily a contradiction for
either case b = ±i.
14 TOBIAS J. HAGGE AND SEUNG-MOON HONG
x y z
⊕sRsx,yV
x y z
αty,x,zoo
x y z
Htx,y,z
x y z
⊕sRsx,zV
]];;;;;;;;
αty,z,x����
x y z
αtx,y,z
]];;;;;;;;
x y z
⊕sV syzR
x,soo
x y z
⊕sR̄sx,yV
� x y z
αty,x,zoo
x y z
H̄tx,y,z
x y z
⊕sR̄sx,zV
αty,z,x����
x y z
αtx,y,z
]];;;;;;;;α
x,y,z
]];;;;;;;;
x y z
⊕sV syzR̄
x,soo
Figure 6. Equivalent hexagon equalities
7. Pivotal structures and sphericity
Let C be a rigid monoidal category. A pivotal structure for C is a monoidal natural
isomorphism π from ∗∗ to Id. A strict pivotal structure is a pivotal structure
which is the identity. In a pivotal monoidal category, the right trace trr of an
endomorphism f : x → x is given by trr(f) = bx ◦ (f ⊗ Idx∗) ◦ (π−1x ⊗ Idx∗) ◦ dx∗ ∈
End(1) ∼= C. The left trace trl is given by trl(f) = bx∗ ◦ (f∗ ⊗ Idx∗∗) ◦ ((πx)∗ ⊗
Idx∗∗) ◦ dx∗∗ . A pivotal monoidal category is spherical if trr = trl.
Pivotal structures may not be unique. For example, in a fusion category with
object types given by a finite group G, group multiplication as tensor product and
trivial associativity matrices, any group homomorphism G → C induces a pivotal
structure. Furthermore, pivotal structures depend on choices of rigidity. However, if
one chooses a new rigidity structure with b′x = cbx and d
x = c
−1dx, then π
x = c
gives a new pivotal structure π′ inducing the same traces as π.
For a strictified skeletal fusion category, we shall assume the rigidity structures
described in Section 3.3. Then ∗∗ is an object fixing monoidal endofunctor. The
isomorphisms Ja,b : a
∗∗ ⊗ b∗∗ → (a ⊗ b)∗∗ associated with ∗∗ considered as a
SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 15
monoidal functor may be taken to be the identity on a⊗ b. If such a category has
a pivotal structure π, it must take the following form: for each strand (x), there is
a scalar tx such that πx = txIdx, and π(x1,...,xn) = tx1 . . . txnId(x1,...xn). Then for
all sequences (x1, . . . , xm) and (y1, . . . , yn), and all f : (x1, . . . , xm) → (y1, . . . , yn),
f∗∗ = tx1 . . . txmt
. . . t−1yn f . This implies that, in particular, t1 = 1.
Writing out the diagrams for b∗∗x and d
x and applying rigidity gives that b
x = bx
and d∗∗x = dx. Thus one must have txtx∗ = 1, and for self-dual strands, tx = ±1;
in this case tx is called the Frobenius-Schur indicator for x.
Furthermore, in a strictified skeleton the left and right trace on a strand may be
rewritten as follows:
trr(f) = t
x bx ◦ (f ⊗ Idx∗) ◦ dx∗ ,
trl(f) = txbx∗ ◦ (Idx∗ ⊗ f) ◦ dx.
Lemma 2. Every pivotal fusion category with self-dual simple objects is spherical.
Proof. The result holds since if x is a self dual strand then bx = bx∗ , dx = dx∗ and
tx = t
x . Thus for each f : x → x we have trr(f) = trl(f). �
Kitaev has shown in [6] that every unitary category admits a spherical structure.
A more general property called pseudo-unitarity is shown in [2] to guarantee a
spherical structure. However, it is not known if every fusion category admits a
pivotal or spherical structure.
For arbitrary fusion categories, one has that ∗ ∗ ∗∗ ∼= Id. This was shown in [2],
using an analog of Radford’s formula for S4 for representation categories of weak
Hopf algebras, which was developed in [8]. The following theorem shows that, in a
strictified skeletal fusion category, a convenient choice of rigidity makes ∗ ∗ ∗∗ the
identity on the nose. Extending the result to general fusion categories via natural
equivalences gives an elementary proof that ∗ ∗ ∗∗ ∼= Id.
Theorem 3. In a strictified skeletal fusion category, there is a choice of rigidity
structures such that ∗ ∗ ∗∗ = Id.
Proof. The functor ∗ ∗ ∗∗ is the identity on (2)-stranded morphisms by rigidity;
it suffices to prove the result for (2, 1) stranded morphisms. Let V = V zxy be a
(2, 1) stranded morphism space with a basis {vi}, and let {wi} be an algebraically
dual basis for the space W = V xyz , in the sense that wi ◦ vj = δijIdz . For any
simple object z, define the right pseudo-trace ptrr of an endomorphism f : z → z
by ptrr(f) = bz ◦ (f ⊗ Idz∗) ◦ dz∗ , and the left pseudo-trace ptrl by ptrl(f) =
bz∗ ◦ (Idz∗ ⊗f)◦dz. This definition is possible because ∗∗ is the identity on objects.
Scale rigidity morphisms if necessary so that for any strand z, ptrr(Idz) = ptrl(Idz).
Because dz and bz∗ are nonzero elements of one dimensional algebraically dual
morphism spaces, ptrr(Idz) 6= 0. One may now exchange left pseudo-traces for
right pseudo-traces, just like with traces in a graphical calculus for a spherical
category.
Figure 7 gives the proof. On the left side, bending arms and pseudo-sphericity
implies that the algebraic dual basis of the basis {w∗∗i } is {∗∗vi}. However, on the
right side the functoriality of the double dual implies that the algebraic dual basis
of {w∗∗i } is {v∗∗i }. Since the left and right double dual are inverse functors, ∗ ∗ ∗∗
is the identity. �
16 TOBIAS J. HAGGE AND SEUNG-MOON HONG
Figure 7. In a strictified skeletal fusion category, with the right
choice of rigidity structures the quadruple dual is the identity.
Even if a category admits a pivotal structure it is not known whether it admits
a spherical pivotal structure. Pictorial considerations do not readily provide an
answer. It is possible, however, to partially describe what a pivotal strictified
skeleton which did not admit a spherical structure would look like.
Let C be a pivotal strictified skeletal fusion category which does not admit a
spherical structure. Choose rigidity morphisms which give a pseudo-spherical struc-
ture as above, and a matching pivotal structure. Then for any object x, one has
the following:
trl(Idx)
trr(Idx)
txptrl(x)
t−1x ptrr(x)
= t2x.
Therefore, C is spherical iff there exists a pivotal structure such that all of the tx
are ±1. Thus there must be some strand x such that tx 6= ±1.
For strands u and v, u ⊗ v has a nontrivial morphism to some object w, and
tutv(tw)
−1 = ±1, since ∗ ∗ ∗∗ = Id. Thus the set of scalars t and their additive
inverses forms a finite subgroup G of C. Note that we can apply any group ho-
momorphism that preserves ±1 to the set of scalars t and get a new set of scalars
t′ which also gives a pivotal structure. At least one product tutv(tw)
−1 must be
equal to −1, or else we could apply the trivial homomorphism to the set of scalars
t to get a new pivotal structure with t′u = 1 for all strands u, which would make C
spherical.
Every finite subgroup of C is a cyclic group of roots of unity. We have |G| = 2k
for some k, and since C is not spherical, |G| ≥ 4. Using a homomorphism which
preserves −1 we may switch to a new pivotal structure which gives |G| = 2k for
some k, where k ≥ 2 to contradict sphericity. Pick an object v with t2v = −1. Then
v is not self dual, and for a simple summand w in v⊗ v, one has w 6= 1 and t2w = 1.
Therefore, C has at least four objects, v, v∗, w and 1. The set of objects u such
that t2u = 1 generates a spherical subcategory C′ with at least two simple objects,
and missing at least two.
SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 17
Lemma 4. Any fusion category which is pivotal but admits no spherical structure
contains at least five simple object types.
Proof. Assume that C has four simple object types, 1, w, v and v′ as above. Then
C′ has two simple object types, and by the classification of fusion categories with
two simple object types in [10], its fusion rules are given by w⊗w ∼= nw⊕1, where
n ∈ {0, 1}. The pivotal structure places limitations on the fusion rules, for example
v⊗w ∼= av⊕bv′ for some a and b in N. An easy calculation shows that C admits only
one associative fusion ring, in which objects and tensor products are given by the
group Z4. Any such category is pseudo-unitary and therefore spherical, as described
in [2], which contradicts the assumption. Therefore, a pivotal fusion category which
can’t be made spherical must have at least five simple object types. �
8. Proof of Theorem 1 part 4: spherical structure calculations
In this section we explicitly compute pivotal structures for the categories found
in Section 4. Since these categories have self dual simple objects, Lemma 2 implies
that they are spherical.
It is not hard to determine whether or not a fusion category is pivotal once
a set of associativity matrices is known. One way is to perform the calculations
directly using the associativity matrices, but there is an easier calculation. In order
to explain this calculation, it is convenient to extend the definition of composition
of morphisms over extra-categorical direct sums of morphism spaces. Suppose f ∈
Mor(a, b) and g ∈ Mor(c, d). Define f◦g as usual if b = c, and f◦g = 0 ∈ Mor(a, d)
otherwise. Extend this definition over direct sums of morphism spaces, distributing
composition over direct sum.
Given a strictified skeletal fusion category C and a set of associativity matrices,
choose bases for the (2, 0) and (2, 1)-stranded morphism spaces compatible with
the associativity matrices and choose rigidity so that for each strand x, the basis
element for V 1x∗x is dx. Define morphisms b = ⊕xbx, d = ⊕xdx, and I = ⊕xIdx,
taking sums over the strands.
Then B acts on
x,y,z V
xy as follows:
B(f) = (I ⊗ I ⊗ b) ◦ (I ⊗ f ⊗ I) ◦ (d⊗ I)
For a single (2, 1)-stranded morphism space, this action amounts to “bending
arms”. The cube of B is the double dual. The action of B on a morphism f ∈ V zxy
is given by the associativity matrix a1z∗,x,y, since (Idz∗ ⊗ f) ◦ dz = (g⊗ idy) ◦ dy for
some g ∈ V y
z∗x implies that B(f) = (Idz∗⊗Idx⊗by)◦(Idz∗⊗f⊗Idy∗)◦(dz⊗Idy∗) =
(Idz∗ ⊗ Idx ⊗ by) ⊗ (g ⊗ Idy ⊗ Idy∗) ◦ (dy ⊗ Idy∗) = g by rigidity. For the fusion
rules at hand, the matrix for B is as follows:
v1 v2 v
v1 (a
xxx)1,1 (a
xxx)1,2 0 0 0
v2 (a
xxx)2,1 (a
xxx)2,2 0 0 0
vyxx 0 0 0 a
yxx 0
vxyx 0 0 0 0 a
vxxy 0 0 a
xxy 0 0
For all of the solutions given in Section 4, B3 is the identity matrix, so the corre-
sponding strictified categories have a strict pivotal structure. Non-strict pivotality
18 TOBIAS J. HAGGE AND SEUNG-MOON HONG
would mean that B3 is a diagonal matrix with eigenvalues determined by a family
of invertible scalars t, coherent as described in Section 7.
Appendix A. Associativity matrices
In this section, we give explicit associativity matrices for the categorical realiza-
tion given in Section 4.
ayy,y,y = a
x,y,y = a
y,y,x = a
x,y,x = a
x,x,y = a
x,x,y = a
y,x,x = a
y,x,x = 1,
ayx,y,x = a
y,x,y = −1,
axx,y,x =
axx,x,y =
axy,x,x =
a1x,x,x =
e7πi/12
ayx,x,x =
e7πi/12
axx,x,x =
eπi/6 1−
e2πi/3 1−
e2πi/3 1−
eπi/6
eπi/6 1−
e2πi/3 − 1−
e2πi/3 − 1−
eπi/6
1 1 − 1
(eπi/6−1) 1
e5πi/6 1
(e−πi/3+i) 1
eπi/3
1 1 1
eπi/3 1
(e−πi/3+i) 1
e5πi/6 − 1
(eπi/6−1)
1 −1 − 1
(eπi/6−1) 1
e5πi/6 − 1
(e−πi/3+i) − 1
eπi/3
−1 1 − 1
eπi/3 − 1
(e−πi/3+i) 1
e5πi/6 − 1
(eπi/6−1)
References
[1] Bruno Buchberger. A theoretical basis for the reduction of polynomials to canonical forms.
SIGSAM Bull., 10(3):19–29, 1976.
[2] Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik. On fusion categories. Ann. of Math.,
162(2):581–642, 2005.
[3] J. Frohlich and T. Kerler. Quantum groups, quantum categories, and quantum field theory,
chapter 4. Number 1542 in Lecture Notes in Mathematics. 1993.
[4] Christian Kassel. Quantum Groups. Springer-Verlag, 1995.
[5] Kazhdan and Hans Wenzl. Reconstructing monoidal categories. Adv. Soviet Math., 16:111–
136, 1993.
[6] Alexei Kitaev. Anyons in an exactly solved model and beyond. Annals of Physics, 321(1):2–
111, 2006.
[7] Saunders Mac Lane. Categories for the Working Mathematician, Second Edition. Springer-
Verlag, 1978.
[8] Dmitri Nikshych. On the structure of weak hopf algebras. And. Math., 170:257–286, 2002.
[9] Victor Ostrik. Pre-modular categories of rank 3. math.CT/0507349.
[10] Victor Ostrik. Fusion categories of rank 2. Math. Res. Lett., 10(2-3):177–183, 2003.
[11] Eric Rowell, Richard Stong, and Zhenghan Wang. in preparation.
[12] Daisuke Tambara and Shigeru Yamagami. Tensor categories with fusion rules of self-duality
for finite abelian groups. J. Algebra, 209:692–707, 1998.
[13] Hans Wenzyl and Imre Tuba. On braided tensor categories of type bcd. J. Reine. Angew.
Math., 581:31–69, 2005.
Department of Mathematics, Indiana University, Bloomington, Indana
E-mail address: thagge@indiana.edu,seuhong@indiana.edu
	1. Introduction
	2. Main theorem and outline
	3. Preliminaries and notational conventions
	3.1. Skeletization
	3.2. Strictification
	3.3. Strictified skeletal fusion categories
	3.4. Remarks
	4. Proof of Theorem ?? part  ??:possible tensor category structures
	4.1. Setting up the pentagon equations
	4.2. Normalizations
	4.3. Associativity matrices
	4.4. Pentagon equations with 1 1 matrices
	4.5. Pentagon equations with 2 2 or 6 6 matrices
	4.6. The pentagon equation with 16 16 matrices
	4.7. Solutions
	4.8. Inequivalence of the solutions
	5. Proof of Theorem ?? part ??: rigidity structures
	6. Proof of Theorem ?? part ??: the absence of braidings
	7. Pivotal structures and sphericity
	8. Proof of Theorem ?? part ??: spherical structure calculations
	Appendix A. Associativity matrices
	References

  We classify all fusion categories for a given set of fusion rules with three
simple object types. If a conjecture of Ostrik is true, our classification
completes the classification of fusion categories with three simple object
types. To facilitate the discussion we describe a convenient, concrete and
useful variation of graphical calculus for fusion categories, discuss
pivotality and sphericity in this framework, and give a short and elementary
re-proof of the fact that the quadruple dual functor is naturally isomorphic to
the identity.

Introduction
Let k be an algebraically closed field. A fusion category C over k is a k-linear
semi-simple rigid monoidal category with finitely many (isomorphism classes of)
simple objects, finite dimensional morphism spaces, and End(1) ∼= k. See [7] or [4]
for definitions, and [2] for many of the known results about fusion categories.
The rank r of C is the number of isomorphism classes of simple objects in C. Let
{xi}1≤i≤r be a set of simple object representatives. The fusion rules for C are a
set of r × r N-valued matrices N = {Ni}1≤i≤r, with (Ni)j,k denoted Nkij or, when
convenient, Nxkxixj , such that xi⊗xj ∼=
1≤k≤r N
ijxk. In the sequel, assume k = C.
Fusion categories appear in representation theory, operator algebras, conformal
field theory, and in constructions of invariants of links, braids, and higher dimen-
sional manifolds. There is currently no general classification of them. Classifications
of fusion categories for various families of fusion rules have been given in work by
Kerler ([3]), Tambara and Yamagami ([12]), Kazhdan and Wenzl ([5]), and Wenzl
and Tuba ([13]).
For a given set of fusion rules, there are only finitely many monoidal natural
equivalence classes of fusion categories. This property is called Ocneanu rigidity (see
[2]). It is not known whether or not the number of fusion categories of a given rank
is finite. If one assumes a modular structure, the possibilities up to rank four have
been classified by Belinschi, Rowell, Stong and Wang in [11]. Ostrik has classified
fusion categories up to rank two in [10], and constructed a finite list of realizable
fusion rules for braided categories up to rank three in [9], in which the number of
categories for each set of fusion rules is known. The rank two classification relies in
an essential way on the theory of modular tensor categories; Ostrik shows that the
quantum double of a rank two category must be modular, and uses the theory of
modular tensor categories to eliminate most of the possibilities. The classification
of modular tensor categories is of independent interest; in many contexts one must
assume modularity.
We consider the only set of rank three fusion rules which is known to be realizable
as a fusion category but which has no braided realizations. Ostrik conjectured in
http://arxiv.org/abs/0704.0208v2
2 TOBIAS J. HAGGE AND SEUNG-MOON HONG
[9] that a classification for this rule set completes the classification of rank three
fusion categories.
The axioms for fusion categories over C reduce to a system of polynomial equa-
tions over C. In this context, Ocneanu rigidity, roughly translated, says that nor-
malization of some of the variables in the equations gives a finite solution set. In
this case, one can compute a Gröbner basis for the system and obtain the solutions
(see [1]). However, normalization becomes complicated when there are i, j, k such
that Nkij > 1. The fusion rules we consider are the smallest realizable set with this
property.
2. Main theorem and outline
Theorem 1 (Main Theorem). Consider the set of fusion rules with three simple
object types, x, y and 1. Let 1 be the trivial object, and let x⊗ x ∼= x⊕ x⊕ y⊕ 1,
x⊗ y ∼= y ⊗ x ∼= x and y ⊗ y ∼= 1. Then the following hold:
(1) Up to monoidal natural equivalence, there are four semisimple tensor cat-
egories with the above fusion rules. A set of associativity matrices for one
of these categories is given in Appendix A. Applying a nontrivial Galois
automorphism to all of the coefficients gives a set of matrices for any one
of the other three categories.
(2) The categories in part 1 are fusion categories.
(3) The categories in part 1 do not admit braidings.
(4) The categories in part 1 are spherical.
The structure of the remainder of the paper is as follows:
Section 3 describes the notation and categorical preliminaries used in later parts
of the paper. It constructs a canonical representative for each monoidal natural
equivalence class of fusion categories. This construction is really just two well
known constructions, skeletization and strictification, applied in sequence. These
constructions, taken together, form a bridge between the category theoretic lan-
guage in the statements of the theorems and the algebra appearing in the proofs.
For some of the calculations in this paper the translation between the category the-
ory and the algebra is already widely known, but there are some subtleties when
discussing pivotal structure that justify the treatment. Section 3 concludes by de-
scribing the algebraic equations corresponding to the axioms for a fusion category,
using the language of strictified skeletons.
Section 4 proves part 1 of Theorem 1. The proof amounts to solving the variety
of polynomial equations defined in the previous section, performing normalizations
along the way in order to simplify calculations. The section ends by arguing that
the nature of the normalizations guarantees that the solutions obtained really are
monoidally inequivalent. Section 5 proves part 2 of Theorem 1 by explicitly com-
puting rigidity structures.
Part 3 of Theorem 1 follows from Ostrik’s classification of rank three braided
fusion categories in [9]. Section 6 gives a direct proof by showing that there are no
solutions to the hexagon equations.
Section 7 defines pivotal and spherical structures and discusses their properties.
The focus is on the question of whether every fusion category is pivotal and spher-
ical. A novel and elementary proof that the quadruple dual functor is naturally
isomorphic to the identity functor is given. This proof makes use of the strictified
skeleton construction developed in Section 3. The section concludes by describing
SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 3
what a pivotal category which does not admit a spherical structure would look like.
In particular, it must have at least five simple objects.
Section 8 proves part 4 of Theorem 1 by computing explicit pivotal structures
for the four categories given in Section 4, and invoking a lemma from Section 7 for
sphericity.
3. Preliminaries and notational conventions
This paper uses the “composition of morphisms” convention for functions as
well as morphisms, and left to right matrix multiplication. For calculations of the
fusion rules, our treatment is similar to [12], but the notation differs superficially
for typographic reasons. The notation captures algebraic data sufficient to classify
a fusion category up to monoidal natural equivalence, and is reviewed later in this
section.
We assume that the reader is familiar with the notions of a monoidal category,
a monoidal functor and a monoidal equivalence; for precise definitions, see [7].
Recall that a monoidal category is equipped with an associative bifunctor ⊗ and
a distinguished object 1. Reassociation of tensor factors in a monoidal category is
described by a natural isomorphism of trifunctors α : (−⊗−)⊗− → −⊗ (−⊗−).
Tensor products with 1 have natural isomorphisms ρ : −⊗1 → − and λ : 1⊗− → −.
These isomorphisms are subject to a coherency condition, namely that for any pair
of multifunctors there is at most one natural isomorphism between them which may
be constructed from λ, ρ, α and their inverses, along with Id and ⊗. This coherency
condition is well known to be equivalent to the statement that the category satisfies
the pentagon and triangle axioms (see [7] for a proof).
The triangle equations are the equations ρx⊗y = αx,1,y ◦ (x⊗λy) for all ordered
pairs (x, y) of objects. Here, and in the sequel when the context is unambiguous,
the name of an object is used as a shorthand for the identity morphism on that
object. The pentagon equations, defined for all tuples of objects (w, x, y, z), are as
follows (see also Figure 1):
(αx,y,z ⊗ w) ◦ αx,y⊗z,w ◦ (x⊗ αy,z,w) = αx⊗y,z,w ◦ αx,y,z⊗w,
When studying fusion categories up to monoidal equivalence, one may choose
categories within an equivalence class which have desirable attributes. Since the
categorical properties considered in this paper (fusion rule structure, monoidality,
pivotality, sphericity, presence of braidings) are all well known to be preserved un-
der monoidal equivalence, the desirable attributes may be assumed without loss. In
particular, one may construct, given an arbitrary fusion category, a canonical rep-
resentative for that category’s equivalence class in which one may replace instances
of the words “is isomorphic to” with “equals”.
3.1. Skeletization. The skeleton CSKEL of an arbitrary category C is any full
subcategory of C containing exactly one object from each isomorphism class in
C. If C is semi-simple, every object in C is isomorphic to a direct sum of simple
objects in CSKEL. One may then assume without loss that the objects of CSKEL
consist of simple object representatives and direct sums of such.
It is a well known fact that CSKEL may be given a monoidal structure such
that CSKEL and C are monoidally equivalent. The proof is a straightforward but
tedious extension of Maclane’s proof of the natural equivalence an ordinary category
4 TOBIAS J. HAGGE AND SEUNG-MOON HONG
and its skeleton (see section IV.4 in [7]). In that proof, one defines a family of
isomorphisms ix from objects x to their isomorphic representatives in CSKEL and
uses it to construct a pair of functors F and G which give a natural equivalence.
For the extension, the ix are used to define the tensor product functor on CSKEL,
as well as α, λ, ρ and the monoidal structures for F and G. One then writes out
all of the relevant commutative diagrams and removes any compositions ix ◦ i−1x .
The result in each case is a commutative diagram in C.
3.2. Strictification. Given a monoidal category C, one may construct a strict
monoidal category CST R equivalent to C. In a strict monoidal category, α, λ and ρ
are the identity. It is common practice to assume that a monoidal category is strict
without explicit reference to the construction. However, by using the construction
explicitly we will be able to pick a canonical representative for an equivalence class of
monoidal categories and provide a natural interpretation of the graphical calculus.
Strictification of a monoidal category is analogous to the construction of a tensor
algebra; it gives an equivalent strict category CSTR by replacing the tensor product
with a strictly associative formal tensor product. The objects of CSTR are finite
sequences of objects in C. Morphism spaces of the form
Mor((a1, a2, . . . , am−1, am), (b1, b2, . . . , bn−1, bn))
are given by
Mor(a1 ⊗ (a2 ⊗ . . . (am−1 ⊗ am) . . . ), b1 ⊗ (b2 ⊗ . . . (bn−1 ⊗ bn) . . . ))
in C. The tensor product on objects is just concatenation of sequences, for mor-
phisms it is the tensor product in C pre and post-composed with appropriate asso-
ciativity morphisms. Monoidal equivalence of C with CST R is proven in section XI.3
of [7].
It is not usually possible to make a fusion category strict and skeletal at the
same time. However, the category (CSKEL)ST R, while not a skeleton, is still unique
up to strict natural equivalence. Also, it is a categorical realization of a graphical
calculus, as will shortly become clear. The next subsection describes what strictified
skeleta of fusion categories look like, up to strict equivalence.
3.3. Strictified skeletal fusion categories. A strictified skeletal fusion category
C is as follows: Let N be a set of fusion rules for a set of objects S. Then the objects
in C are multisets of finite sequences of elements of S. C has a tensor product ⊗,
which is defined on objects by pairwise concatenation of sequences, distributed over
elements of multisets. Direct sum of objects is given by multiset disjoint union.
A strand is an object which is a sequence of length one. Strands correspond
to simple object types. If x, y and z are strands, define Mor(x ⊗ y, z) to be a k
vector space isomorphic to kN
xy . For brevity, V yx will denote Mor(x, y), and tensor
products will be omitted when the context is clear. A morphism is (n,m)-stranded
if its source and target are sequences of length n and m, respectively. A morphism
is (n)-stranded if it is (m,n−m)-stranded for some 0 ≤ m ≤ n.
Semi-simplicity of C means that for all objects w, x, y and z there are vector space
isomorphisms
v∈S V
xy ⊗ V zwv ∼= V zwxy ∼=
v∈S V
wx ⊗ V zvy . The first isomorphism
is given by f ⊗ g → (Idw ⊗ f) ◦ g, the inverse of the second by h⊗ l → (h⊗ Idy) ◦ l.
The composition of the two isomorphisms is denoted αzw,x,y. Additionally, each
morphism space V zxy has an algebraically dual space V
z , in the sense that there
are bases {vi}i ⊂ V zxy and {wi}i ⊂ V xyz such that wi ◦ vj = δijIdz.
SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 5
In a strictified skeleton, the trivial object 1 is the zero length sequence. There
is a strand which is isomorphic to the trivial object, but not equal. This strand
shall be taken to be 1 in the sequel. This choice makes the category non-strict,
since λ and ρ are no longer the identity, but it is convenient for graphical calculus
purposes.
(Right) rigidity in a strictified skeleton means that there is a set involution ∗ on
the strands, and each strand x has morphisms bx : 1 → x⊗ x∗ and dx : x∗ ⊗ x → 1
such that (Idx∗ ⊗ bx) ◦ (dx ⊗ Idx∗) = Idx∗ and (bx ⊗ Idx) ◦ (Idx ⊗ dx) = Idx.
This implies that Nzxy = N
z∗x. Left rigidity is similar, and in the sequel, the right
rigidity morphism for x∗ will be defined to be the left rigidity morphism for x.
Define ∗, b, and d on concatenations of strands such that dx⊗y = (Idy∗ ⊗ dx ⊗
Idy) ◦ dy and extend to direct sums. Then there is a contravariant (right) dual
functor ∗ which sends f ∈ V yx to f∗ = (Idy∗ ⊗ bx)◦ (Idy∗ ⊗f ⊗ Idx∗)◦ (dy ⊗ Idx∗) ∈
y∗ . The definition of a left dual functor is similar, and the two duals are inverse
functors by rigidity.
Monoidality for a strictified skeletal fusion category implies that the α are iden-
tity morphisms. For this to be true, it is necessary and sufficient that the following
equation holds for all objects x, y, z, w, and u. Each instance will be referred to as
Pux,y,z,w in the sequel.
αty,z,wV
V syzα
x,s,w ◦
αtx,y,zV
V szwV
V syzV
V syzV
V sxyV
is equal to
V szwα
x,y,s ◦ τ ◦
V sxyα
s,z,w :
V szwV
V szwV
V sxyV
V sxyV
Here ⊗ for vector spaces and morphisms are omitted, and τ is the isomorphism
interchanging the first and the second factors of vector space tensor products (see
Figure 1).
3.4. Remarks.
(1) Every fusion category is monoidally naturally equivalent to a strictified
skeleton. Also, two naturally equivalent strictified skeleta have an invert-
ible equivalence functor that takes strands to strands. This implies that
equivalences are given by permutations of strands along with changes of
basis on the (2, 0) and (2, 1)-stranded morphism spaces.
(2) The functor ∗∗ fixes objects. The isomorphisms Jx,y : x∗∗⊗y∗∗ → (x⊗y)∗∗
associated with ∗∗ in the definition of a monoidal functor (see [7]) may be
taken to be trivial. There is an invertible scalar worth of freedom in the
choice of each bx, dx pair.
(3) Semi-simplicity allows every morphism to be built up from (3)-stranded
morphisms. Choosing bases for the (3)-stranded morphisms allows mor-
phisms in C to be characterized as undirected trivalent graphs with labeled
6 TOBIAS J. HAGGE AND SEUNG-MOON HONG
((x⊗ y)⊗ z)⊗ w
αx,y,zw
uukkk
αxy,z,w
(x⊗ (y ⊗ z))⊗ w
αx,yz,w
(a) (x ⊗ y)⊗ (z ⊗ w)
αx,y,zw
x⊗ ((y ⊗ z)⊗ w)
xαy,z,w // x⊗ (y ⊗ (z ⊗ w))
s,t V
s,t V
V sxyα
s,z,woo
s,t V
αtx,y,zV
s,t V
s,t V
V syzα
x,s,w
s,t V
y,z,wV
V szwα
x,y,s
Figure 1. (a) Pentagon equality and (b) corresponding equality
edges and vertices, subject to associativity relations given by the penta-
gon equations. The labels for the edges are isomorphism types of simple
objects; the labels for the vertices are basis vectors for the corresponding
morphism spaces. This gives a categorically precise interpretation of an
arrowless graphical calculus for C.
(4) If C is pivotal (see Section 7 for the definition), a well known construction
allows one to add a second copy of each object and get a strict pivotal
category. This construction gives a graphical calculus with arrows on the
strands.
(5) Strictified skeleta give any categorical structure preserved under natural
equivalence (and any functorial property preserved under natural isomor-
phism) a purely algebraic description.
4. Proof of Theorem 1 part 1:possible tensor category structures
In this section we classify, up to monoidal equivalence, all C-linear semisimple
tensor categories with fusion rules given in Theorem 1. This amounts to solving
the matrix equations described in the previous section. The simplest equations
(those involving 1× 1 matrices) are solved first, and normalizations are performed
as necessary in order to simplify the equations.
4.1. Setting up the pentagon equations. The fusion rules are given by x⊗x ∼=
1⊕y⊕x⊕x, x⊗y ∼= y⊗x ∼= x, and y⊗y ∼= 1. The non-trivial vector spaces are V 111,
V x1x ,V
x1, V
1y , V
y1, V
xy, V
yx, V
yy, V
xx, V
xx,and V
xx, and they are all 1-dimensional
except the last space which is 2-dimensional.
Let’s choose basis vectors in each space. If we fix any non-zero vector v111 ∈ V 111,
then there are unique vectors vx1x ∈ V x1x, vxx1 ∈ V xx1,v
1y ∈ V
1y, and v
y1 ∈ V
y1 such
that the triangle equality holds. For the other spaces, choose any non-zero vectors
SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 7
in each space and denote them by vxxy ∈ V xxy, vxyx ∈ V xyx, v1yy ∈ V 1yy, v1xx ∈ V 1xx,
vyxx ∈ V yxx, v1 and v2 ∈ V xxx where the two vectors v1 and v2 are linearly independent.
There are 30 associativities. It is a well known fact that if at least one of
the bottom objects is 1 then the associativity is trivial. That is, with the above
basis choices the matrix for αzu,v,w is trivial if at least one of the u, v and w is 1.
Now we have ten non-trivial 1-dimensional associativities, αyy,y,y,α
x,y,y,α
y,y,x,α
x,y,x,
αyx,y,x,α
y,x,y,α
x,x,y,α
x,x,y, α
y,x,x, and α
y,x,x, five non-trivial 2-dimensional ones,
αxx,y,x,α
x,x,y,α
y,x,x,α
x,x,x, and α
x,x,x, and one 6-dimensional one, α
x,x,x.
4.2. Normalizations. With the above basis choices we obtain a basis for each
tensor product of vector spaces in a canonical way and can parameterize each
associativity and pentagon equation. However, at this point our basis elements
have not been uniquely specified, and we should expect to obtain solutions with
free parameters. As the calculation progresses it will be convenient to simplify the
pentagon equations by requiring certain coefficients of certain associativity matrices
to be 1 or 0. These normalizations should be thought of as restrictions on the basis
choices made above. Normalizations simplify the equations and have an additional
advantage: once the set of possible bases is sufficiently restricted, Ocneanu rigidity
[2] guarantees a finite set of possibilities for the associativity matrices of fusion
categories, which can be found algorithmically by computing a Gröbner basis.
4.3. Associativity matrices. The following are the 1-dimensional associativities:
αyy,y,y : v
y1 7→ ayy,y,yv1yyv
αxx,y,y : v
x1 7→ axx,y,yvxxyvxxy
αxy,y,x : v
yx 7→ axy,y,xv1yyvx1x
α1x,y,x : v
xx 7→ a1x,y,xvxxyv1xx
αyx,y,x : v
xx 7→ ayx,y,xvxxyvyxx
αxy,x,y : v
yx 7→ axy,x,yvxyxvxxy
α1x,x,y : v
xx 7→ a1x,x,yvyxxv1yy
αyx,x,y : v
xx 7→ ayx,x,yv1xxv
α1y,x,x : v
yy 7→ a1y,x,xvxyxv1xx
αyy,x,x : v
y1 7→ ayy,x,xvxyxvyxx
where associativity coefficients are all non-zero.
For 2-dimensional and 6-dimensional associativities we need to fix the ordering
of basis elements in each Hom vector space. The orderings are as follows:
{vxyxv1, vxyxv2} for V xx(yx), {vxxyv1, vxxyv2} for V x(xy)x,
{vxxyv1, vxxyv2} for V xx(xy), {v1v
xy, v2v
xy} for V x(xx)y,
{v1vxyx, v2vxyx} for V xy(xx), {v
yxv1, v
yxv2} for V x(yx)x,
{v1v1xx, v2v1xx} for V 1x(xx), {v1v
xx, v2v
xx} for V 1(xx)x,
{v1vyxx, v2vyxx} for V
x(xx)
, {v1vyxx, v2vyxx} for V
(xx)x
{v1xxvxx1, vyxxvxxy, v1v1, v1v2, v2v1, v2v2} for V xx(xx),
and {v1xxvx1x, vyxxvxyx, v1v1, v1v2, v2v1, v2v2} for V x(xx)x.
With these ordered bases, each associativity has a matrix form (recall that we are
using the right multiplication convention). That is, αxx,y,x is given by the invertible
2 × 2 matrix axx,y,x, and αxx,x,y is given by the invertible 2 × 2 matrix axx,x,y,etc.,
and finally αxx,x,x is given by the invertible 6× 6 matrix axx,x,x.
8 TOBIAS J. HAGGE AND SEUNG-MOON HONG
4.4. Pentagon equations with 1× 1 matrices. Considering only nontrivial as-
sociativities, there are 17 1-dimensional pentagon equations, 14 2-dimensional pen-
tagon equations, 6 6-dimensional ones, and 1 16-dimensional one. Without redun-
dancy, the following are the 1-dimensional equations:
P xx,y,y,y : a
y,y,y a
x,y,y = a
x,y,y.
P 1x,x,y,y : a
x,y,ya
x,x,y a
x,x,y = 1 ,
P 1x,y,x,y : a
y,x,y a
x,y,x = a
x,y,x ,
P yx,y,x,y : :a
y,x,y a
x,y,x = a
x,y,x ,
P 1x,y,y,x : a
y,y,x a
x,y,y = (a
x,y,x)
P yx,y,y,x : a
y,y,x a
x,y,y = (a
x,y,x)
P xy,x,y,y :(a
y,x,y)
2 = 1 ,
P 1y,y,x,x : a
y,x,x a
y,x,x a
y,y,x = 1 ,
P 1y,x,x,y : a
x,x,y a
y,x,x = a
y,x,x a
x,x,y
If we normalize the basis we may assume axy,y,x, a
x,y,x and a
x,x,y to be 1 (for
normalization see [12] or [6]), and we can solve the above 1-dimensional equations.
Here is the solution:
ayy,y,y = a
x,y,y = a
x,x,y = 1, a
y,x,y = a
x,y,x = ±1, a1y,x,x = ayy,x,x = ±1.
Let’s say g := axy,x,y = a
x,y,x and h := a
y,x,x = a
y,x,x in the sequel. Also let
A := axx,y,x, B := a
x,x,y, D := a
x,x,x, E := a
x,x,x, F := a
y,x,x and Φ := a
x,x,x for
brevity.
4.5. Pentagon equations with 2× 2 or 6× 6 matrices. Now, the following are
the 2-dimensional pentagon equations using the above 1-dimensional solutions:
P xy,y,x,x : F
2 = Id2
P xy,x,y,x : gAF = FA
P xx,y,y,x : A
2 = Id2
P xy,x,x,y : gBF = FB
P xx,y,x,y : gBA = AB
P xx,x,y,y : B
2 = Id2
P 1y,x,x,x : EF = D
P yy,x,x,x : DF = E
P 1x,y,x,x : FDA = D
P yx,y,x,x : FEA = gE
P 1x,x,y,x : ADB = D
P yx,x,y,x : AEB = gE
P 1x,x,x,y : BE = D
P yx,x,x,y : BD = E
It should be noted that for this particular category the large number of one
dimensional morphism spaces gives us q-commutativity relations and matrices with
±1 eigenvalues, which are of great help when simplifying the pentagon equations
by hand.
To analyze 2-dimensional and 6-dimensional pentagon equations, at first let’s
look at the isomorphism τ interchanging the first and the second factors of tensor
products. This change of basis is necessary for 6-dimensional pentagon equations
because the image basis of the matrix for αux,y,zw and the domain basis of the matrix
for αuxy,z,w may not be the same. For P
x,y,x,x, τ is an isomorphism from the space
V 1xxV
x1⊕V yxxV xxyV xxy⊕V xxxV xxyV xxx to V xxyV 1xxV xx1⊕V xxyV yxxV xxy⊕V xxyV xxxV xxx, both of
SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 9
which correspond to Hom((x⊗ y)⊗ (x⊗x), x). With the canonically ordered basis
{v1xxvxxyvxx1, vyxxvxxyvxxy, vivxxyvj} and {vxxyv1xxvxx1, vxxyvyxxvxxy, vxxyvivj}, respectively,
τ turns out to be I6. For P
y,x,x,x, P
x,x,y,x and P
x,x,x,y, τ is also I6. But for P
x,x,x,x,
it is τ1, and for P
x,x,x,x, it is τ2, defined as follows:
τ1 :=
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 0 1 0
0 0 0 1 0 0
0 0 0 0 0 1
, τ2 :=
0 1 0 0 0 0
1 0 0 0 0 0
0 0 1 0 0 0
0 0 0 0 1 0
0 0 0 1 0 0
0 0 0 0 0 1
Here are the 6 6-dimensional pentagon equations:
P xy,x,x,x ; Φ(I2 ⊕ I2 ⊗ F )
⊕ F ⊗ I2
⊕ I2 ⊗ F
P xx,y,x,x ;
⊕ F ⊗ I2
⊕A⊗ I2
= (I2 ⊕ I2 ⊗A)Φ
P xx,x,y,x ;
⊕A⊗ I2
⊕B ⊗ I2
= Φ(I2 ⊕ I2 ⊗A)
P xx,x,x,y ;
⊕B ⊗ I2
(I2 ⊕ I2 ⊗ B)Φ = Φ
⊕ I2 ⊗B
P 1x,x,x,x ; (I2 ⊕ I2 ⊗D)Φ = (I2 ⊕ I2 ⊗D)τ1
⊕ I2 ⊗D
P yx,x,x,x ; Φ
⊕ I2 ⊗ E
Φ = (I2 ⊕ I2 ⊗ E)τ2
⊕ I2 ⊗ E
If we normalize the basis {v1, v2} of V xxx, we may assume A is of the form
and then get g = −1 and h = 1 using the above equations. Following is the
computation for this:
At first we may assume that matrix A is of the Jordan canonical form, then
A = ±I2 or
from P xx,y,y,x. We eliminate the possibility A = ±I2 from
P xy,x,y,x, P
x,y,x,x and P
y,x,x,x which imply respectively that g = 1, F = ±I2 and
then det(Φ) = 0, since the first two columns of Φ are scalar multiples of each other.
So we conclude A =
. Now we eliminate the possibility g = 1 using P xy,x,y,x,
P xy,y,x,x and P
y,x,x,x, which imply F is a diagonal matrix, with entries ±1 and then
det(Φ) = 0, respectively. For the case A =
and g = −1, F is of the form
[ 0 f
1/f 0
from P xy,x,y,x and P
y,y,x,x, and B is of the form
1/b 0
from P xx,y,x,y and
P xx,x,y,y. If h = −1, the first column of Φ has to be zero by comparing the first and
the second columns of P xy,x,x,x, P
x,y,x,x, P
x,x,y,x, and P
x,x,x,y.
At this point we have fixed all 1-dimensional associativity matrices.
From the above equations, we get
[ 0 f
1/f 0
for F and
1/b 0
for B with the
relation f2 + b2 = 0 from P xy,x,x,y. We note that the diagonalization of A defines
each basis element v1 and v2 only up to choice of a nonzero scalar. By using up
one of these degrees of freedom, we may assume f = 1. Then from the above
6-dimensional equations, we get the following:
D = d
E = d
φ φ −wb w w −wb
φ −φ −wb w −w wb
x x −yb z y −zb
x x −zb y z −yb
x −x −yb z −y zb
−x x zb −y z −yb
10 TOBIAS J. HAGGE AND SEUNG-MOON HONG
4.6. The pentagon equation with 16 × 16 matrices. Now we analyze the 16-
dimensional pentagon equation P xx,x,x,x. It is convenient to express each Hom vector
space in two different ways and put basis permutation matrices into the pentagon
equation. The following are two expressions with ordered direct sum.
Hom(x(x(xx)), x) :
V xxxV
x1 ⊕ V xxxV yxxV xxy ⊕ V 1xxV xx1V xxx ⊕ V yxxV xxyV xxx ⊕ V xxxV xxxV xxx, and
V 1xxV
xx ⊕ V yxxV xxyV xxx ⊕ V xxxV 1xxV xx1 ⊕ V xxxV yxxV xxy ⊕ V xxxV xxxV xxx
Hom(x((xx)x)), x) :
V xxxV
x1 ⊕ V xxxV yxxV xxy ⊕ V 1xxV x1xV xxx ⊕ V yxxV xyxV xxx ⊕ V xxxV xxxV xxx, and
V 1xxV
xx ⊕ V yxxV xyxV xxx ⊕ V xxxV 1xxV xx1 ⊕ V xxxV yxxV xxy ⊕ V xxxV xxxV xxx
Hom((x(xx))x), x) :
V 1xxV
xx ⊕ V yxxV xxyV xxx ⊕ V xxxV 1xxV x1x ⊕ V xxxV yxxV xyx ⊕ V xxxV xxxV xxx, and
V xxxV
1x ⊕ V xxxV yxxV xyx ⊕ V 1xxV xx1V xxx ⊕ V yxxV xxyV xxx ⊕ V xxxV xxxV xxx
Hom((((xx)x)x), x) :
V xxxV
1x ⊕ V xxxV yxxV xyx ⊕ V 1xxV x1xV xxx ⊕ V yxxV xyxV xxx ⊕ V xxxV xxxV xxx, and
V 1xxV
xx ⊕ V yxxV xyxV xxx ⊕ V xxxV 1xxV x1x ⊕ V xxxV yxxV xyx ⊕ V xxxV xxxV xxx
Hom((xx)(xx), x) :
V 1xxV
x1 ⊕ V yxxV xxxV xxy ⊕ V xxxV 1xxV x1x ⊕ V xxxV yxxV xyx ⊕ V xxxV xxxV xxx, and
V 1xxV
1x ⊕ V yxxV xxxV xyx ⊕ V xxxV 1xxV xx1 ⊕ V xxxV yxxV xxy ⊕ V xxxV xxxV xxx
where each direct summand space has canonical ordered basis. For exam-
ple V xxxV
xx has basis {vivjvk} where (i, j, k) range from 1 to 2 in the order
(1, 1, 1), (1, 1, 2), (1, 2, 1), etc., and V xxxV
1x has {v1v1xxvx1x, v2v1xxvx1x}.
τ3 :=
⊕ I8 and
τ4 :=
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
Then the pentagon equation P xx,x,x,x is of the form:
(D ⊕ E ⊕ Φ⊗ I2)τ3(I2 ⊕A⊕ Φ̃)τ3(D ⊕ E ⊕ Φ⊗ I2)τ3
= τ3(I2 ⊕B ⊕ Φ̃)τ4(I2 ⊕ F ⊕ Φ̃)
where
φ 0 φ 0 −wb w w −wb 0 0 0 0
0 φ 0 φ 0 0 0 0 −wb w w −wb
φ 0 −φ 0 −wb w −w wb 0 0 0 0
0 φ 0 −φ 0 0 0 0 −wb w −w wb
x 0 x 0 −yb z y −zb 0 0 0 0
x 0 x 0 −zb y z −yb 0 0 0 0
x 0 −x 0 −yb z −y zb 0 0 0 0
−x 0 x 0 zb −y z −yb 0 0 0 0
0 x 0 x 0 0 0 0 −yb z y −zb
0 x 0 x 0 0 0 0 −zb y z −yb
0 x 0 −x 0 0 0 0 −yb z −y zb
0 −x 0 x 0 0 0 0 zb −y z −yb
SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 11
4.7. Solutions. We may assume x = 1 once we normalize basis vector vxxy. Then
from the equations, we get four explicit solution sets for the parameters b, φ, d, w, y,
and z. We list one solution here. All of its values lie in the field Q(
3, i); the
other solutions are obtained by applying Galois automorphisms. The full set of
associativity matrices for this solution is given in Appendix A.
b = i, φ =
, d =
e7πi/12, w =
e2πi/3,
(e−πi/3 + i), z =
e5πi/6.
4.8. Inequivalence of the solutions. To see that these solutions are monoidally
inequivalent, recall from the previous section that for strictified skeletons a natural
equivalence between two solutions to the pentagon equations is limited to change of
basis on the (2, 0) and (2, 1)-stranded morphism spaces, along with permutation of
the strands. In our case permutation of strands does not preserve the fusion rules.
Therefore, we must show that it is not possible to replicate the effect of a nontrivial
Galois automorphism by change of basis choices for the (2, 0) and (2, 1)-stranded
morphism spaces.
The Galois automorphism that fixes
3 and sends i to −i changes the eigenvalues
of the matrix a1x,x,x. However, a
x,x,x is determined by the basis choices v1 and v2,
and its rows and columns are indexed by v1 and v2. Thus changes to v1 and v2
conjugate a1x,x,x by a change of basis matrix, which doesn’t affect its eigenvalues.
Therefore this automorphism does not correspond to a change of basis.
The other two Galois automorphisms send
3 to −
3 and thus change the
value of the (1, 1) entry of axx,x,x. But this entry is invariant under change of basis.
Therefore no Galois automorphism corresponds to a change in basis, and the four
solutions given above are mutually monoidally inequivalent.
5. Proof of Theorem 1 part 2: rigidity structures
This section explicitly computes rigidity structures for the categories given in
the previous section. Rigidity implies that these categories are fusion categories.
Given v1xx ∈ V 1xx, choose a vector vxx1 ∈ V xx1 such that vxx1 ◦v1x,x = id1(see Figure
2). Now we define right death and birth, dx := v
xx : x ⊗ x → 1, bx := 1φv
1 : 1 →
x⊗ x (see Figure 3).
With these definitions, right rigidity is an easy consequence by direct computa-
tion. The following is a graphical version of it:
= = = idx,
= = = idx
where the first and the third equalities are from the definitions above, and the
second equalities are the associativity αxxxx and (α
−1, respectively.
12 TOBIAS J. HAGGE AND SEUNG-MOON HONG
, = 1
Figure 2. Graphical notation of v1xx and v
1 and property
:= =: , := 1
= = 1
Figure 3. Definitions of bx and dx, and elementary properties
(yx)z
αy,x,z // y(xz)
ycx,z
(xy)z
cx,yz
;;wwwwwwwww
αx,y,z
y(zx)
x(yz)
cx,yz // (yz)x
αy,z,x
;;wwwwwwwww
(yx)z
αy,x,z // y(xz)
yc−1z,x
(xy)z
c−1y,xz
;;wwwwwwwww
αx,y,z
y(zx)
x(yz)
c−1yz,x // (yz)x
αy,z,x
;;wwwwwwwww
Figure 4. Hexagon equalities
The same morphisms give a left rigidity structure when treated as left birth and
left death. Treat the objects y and 1 analogously by replacing φ with 1.
6. Proof of Theorem 1 part 3: the absence of braidings
The categories under consideration are known not to be braided (see [9]). How-
ever, once associativity matrices are known it is in principle not difficult to classify
braidings by direct computation. In this section we perform this computation and
show that no braidings are possible.
A braiding consists of a natural family of isomorphisms {cx,y : x ⊗ y → y ⊗ x}
such that two hexagon equalities hold:
(cx,y ⊗ z) ◦ αy,x,z ◦ (y ⊗ cx,z) = αx,y,z ◦ cx,yz ◦ αy,z,x and
((cy,x)
−1 ⊗ z) ◦ αy,x,z ◦ (y ⊗ (cz,x)−1) = αx,y,z ◦ (cyz,x)−1 ◦ αy,z,x.
SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 13
= rzx,y
= r̄zx,y
Figure 5. Isomorphisms Rzx,y and R̄
We define isomorphisms Rzx,y : V
yx → V zxy by f 7→ cx,y ◦ f and R̄zx,y : V zyx → V zxy
by f 7→ (cx,y)−1 ◦ f for any f ∈ V zyx. Figure 5 shows the 1-dimensional case
where rzx,y is nonzero and r̄
x,y = (r
−1. For higher dimensional spaces it can
be expressed as an invertible matrix, also denoted rzx,y on the canonically ordered
basis as before.
With this linear isomorphism, the hexagon equations are equivalent to the equa-
tions
⊕sRsx,zV tys ◦ αty,x,z ◦ ⊕sRsx,yV tsz = αty,z,x ◦ ⊕sV syzRtx,s ◦ αtx,y,z, and
⊕sR̄sx,zV tys ◦ αty,x,z ◦ ⊕sR̄sx,yV tsz = αty,z,x ◦ ⊕sV syzR̄tx,s ◦ αtx,y,z, which we still call
hexagon equations, referred to as Htx,y,z and H̄
x,y,z, respectively. These are illus-
trated graphically in Figure 6).
We show the absence of a braiding by assuming the existence and deriving a
contradiction.
We need five 2-dimensional hexagon equations as follows:
Hxy,x,x : R
y,x ⊗ I2 ◦ αxx,y,x ◦Rxy,x ⊗ I2 = αxx,x,y ◦ I2 ⊗Rxy,x ◦ αxy,x,x
H̄xy,x,x : R̄
y,x ⊗ I2 ◦ αxx,y,x ◦ R̄xy,x ⊗ I2 = αxx,x,y ◦ I2 ⊗ R̄xy,x ◦ αxy,x,x
Hxx,y,x : R
x,x ⊗ 1 ◦ αxy,x,x ◦Rxx,y ⊗ I2 = αxy,x,x ◦ 1⊗Rxx,x ◦ αxx,y,x
H̄xx,y,x : R̄
x,x ⊗ 1 ◦ αxy,x,x ◦ R̄xx,y ⊗ I2 = αxy,x,x ◦ 1⊗ R̄xx,x ◦ αxx,y,x
H1x,x,x : R
x,x ⊗ 1 ◦ α1x,x,x ◦Rxx,x ⊗ 1 = α1x,x,x ◦ I2 ⊗R1x,x ◦ α1x,x,x
These are of the following forms, respectively:
(rxy,x)
= rxy,x
1/b 0
(rxx,y)
= (rxx,y)
1/b 0
rxx,y
(rxy,x)
]−1 [
]−1 [
= d2r1x,x
where
represents the matrix rxx,x.
From the first four equations, we get rxy,x = b, r
x,y = 1/b, −n = rxx,yk, m = rxx,yl,
−n = rxy,xk, which imply k = n = 0 since rxx,y 6= rxy,x as above. Now from the final
one we get l2 = dr1x,x(b+1) and −blm = dr1x,x(1+ b), and the later equality means
l2 = −dr1x,x(1 + b) by substituting m = rxx,yl. We get easily a contradiction for
either case b = ±i.
14 TOBIAS J. HAGGE AND SEUNG-MOON HONG
x y z
⊕sRsx,yV
x y z
αty,x,zoo
x y z
Htx,y,z
x y z
⊕sRsx,zV
]];;;;;;;;
αty,z,x����
x y z
αtx,y,z
]];;;;;;;;
x y z
⊕sV syzR
x,soo
x y z
⊕sR̄sx,yV
� x y z
αty,x,zoo
x y z
H̄tx,y,z
x y z
⊕sR̄sx,zV
αty,z,x����
x y z
αtx,y,z
]];;;;;;;;α
x,y,z
]];;;;;;;;
x y z
⊕sV syzR̄
x,soo
Figure 6. Equivalent hexagon equalities
7. Pivotal structures and sphericity
Let C be a rigid monoidal category. A pivotal structure for C is a monoidal natural
isomorphism π from ∗∗ to Id. A strict pivotal structure is a pivotal structure
which is the identity. In a pivotal monoidal category, the right trace trr of an
endomorphism f : x → x is given by trr(f) = bx ◦ (f ⊗ Idx∗) ◦ (π−1x ⊗ Idx∗) ◦ dx∗ ∈
End(1) ∼= C. The left trace trl is given by trl(f) = bx∗ ◦ (f∗ ⊗ Idx∗∗) ◦ ((πx)∗ ⊗
Idx∗∗) ◦ dx∗∗ . A pivotal monoidal category is spherical if trr = trl.
Pivotal structures may not be unique. For example, in a fusion category with
object types given by a finite group G, group multiplication as tensor product and
trivial associativity matrices, any group homomorphism G → C induces a pivotal
structure. Furthermore, pivotal structures depend on choices of rigidity. However, if
one chooses a new rigidity structure with b′x = cbx and d
x = c
−1dx, then π
x = c
gives a new pivotal structure π′ inducing the same traces as π.
For a strictified skeletal fusion category, we shall assume the rigidity structures
described in Section 3.3. Then ∗∗ is an object fixing monoidal endofunctor. The
isomorphisms Ja,b : a
∗∗ ⊗ b∗∗ → (a ⊗ b)∗∗ associated with ∗∗ considered as a
SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 15
monoidal functor may be taken to be the identity on a⊗ b. If such a category has
a pivotal structure π, it must take the following form: for each strand (x), there is
a scalar tx such that πx = txIdx, and π(x1,...,xn) = tx1 . . . txnId(x1,...xn). Then for
all sequences (x1, . . . , xm) and (y1, . . . , yn), and all f : (x1, . . . , xm) → (y1, . . . , yn),
f∗∗ = tx1 . . . txmt
. . . t−1yn f . This implies that, in particular, t1 = 1.
Writing out the diagrams for b∗∗x and d
x and applying rigidity gives that b
x = bx
and d∗∗x = dx. Thus one must have txtx∗ = 1, and for self-dual strands, tx = ±1;
in this case tx is called the Frobenius-Schur indicator for x.
Furthermore, in a strictified skeleton the left and right trace on a strand may be
rewritten as follows:
trr(f) = t
x bx ◦ (f ⊗ Idx∗) ◦ dx∗ ,
trl(f) = txbx∗ ◦ (Idx∗ ⊗ f) ◦ dx.
Lemma 2. Every pivotal fusion category with self-dual simple objects is spherical.
Proof. The result holds since if x is a self dual strand then bx = bx∗ , dx = dx∗ and
tx = t
x . Thus for each f : x → x we have trr(f) = trl(f). �
Kitaev has shown in [6] that every unitary category admits a spherical structure.
A more general property called pseudo-unitarity is shown in [2] to guarantee a
spherical structure. However, it is not known if every fusion category admits a
pivotal or spherical structure.
For arbitrary fusion categories, one has that ∗ ∗ ∗∗ ∼= Id. This was shown in [2],
using an analog of Radford’s formula for S4 for representation categories of weak
Hopf algebras, which was developed in [8]. The following theorem shows that, in a
strictified skeletal fusion category, a convenient choice of rigidity makes ∗ ∗ ∗∗ the
identity on the nose. Extending the result to general fusion categories via natural
equivalences gives an elementary proof that ∗ ∗ ∗∗ ∼= Id.
Theorem 3. In a strictified skeletal fusion category, there is a choice of rigidity
structures such that ∗ ∗ ∗∗ = Id.
Proof. The functor ∗ ∗ ∗∗ is the identity on (2)-stranded morphisms by rigidity;
it suffices to prove the result for (2, 1) stranded morphisms. Let V = V zxy be a
(2, 1) stranded morphism space with a basis {vi}, and let {wi} be an algebraically
dual basis for the space W = V xyz , in the sense that wi ◦ vj = δijIdz . For any
simple object z, define the right pseudo-trace ptrr of an endomorphism f : z → z
by ptrr(f) = bz ◦ (f ⊗ Idz∗) ◦ dz∗ , and the left pseudo-trace ptrl by ptrl(f) =
bz∗ ◦ (Idz∗ ⊗f)◦dz. This definition is possible because ∗∗ is the identity on objects.
Scale rigidity morphisms if necessary so that for any strand z, ptrr(Idz) = ptrl(Idz).
Because dz and bz∗ are nonzero elements of one dimensional algebraically dual
morphism spaces, ptrr(Idz) 6= 0. One may now exchange left pseudo-traces for
right pseudo-traces, just like with traces in a graphical calculus for a spherical
category.
Figure 7 gives the proof. On the left side, bending arms and pseudo-sphericity
implies that the algebraic dual basis of the basis {w∗∗i } is {∗∗vi}. However, on the
right side the functoriality of the double dual implies that the algebraic dual basis
of {w∗∗i } is {v∗∗i }. Since the left and right double dual are inverse functors, ∗ ∗ ∗∗
is the identity. �
16 TOBIAS J. HAGGE AND SEUNG-MOON HONG
Figure 7. In a strictified skeletal fusion category, with the right
choice of rigidity structures the quadruple dual is the identity.
Even if a category admits a pivotal structure it is not known whether it admits
a spherical pivotal structure. Pictorial considerations do not readily provide an
answer. It is possible, however, to partially describe what a pivotal strictified
skeleton which did not admit a spherical structure would look like.
Let C be a pivotal strictified skeletal fusion category which does not admit a
spherical structure. Choose rigidity morphisms which give a pseudo-spherical struc-
ture as above, and a matching pivotal structure. Then for any object x, one has
the following:
trl(Idx)
trr(Idx)
txptrl(x)
t−1x ptrr(x)
= t2x.
Therefore, C is spherical iff there exists a pivotal structure such that all of the tx
are ±1. Thus there must be some strand x such that tx 6= ±1.
For strands u and v, u ⊗ v has a nontrivial morphism to some object w, and
tutv(tw)
−1 = ±1, since ∗ ∗ ∗∗ = Id. Thus the set of scalars t and their additive
inverses forms a finite subgroup G of C. Note that we can apply any group ho-
momorphism that preserves ±1 to the set of scalars t and get a new set of scalars
t′ which also gives a pivotal structure. At least one product tutv(tw)
−1 must be
equal to −1, or else we could apply the trivial homomorphism to the set of scalars
t to get a new pivotal structure with t′u = 1 for all strands u, which would make C
spherical.
Every finite subgroup of C is a cyclic group of roots of unity. We have |G| = 2k
for some k, and since C is not spherical, |G| ≥ 4. Using a homomorphism which
preserves −1 we may switch to a new pivotal structure which gives |G| = 2k for
some k, where k ≥ 2 to contradict sphericity. Pick an object v with t2v = −1. Then
v is not self dual, and for a simple summand w in v⊗ v, one has w 6= 1 and t2w = 1.
Therefore, C has at least four objects, v, v∗, w and 1. The set of objects u such
that t2u = 1 generates a spherical subcategory C′ with at least two simple objects,
and missing at least two.
SOME NON-BRAIDED FUSION CATEGORIES OF RANK 3 17
Lemma 4. Any fusion category which is pivotal but admits no spherical structure
contains at least five simple object types.
Proof. Assume that C has four simple object types, 1, w, v and v′ as above. Then
C′ has two simple object types, and by the classification of fusion categories with
two simple object types in [10], its fusion rules are given by w⊗w ∼= nw⊕1, where
n ∈ {0, 1}. The pivotal structure places limitations on the fusion rules, for example
v⊗w ∼= av⊕bv′ for some a and b in N. An easy calculation shows that C admits only
one associative fusion ring, in which objects and tensor products are given by the
group Z4. Any such category is pseudo-unitary and therefore spherical, as described
in [2], which contradicts the assumption. Therefore, a pivotal fusion category which
can’t be made spherical must have at least five simple object types. �
8. Proof of Theorem 1 part 4: spherical structure calculations
In this section we explicitly compute pivotal structures for the categories found
in Section 4. Since these categories have self dual simple objects, Lemma 2 implies
that they are spherical.
It is not hard to determine whether or not a fusion category is pivotal once
a set of associativity matrices is known. One way is to perform the calculations
directly using the associativity matrices, but there is an easier calculation. In order
to explain this calculation, it is convenient to extend the definition of composition
of morphisms over extra-categorical direct sums of morphism spaces. Suppose f ∈
Mor(a, b) and g ∈ Mor(c, d). Define f◦g as usual if b = c, and f◦g = 0 ∈ Mor(a, d)
otherwise. Extend this definition over direct sums of morphism spaces, distributing
composition over direct sum.
Given a strictified skeletal fusion category C and a set of associativity matrices,
choose bases for the (2, 0) and (2, 1)-stranded morphism spaces compatible with
the associativity matrices and choose rigidity so that for each strand x, the basis
element for V 1x∗x is dx. Define morphisms b = ⊕xbx, d = ⊕xdx, and I = ⊕xIdx,
taking sums over the strands.
Then B acts on
x,y,z V
xy as follows:
B(f) = (I ⊗ I ⊗ b) ◦ (I ⊗ f ⊗ I) ◦ (d⊗ I)
For a single (2, 1)-stranded morphism space, this action amounts to “bending
arms”. The cube of B is the double dual. The action of B on a morphism f ∈ V zxy
is given by the associativity matrix a1z∗,x,y, since (Idz∗ ⊗ f) ◦ dz = (g⊗ idy) ◦ dy for
some g ∈ V y
z∗x implies that B(f) = (Idz∗⊗Idx⊗by)◦(Idz∗⊗f⊗Idy∗)◦(dz⊗Idy∗) =
(Idz∗ ⊗ Idx ⊗ by) ⊗ (g ⊗ Idy ⊗ Idy∗) ◦ (dy ⊗ Idy∗) = g by rigidity. For the fusion
rules at hand, the matrix for B is as follows:
v1 v2 v
v1 (a
xxx)1,1 (a
xxx)1,2 0 0 0
v2 (a
xxx)2,1 (a
xxx)2,2 0 0 0
vyxx 0 0 0 a
yxx 0
vxyx 0 0 0 0 a
vxxy 0 0 a
xxy 0 0
For all of the solutions given in Section 4, B3 is the identity matrix, so the corre-
sponding strictified categories have a strict pivotal structure. Non-strict pivotality
18 TOBIAS J. HAGGE AND SEUNG-MOON HONG
would mean that B3 is a diagonal matrix with eigenvalues determined by a family
of invertible scalars t, coherent as described in Section 7.
Appendix A. Associativity matrices
In this section, we give explicit associativity matrices for the categorical realiza-
tion given in Section 4.
ayy,y,y = a
x,y,y = a
y,y,x = a
x,y,x = a
x,x,y = a
x,x,y = a
y,x,x = a
y,x,x = 1,
ayx,y,x = a
y,x,y = −1,
axx,y,x =
axx,x,y =
axy,x,x =
a1x,x,x =
e7πi/12
ayx,x,x =
e7πi/12
axx,x,x =
eπi/6 1−
e2πi/3 1−
e2πi/3 1−
eπi/6
eπi/6 1−
e2πi/3 − 1−
e2πi/3 − 1−
eπi/6
1 1 − 1
(eπi/6−1) 1
e5πi/6 1
(e−πi/3+i) 1
eπi/3
1 1 1
eπi/3 1
(e−πi/3+i) 1
e5πi/6 − 1
(eπi/6−1)
1 −1 − 1
(eπi/6−1) 1
e5πi/6 − 1
(e−πi/3+i) − 1
eπi/3
−1 1 − 1
eπi/3 − 1
(e−πi/3+i) 1
e5πi/6 − 1
(eπi/6−1)
References
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Department of Mathematics, Indiana University, Bloomington, Indana
E-mail address: thagge@indiana.edu,seuhong@indiana.edu
	1. Introduction
	2. Main theorem and outline
	3. Preliminaries and notational conventions
	3.1. Skeletization
	3.2. Strictification
	3.3. Strictified skeletal fusion categories
	3.4. Remarks
	4. Proof of Theorem ?? part  ??:possible tensor category structures
	4.1. Setting up the pentagon equations
	4.2. Normalizations
	4.3. Associativity matrices
	4.4. Pentagon equations with 1 1 matrices
	4.5. Pentagon equations with 2 2 or 6 6 matrices
	4.6. The pentagon equation with 16 16 matrices
	4.7. Solutions
	4.8. Inequivalence of the solutions
	5. Proof of Theorem ?? part ??: rigidity structures
	6. Proof of Theorem ?? part ??: the absence of braidings
	7. Pivotal structures and sphericity
	8. Proof of Theorem ?? part ??: spherical structure calculations
	Appendix A. Associativity matrices
	References
