A complete classification of unitary fusion categories tensor generated by an object of dimension 1+ 5 √ 2
AA complete classification of unitary fusion categoriestensor generated by an ob ject of dimension +√ Cain Edie-Michell
Abstract
In this paper we give a complete classification of unitary fusion categories ⊗ -generated by an object of dimension +√ . We show that all such categories ariseas certain wreath products of either the Fibonacci category, or of the dual even partof the D subfactor. As a by-product of proving our main classification result weproduce a classification of finite unitarizable quotients of Fib ∗ N satisfying a certainsymmetry condition. Fusion categories are rich algebraic objects providing connections between various areasof mathematics such as representation theory [29], operator algebras [7, 4], and quantumfield theories in physics [9, 19]. Therefore, classification results regarding fusion categorieshave important implications for these various subjects. While a full classification resultremains the distant goal, such a task is hopelessly out of reach with current availabletechniques. Instead research focuses on classifying “small” fusion categories, where smallcan have a variety of different meanings. For example one can attempt to classify fusioncategories with a small number of simple objects [31].Inspired by the successful classification of low index subfactors [24, 1] one can attempt aprogram to classify unitary fusion categories ⊗ -generated by an object of small dimension.Given the success the low index subfactor classification in finding exotic examples [2], wehope that similar success can be obtained in finding exotic examples of fusion categoriesthrough this program. One of the earliest results in the field is the classification of unitaryfusion categories generated by a self-dual object of dimension less than 2 [30, 3, 20, 21, 23,25, 34], which contained the exceptional E and E examples. This result was generalisedin [11], replacing the self-dual condition with a mild commutativity condition. Here againexceptional examples were found, namely the quantum subgroups E of sl and E , of sl ⊕ sl . Thus the program to classify unitary fusion categories ⊗ -generated by an objectof small dimension appears fruitful for discovering interesting new examples.While the results [30] and [11] do provide a broad classification, it is somewhat unsat-isfying having conditions on the ⊗ -generating object of small dimension. However verylittle is known about the classification of categories ⊗ -generated by an arbitrary object ofsmall dimension. Outside of the trivial result for objects of dimension , the easy proof1 a r X i v : . [ m a t h . QA ] M a r f which has been known as folklore since the earliest days of the field (details can befound in [10, Section 9]), such classification results are non-existent in the literature. Thepurpose of this paper is to provide the first non-trivial such classification. The main resultof this paper gives a complete classification of unitary fusion categories ⊗ -generated byan object of dimension +√ . Theorem 1.1.
Let C be a unitary fusion category ⊗ -generated by an object of dimension +√ . Then C is unitarily monoidally equivalent to either Fib ⊠ N ω ⋊ Z NM where N, M ∈ N and ω ∈ H ( Z NM , S ) , where the action of Z NM factors through the action of Z N on Fib ⊠ N that cyclically per-mutes the N factors, or T T ⊠ N ω ⋊ Z NM where N, M ∈ N and ω ∈ H ( Z NM , S ) , where the Z NM action factors through the action of Z N on T T ⊠ N described in Re-mark 4.9. To summarise the above theorem, we find all unitary fusion categories ⊗ -generated byan object of dimension +√ are constructed as wreath products of the category Fib orthe category
T T , which is perhaps better known as the non-trivial “fish-like” quotientof Fib ∗ Fib , or as the dual even part of the D subfactor. While this classification doesnot reveal the existence of any truly exotic new fusion categories, it does find T T ⋊ Z as a unitary fusion category ⊗ -generated by an object of dimension +√ . To the authorsbest knowledge, this example of a category ⊗ -generated by an object of small dimensionwas not previously known.Our arguments to prove Theorem 1.1 reduce quickly to classifying finite unitarizablequotients of Fib ∗ N that have a symmetry between the N distinct Fib generators. Hence asa by-product of our main theorem, we produce a classification of such quotients, extendingresults of Liu [26] and Izumi-Morrison-Penneys [22]. The lack of truly exotic categoriesin Theorem 1.1 boils down to the surprising lack of interesting unitarizable quotients of
Fib ∗ N for N ≥ . Towards generalising Theorem 1.1, it appears that classifying cyclicquotients of C ∗ N for C will play a key role, as from such a quotient one can construct anew category ⊗ -generated by an object of the same dimension as the ⊗ -generator of C . Inparticular it would be interesting to classify finite quotients of free products of the Isingcategory, or of free products of the “tadpole” categories [28]. Remark 1.2.
We wish to point out that the assumption that the category C is unitaryin the above theorem can almost certainly be removed. The new theorem would classifyfusion categories ⊗ -generated by an object of Frobenius-Perron dimension +√ . The stick-ing point for proving this more general theorem, is that there only exists a classificationof finite unitary quotients of Fib ∗ Fib in the current literature. If one could remove theunitary assumption on this classification result, and obtain a classification of all finitequotients of
Fib ∗ Fib , then the methods of this paper would directly generalise. The ex-panded classification theorem would now include the Galois conjugates of the categoriesin the above theorem, under the map √ ↦ −√ . hile the main Theorem of this paper is a statement of unitary fusion categories,for the majority of the proofs in this paper, we work in the unitarizable setting, that isfusion categories which have a unitary structure but where we have not chosen the unitarystructure. This is for two reasons. We can’t work in the purely algebraic setting becausethe results of Liu [26] only apply to unitary categories. However, we also can’t work inthe purely unitary setting because the results of Galindo [14, Theorem 4.1] only apply inthe algebraic setting. Adapting Liu’s results to the purely algebraic setting might be quitedifficult, while adapting Galindo’s result to the unitary setting seems seems tractable. As every unitary fusion category embeds uniquely into bimodules of the hyperfinitetype II factor R [17, 32], we thus have as a corollary to Theorem 1.1 a (non-constructive)classification of finite depth bimodules of R with index +√ . The Jones index theoremimplies that the three smallest dimensions of a finite depth bimodule of R are , √ , and +√ . Hence the results of this paper, coupled with the known classification of unitaryfusion categories generated by an object of dimension 1, give a complete classification ofall finite depth bimodules of R for two of these three smallest dimensions. This motivatesthe classification of unitary fusion categories ⊗ -generated by an object of dimension √ ,which would give as a corollary a classification of finite depth bimodules of R in the range [ , +√ ] .Our paper is structured as follows.In Section 2 we give the necessary definitions for this paper. In particular we definesemi-direct product categories, free product categories and quotients, and introduce thecategories Fib and
T T .In Section 3 we prove that every unitarizable fusion category ⊗ -generated by an objectof dimension +√ is a semi-direct product of a certain quotient of Fib ∗ N . The argumentis fairly straightforward, consisting of fusion ring manipulation and Frobenius reciprocity.We end the section by discussing the feasibility of possible generalisations, includingremoving the unitarizable condition, and changing the dimension of the ⊗ -generator.Section 4 represents the bulk of this paper. Here we classify finite unitarizable quo-tients of Fib ∗ N satisfying a certain symmetry condition. Using fusion ring arguments in amarathon case by case analysis, we show the surprising result that the only such quotientsare Fib ⊠ N and T T ⊠ N if N is even. Further, we show that any finite unitarizable quotientof Fib ∗ N factors through Fib ⊠ n ⊠T T ⊠ m where n + m = N . Using this result it should bepossible to classify all finite unitarizable quotients of Fib ∗ N . We neglect to follow up onthis generalised result as it goes beyond the scope of this paper.We end the paper with Section 5, which ties together the results of this paper to proveTheorem 1.1. Acknowledgements
The author would like to thank Dietmar Bisch and Vaughan Jones for useful discussionsregarding the problem tackled in this paper. We thank Corey Jones, Scott Morrison,and Dave Penneys for conversations that inspired the author to begin thinking about thisproblem. We thank Noah Snyder for pointing out a crucial error in a previous version ofthis paper. 3
Preliminaries
For the basic theory of fusion categories we direct the reader to [12]. Unless explicitlystated, all categories in this paper can be assumed to be fusion categories. That is,semisimple rigid tensor categories, with a finite number of simple objects, and simpleunit.Given an object X in a fusion category C , then the category ⊗ -generated by X isthe full subcategory of C containing all summands of the objects X ⊗ n for all n ∈ N .The category ⊗ -generated by X is a fusion category by [12, Lemma 3.7.6]. We say C is ⊗ -generated by X if the category generated by X ∈ C is C . The Fibonacci category
A recurring theme in this paper will be the appearance of the Fibonacci categories. AFibonacci fusion category has two simple objects and τ , satisfying the fusion rule τ ⊗ τ ≅ ⊕ τ. There exist two associators for this fusion rule, both of which can be found in [6]. Onlyone of these fusion categories has a unitary structure, which is unique.
Definition 2.1.
We write
Fib for the unitary fusion category with Fibonacci fusion rules,and we write
Fib for the non-unitary fusion category with Fibonacci fusion rules.The categories
Fib and
Fib can be realised in a variety of different ways. For example
Fib is monoidally equivalent to: the even part of the A planar algebra, the adjoint cate-gory of the category of level integrable representations of ̂ sl , and the semi-simplificationof the category of U e πi ( g ) modules. Each of these categories is defined over Q [√ ] . The Fib category can be realised as the Galois conjugate of
Fib by √ ↦ −√ .The category Fib is the prototypical example of a unitary fusion category ⊗ -generatedby an object of dimension +√ . We will see later in this paper that every unitary fusioncategory ⊗ -generated by an object of dimension +√ can be constructed from Fib , thoughin surprisingly complicated ways. As such, the category
Fib will appear frequently in thispaper. To help with computations we introduce the following lemma, allowing a sort ofcancellation of Fibonacci objects.
Lemma 2.2.
Let C be a fusion category, and τ ∈ C an object satisfying the Fibonaccifusion rule τ ⊗ τ ≅ ⊕ τ . Then for any objects X, Y ∈ C we have τ ⊗ X ≅ τ ⊗ Y (cid:212)⇒ X ≅ Y. Proof.
Suppose τ ⊗ X ≅ τ ⊗ Y, then we can left multiply by τ to get X ⊕ τ ⊗ X ≅ Y ⊕ τ ⊗ Y ≅ Y ⊕ τ ⊗ X. W ∈ C , we have dim ( W → X ⊕ τ ⊗ X ) = dim ( W → X ) + dim ( W → τ ⊗ X ) and dim ( W → Y ⊕ τ ⊗ X ) = dim ( W → Y ) + dim ( W → τ ⊗ X ) As the objects X ⊕ τ ⊗ X and Y ⊕ τ ⊗ X are isomorphic, we have dim ( W → X ⊕ τ ⊗ X ) = dim ( W → Y ⊕ τ ⊗ X ) . Therefore, for each simple W ∈ C , we have dim ( W → X ) = dim ( W → Y ) . As C is a semi-simple category, every object is uniquely determined, up to isomorphism,by the multiplicities of isomorphism classes of simple objects in its decomposition. Thus X ≅ Y . Free products and quotients
Given two fusion categories C and D , one can form the free product category C ∗ D . In
C ∗ D the objects consist of words of the objects in C and D , and the morphisms consistof non-crossing morphisms of both C and D . Additional details can be found in [22, 5].For this paper we will be interested in the free product category Fib ∗ N for N ∈ N ≥ . Ifwe write { τ i ∶ ≤ i ≤ N } for the N generators of the component Fib categories, then anobject of
Fib ∗ N will be of the form τ a τ b τ c ⋯ where a, b, c, ⋯ ∈ { , , ⋯ N } . It is obvious that the category
Fib ∗ N has infinitely many simple objects if N ≥ , andhence is not fusion. However it is possible to take quotients of Fib ∗ N to obtain unitarizablefusion categories. Definition 2.3.
A unitarizable quotient of
Fib ∗ N is a rigid unitarizable tensor category C , along with a faithful dominant functor Fib ∗ N → C . Following [22] it is helpful to work with the subcategory of a unitarizable quotient of
Fib ∗ N which lies in the image of the faithful dominant functor. Taking the idempotentcompletion of this image category recovers the unitarizable quotient, hence there is no costto working with this image category. With this perspective, we can think of a unitarizablequotient of Fib ∗ N as having the same objects as Fib ∗ N , i.e words in { τ i ∶ ≤ i ≤ N } , butwith more morphisms. These additional morphisms give relations between the objects ofthe quotient, causing non-isomorphic objects in Fib ∗ N to become isomorphic, and simpleobjects to become non-simple. In particular we will be interested in finite unitarizablequotients of Fib ∗ N . 5 efinition 2.4. A finite unitarizable quotient of
Fib ∗ N is a unitarizable quotient thathas finitely many simple objects.A simple example of a unitarizable quotient of Fib ∗ N is the N -fold Deligne product Fib ⊠ N , where we impose the relation that the generators { τ i ∶ ≤ i ≤ N } pairwise com-mute. The simple objects of Fib ⊠ N are all words consisting of unordered non-repeatingcombinations of letters in { τ i ∶ ≤ i ≤ N } . Thus the rank of Fib ⊠ N is equal to N , so Fib ⊠ N gives our first example of a finite unitarizable quotient of Fib ∗ N .A more complicated example of a finite unitarizable quotient is the category T T ,constructed in [22, 26] and unpublished work of Izumi. Definition 2.5.
The unitary fusion category
T T has six simple objects , f ( ) , ρ, σ, σ ,and µ , with dimensions , ( +√ ) , +√ , ( +√ ) , ( +√ ) , and +√ respectively. Thefusion rules are given by ⊗ f ( ) ρ σ σ µf ( ) + f ( ) + σ + σ + ρ + µ f ( ) + σ f ( ) + σ + σ + µ f ( ) + σ + σ + ρ f ( ) + σρ f ( ) + σ + ρ f ( ) σ + µ σσ f ( ) + σ + σ + ρ σ + µ f ( ) + σ + f ( ) + µ f ( ) σ f ( ) + σ + σ + µ f ( ) + f ( ) + ρ f ( ) + σ σ + ρµ f ( ) + σ σ σ + ρ f ( ) + µ The objects ρ and µ in T T both satisfy Fibonacci fusion, and together they ⊗ -generateall of T T . Thus T T is a finite quotient of Fib ∗ .A theorem of Liu [26] shows that the only finite unitarizable quotients of Fib ∗ are Fib , Fib ⊠ , and T T . Outside of this result little is known of finite unitarizable quotientsof Fib ∗ N . In Section 4 of this paper we will extend Liu’s result to classify finite rankpreserving cyclic unitarizable quotients of Fib ∗ N for all N . Definition 2.6.
We say a unitarizable quotient of
Fib ∗ N is rank preserving if τ i ≅ τ j implies i = j . Definition 2.7.
We say a unitarizable quotient of
Fib ∗ N is cyclic if there is a monoidalauto-equivalence mapping the generators τ ↦ τ ↦ ⋯ ↦ τ N ↦ τ . Example 2.8.
Consider the finite quotient
T T of Fib ∗ . As ρ ≇ µ we have that T T is a rank preserving quotient. Further, one has from [27, Lemma 3.4] that the planaralgebra for T T is generated by a single -box T . From the relations for T given in thesame paper, we can see that the planar algebra for T T has a *-automorphism mapping T ↦ − T . Thus by [18, Theorem A] we get a monoidal auto-equivalence of T T that sends ρ ↔ µ, and σ ↔ σ. Thus
T T is a finite rank preserving cyclic unitarizable quotient of Fib ∗ .Our motivation for classifying such quotients will become apparent in Section 3. Weremark that the techniques of Section 4 should extend in a fairly straightforward mannerto classify all finite unitarizable quotients of Fib ∗ N , not just finite rank preserving cyclicquotients. Infinite quotients of Fib ∗ N however still remain mysterious.6 -graded categories Let G be a finite group, and C a fusion category. We say C is G -graded if we have anabelian decomposition C ≃ ⊕ G C g , with each C g a non-trivial abelian subcategory of C , such that the tensor product of C restricted to C g × C h has image in C gh .Consider the following monoidal subcategory of C , which we call the adjoint subcate-gory of C . Ad (C) ∶= ⟨ X ⊗ X ∶ X ∈ Irr (C)⟩ . The category C is always graded by some group G , such that C e = Ad (C) . Semi-direct product categories
An important construction for this paper will be the semi-direct product of a unitary fusioncategory C with a finite group. The key ingredient for this construction is a categoricalaction of C . Definition 2.9.
Let G be a finite group, and C a unitary fusion category. A categoricalaction of G on C is a monoidal functor ρ ∶ G → Eq (C) , where G is the monoidal category whose objects are the objects of G , and whose mor-phisms are identities, and Eq (C) is the monoidal category of monoidal *-auto-equivalencesof C .Given a unitary fusion category with a categorical action we can construct the semi-direct product, a new unitary fusion category. Definition 2.10.
Let C be a unitary fusion category with a categorical G action ρ , and let ω ∈ H ( G, S ) . We define the semi-direct product category C ω ⋊ G as the abelian category C ω ⋊ G ∶= ⊕ G C , with tensor product ( X , g ) ⊗ ( X , g ) ∶= ( X ⊗ ρ ( g )[ X ] , g g ) , and associator [( X , g ) ⊗ ( X , g )] ⊗ ( X , g ) → ( X , g ) ⊗ [( X , g ) ⊗ ( X , g )] given by the isomorphism ω g ,g ,g ( id X ⊗ τ g X ,ρ ( g )[ X ] ) ○ ( id X ⊗ id ρ ( g )[ X ] ⊗ µ g ,g ) . Here τ g is the tensorator for the monoidal functor ρ ( g ) , and µ is the tensorator for themonoidal functor ρ . The ∗ -structure for C ω ⋊ G is directly inherited from the ∗ -structurefor C . 7t is clear from the definition that C ω ⋊ G is a G -graded category, and further, eachgraded piece contains an invertible element. The following theorem, due to Galindo, givesa converse to this observation, showing that every graded category with an invertibleelement in each graded piece must be a semi-direct product. Theorem 2.11. [14, Theorem 4.1] Let C be a G -graded fusion category such that eachgraded piece C g contains an invertible element U g . Then C is monoidally equivalent to C e ω ⋊ G for some ω ∈ H ( G, C × ) , where the categorical action ρ of G on C e is defined on objectsby ρ g ( X ) ∶= U g ⊗ X ⊗ U g − . The main aim of this paper is to provide a complete classification of unitary fusion cate-gories ⊗ -generated by an object of dimension +√ . While initially this may seem like ahopelessly difficult task, we shall see in this Section that such categories all share the sameuniform structure. That is they are all semi-direct products of certain finite unitarizablequotients of free products of arbitrarily many copies of Fib . As semi-direct product cat-egories are well understood, we have reduced our classification problem to the somewhatsimpler task of classifying finite unitarizable quotients of free products. We will deal withsuch quotients in the next Section of this paper. For now we focus on proving the fol-lowing theorem. The proof is fairly straightforward, and boils down to basic fusion ringarguments, and a lot of Frobenius reciprocity.
Theorem 3.1.
Let C be a unitarizable fusion category ⊗ -generated by an object of di-mension +√ . Then C is monoidally equivalent to a semi-direct product of a finite rankpreserving cyclic unitarizable quotient of Fib ∗ N by a cyclic group, with the generator ofthe cyclic group acting on the quotient by factoring through the given cyclic action.Proof. Let C be a unitarizable fusion category ⊗ -generated by an object X of dimension +√ . As X ⊗ X must contain the tensor unit, we have that X ⊗ X ≅ ⊕ τ with τ some object in C . A dimension count shows τ has dimension +√ . In particular τ is a simple object. Further, as both and X ⊗ X are self-dual, we also have that τ isself-dual. Hence τ generates a Fib subcategory of C , and in particular satisfies the fusionrule τ ⊗ τ ≅ ⊕ τ .As dim ( X ⊗ X → τ ) = , we can apply Frobenius reciprocity to get dim ( X → τ ⊗ X ) = .As X is simple, we get X ⊆ τ ⊗ X and so τ ⊗ X ≅ X ⊕ g g an object of C . Another dimension count shows g is a simple object of dimension . In particular dim ( τ ⊗ X → g ) = , and another application of Frobenius reciprocitygives dim ( τ ⊗ g → X ) = . As both τ ⊗ g and X are simple, we thus have X ≅ τ ⊗ g. (3.1)As C is a fusion category, the order of g is finite. Let N be the smallest integer (whichmust exist as the order of g is finite) such that g N ⊗ τ ⊗ g N = τ . Consider the N distinctsimple objects τ n ∶= g n ⊗ τ ⊗ g n ∶ n = , , ⋯ , N − . It is straightforward to verify that each of these distinct objects satisfies the fusion τ n ⊗ τ n ≅ ⊕ τ n , and further each τ n has dimension +√ . Hence each τ n generates a Fib subcategory of C , and so the collection of objects { τ n ∶ n = , , ⋯ , N } together ⊗ -generate a subcategoryof C that is equivalent to a finite rank preserving quotient of Fib ∗ N . As C is unitarizable,this finite rank preserving quotient must also be unitarizable.As X ⊗ -generates C , the adjoint subcategory of C is ⊗ -generated by the objects X n ⊗ X n for n ∈ N . Using Equation (3.1) we can write X n ⊗ X n ≅ ( τ ⊗ g ) n ⊗ ( g ⊗ τ ) n ≅ ( n − ⊗ i = τ i ) ⊗ ( ⊗ i = n − τ i ) . Hence, the adjoint subcategory of C is ⊗ -generated by the N simple objects { τ n } , and thusthe adjoint subcategory of C is equivalent to a finite rank preserving quotient of Fib ∗ N .As C is ⊗ -generated by X , the category C is a cyclic extension of Ad (C) , with X living in the 1-graded component of this cyclic grading (written additively). Furthermore,Equation (3.1) shows that g also lives in the 1-graded component of the cyclic grading,which implies that every graded component of the grading has an invertible object. ThusTheorem 2.11 implies that C is a semi-direct product of its adjoint subcategory by a cyclicgroup, and hence is a semi-direct product of a finite rank preserving unitarizable quotientof Fib ∗ N by a cyclic group. The same theorem shows the generator of the cyclic groupacts on the generators of Fib ∗ N by sending τ i ↦ τ i + . Hence the finite rank preservingunitarizable quotient of Fib ∗ N must also be cyclic.Before we move on to classifying finite rank preserving cyclic quotients of Fib ∗ N , letus discuss possible generalisations of the above theorem. The key step in the above proofis finding the invertible object g , which lives in the same graded component of C as thegenerator X . With this object in hand we know we are looking at a semi-direct productcategory, and hence classification is possible. If we consider a generator of dimension √ then it is fairly easy to convince oneself that we can never get such an invertible object,the problem being that we run into issues with the grading. However if we restrict toobjects of dimension ( π N + ) then these grading issues seem to not appear. Hence9t should be possible to prove a generalisation of the above theorem for fusion categories ⊗ -generated by an object of dimension ( π N + ) . We stress that we have not workedthrough the details of this generalisation, and do not claim it as conjecture.Our lack of current interest in the mentioned generalisation stems from the fact wewould have to classify finite unitarizable quotients of T N ∗ T N for all N to provide ageneralised classification result. Given the difficulty of the T ∗ T case [26, 22], thegeneral N case appears completely out of reach. Supposing a breakthrough insight isdiscovered, allowing for the classification of finite unitarizable quotients of T N ∗ T N for all N , then it is likely the techniques from this paper would generalise to provide a generalisedclassification result.The other generalisation one can consider is to relax the unitarizable condition onthe category C , and to instead consider fusion categories ⊗ -generated by an object ofFrobenius-Perron dimesion +√ . The above theorem has a straightforward generalisation,now showing that any such category is a semi-direct product of either Fib ∗ N or of theGalois conjugate Fib ∗ N . The issue with pursuing this generalisation is that there iscurrently no classification of non-unitarizable quotients of Fib ∗ Fib . Thus the resultsof the following Section can not be extended to give a classification of not necessarilyunitarizable finite cyclic quotients of
Fib ∗ N . However, this is the only stumbling block.Assuming the expected result that every quotient of Fib ∗ Fib is unitarizable, then the restof this paper would generalise directly to give a complete classification of fusion categories ⊗ -generated by an object of Frobenius-Perron dimension +√ . Fib ∗ N With Theorem 3.1 in mind, we have motivation to classify finite rank preserving cyclicunitarizable quotients of
Fib ∗ N . This is a priori a very difficult task, given the complexitythat was required to classify finite unitarizable quotients of Fib ∗ Fib . However, mirac-ulously we are able to prove the following classification result, purely using fusion ringarguments.
Theorem 4.1.
Let C be a rank preserving finite cyclic unitarizable quotient of Fib ∗ N ,then C is monoidally equivalent to either Fib ⊠ N , or, if N is even, T T ⊠ N . Essentially this theorem shows that the only interesting finite cyclic unitarizable quo-tients of
Fib ∗ N are rank 2. While we did find this result surprising (and somewhatdisappointing, as it excludes the possibility of potential new exotic unitarizable fusioncategories), it does have some parallels to a classical group theory result.One can consider the category Fib as being the higher categorical analogue of thegroup Z , in the sense that Fib has two simple objects, but with non-pointed fusion.10ith this analogy in mind, we can draw a further analogy between the classification offinite quotients of
Fib ∗ N , and the Coxeter classification of finite quotients of Z ∗ N [8].In the case where N = , the classical Coxeter classification simply gives the dihe-dral groups, while the quantum analogue follows from Liu’s classification of quotients of Fib ∗ Fib . While there are an infinite family of dihedral groups, Liu’s classification showsthat only the first two of the quantum counterparts exist. Hence, it seems reasonable toexpect the that the classification of finite quotients of
Fib ∗ N should be more significantlymore restrictive than the classification of finite quotients of Z ∗ N . Given that the examplesappearing in Coxeter’s classification were sparse to begin with, this point of view helpsto explain the surprising lack of interesting examples in Theorem 4.1.Let us briefly sketch the proof of Theorem 4.1.We begin by considering the finite unitarizable quotients of Fib ∗ , with generators a , b , and c . We are able to show that we must have a relation of the form abcab ⋯·„„„„„„„„‚„„„„„„„„„¶ length n ≅ bcabc ⋯·„„„„„„„„‚„„„„„„„„¶ length n , for some finite n . Further, as each pair of generators must generate a finite unitarizablequotient of Fib ∗ , we must have the further three relations ab ≅ ba or aba ≅ bab and bc ≅ cb or bcb ≅ cbc and ac ≅ ca or aca ≅ cac. A marathon case by case analysis shows that in fact, one of the generators commuteswith the other two. With this result in hand one can quickly prove that any rank pre-serving finite unitarizable quotient of
Fib ∗ N must factor through Fib ⊠ n ⊠T T ⊠ m where n + m = N .At this point we use our cyclic quotient assumption to see that a rank preserving cyclicunitarizable quotient of Fib ∗ N must be a quotient of either Fib ⊠ N , or T T ⊠ N if N is even.Finally we show that only the trivial quotients of these categories exist.The results of this Section, while not particularly technical, are extremely long andmessy. A “better” proof could involve generalising the results of [26] to give a finite boundon the n such that abcab ⋯·„„„„„„„„‚„„„„„„„„„¶ length n ≅ bcabc ⋯·„„„„„„„„‚„„„„„„„„¶ length n . From here, one would now just have a finite number of cases to consider to show that oneof the generators commutes with the other two, significantly shortening the length of theproof.For now we just have our marathon brute force proof. Before we begin, we introducesome notation 11 efinition 4.2.
Let C be a unitarizable quotient of Fib ∗ . We say a word in C is cyclicif the letters are tri-alternating. Example 4.3.
Consider a unitarizable quotient of
Fib ∗ = ⟨ τ , τ , τ ⟩ . The words τ τ τ τ τ and τ τ τ τ are cyclic, but the word τ τ τ τ is not.We begin with the proofs of this Section with the following lemma, inspired by [22,Proposition 3.2]. Lemma 4.4.
Let C be a unitarizable quotient of Fib ∗ with generators a , b , and c , andlet n ∈ N be the smallest such that either abcabc ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≅ bcabca ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n or bcabca ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≅ cabcab ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n or cabcab ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≅ abcabc ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n . then all cyclic words of length n or less are simple. If there is no such n , then all cyclicwords are simple.Proof. This proof inducts on the length k of the cyclic word. k = : As C is a tensor category, the tensor unit is simple. k = : As the objects a , b , and c have dimension +√ , they must be simple.Inductive step for k ≤ n :Suppose that all cyclic words of length k − or less are simple. Consider a cyclicword of length k , we consider nine cases, depending on if the word starts with a , b ,or c , and if the word ends in a , b , or c . We work though the case where the cyclicword begins with a and ends with b , and leave the other eight cases to the reader, asthey are near identical. We compute dim ( abc ⋯ cab ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length k → abc ⋯ cab ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length k ) = dim ( aabc ⋯ bca ·„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„¶ length k → bca ⋯ cabb ·„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„¶ length k ) dim ( abc ⋯ bca ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length k-1 ⊕ bca ⋯ bca ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length k-2 → bca ⋯ bca ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length k-2 ⊕ bca ⋯ cab ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length k-2 ) )= dim ( abc ⋯ bca ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length k-1 → bca ⋯ bca ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length k-2 ) + dim ( bca ⋯ bca ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length k-2 → bca ⋯ bca ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length k-2 )+ dim ( abc ⋯ bca ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length k-1 → bca ⋯ cab ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length k-1 ) + dim ( bca ⋯ bca ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length k-2 → bca ⋯ cab ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length k-1 )= + + + = . Where the last step uses the inductive hypothesis, along with the assumption in thestatement of the lemma that abc ⋯ bca ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length k-1 ≇ bca ⋯ cab ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length k-1 . Thus abc ⋯ cab ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length k is simple.We can repeat the inductive step until k = n + , hence proving the statement of thelemma.Restricting our attention to finite unitarizable quotients, we get the following Corol-lary. Corollary 4.5.
Let C be a finite unitarizable quotient of Fib ∗ with generators a , b , and c . Then, up to a relabelling of the three generators, there exists an n ∈ N such that abcab ⋯·„„„„„„„„‚„„„„„„„„„¶ length n ≅ bcabc ⋯·„„„„„„„„‚„„„„„„„„¶ length n and that all cyclic words of length n or less are simple.Proof. If there didn’t exist a n ∈ N such that either abcab ⋯·„„„„„„„„‚„„„„„„„„„¶ length n ≅ bcabc ⋯·„„„„„„„„‚„„„„„„„„¶ length n or bcabc ⋯·„„„„„„„„‚„„„„„„„„¶ length n ≅ cabca ⋯·„„„„„„„„„‚„„„„„„„„„¶ length n or cabca ⋯·„„„„„„„„„‚„„„„„„„„„¶ length n ≅ abcab ⋯·„„„„„„„„‚„„„„„„„„„¶ length n , then Lemma 4.4 would imply that C would have an infinite number of simple objects,contradicting the assumption that C is a finite unitarizable quotient. Thus there mustexist such a n ∈ N . Assume that this n is chosen to be minimal, and further, use up a13egree of freedom between the labels of the generators a , b , and c so that n is the smallestinteger such that abcab ⋯·„„„„„„„„‚„„„„„„„„„¶ length n ≅ bcabc ⋯·„„„„„„„„‚„„„„„„„„¶ length n . As n was chosen minimally, we have that all cyclic words of length n − or less are distinct.Thus Lemma 4.4 gives that all cyclic words of length n or less are simple.Hence we have shown that any finite unitarizable quotient of Fib ∗ must have a relationof the form abcab ⋯·„„„„„„„„‚„„„„„„„„„¶ length n ≅ bcabc ⋯·„„„„„„„„‚„„„„„„„„¶ length n for some n . We will now show that this relation is inconsistent with most other relationswe can put on Fib ∗ . Key to finding these inconsistencies will be the following non-isomorphisms. Lemma 4.6.
Let C be a rank preserving unitarizable quotient of Fib ∗ with generators a , b , and c satisfying the relations aba ≅ bab, bcb ≅ cbc, and aca ≅ cac. Then we have for any n ∈ N abcabc ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≇ b abcabc ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − bcabca ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≇ c bcabca ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − cabcab ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≇ a cabcab ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − acbacb ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≇ c acbacb ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − bacbac ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≇ a bacbac ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − cbacba ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≇ b cbacba ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − . Proof.
We induct on the length n . n = : This case follows as C is a rank preserving quotient, so a ≇ b ≇ c . n = : If ab ≅ ba then we can use the relation aba = bab to get b ⊕ ab ≅ a ⊕ ab a ≅ b , contradicting the assumption that C is a rank preserving quotient.Thus ab ≇ ba .A similar argument shows bc ≇ cb and ac ≇ ca .Inductive Step:Suppose the result holds true for n − , and aiming for a contradiction, assume abcabc ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≅ b abcabc ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − . Using the relation aba ≅ bab we can rewrite the right hand side to get abcabc ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≅ aba cabcab ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − . We now apply Lemma 2.2 twice to get cabcab ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − ≅ a cabcab ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − , which is a contradiction to the inductive hypothesis. Hence abcabc ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≇ b abcabc ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − . The other cases follow near identically.We now prove the main technical lemma of this Section, showing that any finitequotient of Fib ∗ must have a large amount of commutativity. Lemma 4.7.
Let C be a rank preserving finite unitarizablequotient of Fib ∗ with generators a , b , and c . Then one of the generators commutes with the other two. As mentioned earlier the proof of this lemma consists of brute force case bashing. Webegin by setting up the cases. Let us write a , b , and c for the three generators of thisquotient. Using Corollary 4.5 we can arrange the generators a, b , and c in such a way sothat there exists a n ∈ N such that abcab ⋯·„„„„„„„„‚„„„„„„„„„¶ length n ≅ bcabc ⋯·„„„„„„„„‚„„„„„„„„¶ length n (4.1)and such that all cyclic words of length n or less are simple.With this fixed labelling of the generators, we can consider the four distinct cases:• None of a , b , or c commute with each other,• Only a and b commute with each other,15 Only b and c commute with each other,• Only a and c commute with each other.We are able to show that each of these four cases can not occur, hence showing thatone of the generators commutes with the other two. To help with readability, we will dealwith each of these cases in separate lemmas. Sublemma 4.7.1.
There are no rank preserving finite unitarizable quotients of
Fib ∗ such that none of a , b , or c commute with each other. Proof.
Consider such a quotient. As none of a , b , or c commute with each other, we musthave from the classification of finite unitary quotients of Fib ∗ Fib the relations aba ≅ bab, bcb ≅ cbc, and aca ≅ cac. (4.2)Here we split the proof up into three cases, depending on n mod 3 . Case: n ≡ ( mod 3 ) If n ≡ ( mod 3 ) , then the cyclic word bcabc ⋯·„„„„„„„„‚„„„„„„„„¶ length n ends in a . Consider the word bcabc ⋯ abc ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − a cba ⋯ cbacb ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − . If n ≡ ( mod 6 ) then we can use the relations (4.2) to write bcabc ⋯ abc ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − a cba ⋯ cbacb ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − ≅ cbacb ⋯ acb ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − a bca ⋯ bcabc ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − , and if n ≡ ( mod 6 ) then we can use the relations (4.2) to write bcabc ⋯ abc ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − a cba ⋯ cbacb ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − ≅ bacba ⋯ cba ·„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„¶ length n − c abc ⋯ abcab ·„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„¶ length n − . Hence we further break the proof up into two more sub-cases.
Case: n ≡ ( mod 6 ) As n ≡ ( mod 6 ) we have the relation bcabc ⋯ abc ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − a cba ⋯ cbacb ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − ≅ cbacb ⋯ acb ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − a bca ⋯ bcabc ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − . We can use relation (4.1) on the first n letters of the left hand side of the aboveequation to get abcab ⋯ cab ·„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„¶ length n − c cba ⋯ cbacb ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − ≅ cbacb ⋯ acb ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − a bca ⋯ bcabc ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − . (4.3)16valuating the left hand side using the Fib fusion rules reveals that a is a simplesummand. Hence dim ( a → cbacb ⋯ acb ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − a bca ⋯ bcabc ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − ) > , and an application of Frobenius reciprocity gives dim ( a cbacb ⋯ acb ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − → cba ⋯ cbacb ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − a ) > . As cyclic words of length n are simple, we thus have cba ⋯ cba ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≅ acb ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n . Returning to equation (4.3) we can now substitute in the above relation to thefirst n letters of the right hand side to get abcab ⋯ cab ·„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„¶ length n − c cba ⋯ cbacb ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − ≅ a cbacb ⋯ acb ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − bca ⋯ bcabc ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − , to which we can apply Lemma 2.2 to get bcab ⋯ cab ·„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„¶ length n − c cba ⋯ cbacb ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − ≅ cbacb ⋯ acb ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − bca ⋯ bcabc ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − . Expanding the left hand of this equation reveals a simple b summand, andexpanding the right hand side completely gives b ⊂ ⎛⎜⎝ n − ⊕ i = cbacba ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length i ⊗ ⋯ abcabc ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length i − ⎞⎟⎠ ⊕ . Thus there exists an i such that dim ( b → cbacba ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length i ⊗ ⋯ abcabc ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length i − ) > . Applying Frobenius reciprocity gives dim ( b cbacba ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length i − → cbacba ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length i ) > , which, as cbacba ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length i is simple, implies that b cbacba ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length i − ≅ cbacba ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length i . However this is a contradiction to Lemma 4.6.17 ase: n ≡ ( mod 6 ) As n ≡ ( mod 6 ) we have the relation bcabc ⋯ abc ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − a cba ⋯ cbacb ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − ≅ bacba ⋯ cba ·„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„¶ length n − c abc ⋯ abcab ·„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„¶ length n − . We can use relation (4.1) on the first n letters of the left hand side of the aboveequation to get abcab ⋯ cab ·„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„¶ length n − c cba ⋯ cbacb ·„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„¶ length n − ≅ bacba ⋯ cba ·„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„¶ length n − c abc ⋯ abcab ·„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„¶ length n − . Evaluating the left hand side using the
Fib fusion rules reveals that a is a simplesummand. Hence dim ( a → bacba ⋯ cba ·„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„¶ length n − c abc ⋯ abcab ·„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„¶ length n − ) > , and an application of Frobenius reciprocity gives dim ( a bacba ⋯ cba ·„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„¶ length n − → bacba ⋯ cbac ·„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„¶ length n ) > . As bacba ⋯ cbac ·„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„¶ length n is simple we thus have a bacba ⋯ cba ·„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„¶ length n − ≅ bacba ⋯ cbac ·„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„¶ length n . However this is a contradiction to Lemma 4.6.
Case: n ≡ ( mod 3 ) If n ≡ ( mod 3 ) , then the cyclic word bcabc ⋯·„„„„„„„„‚„„„„„„„„¶ length n ends in b . Consider the word bca ⋯ bca ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − b acb ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − . If n ≡ ( mod 6 ) then we can use the relations (4.2) to write bca ⋯ bca ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − b acb ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − ≅ bac ⋯ bac ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − b cab ⋯ cab ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − , and if n ≡ ( mod 6 ) then we can use the relations (4.2) to write bca ⋯ bca ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − b acb ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − ≅ cba ⋯ cba ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − c abc ⋯ abc ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − . Hence we further break the proof up into two more sub-cases.18 ase: n ≡ ( mod 6 ) If n ≡ ( mod 6 ) then we have bca ⋯ bca ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − b acb ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − ≅ bac ⋯ bac ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − b cab ⋯ cab ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − , to which we can apply relation (4.1) on the first n letters of the left hand sideof the above equation to get abc ⋯ abc ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − a acb ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − ≅ bac ⋯ bac ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − b cab ⋯ cab ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − . Evaluating the left hand side reveals that a ⊂ bac ⋯ bac ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − b cab ⋯ cab ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − , from which we can use a similar argument as in the n ≡ ( mod 6 ) case to seethat a bac ⋯ bac ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − ≅ bac ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n . However this is a contradiction to Lemma 4.6.
Case: n ≡ ( mod 6 ) If n ≡ ( mod 6 ) then we have bca ⋯ bca ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − b acb ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − ≅ cba ⋯ cba ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − c abc ⋯ abc ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − , to which we can apply relation (4.1) on the first n letters of the left hand sideof the above equation to get abc ⋯ abc ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − a acb ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − ≅ cba ⋯ cba ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − c abc ⋯ abc ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − . (4.4)Evaluating the left hand side reveals that a ⊂ cba ⋯ cba ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − c abc ⋯ abc ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − , from which we can use a similar argument as in the n ≡ ( mod 6 ) case to seethat acb ⋯ cba ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≅ cba ⋯ bac ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n . We can apply the above relation to the first n letters on the right hand side ofEquation (4.4) to get abc ⋯ abc ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − a acb ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − ≅ acb ⋯ cba ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n abc ⋯ abc ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − , bca ⋯ abc ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − a acb ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − ≅ cba ⋯ cba ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − abc ⋯ abc ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − Evaluating the both sides of the above equation shows b ⊂ ⎛⎜⎝ n − ⊕ i = cbacba ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length i ⊗ ⋯ abcabc ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length i − ⎞⎟⎠ ⊕ . Thus there exists an i such that dim ( b → cbacba ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length i ⊗ ⋯ abcabc ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length i − ) > . Applying Frobenius reciprocity gives dim ( b cbacba ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length i − → cbacba ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length i ) > , which, as cbacba ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length i is simple, implies that b cbacba ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length i − ≅ cbacba ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length i . However this is a contradiction to Lemma 4.6.
Case: n ≡ ( mod 3 ) If n ≡ ( mod 3 ) , then the cyclic word bcabc ⋯·„„„„„„„„‚„„„„„„„„¶ length n ends in c . Consider the word bca ⋯ cab ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − c bac ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − . If n ≡ ( mod 6 ) then we can use the relations (4.2) to write bca ⋯ cab ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − c bac ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − ≅ cba ⋯ bac ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − b cab ⋯ abc ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − , and if n ≡ ( mod 6 ) then we can use the relations (4.2) to write bca ⋯ cab ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − c bac ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − ≅ bac ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − a bca ⋯ cab ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − . Hence we further break the proof up into two more sub-cases.20 ase: n ≡ ( mod 6 ) If n ≡ ( mod 6 ) then we have bca ⋯ cab ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − c bac ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − ≅ cba ⋯ bac ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − b cab ⋯ abc ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − , to which we can apply relation (4.1) on the first n letters of the left hand sideof the above equation to get abc ⋯ bca ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − b bac ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − ≅ cba ⋯ bac ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − b cab ⋯ abc ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − . Evaluating the left hand side reveals that a ⊂ cba ⋯ bac ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − b cab ⋯ abc ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − , from which we can use a similar argument as in the n ≡ ( mod 6 ) case to seethat acb ⋯ bac ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≅ cba ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n . One can now proceed as in the n ≡ ( mod 6 ) or n ≡ ( mod 6 ) case to obtaina contradiction. Case: n ≡ ( mod 6 ) If n ≡ ( mod 6 ) then we have bca ⋯ cab ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − c bac ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − ≅ bac ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − a bca ⋯ cab ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − , to which we can apply relation (4.1) on the first n letters of the left hand sideof the above equation to get abc ⋯ bca ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − b bac ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − ≅ bac ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − a bca ⋯ cab ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − . Evaluating the left hand side reveals that a ⊂ bac ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − a bca ⋯ cab ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − , from which we can use a similar argument as in the n ≡ ( mod 6 ) case to seethat a bac ⋯ acb ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n − ≅ bac ⋯ cba ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n . However this is a contradiction to Lemma 4.6.21 ublemma 4.7.2.
There are no rank preserving finite unitarizable quotients of
Fib ∗ such that only a and b commute with each other. Proof.
Consider such a quotient. As only a and b commute, we must have the otherrelations bcb ≅ cbc and aca ≅ cac. Let us consider the equation (4.1) abcab ⋯·„„„„„„„„‚„„„„„„„„„¶ length n ≅ bcabc ⋯·„„„„„„„„‚„„„„„„„„¶ length n . If n ≥ then we have bcabca ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≅ bcbaca ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≅ cbcaca ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„„¶ length n ≅ cbccac ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n Thus abcabc ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≅ cbccac ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n . However we can expand the cc on the right hand side to see that abcabc ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n is not simple,which is a contradiction.If n = , , ⋯ then one can quickly deduce a contradiction from equation (4.1). Wework through the n = case, and leave the other four cases to the reader, as they are allfairly similar.If n = , then we have the relation abcab ≅ bcaba. We can write abcab ≅ bacab ≅ bcacb, thus bcacb ≅ bcaba. Applying Lemma 2.2 three times gives cb ≅ ba. a and b commute, we get cb ≅ ab. Finally, using Lemma 2.2 gives c ≅ a which is a contradiction. Sublemma 4.7.3.
There are no rank preserving finite unitarizable quotients of
Fib ∗ such that only b and c commute with each other. Proof.
As only b and c commute, we must have the other relations aba ≅ bab and aca ≅ cac. Let us consider the equation (4.1) abcab ⋯·„„„„„„„„‚„„„„„„„„„¶ length n ≅ bcabc ⋯·„„„„„„„„‚„„„„„„„„¶ length n . If n ≥ then we have bcabcab ⋯·„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„¶ length n ≅ bcacbab ⋯·„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„¶ length n ≅ bacabab ⋯·„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„¶ length n ≅ bacaaba ⋯·„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„¶ length n . Thus abcabca ⋯·„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„¶ length n ≅ bacaaba ⋯·„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„¶ length n . However we can expand the aa on the right hand side to see that abcabca ⋯·„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„¶ length n is not simple,which is a contradiction.Now we work through the cases n = , , , , , and derive a contradiction in eachcase. Here the cases n = and n = are somewhat difficult, and we leave the easier casesto the readerIf n = then we have the relation abc ≅ bca. We can left multiply by b and right multiply by c to get bab ⊕ babc ≅ cac ⊕ bcac. A direct computation shows dim ( bab → bab ) = , thus bab is simple, and so either bab ≅ cac or bab ⊂ bcac . The former case we can rewrite as aba ≅ aca,
23o which we can apply Lemma 2.2 to show b = c , a contradiction. The later case impliesthat dim ( bab → bcac ) > , to which we can use Frobenius reciprocity to see dim ( bbab → cac ) > which implies either bab ≅ cac or ab ⊂ cac , each of which is a contradiction.If n = then we have the relation abcabc ≅ bcabca. We can write abcabc ≅ abcacb ≅ abacab ≅ babcab, thus babcab ≅ bcabca. We apply Lemma 2.2 to see abcab ≅ cabca. Right multiplying by b gives abcab ⊕ abca ≅ cabcab. which contradicts the fact that cyclic words of length or less are simple. Sublemma 4.7.4.
There are no rank preserving finite unitarizable quotients of
Fib ∗ such that only a and c commute with each other. Proof.
Consider such a quotient. As only a and c commute, we must have the otherrelations aba ≅ bab and bcb ≅ cbc. Let us consider the equation (4.1) abcab ⋯·„„„„„„„„‚„„„„„„„„„¶ length n ≅ bcabc ⋯·„„„„„„„„‚„„„„„„„„¶ length n . If n ≥ then we have abcabc ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≅ abacbc ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≅ babcbc ⋯·„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≅ babbcb ⋯·„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n . babbcb ⋯·„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n ≅ bcabca ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n . However we can expand the bb on the left hand side to see that bcabca ⋯·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length n is not simple,which is a contradiction.Now we work through the cases n = , , , , and derive a contradiction in each case.Here the case n = is difficult, and we leave the easier cases to the reader.If n = then we have the relation ab ≅ bc. Left multiplying by b , and right multiplying by b gives babb ≅ bbcb which we expand to see ba ⊕ bab ≅ cb ⊕ bcb. As ba ≅ cb we must have bab ≅ bcb , to which we can apply Lemma 2.2 to show a ≅ c , acontradiction.With the technical Lemma 4.7 in hand, we can now make a significant reduction tothe problem of classifying finite unitarizable quotients of Fib ∗ N . We show that any suchquotient must factor through a certain Deligne product of Fib and
T T categories. Lemma 4.8.
Let C be a rank preserving finite unitarizable quotient of Fib ∗ N , then C is arank preserving unitarizable quotient of Fib ⊠ n ⊠T T ⊠ m where n + m = N .Proof. We prove by induction on N . N = :The only quotient of Fib is Fib itself. N = :By [26] the only finite unitarizable quotients of Fib ∗ Fib are
Fib , Fib ⊠ Fib and
T T . Only the latter two of these are rank preserving quotients.Inductive step:Suppose the result holds for N − , and let C be a finite rank preserving unitarizablequotient of Fib ∗ N . Consider an arbitrary generator τ of C . We have two cases toconsider, either τ commutes with all the other generators, or there exists anothergenerator τ with which it doesn’t commute.25f τ commutes with all the other generators, then C must be a unitarizable quo-tient of Fib ⊠D , where D is a finite rank preserving unitarizable quotient of Fib ∗( N − ) .Thus we can use the inductive hypothesis to see that C is a rank preserving unitariz-able quotient of Fib ⊠ Fib ⊠ n ⊠T T ⊠ m where n + m = N − .If there exists a generator τ for which τ does not commute, then they mustgenerate a T T subcategory of C . Let τ be a third generator of C , then the gener-ators τ , τ , and τ generate a finite rank preserving unitarizable quotient of Fib ∗ .By Lemma 4.7 we must have that τ commutes with τ and τ . As τ was chosenarbitrarily we have shown that τ and τ commute with all the other N − generatorsof Fib ∗ N . Thus C is a rank preserving unitarizable quotient of T T ⊠ D , where D isa finite rank preserving unitarizable quotient of Fib ∗( N − ) . Therefore we can use theinductive hypothesis to see that C is a rank preserving unitarizable quotient of T T ⊠ Fib ⊠ n ⊠T T ⊠ m where n + m = N − .Irrespective on whether τ commutes with all the other generators or not, we haveshown that C is a rank preserving unitarizable quotient of Fib ⊠ n ⊠T T ⊠ m where n + m = N .Recall that we are interested in classifying finite cyclic unitarizable quotients of Fib ∗ N .So far our lemmas have just required the quotient be finite. While we could continue with-out a symmetry condition, and obtain a classification of all finite unitarizable quotients of Fib ∗ N , such a result is beyond the scope of this paper, and the proof of such a generalisedresult appears tedious, if straightforward. In the following Remark we discuss which ofthe categories Fib ⊠ n ⊠T T ⊠ m are cyclic, and thus could have cyclic quotients. Remark 4.9.
Recall a finite cyclic unitarizable quotient of
Fib ∗ N is a finite quotient thathas an action of Z N such that the generator of Z N maps τ ↦ τ ↦ ⋯ ↦ τ N ↦ τ . Clearly the finite quotient
Fib ⊠ N is cyclic, with the symmetry coming from the canonical Z N action permuting the factors.When N is even, we also have the quotient T T ⊠ N is cyclic. In this case the symmetryis more involved. Let τ and φ be the Fibonacci generators of T T , then the generator of Z N acts on T T ⊠ N by τ ↦ τ ↦ ⋯ τ N ↦ φ ↦ φ ↦ ⋯ ↦ φ N ↦ τ . s both the categories Fib ⊠ N and T T ⊠ N have trivial universal grading group, and haveno non-trivial invertible objects, we have from [15, Theorem 3.4] along with [13, Theorem5.5] that both of the above Z N actions lift to categorical actions of Z N on the quotients. It is clear by the definition of a cyclic quotient that the categories mentioned in theabove Remark are the only possible cyclic unitarizable quotients of
Fib ∗ N . Thus we havethe following Corollary 4.10.
Let C be a rank preserving finite cyclic unitarizable quotient of Fib ∗ N .Then C is a cyclic unitarizable quotient of either Fib ⊠ N , or, if N is even, T T ⊠ N . Finally we need to determine if there are any further unitarizable quotients of
Fib ⊠ N or T T ⊠ N . This is equivalent to classifying central commutative algebras in these categories,a hard problem in general [16]. However we are able to find arguments specific to thetwo above categories that bypass having to classify such algebras. The idea behind thesearguments is to simply compute all the simple objects in such a quotient. We find thatthe simple objects of Fib ⊠ N or T T ⊠ N remain simple and distinct in any quotient. Thusonly the trivial quotients exist. This argument was inspired by a similar technique usedin [22].We begin with the Fib ⊠ N case. Lemma 4.11.
Let C be a rank preserving unitarizable quotient of Fib ⊠ N , then C ismonoidally equivalent to Fib ⊠ N .Proof. As C is a rank preserving finite unitarizable quotient of Fib ⊠ N , it contains thecommuting distinct simple objects τ , τ , ⋯ , τ N each individually satisfying the Fibonaccifusion rule.Consider the words of C that contain at most one of each of the objects τ , τ , ⋯ , τ N .We will call such words source-simple words. We say two source-simple words are identicalif they share the same letter set.To prove the claim of this lemma, we will show that all source-simple words are simple,and two source-simple words are distinct if they are non-identical. With this result, wecount the global dimension of C as at least ( +√ ) N , hence C must be the trivial quotientof Fib ⊠ N .We begin by showing all source-simple words are simple.We induct on the length of the word.As the objects τ , τ , ⋯ , τ N are simple, we have that the result holds for source-simplewords of length .Suppose the result holds for source-simple words of length k − for k ≤ N . That is,suppose all source-simple words of length k − or less are simple. Consider a source-simpleword of length k . As the labels for the N generators of Fib ⊠ N were chosen arbitrarily we27an assume that this words begins with τ and ends with τ . We compute the endomor-phisms of this word. dim ( τ ⋯ τ † length k → τ ⋯ τ † length k ) = dim ( τ τ ⋯† length k → ⋯ τ τ † length k )= dim ( τ ⋯(cid:176) length k − ⊕ ⋯fi length k − → ⋯ τ (cid:176) length k − ⊕ ⋯fi length k − )= dim ( τ ⋯(cid:176) length k − → ⋯ τ (cid:176) length k − ) + dim ( ⋯fi length k − → ⋯ τ (cid:176) length k − )+ dim ( τ ⋯(cid:176) length k − → ⋯fi length k − ) + dim ( ⋯fi length k − → ⋯fi length k − )= + dim ( τ ⋯(cid:176) length k − → ⋯ τ (cid:176) length k − ) . Here the last step uses the inductive hypothesis.As both τ ⋯(cid:176) length k − and ⋯ τ (cid:176) length k − are source-simple words, we have that they are simpleby the inductive hypothesis. Thus dim ( τ ⋯(cid:176) length k − → ⋯ τ (cid:176) length k − ) is either equal to or .If dim ( τ ⋯(cid:176) length k − → ⋯ τ (cid:176) length k − ) = then we have the equality τ ⋯(cid:176) length k − ≅ ⋯ τ (cid:176) length k − as both words are simple. Using thecommutativity of the τ i ’s we can write τ ⋯(cid:176) length k − ≅ τ ⋯(cid:176) length k − to which we can apply Lemma 2.2 k − times (as the rightmost k − letters of both wordsare identical up to order) to see τ ≅ τ . Thus we have a contradiction, so dim ( τ ⋯(cid:176) length k − → ⋯ τ (cid:176) length k − ) = . Therefore the endomorphism space of τ ⋯ τ † length k is 1 dimensional, and hence is simple. Thus,all source-simple words of length k are simple, completing the induction.We now prove that all non-identical source-simple words are distinct. Again we inducton the length of the word. 28s the objects τ , τ , ⋯ , τ N are distinct, we have that the result holds for source-simplewords of length .Suppose the result holds for source-simple words of length k − for k ≤ N . That is,suppose all non-identical source-simple words of length k − are distinct. Consider twosource-simple words, and suppose they are equal. We consider two cases, either these twowords share a letter, or they do not share a letter.If the two words do not share a letter then consider the word obtained by concatenatingthese two words together. As the two component words share no letters, we have theconcatenated word is source-simple, and thus simple by the earlier result. However as thetwo component words are equal, we can use Frobenius reciprocity to show that there isa non-trivial morphism from the tensor unit to the concatenated word, a contradiction.Thus this case can not occur.Now suppose the two words share a letter, say τ . We can use the commutativity ofthe τ i ’s to bring the shared τ to the left hand side of each word. Then we can applyLemma 2.2 to see that the two words of length k − obtained by removing τ are equal.The inductive hypothesis then gives that these two words of length k − are identical.As the two words of length k are obtained from this length k − word by appending a τ , we thus have these two source-simple words of length k are identical, completing theinduction.Next we consider the T T ⊠ N case. The arguments here are slight alterations of the Fib ⊠ N case. Lemma 4.12.
Let C be a rank preserving unitarizable quotient of T T ⊠ N . Then C ismonoidally equivalent to T T ⊠ N .Proof. As C is a rank preserving finite quotient of T T ⊠ N , it contains the distinct simpleobjects τ , φ , τ , φ ⋯ , τ N , φ N each individually satisfying the Fib fusion rules, and τ i φ i τ i ≅ φ i τ i φ i , τ i τ j ≅ τ j τ i , φ i τ j ≅ τ j φ i , φ i φ j ≅ φ j φ i , for i ≠ j .Consider the words of C that for each i , contain either at most one τ i and at most one φ i , or at most one of the words τ i φ i τ i . We will call such words source-simple words. Wesay two source-simple words are identical if for each i , the two subwords consisting of τ i ’sand φ i ’s are equal in T T .To prove the claim of this lemma, we will show that all source-simple words are simple,and two source-simple words are distinct if they are non-identical. With this result, wecount the global dimension of C as at least ( + √ ) N , hence C must be the trivialquotient of T T ⊠ N .As in the Fib ⊠ N case we begin by proving first proving that all source-simple wordsare simple. We proceed by induction.As the objects τ , φ , τ , φ , ⋯ , τ N , φ N are all simple, we have that the result holds forsource-simple words of length .Suppose the result holds for words of length k − for k < N . That is, suppose allsource-simple words of length k − are simple. Consider a source-simple word of length k .29e can assume that this word begins with the letter τ . We have four cases to consider,either the word ends in τ , τ , φ , or φ . Case: The source-simple word ends in τ If the source-simple word begins and ends in τ then the word must also containa φ and no additional τ ’s, by the definition of source-simple. Using the inductivehypothesis we compute the endomorphism space of this word dim ( τ ⋯ τ † length k → τ ⋯ τ † length k ) = + dim ( τ ⋯(cid:176) length k − → ⋯ τ (cid:176) length k − ) . As both τ ⋯(cid:176) length k − and ⋯ τ (cid:176) length k − are source-simple, they are simple by the inductivehypothesis. Thus dim ( τ ⋯(cid:176) length k − → ⋯ τ (cid:176) length k − ) is equal to either or .If dim ( τ ⋯(cid:176) length k − → ⋯ τ (cid:176) length k − ) = then τ ⋯(cid:176) length k − ≅ ⋯ τ (cid:176) length k − . As the parent length k word only contained two τ ’s and a single φ we have thateach of these words contains a τ φ and φ τ subword respectively, and no other τ ’sor φ ’s. Thus we can commute to the right to get ⋯ τ φ ·„„„„‚„„„„„¶ length k − ≅ ⋯ φ τ ·„„„„‚„„„„„¶ length k − . The leftmost k − letters of both these words are the same, so we can apply Lemma 2.2 k − times to get τ φ ≅ φ τ , a contradiction. Thus dim ( τ ⋯(cid:176) length k − → ⋯ τ (cid:176) length k − ) = and so τ ⋯ τ † length k is simple. Case: The source-simple word ends in τ Suppose we have a source-simple word that begins with τ and ends in τ . Usingthe inductive hypothesis we compute the endomorphism space of this word dim ( τ ⋯ τ † length k → τ ⋯ τ † length k ) = + dim ( τ ⋯(cid:176) length k − → ⋯ τ (cid:176) length k − ) .
30s both τ ⋯(cid:176) length k − and ⋯ τ (cid:176) length k − are source-simple, they are simple. Thus dim ( τ ⋯(cid:176) length k − → ⋯ τ (cid:176) length k − ) is equal to either or .If dim ( τ ⋯(cid:176) length k − → ⋯ τ (cid:176) length k − ) = then τ ⋯(cid:176) length k − ≅ ⋯ τ (cid:176) length k − . We have three further sub-cases to consider, either the source-simple word oflength k contains only a single τ , and no additional τ ’s or φ ’s, or the source-simpleword of length k contains a τ φ , and no additional τ ’s or φ ’s, or the source-simpleword of length k contains a τ φ τ , and no additional τ ’s or φ ’s. Sub-case: The source-simple word of length k contains only a single τ , and no additional τ ’s or φ ’s In this case we can commute the τ to the far right of τ ⋯(cid:176) length k − , and cancelthe shared leftmost k − letters in the equation ⋯ τ (cid:176) length k − ≅ ⋯ τ (cid:176) length k − , to obtain τ ≅ τ , a contradiction. Sub-case: The source-simple word of length k contains a τ φ , and noadditional τ ’s or φ ’s In this case we can commute the φ to the far right of the word ⋯ τ (cid:176) length k − ,and we can commute the τ and φ to the far right of the word τ ⋯(cid:176) length k − to getthe equality ⋯ τ φ ·„„„„‚„„„„„¶ length k − ≅ ⋯ τ φ ·„„„„‚„„„„„¶ length k − . Cancelling the shared leftmost k − letters in the above equation, and the sharedrightmost letter gives τ ≅ τ , a contradiction. Sub-case: The source-simple word of length k contains a τ φ τ , andno additional τ ’s or φ ’s
31n this case we can commute the φ τ to the far right of the word ⋯ τ (cid:176) length k − ,and we can commute the τ φ τ to the far right of the word τ ⋯(cid:176) length k − to get theequality ⋯ τ φ τ ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length k − ≅ ⋯ τ φ τ ·„„„„„„„„„„„„‚„„„„„„„„„„„„¶ length k − . Cancelling the shared leftmost k − letters in the above equation, and the twoshared rightmost letter gives τ ≅ τ , a contradiction.In all cases we get a contradiction, thus dim ( τ ⋯(cid:176) length k − → ⋯ τ (cid:176) length k − ) = . Hence τ ⋯ τ † length k is simple. Case: The source-simple word ends in φ If the source-simple word begins with τ and ends in φ then the word mustcontain no additional τ ’s or φ ’s, by the definition of source-simple. Using theinductive hypothesis we compute the endomorphism space of this word dim ( τ ⋯ φ ·„„„„‚„„„„„¶ length k → τ ⋯ φ ·„„„„‚„„„„„¶ length k ) = + dim ( τ ⋯(cid:176) length k − → ⋯ φ – length k − ) . As both τ ⋯(cid:176) length k − and ⋯ φ – length k − ) are source-simple, they are simple. Thus dim ( τ ⋯(cid:176) length k − → ⋯ φ – length k − ) is equal to either or .If dim ( τ ⋯(cid:176) length k − → ⋯ φ – length k − ) = then τ ⋯(cid:176) length k − ≅ ⋯ φ – length k − . As both these words contain no additional τ ’s or φ ’s we can commute the τ to thefar right of the left word to get ⋯ τ (cid:176) length k − ≅ ⋯ φ – length k − , k − letters of both words being the same. Applying Lemma 2.2 k − times gives τ ≅ φ , a contradiction. Thus dim ( τ ⋯(cid:176) length k − → ⋯ φ – length k − ) = and so τ ⋯ φ ·„„„„‚„„„„„¶ length k is simple. Case: The source-simple word ends in φ This case is near identical to the case where the word ends in τ .This case by case analysis shows all source-simple words of length k are simple, com-pleting the induction.We now prove that all non-identical source-simple words are distinct. Again we inducton the length of the word.As the objects τ , φ , τ , φ ⋯ , τ N , φ N are all distinct, we have that the result holds forsource-simple words of length .Suppose the result holds for source-simple words of length k − for k ≤ N . That is,suppose all non-identical source-simple words of length k − are distinct. Consider twoequal source-simple words of length k − . We have two cases to consider, either bothwords share a letter, or they don’t share a letter.Suppose the two words don’t share a letter. Then the concatenation of these twowords is another source-simple word, and is thus simple. However as the two componentwords are equal, we can use Frobenius reciprocity to show that the concatenated wordhas a non-trivial morphism to the tensor unit. Thus we have a contradiction, so this casecan not occur.Suppose the two words share a letter τ . Considering all possible positions of the τ ’sand φ ’s in both words, in all but one bad case we can simultaneously commute (by thestandard commuting relations, and with the relation φ τ φ = τ φ τ ) either a τ or a φ tothe same side of both words and then cancel with Lemma 2.2 and appeal to the inductivehypothesis. The bad case is if one word contains just a φ τ and the other contains just a τ φ . For this bad case we can commute these blocks to the far left of both words to getan equality of the form φ τ ⋯fi length k − ≅ τ φ ⋯fi length k − . We thus have = dim ( φ τ ⋯fi length k − → τ φ ⋯fi length k − ) = dim ( φ τ φ τ ⋯fi length k − → ⋯fi length k − )= dim ( φ φ τ φ ⋯fi length k − → ⋯fi length k − ) dim ( φ τ φ ⋯fi length k − → ⋯fi length k − )+ dim ( τ φ ⋯fi length k − → ⋯fi length k − )= dim ( τ φ ⋯fi length k − → φ ⋯fi length k − )+ dim ( φ ⋯fi length k − → τ ⋯fi length k − )= . Here the last step follows from the fact that the words τ φ ⋯fi length k − , φ ⋯fi length k − , and τ ⋯fi length k − are all source-simple. Thus dim ( τ φ ⋯fi length k − → φ ⋯fi length k − ) = because it is a morphism space between simples of different dimension, and dim ( φ ⋯fi length k − → τ ⋯fi length k − ) = by the inductive hypothesis. Thus we have a contradiction, thus this single bad case cannot occur. Hence all non-identical source-simple words of length k are distinct, completingthe induction.Summing up the results of this section, we have Theorem 4.1. Seeing that we have classified finite rank preserving cyclic unitarizable quotients of
Fib ∗ N in the previous section, we can now apply Theorem 3.1 to give a classification of unitaryfusion categories ⊗ -generated by an object of dimension +√ . Proof of Theorem 1.1.
Let C be a unitary fusion category ⊗ -generated by an object ofdimension +√ . Then in particular C is unitarizable, so by Theorem 3.1 we have that C is monoidally equivalent to a semi-direct product of a finite rank preserving cyclicunitarizable quotient of Fib ∗ N by a cyclic group that factors through the cyclic Z N action.From Theorem 4.1 we know that the finite cyclic unitarizable quotients of Fib ∗ N are Fib ⊠ N for any N , and T T ⊠ N for N even, with the cyclic actions given by Remark 4.9. For acyclic group to factor through the cyclic Z N action, the order must be a multiple of N .Hence C is monoidally equivalent to either• A crossed product of Fib ⊠ N by Z NM , with the generator of Z NM acting by cyclicallypermuting the factors, or 34 A crossed product of T T ⊠ N by Z NM , with the group Z NM acting by factoringthrough the Z N action described in Remark 4.9.Thus C is monoidally equivalent to either• Fib ⊠ N ω ⋊ Z NM where N, M ∈ N and ω ∈ H ( Z NM , C × ) , or• T T ⊠ N ω ⋊ Z NM where N, M ∈ N and ω ∈ H ( Z NM , C × ) . Each of these categories has a unitary structure, and further, this unitary structure isunique by [33]. Finally [33, Theorem 1] shows that the monoidal equivalence above isnaturally isomorphic to a unitary monoidal equivalence.
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