Type II quantum subgroups of \mathfrak{sl}_N. I: Symmetries of local modules
TTYPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCALMODULES CAIN EDIE-MICHELL
Abstract.
This paper is the first of a pair that aims to classify a large number of the type II quantum subgroups of the categories C( sl r + , k ) . In this work we classify the braided auto-equivalences of the categories of local modules for all known type I quantum subgroups of C( sl r + , k ) , barring a single unresolved case for an orbifold of C( sl , ) . We find that thesymmetries are all non-exceptional except for four possible cases (up to level-rank duality).These exceptional cases are the orbifolds C( sl , ) ( Z ) , C( sl , ) ( Z ) , C( sl , ) ( Z ) , and C( sl , ) ( Z ) .We develop several technical tools in this work. We give a skein theoretic description of theorbifold quantum subgroups of C( sl r + , k ) . Our methods here are general, and the techniquesdeveloped will generalise to give skein theory for any orbifold of a braided tensor category. Wealso uncover an unexpected connection between quadratic categories and exceptional braidedauto-equivalences of the orbifolds. We use this connection to construct two of the four excep-tionals, and to reduce the C( sl , ) ( Z ) case to a concrete finite computation.In the sequel to this paper we will use the classified braided auto-equivalences to constructthe corresponding type II quantum subgroups of the categories C( sl r + , k ) . When paired withGannon’s type I classification for r ≤
6, this will complete the type II classification for thesesame ranks, excluding the one exception at C( sl , ) . Introduction
One of the oldest open problems in quantum algebra has been the program to classify the quantum subgroups (or module categories) of the categories C( g , k ) . This program was initiallyinvestigated in the language of conformal field theory by Cappelli, Itzykson, and Zuber [5].They used physical reasoning to argue that a quantum subgroup of C( g , k ) is precisely the dataneeded to extend a chiral Wess-Zumino-Witten conformal field theory (constructed from g and k ) up to a full conformal field theory . With this motivation in hand they were then able togive a combinatorial classification of the quantum subgroups of C( sl , k ) . Their results wereunexpected and exciting, falling into an A − D − E pattern. The two infinite families A and D were expected, but far more intriguing were the three exceptional examples E , E , and E .Inspired by the richness of the sl classification, there was a flurry of activity to give classifi-cation results for the higher rank Lie algebras [44, 7, 8]. However this proved far more difficultthan the rank one case. Despite the intense research activity directed towards the problem,very few new classification results were achieved. Once the dust had settled, a combinatorialclassification for sl had been given by Gannon [25], and sl had been claimed by Ocneanu [45],but without supplied proof. It was here that the project stagnated, with many considering it tobe intractable.In a more general setting, the problem of extending chiral conformal field theory up to fullconformal field theory was studied rigorously by Fuchs, Runkel, and Schweigert [20, 21, 22, 23,18]. They were able to mathematically confirm the physical arguments of Cappelli, Itzykson, andZuber. It was proven that the data to extend a chiral conformal field theory is precisely a modulecategory over the representation category of the chiral theory. However, a module category ismore than just its combinatorics, which is what was classified in [5] and [25]. There is also thecategorical data of the module category, which is captured by the 6-j symbols, or equivalently a r X i v : . [ m a t h . QA ] F e b CAIN EDIE-MICHELL the associator, of the module. Thus classification for sl and sl was incomplete. The categoricaldata for the sl case was worked out in the subfactor language in [1, 32, 33, 38, 39, 51, 46], andin the categorical language in [46]. For the sl case the categorical data was worked out in [17].Recently there has been a massive revitalisation in the program to classifying quantum sub-groups of the higher rank Lie algebras. This began with work of Schopieray [49], which gavelevel bounds on which categories C( g , k ) could have exceptional type I quantum subgroups forthe rank two Lie algebras. These techniques were then drastically improved upon by Gannon[24], where effective level bounds were determined for all Lie algebras. In short, this allowed fora computer search to find all type I quantum subgroups for any Lie algebra. These computersearches were performed by Gannon, and type I classification was given for all ranks less than7, a dramatic improvement on the state of knowledge. For these examples it was found thatthere are the expected infinite families of de-equivariantisation (or orbifold) type I quantumsubgroups, a finite number of type I quantum subgroups coming from conformal inclusions ofLie groups [51], and four new truly exceptional examples.Thus the type I case has essentially been solved, and classification up to higher ranks is now amatter of computer power, rather than mathematical insight. However, the type II case (whichcomprise all remaining examples) still remains entirely open. This paper is the first in a pairto classify the type II quantum subgroups for sl n . The techniques developed in these paperswill generalise to the other classical algebras. However we restrict our attention now to the type A case for three reasons. First is that the details of working through the generalisation willrequire substantial effort that would push the length of these papers beyond a readable limit.Second is that combinatorial evidence suggests that type A has the richest behaviour with type II quantum subgroups, so we can expect to find the most interesting results by studying thiscase. Finally, historically the type A case had received the most attention, and thus results intype A will attract more interest than the other classical Lie algebras.Our main tool to classify type II quantum subgroups of the categories C( sl r + , k ) is thefollowing theorem due to Davydov, Nikshych, and Ostrik, which gives a bijective correspon-dence between all quantum subgroups, and pairs of type I quantum subgroups, and a braidedequivalence between their categories of local modules . Theorem 1.1. [10]
Let C be a modular category. There is a bijective correspondence { Irreducible modules over
C} ↔ ⎧⎪⎪⎪⎨⎪⎪⎪⎩
Triples (M , M , F ) , where M and M are irreducible type I modules over C , and F ∶ M → M is a braided equivalence. ⎫⎪⎪⎪⎬⎪⎪⎪⎭ The work of Gannon has classified the type I modules of sl n for n ≤
7. Thus to give theclassification of the type II modules, and hence complete the classification of all quantumsubgroups, we need to determine all braided equivalences between their local modules. Gannonfinds that there are three kinds of type I modules [24]. The first class (and most exciting astype I modules) are the four truly exceptional examples, with two occurring at C( sl , ) andtwo at C( sl , ) . These quantum subgroups have categories of local modules equivalent to: C( so , ) , C( sl , ) ζ , Vec , andVec , where C( sl , ) ζ is a Galois conjugate of the category C( sl , ) . For all of these examples, thecategories of local modules are completely understood. The second class consists of the modulecategories constructed from conformal inclusions of Lie groups. These can be found in [9], and YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 3 for the type A case they are: A , ⊂ B , ,A , ⊂ G , ,A , ⊂ E , ,A , ⊂ E , ,A , ⊂ D , A , ⊂ C , ,A , ⊂ D , ,A n,n − ⊂ A ( n − )( n − ) , ,A n,n + ⊂ A n ( n + ) , ,A n + , n + ⊂ B n + n + , ,A n, n + ⊂ D n ( n + ) , , and A n + , n + ⊂ B n + n + , . For all of these examples, the category of local modules is braided equivalent to C( g , ) , where g is the corresponding Lie algebra of the super group. The third class consists of the infinitenumber of orbifold modules, constructed via de-equivariantisation. For C( sl r + , k ) these areparametrised by m a divisor of r +
1, such that m ∣ k ( r + ) is r is even, or 2 m ∣ k ( r + ) is r is odd. We write C( sl r + , k ) Rep ( Z m ) for these type I modules. The category of local modules C( sl r + , k ) ( Z m ) for these examples is described in the bulk of the paper.It is extremely rare that the categories of local modules for any of these type I modulescoincide. Thus the interesting type II module categories of C( sl r + , k ) come from exceptionalbraided auto-equivalences of these categories of local modules. The goal of this paper is todetermine the braided auto-equivalences of the categories of local modules for all known type I quantum subgroups. In the sequel to this paper we will identify the small number of exceptionswhere the categories of local modules coincide, and work through the details of Theorem 1.1 inorder to explicitly construct and classify the corresponding type II quantum subgroups. Pairedwith Gannon’s classification of type I quantum subgroups, this will give type II classificationfor n ≤ sl r + , there is a bound on k for which exceptional type II quantumsubgroups of C( sl r + , k ) can occur. These results will put us in a strong position to classify type II modules for larger n ≥
8, once the type I classification has been sorted for these n .Let us examine the braided auto-equivalences of the local modules for the known type I quantum subgroups. For the four truly exceptional examples found by Gannon we can quicklycompute that the auto-equivalence groups are all trivial, except for the Galois conjugate of C( sl , ) which has auto-equivalence group Z [12, Theorem 1.2]. For the type I quantum sub-groups coming from conformal inclusions of Lie groups, the group of braided auto-equivalenceshas been computed in earlier works of the author [13, Theorem 1.1]. For completeness, we collectthe results here. Theorem 1.2.
We have
EqBr (C( so , )) = { e } , EqBr (C( g , )) = { e } , EqBr (C( e , )) = Z , EqBr (C( e , )) = { e } , EqBr (C( so , )) = Z EqBr (C( sp , )) = Z , EqBr (C( so , )) = Z , EqBr (C( sl ( n − )( n − ) , )) = Z p + t , EqBr (C( sl n ( n + ) , )) = Z p + t , EqBr (C( so ( n + n + )+ , )) = { e } , EqBr (C( so n ( n + ) , )) = ⎧⎪⎪⎨⎪⎪⎩ Z if n ≡ { , } ( mod 4 ) S if n ≡ { , } ( mod 4 ) , and EqBr (C( so ( n + n + )+ , )) = { e } , where p is the number of distinct odd primes that divide the rank plus one, and t is equal to 1 ifthe rank is equivalent to mod , and 0 otherwise. Finally we have the orbifold type I quantum subgroups. Somewhat paradoxically these havethe most interesting categories of local modules, and hence determining their group of braidedauto-equivalences is highly non-trivial. The remainder of this paper will be devoted to provingthe following theorem, which determines the braided auto-equivalences groups in question. Ex-citingly we find a finite number of cases where the braided auto-equivalence group is exceptional,which corresponds to the existence of exceptional type II quantum subgroups. These excep-tional type II quantum subgroups will be explicitly constructed in the sequel. In this case of C( sl , ) ( Z ) we are only able to pin down the braided auto-equivalence group to one of twooptions, corresponding to the existence, or non-existence, of an exceptional auto-equivalence.This one case is particularly hard to determine, and has resisted the methods we develop in thispaper. Still we are able to reduce the existence of the exceptional auto-equivalence down to afinite concrete computation. This case will hopefully be addressed with new methods in futurework. Theorem 1.3.
Let r ≥ and k ≥ and m a divisor of r + satisfying m ∣ k ( r + ) if r is even,and m ∣ k ( r + ) if r is odd. Then except for the cases ● C( sl , ) ( Z ) , ● C( sl , ) ( Z ) , ● C( sl , ) ( Z ) , ● C( sl , ) ( Z ) , ● C( sl , ) ( Z ) , ● C( sl , ) ( Z ) , ● C( sl , ) ( Z ) , and ● C( sl , ) ( Z ) we have that EqBr (C( sl r + , k ) ( Z m ) ) = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩{ e } if k = and r = Z m ′ × Z p + t if k = or r = D m ′ × Z p + t otherwisewhere ● m ′ = gcd ( m, k ) , ● m ′′ = mm ′ , ● p is the number of distinct odd primes dividing r + mm ′′ but not km ′ , and YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 5 ● t = ⎧⎪⎪⎨⎪⎪⎩ if r + mm ′′ is odd, or if km ′ ≡ ( mod 4 ) , or if both km ′ is odd, and r + mm ′′ ≡ ( mod 4 ) otherwise.For the remaining exceptional cases we have that EqBr (C( sl , ) ( Z ) ) = S , EqBr (C( sl , ) ( Z ) ) = S EqBr (C( sl , ) ( Z ) ) ∈ { D , S } , EqBr (C( sl , ) ( Z ) ) = A , EqBr (C( sl , ) ( Z ) ) ∈ { D , S } , EqBr (C( sl , ) ( Z ) ) = S × Z , EqBr (C( sl , ) ( Z ) ) = S × Z , and EqBr (C( sl , ) ( Z ) ) = S . With this theorem in hand, we are now placed to classify all type II quantum subgroupswhose type I parents are in the known list. In particular this will allow us to classify all type II quantum subgroups of sl n for n ≤ C( sl , ) . This will completethe classification of all quantum subgroups for these examples. As mentioned earlier, this type II classification will be dealt with in the sequel to this paper. Extrapolating from the work ofGannon, we can expect the truly exceptional type I quantum subgroups of the higher rank sl n to be exceedingly rare, and when they do occur, we can expect their categories of local modulesto be somewhat trivial. This means that when the type I classification has been extended tohigher rank, the results of this paper will allow the type II classification to nearly immediatelyfollow.With the motivation and main theorem of this paper described, let us move on to describingthe structure of the article.In section 2 we introduce the background required to begin this paper. We introduce thecombinatorics of the categories C( sl r + , k ) . In particular we give the formula for the dimensionsof the simples, and prove useful inequalities which they obey. We describe the structure of theorbifold C( sl r + , k ) Rep ( Z m ) , and of the local modules C( sl r + , k ) ( Z m ) . We explicitly determineuseful structure of the category C( sl r + , k ) ( Z m ) , including the parametrisation of the simples,the group of invertibles, and the adjoint subcategory.In section 3 we determine the so called non-exceptional braided auto-equivalences of C( sl r + , k ) ( Z m ) .These are the braided auo-equivalences which fix the image of the adjoint representation underthe free module functor. The end result is the expected one, i.e. we show all non-exceptionalbraided auto-equivalences are either charge conjugation, simple currents, or come from thecanonical Z m -action. However proving this result is highly technical, and requires several pow-erful techniques. The difficulty here is not surprising, as determining the non-exceptional braidedauto-equivalences has troubled researchers working on this same problem in the past. To beginwe develop skein theory for the adjoint subcategory of C( sl r + , k ) ( Z m ) . Our methods here aregeneral, and will allow one to find skein theory for de-equivariantisation by an abelian group ofany braided category, given that skein theory of the original category is known. With this skeintheory in hand we can then use standard planar algebra techniques to find the non-exceptionalbraided auto-equivalence group of the adjoint subcategory. To extend these auto-equivalencesto the entire category we use the techniques developed by the author in [15]. These techniquesgive an upper bound on the number of auto-equivalences which may extend an auto-equivalenceon the adjoint subcategory. By a happy coincidence, this upper bound is precisely realised by CAIN EDIE-MICHELL simple current auto-equivalences, introduced in the authors work [14], which was inspired bycombinatorics from conformal field theory. This happy coincidence suggests the potential for ageneral theorem.
Conjecture 1.4.
Let C be a modular tensor category, C ad its adjoint subcategory, and F anauto-equivalence of C which restricts to the identity on C ad . Then F is isomorphic to a simplecurrent auto-equivalence. The validity of this general conjecture remains to be investigated. All together, the re-sults of this section fully classify all non-exceptional braided auto-equivalences of the categories C( sl r + , k ) ( Z m ) .In section 4 we investigate the combinatorics of the exceptional braided auto-equivalences of C( sl r + , k ) ( Z m ) . We show that with a finite number of exceptions, that every braided auto-equivalence of C( sl r + , k ) ( Z m ) is non-exceptional, and hence is covered by the results of theprevious section. Our main observation here is simple. If there were an exceptional braided auto-equivalence of C( sl r + , k ) ( Z m ) , then its image of the adjoint would have the same dimensionand twist. This puts massive combinatorial restrictions on the objects of the category C( sl r + , k ) .By studying these restrictions in a fairly nasty case by case analysis, we are able to obtain aseries of inequalities which imply that both the rank and level must be small. From here wecan computer search to find the finite cases where C( sl r + , k ) ( Z m ) has an exceptional braidedauto-equivalence, at the level of the fusion ring and twists. Up to level-rank duality we findfour possible candidates for exceptional braided auto-equivalences. These are C( sl , ) ( Z ) , C( sl , ) ( Z ) , C( sl , ) ( Z ) , and C( sl , ) ( Z ) . While the case by case analysis is messy,and a uniform approach to this section would be desired, the exceptional examples which arediscovered mean that such a uniform approach is unlikely to exist.In section 5 we finish up by attempting to realise all of the exceptional braided auto-equivalencesof C( sl r + , k ) ( Z m ) for the finite number of remaining cases identified in the previous section.We see two situations at hand. The first has already been observed in the literature [41], andconcerns the category C( sl , ) ( Z ) . Here the exceptional braided auto-equivalence existsdue to a coincidence of categories. The second situation is much more extremely interesting.We show a connection between the three remaining examples, and three explicit quadratic cat-egories . This connection is sufficiently explicit, so that having a construction of the quadraticcategories allows us the construction of the exceptional braided auto-equivalences. Two of thethree quadratic categories have been constructed by Izumi [34, 35], which allows these cases tobe solved. For the remaining case, the quadratic category is too large for the standard Cuntzalgebra method to be efficient, as the equations that are required to be solved are too compli-cated for the author, or computer, to solve. This unsolved remaining case is the reason why ourmain theorem has an unresolved case for C( sl , ) . It appears a fresh method is needed to solvethis case.The connection between type II quantum subgroups, and quadratic categories appears to bemore than just a convenient coincidence. It occurs for other Lie algebras outside the A series,and the author will weakly conjecture that every exceptional type II quantum subgroup fora simple Lie algebra comes from either a coincidence of categories, or from a connection to aquadratic category. We will not say much more on this to avoid spoiling future work. Acknowledgements.
We would like to thank Dietmar Bisch for many clarifying conversationsthroughout the duration of this work. We would also like to thank Scott Morrison for help-ful conversations on skein theory for orbifolds, Terry Gannon for helpful conversations on thecombinatorics of the categories C( sl r + , k ) and on his results in the type I case, and PinhasGrossman and Masaki Izumi for helpful conversations on the Cuntz algebra construction ofquadratic categories. YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 7 This material is based work supported by the National Science Foundation under Grant No.DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institutein Berkeley, California, during the Spring 2020 semester. In addition, the author gratefullyacknowledges support from the AMS-Simons travel grant2.
Preliminaries
We refer the reader to [16] for the basics of fusion categories.2.1.
Quantum Integers, Dimensions, and Inequalities.
The main object of study in thispaper will be the modular tensor categories C( sl r + , k ) , the category of level k integrable rep-resentations of ˆ sl n . For an overview us these categories see [50]. For our purposes we willonly require some basic combinatorics of these categories. The simple objects of C( sl r + , k ) areparametrised by r ∑ i = λ i Λ i where λ i ∈ N and r ∑ i = λ i = k. Often we will omit the λ term of a simple object, as its value can be deduced from the remaining λ i ’s. For example, the vector representation ( k − ) Λ + Λ will usually be written simply asΛ . A special subset of these simples are the r + { k Λ i ∶ i ∈ Z r + } . To describe the quantum dimensions of the simple objects of C( sl r + , k ) we will need twoingredients. The first are the quantum integers. Definition 2.1.
We define the n -th quantum integer (as a function of r and k ) as [ n ] r,k ∶= q n − q − n q − q − and q = e πi ( + k + r ) . The second ingredient is the hook formula, which gives the quantum dimension of a simpleof C( sl r + , k ) in terms of quantum integers. To describe this formula, we have to introduce thetableaux of a simple object. Let X = ∑ ri = λ i Λ i be a simple object, and define a r × k tableaux T ( X ) whose j -th row contains ∑ ri = j λ i boxes. For each box ( x, y ) in the tableaux T ( X ) we candefine the content, which is the quantum integer [ r + + − x + y ] r,k , and the hook length, whichis the quantum integer [ h ] r,k , where h is the number of boxes with the same x or y coordinate.The quantum dimension of X is the product over all the boxes of T ( X ) of the contents dividedby the hooks. For a quick example, we have that the tableaux for the object Λ + Λ ∈ C( sl r + , k ) has two boxes in row one, and one box in row two. Thus the contents are [ r + ] r,k , [ r ] r,k , and [ r + ] r,k , and the hooks are [ ] r,k , [ ] r,k , and [ ] r,k . Therefore the hook formula tells us that the quantum dimension of Λ + Λ is [ r ] r,k [ r + ] r,k [ r + ] r,k [ ] r,k .There are two natural actions of the simples of C( sl r + , k ) that preserve the dimensions. Theseare charge-conjugation which sends r ∑ i = λ i Λ i ↦ r ∑ i = λ i Λ − i . The fact that this map preserves dimensions can be deduced from the hook formula. The otheraction comes from simple currents, which sends r ∑ i = λ i Λ i ↦ r ∑ i = λ i Λ i + a for a ∈ Z r + . CAIN EDIE-MICHELL
This map preserves dimensions as it is simply tensoring by the invertible k Λ a . For an object X ∈ C( sl r + , k ) we write [ X ] for its orbit under the action of simple currents.The dimensions of the simples of C( sl r + , k ) respect the geometry of the truncated Weylchamber in a nice manner. Namely if one draws a convex hull in the truncated Weyl chamber,then the minimum of the dimensions in this hull will occur at the corners. Lemma 2.2. [27]
For ≤ i ≤ N , let X i ∈ C( sl r + , k ) simple objects, and t i ∈ [ , ] such that ∑ Ni = t i = . Then dim ( N ∑ i = t i X i ) ≥ min { dim ( X i ) ∶ ≤ i ≤ N } , with equality occurring exactly at the corners of the convex hull. We also have the following inequalities of quantum integers.
Lemma 2.3.
For all ≤ n ≥ r + k we have [ n ] r,k < [ n ] r + ,k and [ n ] r,k < [ n ] r,k + . Proof.
The second inequality holds as [ n ] r,k is equal to [ n ] n − ,r + k − n + . The value [ n ] n − ,r + k − n + is precisely the graph norm of the fusion graph for Λ ∈ C( sl n , r + k − n + ) . This fusion graphembeds in the fusion graph for Λ ∈ C( sl n , r + k − n + ) , which has graph norm [ n ] n − ,r + k − n + .As graph norms respect inclusions, we get that [ n ] n − ,r + k − n + < [ n ] n − ,r + k − n + , which is equivalent to [ n ] r,k < [ n ] r,k + . The first inequality now holds as [ n ] r,k = [ n ] k,r . (cid:3) Often it will be useful to bound a quantum integer by a simpler function of n . The followinginequalities allow us exactly that. The first bounds the quantum integer above. Lemma 2.4. [49]
For all n ≥ we have [ n ] r,k ≤ n. The second bounds the quantum integer below.
Lemma 2.5. [49]
Suppose that ≤ n ≤ c − c ( + r + k ) for some c ∈ N , then [ n ] r,k ≥ c . With these general inequalities in hand, we can now prove a collection of useful inequalitieson the dimensions of the simples of C( sl r + , k ) . While these inequalities may seem out of placefor now. Their use will become apparent in Section 4. Lemma 2.6.
For all ≤ h ≤ k and ≤ a ≤ r we have dim ( h Λ a ) = dim (( k − h ) Λ a ) . Proof.
By applying a simple current symmetry, we see that the objects ( h Λ a ) and (( k − h ) Λ − a ) have the same dimension. Applying charge conjugation then gives the result. (cid:3) Our first true inequality gives bounds on the symmetric powers of the fundamental represen-tations.
Lemma 2.7.
Let ≤ a ≤ r + and ≤ λ a ≤ k − . If λ a ≤ j ≤ k − λ a then dim ( j Λ a ) ≥ dim ( λ a Λ a ) . YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 9 Proof.
From Lemma 2.6 we havedim ( λ a Λ a ) = dim (( k − λ a ) Λ a )) . We can write j Λ a = ( − j − λ a k − λ a ) ⋅ λ a Λ a + j − λ a k − λ a ⋅ ( k − λ a ) Λ a . Thus the result follows from Lemma 2.2. (cid:3)
Applying level-rank duality to the above bound, we can also obtain the following. Together,these bounds allow us to understand the ordering on the dimensions of the symmetric powersof the fundamental representations.
Lemma 2.8.
Let ≤ a ≤ r and ≤ λ a ≤ k . If a ≤ j ≤ r + − a then dim ( λ a Λ j ) ≥ dim ( λ a Λ a ) . Proof.
Via a level-rank duality, we have that the dimension of ( λ a Λ j ) in C( sl r + , k ) is equal tothe dimension of ( j Λ λ a ) in C( sl k , r + ) . The result then follows from Lemma 2.7. (cid:3) Our next few bounds will apply to objects X ∈ C( sl r + , k ) satisfying certain properties. Thefirst applies to objects which are “far away” from symmetric powers of fundamental representa-tions. Lemma 2.9.
Suppose X = ∑ ri = λ i Λ i ∈ C( sl r + , k ) with each λ i ≤ k − . Then dim ( X ) ≥ dim ( ) . Proof.
By applying a simple current symmetry, we can assume λ is maximal among all other λ i . Define w ∶= r ∑ i = λ i = k − λ . As λ ≤ k −
3, we have w ≥
3. We now break into cases:If w ≤ k −
3, then we can write X = r ∑ i = λ i w ( w Λ i ) . As ∑ ri = λ i w =
1, we can apply Lemma 2.2 to seedim ( X ) ≥ min { dim ( w Λ i ) ∶ ≤ i ≤ r } . We have assumed 3 ≤ w ≤ k −
3, so lemma 2.7 givesmin { dim ( w Λ i ) ∶ ≤ i ≤ r } ≥ dim ( i ) . Now lemma 2.8 gives dim ( i ) ≥ dim ( ) . Thus dim ( X ) ≥ dim ( ) . If w = k −
1, then λ =
1, and as λ is maximal we have λ i ≤
1. Hence X = k − ∑ i = Λ a i , with a i ∈ { , ⋯ , r } and pairwise distinct. Define for a ≠ b the objects Y a,b ∶= a + ( k − ) Λ b . Via a simple current symmetry, we can see thatdim ( Y a,b ) = dim ( a − b ) and hence by lemma 2.8 we have dim ( Y a,b ) ≥ dim ( ) . We can write X = k − ∑ i = Λ a i = k − ∑ i = k − Y a i ,a i + with the understanding that a k = a . Hence by Lemma 2.2 we getdim ( X ) ≥ min { dim ( Y a i ,a i + ) ∶ ≤ i ≤ k } ≥ dim ( ) . If w = k −
2, then λ =
2, and as λ is maximal we have λ i ≤
2. Hence X = k ′ ∑ i = Λ a i + k ′′ ∑ i = b i , with a i , b i ∈ { , ⋯ , r } , the collection { a i , b j } pairwise distinct, and k ′ + k ′′ = k −
2. We can write X = k ′ ∑ i = k − Y a i ,a i + + k ′′ ∑ i = ( k − Y b i ,b i + + k − Y b i ,b i + ) . Hence by Lemma 2.2 we getdim ( X ) ≥ min { dim ( Y a i ,a i + ) ∶ ≤ i ≤ k ′ } ∪ { dim ( Y b j ,b j + ) ∶ ≤ j ≤ k ′′ } ≥ dim ( ) . In all cases we have dim ( X ) ≥ dim ( ) , as in the statement of the Lemma. (cid:3) The last bound we will give applies to objects that are fixed by the invertible objects of C( sl r + , k ) . To make this precise we introduce new notation. Definition 2.10.
Let X ∈ C( sl r + , k ) a simple object. Given Z d a subgroup of the invertibles of C( sl r + , k ) , we define Stab Z d ( X ) ∶= { g ∈ Z d ∶ g ⊗ X ≅ X } . If this stabaliser subgroup of an object is non-trivial, then the following lemma gives strongrestrictions on the dimension of that object.
Lemma 2.11.
Suppose X ∈ C( sl r + , k ) with Stab Z r + ( X ) = Z d . Let m ∈ N , and ≤ a ≤ k . Then dim ( X ) ≥ dim ( a Λ m r + d ) . Proof.
As Stab Z r + ( X ) = Z d we have that X is fixed by k Λ r + d . Thus X is of the form r + d ∑ i = λ i d ∑ j = Λ i + j r + d , with r + d ∑ i = λ i = kd . Let m ∈ N , and 0 ≤ a ≤ k and define for 1 ≤ i ≤ r + P i ∶= ( k − a ) Λ i + a Λ i + m r + d . As k Λ ⊗ P i = P i + , we have that dim ( P i ) = dim ( P j ) for all 1 ≤ i, j ≤ r + X = r + d ∑ i = λ i k d ∑ j = P i + j r + d . YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 11 To see this we count the multiplicity of an arbitary Λ (cid:96) in both sides of the above equation. Inthe object X , the multiplicity of (cid:96) is equal to λ (cid:96) ( mod r + d ) . On the right hand side, Λ (cid:96) will appearin the P (cid:96) term where it appears with multiplicity a λ (cid:96) ( mod r + d ) k , and in the P (cid:96) − m r + d term whereit appears with multiplicity ( k − a ) λ (cid:96) − m r + d ( mod r + d ) k = ( k − a ) λ (cid:96) ( mod r + d ) k . Thus in the entire righthand side, Λ (cid:96) appears with multiplicity λ (cid:96) ( mod r + d ) , and so the claim is valid.As ∑ r + d i = λ i k ∑ dj = =
1, we can use Lemma 2.2 to see thatdim ( X ) ≥ min ( dim ( P i ) ∶ ≤ i ≤ r + ) = dim ( P ) = dim ( a Λ m r + d ) as desired. (cid:3) De-equivariantization.
Our main focus of study in this paper will be the orbifold type I quantum subgroups, and their local modules. These are constructed as de-equivariantisations of the modular categories C( sl r + , k ) .In general let C be a braided tensor category, and choose a distinguished subcategory braidedequivalent to Rep ( G ) for G a finite group. We can consider the function algebra Fun ( G ) ⊂ Rep ( G ) → C , which lifts to a commutative algebra in Z(C) via the braiding. We write ( Fun ( G ) , σ ) for this commutative central algebra object. Definition 2.12.
The de-equivariantisation of C by Rep ( G ) is defined as the category of Fun ( G ) modules, which can be endowed with the structure of a G -crossed braided category via σ . Wewrite C Rep ( G ) for this de-equivariantisation.The category C Rep ( G ) has the canonical structure of a G -crossed braided category. The G -action is given by left translation of the algebra Fun ( G ) i.e. multiplication by group elements.The category of local modules of the commutative central algebra object ( Fun ( G ) , σ ) is thetrivially graded subcategory of C Rep ( G ) with respect to the G -crossed structure.With the generalities out of the way, let us now focus on the specific de-equivariantisations ofinterest for this paper. The Tannakian subcategories of C( sl r + , k ) are completely understoodand classified. These subcategories of C( sl r + , k ) are parametrised by m a divisor of r + m ∣ k ( r + ) if r is even, and 2 m ∣ k ( r + ) if r is odd. The corresponding Tannakiansubcategory is equivalent to Rep ( Z m ) , and is generated by the invertible object k Λ r + m .Our goal is to describe the basic structure of the modular tensor category C( sl r + , k ) ( Z m ) .We begin by looking at the Z m -crossed braided category C( sl r + , k ) Rep ( Z m ) . The objects of C( sl r + , k ) Rep ( Z m ) are of the form ( Y, ρ ) , where Y is an object which is fixed by tensoring by k Λ r + m , and ρ is a choice of isomorphism Y → k Λ r + m ⊗ Y satisfying a standard coherence condition.We have the free module functor F Z m ∶ C( sl r + , k ) → C( sl r + , k ) Rep ( Z m ) given by tensoring with the algebra object Fun ( Z m ) = ⊕ i k Λ i r + m . It is known that the functor F Z m is dominant [3, Proposition 5.5]. The adjoint to F Z m is the lax monoidal functor given byforgetting the isomorphism ρ . That is F ∗ Z m (( Y, ρ )) = Y. In order to simplify our proofs and computations later, it is necessary to give a more elemen-tary description of the simple objects of C( sl r + , k ) Rep ( Z m ) . Lemma 2.13.
The simple objects of C( sl r + , k ) Rep ( Z m ) are parameterised by pairs ( X, χ X ) ,where X is a simple object of C( sl r + , k ) considered up to action by k Λ r + m , and χ X is a characterof the group Stab Z m ( X ) . The dimension of the simple object ( X, χ X ) is given by dim ( X, χ X ) = dim ( X )∣ Stab Z m ( X )∣ The canonical Z m -action on these simples is given by multiplication of χ X by the standardcharacter g j ↦ e πi j ∣ Stab Z m ( X )∣ .Proof. This lemma is known to experts. However the author was unable to find an explicitproof. Our skein theoretic version of the proof can be found in Lemma 3.6. (cid:3)
Under this parametrisation, we can explicitly describe the free module functor F Z m . We have F Z m ( X ) = ⊕ χ ∈ ˆ G ( X, χ ) . To obtain the simple objects of the category C( sl r + , k ) ( Z m ) we must take the objects whichare 0-graded in the Z m -graded category C( sl r + , k ) Rep ( Z m ) . The Z m -grading on the category C( sl r + , k ) Rep ( Z m ) is inherited from the Z r + -grading on the category C( sl r + , k ) . Thus a simpleobject (∑ ri = λ i Λ i , χ ) will live in the ∑ ri = iλ i graded component of C( sl r + , k ) Rep ( Z m ) , takenmodulo m . This gives the following. Lemma 2.14.
The simple objects of C( sl r + , k ) ( Z m ) are parametrised by pairs (∑ ri = λ i Λ i , χ ) where ∑ ri = iλ i ≡ ( mod m ) , and χ is a character of Stab Z m (∑ ri = λ i Λ i ) . We now compute some useful information about the categories C( sl r + , k ) ( Z m ) . Let usdefine m ′ = gcd ( m, k ) and m ′′ ∶= mm ′ .For the remainder of the paper we will constantly encounter three exceptions in nearlyall of our lemmas and proofs. These are the categories C( sl , ) ( Z ) , C( sl , ) ( Z ) , and C( sl , ) ( Z ) . To put them to rest, we deal with them now. Lemma 2.15.
The claims of theorem 1.3 hold for the categories C( sl , ) ( Z ) , C( sl , ) ( Z ) ,and C( sl , ) ( Z ) .Proof. From the formula of the dimensions of the simples of C( sl r + , k ) ( Z m ) we immediatelysee that each of these cases is pointed. By considering twists we find that C( sl , ) ( Z ) ≃ Vec ( Z , { , e πi , e πi )C( sl , ) ( Z ) ≃ Vec ( Z × Z , { , − , − , − )C( sl , ) ( Z ) ≃ Vec ( Z , { , e πi , e πi , e πi , e πi , e πi } . With these explicit presentations, it is straight-forward to verify that they satisfy the claims ofTheorem 1.3. (cid:3)
Remark 2.16.
In order to keep the statements of various lemma tidy, for the remainder ofthis paper we will implicitly assume that the three cases C( sl , ) ( Z ) , C( sl , ) ( Z ) , and C( sl , ) ( Z ) are ignored. Let us study the group of invertibles of the modular category C( sl r + , k ) ( Z m ) . Lemma 2.17.
We have
Inv (C( sl r + , k ) ( Z m ) ) = {( k Λ (cid:96)m ′′ , ) ∶ (cid:96) ∈ Z r + mm ′′ } ≅ Z r + mm ′′ . YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 13 Proof.
From the formula for the dimensions of the simples of C( sl r + , k ) ( Z m ) , it is clear that ( X, χ X ) ∈ C( sl r + , k ) ( Z m ) will be invertible if and only if X has integer dimension, and lives ina graded component of C( sl r + , k ) which is a multiple of m . The objects with integer dimensionin C( sl r + , k ) have been classified [48], and aside from the special cases we have discarded, theonly such objects are the invertibles.The invertible objects of C( sl r + , k ) are of the form k Λ i for i ∈ Z r + . These invertible ob-jects live in the graded component ki of C( sl r + , k ) . Hence, to find the invertible objects of C( sl r + , k ) ( Z m ) , we need to see which i ∈ Z r + satisfy the equation ki = N m for some N ∈ N .We can write this equation as km ′ i = N m ′′ . As a consequence of the definition of m ′ and m ′′ , wehave that km ′ and m ′′ are coprime. Thus, m ′′ divides i , and so i is a multiple of m ′′ . This tellsus the group of invertibles of C( sl r + , k ) ( Z m ) is generated by the object ( k Λ m ′′ , ) , and henceform a group isomorphic to Z r + mm ′′ . (cid:3) The category C( sl r + , k ) ( Z m ) is modular, which implies the universal grading group isisomorphic to the group of invertibles. Hence we get the following corollary. Corollary 2.18.
The universal grading group of C( sl r + , k ) ( Z m ) is the group Z r + mm ′′ . Let us now identify the adjoint subcategory of C( sl r + , k ) ( Z m ) . Knowing this subcat-egory will allow us to use powerful graded category techniques. A natural guess would bethat (C( sl r + , k ) ( Z m ) ) ad ≃ (C( sl r + , k ) ad ) Rep ( Z m ) . However this doesn’t even typecheck, asRep ( Z m ) is not necessarily always a subcategory of (C( sl r + , k ) ad ) i.e. consider C( sl , ) Rep ( Z ) .Instead we find that the adjoint subcategory is equivalent to C( sl r + , k ) adRep ( Z m ′ ) . As m ′ divides k , we have that Rep ( Z m ′ ) is a subcategory of C( sl r + , k ) ad . Lemma 2.19.
We have (C( sl r + , k ) ( Z m ) ) ad ≃ C( sl r + , k ) ad Rep ( Z m ′ ) Proof.
The category C( sl r + , k ) adRep ( Z m ′ ) naturally embeds in C( sl r + , k ) ( Z m ) via the identityfunctor. As C( sl r + , k ) adRep ( Z m ′ ) is generated by the simple object Ω, and dimHom ( Ω ⊗ Ω → Ω ) ≥
1, the adjoint subcategory of C( sl r + , k ) adRep ( Z m ′ ) is itself. Hence the adjoint subcategory of C( sl r + , k ) ( Z m ) contains C( sl r + , k ) adRep ( Z m ′ ) . The global dimension of (C( sl r + , k ) ( Z m ) ) ad and C( sl r + , k ) adRep ( Z m ′ ) are the same, thus we have an equivalence. (cid:3) Finally, we study the invertible objects of the adjoint subcategory C( sl r + , k ) adRep ( Z m ′ ) . Lemma 2.20.
We have
Inv ( Ad (C( sl r + , k )) Rep ( Z m ′ ) ) ≅ Z n ′ m ′ , where n ′ ∶= gcd ( r + , k ) .Proof. This proof is fairly similar to the proof of Lemma 2.17. The same idea shows that anyinvertible of C( sl r + , k ) adRep ( Z m ′ ) will be of the form ( k Λ i , ) where i ∈ Z r + m ′ and ki ≡ ( mod r + ) .This implies that i has to be a multiple of r + n ′ . Thus the invertible objects of C( sl r + , k ) adRep ( Z m ′ ) are of the form ( k Λ j r + n ′ , ) , where j ∈ Z n ′ m ′ . (cid:3) Let us single out a distinguished object of C( sl r + , k ) ( Z m ) . Consider the object ( Λ + Λ r ) ∈C( sl r + , k ) ad . We can take the image of this object under the functor F Z m to obtain an objectin C( sl r + , k ) adRep ( Z m ′ ) ⊂ C( sl r + , k ) ( Z m ) . Definition 2.21.
We define the objectΩ ∶= F Z m ′ ( Λ + Λ r ) ∈ C( sl r + , k ) adRep ( Z m ′ ) . This distinguished object Ω is always simple.
Lemma 2.22.
The object Ω is a simple object in C( sl r + , k ) ( Z m ) .Proof. This lemma is equivalent to showing that Stab Z m ( Λ + Λ r ) is trivial. Let j ∈ Z m , thenthe corresponding invertible object of C( sl r + , k ) is k Λ j r + m . We compute k Λ j r + m ⊗ ( Λ + Λ r ) ≅ (( k − ) Λ j r + m + Λ j r + m + + Λ j r + m − ) . A case by case analysis, where we consider k ≥ k =
3, and k = (cid:3) The invertible objects of C( sl r + , k ) ( Z m ) act transitively on the simple object Ω in all butone special case. Lemma 2.23.
Suppose ( r, k, m ) ≠ ( , , ) , and let ( k Λ (cid:96)m ′′ , ) ∈ Inv (C( sl r + , k ) ( Z m ) ) . Then ( k Λ (cid:96)m ′′ , ) ⊗ Ω ≅ Ω (cid:212)⇒ ( k Λ (cid:96)m ′′ , ) ≅ . Proof.
If we have ( k Λ (cid:96)m ′′ , ) ⊗ Ω ≅ Ω, then this implies k Λ (cid:96)m ′′ ⊗ ( Λ + Λ r ) and ( Λ + Λ r ) live in the same orbit under the action of Z m in C( sl r + , k ) . Thus there exists a j such thatΛ − + (cid:96)m ′′ + ( k − ) Λ (cid:96)m ′′ + Λ (cid:96)m ′′ + = Λ − + j r + m + ( k − ) Λ j r + m + Λ + j r + m . If k >
3, then (cid:96)m ′′ = j r + m , which gives that ( k Λ (cid:96)m ′′ , ) ≅ in C( sl r + , k ) ( Z m ) .If k =
3, then ( k Λ (cid:96)m ′′ , ) is isomorphic to one of or ( k Λ ± , ) . The latter two cases force r =
2. Hence ( k Λ (cid:96)m ′′ , ) ≅ .If k = ( k Λ (cid:96)m ′′ , ) is isomorphic to one of or ( k Λ ± , ) . The latter case forces r = m = (cid:3) The distinguished simple object Ω ∈ C( sl r + , k ) ( Z m ) satisfies several nice properties thatwill make it useful for us in our computations later. Immediately we have that Ω is self-dual, itsdimension is [ r ] r,k [ r + ] r,k , and there exists a map Ω ⊗ Ω → Ω. We will make Ω the base pointof our auto-equivalence computations, and distinguish auto-equivalences based on whether theyfix or move this object.
Definition 2.24.
We say an auto-equivalence of C( sl r + , k ) ( Z m ) is non-exceptional if it mapsΩ to an image of Ω under simple currents. We will say the auto-equivalence is exceptional if itis not non-exceptional.We will see in the bulk of this paper the surprising result that only a finite number of auto-equivalences are exceptional, and that every non-exceptional auto-equivalence comes from eithera simple current auto-equivalence, charge conjugation, or from the canonical Z m -action.While the definition of a non-exceptional auto-equivalence allows for the object Ω to be moved,the following lemma shows this is not the case. Lemma 2.25.
A non-exceptional auto-equivalence of C( sl r + , k ) ( Z m ) must fix Ω .Proof. Let F a non-exceptional auto-equivalence of C( sl r + , k ) ( Z m ) . Then there exists aninvertible element g of C( sl r + , k ) ( Z m ) such that F ( Ω ) ≅ g ⊗ Ω. As g is of the form ( k Λ nm ′′ , ) for n ∈ N , we compute that g ⊗ Ω ≅ (( k − ) Λ nm ′′ + Λ nm ′′ + + Λ nm ′′ − , ) . YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 15 The object Ω is self-dual, and so g ⊗ Ω must be as well, thus g ⊗ Ω ≅ g ∗ ⊗ Ω. Thus byLemma 2.23 we see that g ⊗ ≅ , and so g ≅ ( k Λ j r + m , ) for j ∈ { , } .For the generic case of r ≥ k ≥
3, we know that dim hom ( Ω ⊗ Ω → Ω ) =
2, and thusdim hom ( g ⊗ Ω ⊗ g ⊗ Ω → g ⊗ Ω ) =
2. Using the braiding on the category, along with the fact that g has order two, we see that dim hom ( Ω ⊗ Ω → g ⊗ Ω ) =
2. We explicitly compute the simpledecomposition of Ω ⊗ Ω as ⊕ ⊕ ( Λ + r , ) ⊕ ( Λ + Λ r − , ) ⊕ ( + Λ r − , ) ⊕ ( + r , ) . As g ⊗ Ω = (( k − ) Λ j r + m + Λ j r + m + + Λ j r + m − , ) must appear in this decomposition, we canimmediately deduce that j =
0, i.e. g must be the identity.For the remaining cases, the proof is almost identical, except the decomposition of Ω ⊗ Ωis smaller, and in some cases the stabaliser subgroup of the simples in the decomposition isnon-trivial, so the characters of the stabaliser groups must be changed. (cid:3)
In light of the above result we make the following definition.
Definition 2.26.
We write EqBr (C( sl r + , k ) ( Z m ) ; Ω ) for the group of braided auto-equivalencesof C( sl r + , k ) ( Z m ) which fix Ω, or equivalently, the group of non-exceptional auto-equivalences.2.3. Planar Algebras.
A key tool for the results of this paper are planar algebras . Roughlyspeaking a planar algebra P is a collection of vector spaces {P n ∶ n ∈ N } , along with a multi-linearaction of planar tangles. The full definition can be found in [37], and illuminating examples in[41].We will be interested in planar algebras constructed from symmetrically self-dual objects inpivotal fusion categories. Let X ∈ C be such an object. Then we can define a planar algebra P X by (P X ) n ∶= Hom ( → X ⊗ n ) . Supposing the object X generated C , then we can recover C by taking the idempotent completionof P X . Here the objects are idempotents in the algebras (P X ) n (where we have n legs pointingup, and n legs pointing down) with vertical stacking as the multiplication. The morphisms be-tween two idempotents are elements of the planar algebra which intertwine the two idempotents.The tensor product is given by horizontal juxtaposition, and direct sums are added formally.Additional information on these two constructions can be found in [41].It is proven in [31, Theorem A] that the above bijection between planar algebras and sym-metrically self-dual objects X ∈ C is functorial. That is there is an isomorphism between auto-morphisms of the planar algebra P X , and pivotal auto-equivalences of the category C which fix X . 3. Non-exceptional auto-equivalences of C( sl r + , k ) ( Z m ) In this section we will determine the braided auto-equivalences of C( sl r + , k ) ( Z m ) that fixthe distinguished object Ω. In terms of the notation introduced in this paper, we will determinethe group EqBr (C( sl r + , k ) ( Z m ) ; Ω ) . We show that non-exceptional auto-equivalences (in theformal definition of this paper) are non-exceptional (in the layman terms). That is, every non-exceptional braided auto-equivalence is either charge conjugation, simple current, or comes fromthe canonical Z m -action on C( sl r + , k ) ( Z m ) .Let us outline the arguments of this section. To begin, we initially focus our attention on thedistinguished subcategory C( sl r + , k ) adRep ( Z m ′ ) . The subcategory C( sl r + , k ) adRep ( Z m ′ ) has two nicefeatures that will assist with the results of this section. First is that it has trivial universal gradinggroup, and hence has a unique pivotal structure, and second the category C( sl r + , k ) adRep ( Z m ′ ) is generated by the distinguished object Ω. Together these facts will allow us powerful planaralgebra techniques to determine the non-exceptional symmetries.With the above in mind, we give a presentation of the planar algebra P Ω , i.e. the planaralgebra generated by the object Ω ∈ C( sl r + , k ) adRep ( Z m ′ ) . To achieve this, we observe that P Ω contains the planar algebra P Λ + Λ r , i.e. the planar algebra generated by the object Λ + Λ r ∈C( sl r + , k ) ad . The planar algebra P Λ + Λ r is well understood, and is known to be generated bytwo trivalent vertices. We can then find an additional generator in P Ω , which together with thetwo trivalent vertices, generate all of P Ω . The idea here is that the group Z m ′ is singly generated,which allows us to understand skein theory for de-equivariantisation in terms of the addition ofone additional generator. With the generators of P Ω identified, we can then find relations thatthese generators satisfy. Remark 3.1.
While it is not explicit in this paper, the techniques we have briefly describedabove (and will explain in detail in the remainder of this section), can be used to give skeintheory for any de-equivariantisation by an abelian group.
With the presentation of the planar algebra P Ω in hand, we can use it to give an upper boundfor the group of braided auto-equivalences of C( sl r + , k ) adRep ( Z m ′ ) which fix Ω. We find thatthere are at most 2 m ′ of these auto-equivalences, which compose to form a group isomorphicto D m ′ . Further, we explicitly identify how these potential auto-equivalences act on the simplesof C( sl r + , k ) adRep ( Z m ′ ) . We then construct these 2 m ′ potential auto-equivalences by the chargeconjugation auto-equivalence, which gives us a Z subgroup, and by the canonical Z m ′ -actionon C( sl r + , k ) adRep ( Z m ′ ) which comes from de-equivariantisation.To obtain the auto-equivalences of C( sl r + , k ) ( Z m ) which fix Ω, we appeal to the techniquesdeveloped in [15]. These techniques allow us to give an upper bound for EqBr (C( sl r + , k ) ( Z m ) ; Ω ) in terms of EqBr (C( sl r + , k ) adRep ( Z m ′ ) ; Ω ) and some cohomogical data. While there is no reasonthat this bound should be sharp (the techniques involve verifying that certain obstructions van-ish in order to show that auto-equivalences lift) we are able to show that the theoretical upperbound is realised by simple current auto-equivalences.All together we prove the following theorem. Theorem 3.2.
Let r, k ∈ N , and m a divisor of r + such that m ∣ k ( r + ) . Set m ′ = gcd ( m, k ) and m ′′ = mm ′ . Then we have the following isomorphism of groups EqBr (C( sl r + , k ) ( Z m ) ; Ω ) ≅ ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩{ e } if k = and r = Z m ′ × Z p + t if k = or r = D m ′ × Z p + t otherwisewhere ● p is the number of distinct odd primes dividing r + mm ′′ but not km ′ , and ● t = ⎧⎪⎪⎨⎪⎪⎩ if r + mm ′′ is odd, or if km ′ ≡ ( mod 4 ) , or if both km ′ is odd, and r + mm ′′ ≡ ( mod 4 ) otherwise. With the high-level arguments in mind, let us begin with the details of proving the abovetheorem.Consider the planar algebra P Ω . As Ω generates, and their exists a map Ω ⊗ Ω → Ω, wehave that C( sl r + , k ) adRep ( Z m ′ ) has trivial universal grading group, and thus also has a uniquepivotal structure. Therefore we have that EqBr (C( sl r + , k ) adRep ( Z m ′ ) , Ω ) is isomorphic to thegroup of braided planar algebra automorphisms of P Ω . Our goal is thus to specify as much ofthe structure of this planar algebra P Ω as possible in order to understand its auto-equivalencegroup. YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 17 As the free module functor C( sl r + , k ) ad → C( sl r + , k ) adRep ( Z m ′ ) is dominant, and maps Λ + Λ r to Ω, we obtain a planar algebra embedding P Λ + Λ r → P Ω . The planar algebra P Λ + Λ r is well understood [36, 13]. It is generated by two trivalent verticessatisfying the Thurston relations (see [36, Lemma 3.2]). Hence the planar algebra P Ω alsocontains two trivalent vertices and S satisfying these same Thurston relations. However, there are going to be additional generatorsin this planar algebra. These additional generators come from the de-equivariantization byRep ( Z m ′ ) . Remark 3.3.
For the remainder of this section we will identify C( sl r + , k ) ad as the idempotentcompletion of the planar algebra P Λ + Λ r , and C( sl r + , k ) ad Rep ( Z m ′ ) as the idempotent completion ofthe planar algebra P Ω . This means that we regard simple objects of these categories as minimalidempotents of the planar algebras, and morphisms as elements of the planar algebra whichcommute with the idempotents. Let us write p k Λ r + m ′ for the minimal idempotent of C( sl r + , k ) ad corresponding to the simpleobject k Λ r + m ′ . From the inclusion of planar algebras P Λ + Λ r → P Ω , we have that this idempotent p k Λ r + m ′ also exists in P Ω .The free module functor F Z m ′ ∶ C( sl r + , k ) ad → C( sl r + , k ) adRep ( Z m ′ ) sends k Λ r + m ′ to the tensorunit. Therefore in the planar algebra P Ω , the trivial idempotent and p k Λ r + m ′ are isomorphic.Thus there exists an invertible element S ∈ P Ω (which we draw as a circle to differentiate it fromthe other planar algebra elements) satisfying S − S = S − S = p k Λ r +1 k . The element S lives in the n -box space of P Ω , where n is the smallest n such that k Λ r + m ′ appears in the decomposition of ( Λ + Λ r ) ⊗ n We claim that P Ω is generated by the two trivalent vertices, along with the new element S . Lemma 3.4.
We have that P Ω is generated by the two Thurston trivalent vertices, and theelement S .Proof. Let P S be the sub-planar algebra of P Ω generated by these three elements, and C S thecorresponding category. Then we have a chain of embeddings P Λ + Λ r → P S → P Ω . This gives us dominant monoidal functors F ∶ C( sl r + , k ) ad → C S F ∶ C S → C( sl r + , k ) adRep ( Z m ′ ) , and their adjoints F ∗ ∶ C S → C( sl r + , k ) ad F ∗ ∶ C( sl r + , k ) adRep ( Z m ′ ) → C S . From [4], we have that F ∗ ( C S ) is a commutative central algebra object, and that C S is equivalentto the category of F ∗ ( C S ) -modules in C( sl r + , k ) ad .As these dominant functors F and F are just the inclusions of idempotents, we have thatthe composition of these two dominant functors is equal on the nose to the dominant functor C( sl r + , k ) ad → C( sl r + , k ) adRep ( Z m ′ ) induced by the planar algebra inclusion P Λ + Λ r → P Ω . This induced functor C( sl r + , k ) ad → C( sl r + , k ) adRep ( Z m ′ ) is precisely the free module functor F Z m ′ .Hence we have that F ○ F = F Z m ′ , which implies that F ∗ ○ F ∗ = F ∗ Z m ′ . From this fact we see F ∗ ( C S ) ⊆ F ∗ ○ F ∗ ( C( sl r + ,k ) adRep ( Z m ′) ) = F ∗ Z m ′ ( C( sl r + ,k ) adRep ( Z m ′) ) = Fun ( Z m ′ ) , as a central commutative algebra in C( sl r + , k ) ad . In particular we get that F ∗ ( C S ) ≅ Fun ( Z (cid:96) ) where (cid:96) ∣ m ′ . As Fun ( Z m ′ ) is the central commutative algebra object in C( sl r + , k ) ad corre-sponding to the de-equivariantisation by the Rep ( Z m ′ ) subcategory, the central structure isgiven by the braiding of C( sl r + , k ) ad . Hence the central structure on Fun ( Z (cid:96) ) is also given bythe braiding. This gives that C S is a de-equivariantisation of C( sl r + , k ) ad by Rep ( Z (cid:96) ) , i.e. C S ≃ C( sl r + , k ) adRep ( Z (cid:96) ) . In C S we know that S gives an isomorphism from → p k Λ r + m ′ which implies that m ′ ∣ (cid:96) . Thus C S ≃ C( sl r + , k ) adRep ( Z m ′ ) which gives the desired isomorphism of planar algebras P S ≅ P Ω . (cid:3) In order to study the planar algebra automorphisms of P Ω we need to study the element S further, and deduce further relations that it satisfies. Remark 3.5.
To simplify notation, we will now draw multiple strands of a planar algebra asa single strand in our graphical diagrams. It will be clear from context how many strands aremeant by the diagram.
In the category C( sl r + , k ) ad we have that k Λ ⊗ m ′ r + m ′ ≅ . Thus the object k Λ r + m ′ generates asubcategory with the fusion rules of Z m ′ . We are given that this subcategory is Tannakian (asit is the subcategory we are de-equivariantating by), so it is braided equivalent to Rep ( Z m ′ ) . YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 19 This means we can choose a system of trivalent vertices t n,p = t n,p p k Λ nr +1 m ′ p k Λ pr +1 m ′ p k Λ n + pr +1 m ′ ∶ p k Λ ⊗ nr + m ′ ⊗ p k Λ ⊗ pr + m ′ → p k Λ ⊗ n + pr + m ′ in C( sl r + , k ) ad with trivial 6-j symbols, and such that the charge conjugation auto-equivalence k Λ ⊗ n r + m ′ ↦ k Λ ⊗− n r + m ′ maps t n,p ↦ t − n, − p .We can build an isomorphism j ∶= t , t , ⋯ t m ′ − , = t , p k Λ r +1 m ′ p k Λ r +1 m ′ p k Λ r +1 m ′ t , p k Λ r +1 m ′ t m ′ − , ∶ p k Λ ⊗ m ′ r + m ′ → . Hence we have that j is a map from the idempotent p ⊗ m ′ k Λ r + m ′ to the trivial idempotent in theplanar algebra P Λ + Λ r . In the planar algebra P Ω we have that S ⊗ m ′ is an isomorphism from theidempotent p ⊗ m ′ k Λ r + m ′ to the trivial idempotent. Thus we have that S ⊗ m ′ ○ j lives in the 0-box space of P Ω and is non-zero. This allows us to normalise S so that we get therelation(3.1) S SS = t − , p k Λ r +1 m ′ p k Λ r +1 m ′ p k Λ r +1 m ′ t − , p k Λ r +1 m ′ t − m ′ − , , in the planar algebra P Ω .This explicit presentation of the planar algebra P Ω is sufficient to compute the minimalidempotents up to equivalence, and thus the simple objects of C( sl r + , k ) adRep ( Z m ′ ) . Lemma 3.6.
The simple objects of C( sl r + , k ) ad Rep ( Z m ′ ) are parametrised (up to isomorphism) by ( X, χ X ) where X is a simple object of C( sl r + , k ) ad (up to action by k Λ r + m ′ ) and χ X is a character of thegroup Stab Z m ′ ( X ) . The quantum dimension of ( X, χ X ) is equal to dim ( X )∣ Stab Z m ′ ( X )∣ .Proof. The free module functor F Z m ′ is dominant, therefore every simple object of C( sl r + , k ) adRep ( Z m ′ ) is a sub-object of F Z m ′ ( X ) for some X ∈ C( sl r + , k ) ad . Let p X be the minimal projection in theplanar algebra P Λ + Λ r corresponding to X . As C( sl r + , k ) adRep ( Z m ′ ) is idempotent complete, eachsimple sub-object of X will correspond (up to isomorphism) to a minimal sub-idempotent of p X .As Stab Z m ′ ( X ) ≅ Z d , there exists an isomorphism f X ∶ p X ⊗ p k Λ ⊗ m ′ dr + m ′ in C( sl r + , k ) ad . For each n ∈ Z d we define isomorphisms r X,n ∶ p X → p X in C( sl r + , k ) adRep ( Z m ′ ) by r X,n ∶= f nX ○ S ⊗ n m ′ d = f X S S Sp X p X f X f X . By design we have that r X,n r X,n ′ = r X,n + n ′ . Furthermore, by relation 3.1 we have that r X,d ∶ p X → p X lives in P Λ + Λ r . As p X is simple in C( sl r + , k ) ad , we have that r X,d must be ascaler multiple of p X . We normalise our choice of the isomorphism f X to ensure that r X,d = p X .Thus we have that End ( p X ) in C( sl r + , k ) adRep ( Z m ′ ) is isomorphic to the group algebra C [ Z d ] . Itis a classical result that the minimal idempotents are indexed by characters χ of Z d with p χ = ∣ Stab Z m ′ ( X )∣ ∑ n ∈ Z d χ ( n ) r X,n . The quantum dimension of the minimal idempotent p χ is given by the trace. Note that thetrace of r X,n is 0, unless n =
0, as otherwise we could build a non-trivial morphism k Λ ⊗ nm ′ dr + m ′ → . If n =
0, then the trace of r X,n is the quantum dimension of X . Hence the trace of p χ is equalto the quantum dimension of X divided by ∣ Stab Z m ′ ( X )∣ . (cid:3) Remark 3.7.
For ease of notation, let us fix isomorphisms Z N → ̂ Z N by n ↦ χ n ∶= i ↦ e πi niN . We can now determine an upper bound for the group Aut (P Ω ) , and hence also for the groupEqBr (C( sl r + , k ) adRep ( Z m ′ ) ; Ω ) . YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 21 Lemma 3.8.
We have that
EqBr (C( sl r + , k ) ad Rep ( Z m ′ ) ; Ω ) ⊆ D m ′ , with generators ( X, χ n ) ↦ ( X, χ n + ) and ( X, χ n ) ↦ ( X ∗ , χ − n ) . Proof.
Let φ ∈ Aut (P Ω ) be a braided automorphism. Then φ is determined by where it sends thethree generators. Recall, we have two trivalent vertices satisfying the Thurston relations, andthe generator S which lives in the n -box space, where n is the smallest n such that k Λ r + m ′ appearsin the decomposition of ( Λ + Λ r ) ⊗ n . By explicitly expanding ( Λ + Λ r ) ⊗ we can see that k Λ r + m ′ appears as a summand only in the case C( sl , ) adRep ( Z ) , C( sl , ) adRep ( Z ) , and C( sl , ) adRep ( Z ) .These cases have already been excluded and dealt with previously in the paper.Let us deal with the remaining cases. As S does not live in the three box space, we knowthat there are scalers c , c , c , c ∈ C such that φ ⎛⎜⎜⎝ ⎞⎟⎟⎠ = c + c S φ ⎛⎜⎜⎜⎝ S ⎞⎟⎟⎟⎠ = c + c S . The coefficients c , c , c , c for which φ preserve the Thurston relations are solved for in [13,Lemma 3.1]. With the condition that φ is braided, there are two solutions, which we denote φ id and φ cc . These planar algebra automorphisms on the sub-planar algebra P Λ + Λ r are explicitlyidentified in the cited paper, where it is found that φ cc corresponds to the charge conjugationauto-equivalence of C( sl r + , k ) ad .Now the charge-conjugation auto-equivalence maps k Λ ± r + m ′ ↦ k Λ ∓ r + m ′ , thus we have the fol-lowing in the planar algebra P Λ + Λ r : φ id ( p k Λ ± r + m ′ ) = p k Λ ± r + m ′ and φ cc ( p k Λ ± r + m ′ ) = p k Λ ∓ r + m ′ . As the planar algebra P Λ + Λ r canonically embeds in P Ω , we also have these relations in thelarger planar algebra.To see when these auto-equivalences φ id and φ cc extend to the full planar algebra P Ω we mustdetermine if (and how) these automorphisms act on the generators S . Let us define isomorphisms in P Ω by S n ∶= S ⊗ n ○ t , t , ⋯ t n − , = t , t , p k Λ nr +1 m ′ t n − , S S S S ∶ → p ⊗ nk Λ r + m ′ . Note that trivially we have S = S , and by relation (3.1) we have that S m ′ = φ id extends to P Ω , observe that φ id ( S ) is an isomorphism from → p k Λ r + m ′ . Asthis morphism space is 1-dimensional, we must have that φ id ( S ) = βS for some non-zero scaler β ∈ C . Applying the potential automorphism to relation (3.1) gives that β must be an m ′ -throot of unity.To see when φ cc extends to P Ω , observe that φ cc ( S ) is an isomorphism from → p k Λ − r + m .This implies that φ cc ( S ) = ˆ βS m ′ − for some non-zero scaler ˆ β ∈ C . We apply this potentialautomorphism to relation (3.1) to obtainˆ β m ′ S ⊗ m ′ m ′ − = t − , − m ′ t − , − m ′ ⋯ t − − m ′ , − m ′ . From this equation we expand out the S m ′ − terms to obtain an equation with ( m ′ − ) m ′ of S terms. We then apply relation (3.1) to get an equation purely in terms of the trivalent vertices t . From here we then use that the trivalent vertices t have trivial 6-j symbols to obtain ˆ β m ′ = β must be an m ′ -root of unity.With the explicit presentation of how the 2 m ′ potential automorphisms act on the generator S , it is straight forward to determine that if these automorphisms existed, then they would forma group isomorphic to D m ′ .We now determine how these D m ′ worth of potential automorphisms would act on the simpleobjects of C( sl r + , k ) adRep ( Z m ′ ) . For the planar algebra automorphism sending S to βS let us write β = e πi (cid:96)m ′ for (cid:96) ∈ Z m ′ . We then have ( X, χ n ) ↦ ( X, χ n + (cid:96) ) . For the planar algebra automorphism sending S ↦ ˆ βS m ′ − we have to work a little harder todetermine where it sends the simple object ( X, ( X, χ n ) . Recall that this planar algebra automor-phism restricts to φ cc on the sub-planar algebra P Λ + Λ r . Thus we know how this automorphismacts on the trivalent vertices t , so we can compute that f X ↦ γf m ′ − X ∗ ○ ( t m ′ − , t m ′ − , ⋯ t , ) for some γ ∈ C . By simultaneously rescaling the trivalent vertices t , we can ensure that γ = r X,n ↦ ˆ β n r X ∗ , − n , YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 23 and hence p χ ↦ ∑ n ∈ Z d χ ( n ) ˆ β n r X ∗ , − n . Again let us write ˆ β = e πi (cid:96)m ′ for (cid:96) ∈ Z m ′ . Then this planar algebra automorphism gives rise toan auto-equivalence of C( sl r + , k ) adRep ( Z m ′ ) which sends ( X, χ n ) ↦ ( X ∗ , χ − n + (cid:96) ) . (cid:3) The above lemma gives an upper bound on the braided auto-equivalence group (which fix Ω)for the category C( sl r + , k ) adRep ( Z m ′ ) . In theory we could determine a complete set of relations forthe planar algebra P Ω , and verify that the auto-equivalences exist by checking that they preserveall relations. However this requires additional work which is beyond the scope of this paper.Instead we construct 2 m ′ worth of braided auto-equivalences of the category C( sl r + , k ) adRep ( Z m ′ ) directly, realising the upper bound. Lemma 3.9.
We have
EqBr (C( sl r + , k ) ad Rep ( Z m ′ ) ; Ω ) ≅ ⎧⎪⎪⎨⎪⎪⎩ Z m ′ if k = or r = D m ′ otherwise . Proof.
Let us begin by constructing the Z m ′ worth of braided auto-equivalences. Via construc-tion, we have that Z m ′ acts on C( sl r + , k ) adRep ( Z m ′ ) via the map ( X, χ n ) ↦ ( X, χ n + ) . To obtain the full Z m worth of auto-equivalences we need to show this action is faithful. Thisis equivalent to finding an object X ∈ C( sl r + , k ) ad with Stab Z m ′ ( X ) = Z m ′ . Such an object isgiven by X ∶= m ′ ∑ i = km ′ r + m ′ ∑ j = Λ i r + m ′ + j . To construct the remaining auto-equivalences, we observe that the charge-conjugation auto-equivalences exist for C( sl r + , k ) ad when k ≥ r ≥ ( Z m ′ ) subcategory, and hence descend to auto-equivalences of C( sl r + , k ) adRep ( Z m ′ ) .To finish the proof, we must show that the charge conjugation auto-equivalence never coincideswith the Z m ′ action. Thus we have to find an object X ∈ C( sl r + , k ) ad such that X ∗ is not inthe orbit of the action of k Λ r + m ′ . This object is given by X = Λ + r − . This satisfies the required properties when r ≥ k ≥ r = k ≥
3, then we can use the object X = . If k = r =
1, then m ′ ∈ { , } and the result is given in [11, Theorem 1.2]. (cid:3) Now that we understand the auto-equivalences of the subcategory C( sl r + , k ) adRep ( Z m ′ ) which fixΩ, we can leverage this to determine the auto-equivalences of the full category C( sl r + , k ) ( Z m ) .The idea here is to use the fact that C( sl r + , k ) ( Z m ) is a Z r + mm ′ -graded extension of C( sl r + , k ) adRep ( Z m ′ ) .This allows us to apply the results of [15] to classify the auto-equivalences of C( sl r + , k ) ( Z m ) extending a given auto-equivalence of C( sl r + , k ) adRep ( Z m ′ ) . To convenience the reader, the re-sults of [15] state for a G -graded category ⊕ G C g , the number of auto-equivalences extending F ∈ Eq (C e ) is bounded above by ∣ φ ∈ Aut ( G ) ∶ C g ≃ C φ ( g ) as C e -bimodules for all g ∈ G ∣ ⋅ ∣ H ( G, Inv (Z(C e )))∣ ⋅ ∣ H ( G, C × )∣ . With this bound, we can determine the following result.
Lemma 3.10.
The group of auto-equivalences of C( sl r + , k ) ( Z m ) extending the identity onthe subcategory C( sl r + , k ) ad Rep ( Z m ′ ) is isomorphic to the group { a ∈ Z r + mm ′′ ∶ + a km ′ is coprime to r + mm ′′ } , unless r = , k = , and m = , in which case the group is trivial.Proof. We begin with the group { φ ∈ Aut ( Z r + mm ′′ ) ∶ C g ≃ C φ ( g ) as C e -bimodules for all g ∈ Z r + mm ′′ } .As the C e bimodules form a group, we have that C g ≃ C φ ( g ) as C e -bimodules for all g ∈ Z r + mm ′′ ifand only if C φ ( g ) g − is equivalent to the trivial C e -bimodule, and so only if C φ ( g ) g − contains aninvertible object. Recall that the invertible objects of C( sl r + , k ) ( Z m ) are generated by theobject ( k Λ m ′′ , ) , which lives in the graded component C km ′′ m = C km ′ . Therefore the invertibleobjects of C( sl r + , k ) ( Z m ) live in the graded components C N km ′ for N ∈ N .Let c ∈ Z × r + mm ′′ , then C cg − g contains an invertible object if and only if C c − does. For this tohappen, we need that c ≡ + N km ′ for some N ∈ N . Using Bezout’s identity, this is equiva-lent to having c ≡ ( mod gcd ( km ′ , r + mm ′′ )) . A direct prime by prime computation reveals thatgcd ( km ′ , r + mm ′′ ) = n ′ m ′ , where we recall that n ′ = gcd ( n, k ) . Thus together we have a bound ∣{ φ ∈ Aut ( Z r + mm ′′ ) ∶ C g ≃ C φ ( g ) as C e -bimodules for all g ∈ Z r + mm ′′ }∣ ≤ ∣{ c ∈ Z × r + mm ′′ ∶ c ≡ ( mod n ′ m ′ )}∣ . Now we count the group H ( Z r + mm ′′ , Inv (Z(C( sl r + , k ) adRep ( Z m ′ ) ))) . As a 1-cocycle is deter-mined by its value on the generator, we have that the size of this group is bounded above bythe size of Inv (Z(C( sl r + , k ) adRep ( Z m ′ ) )) . As the universal grading group of C( sl r + , k ) adRep ( Z m ′ ) istrivial, we can use [30] to see that every invertible of C( sl r + , k ) adRep ( Z m ′ ) has at most one liftto the centre. Further, as C( sl r + , k ) adRep ( Z m ′ ) is braided each invertible object has a lift to thecentre via the braiding. ThereforeInv (Z(C( sl r + , k ) adRep ( Z m ′ ) )) ≅ Inv (C( sl r + , k ) adRep ( Z m ′ ) ) ≅ Z n ′ m ′ , and so the size of the group H ( Z r + mm ′′ , Inv (Z(C( sl r + , k ) adRep ( Z m ′ ) ))) is bounded above by n ′ m ′ .It is a classical group theory result that H ( Z r + mm ′′ , C × ) is trivial.Thus there are at most n ′ m ′ ⋅ ∣{ c ∈ Z × r + mm ′′ ∶ c ≡ ( mod n ′ m ′ )}∣ auto-equivalences of C( sl r + , k ) ( Z m ) extending the identity on the C( sl r + , k ) adRep ( Z m ′ ) subcat-egory. We now show this bound is sharp by constructing enough distinct auto-equivalencesof C( sl r + , k ) ( Z m ) to realise the upper bound. We will construct these auto-equivalences assimple current auto-equivalences. For the definition of simple current auto-equivalences we usein this paper, see [13, Lemma 2.4]. YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 25 To construct simple current auto-equivalences, we pick out the invertible object ( k Λ m ′′ , ) ∈C( sl r + , k ) ( Z m ) . This object has order r + mm ′′ , and has self-braiding eigenvalue equal to e πi mm ′′ q ( r + ) where q = rkm ′ . Thus we get simple current auto-equivalences of C( sl r + , k ) ( Z m ) for each elementof the set { a ∈ Z r + mm ′′ ∶ + a rkm ′ is coprime to r + mm ′′ } . To see that these simple current auto-equivalences are distinct, note that they form a group.Therefore, we need to show that for each a ≠
0, the corresponding simple current auto-equivalenceacts non-trivially. Consider ( Λ m ′ , ) ∈ C( sl r + , k ) ( Z m ) . Then we have that the simple currentauto-equivalence sends ( Λ m ′ , ) ↦ ( Λ m ′ , ) ⊗ ( k Λ m ′′ , ) = (( k − ) Λ m ′′ + Λ m ′ + m ′′ ) , ) . To verify that this action in non-trivial, we have to check that ( k − ) Λ m ′′ + Λ m ′ + m ′′ ) does notlive in the orbit of Λ m ′ under the action of Z m . Supposing this was the case, then there wouldexist a t ∈ N such that ( k − ) Λ m ′′ + Λ m ′ + m ′′ ) = ( k − ) Λ t r + m + Λ t r + m + m ′ . Assuming k >
2, we get the equation t r + mm ′′ ≡ ( mod r + ) , which is nonsense, as r + mm ′′ is clearly not invertible in Z r + .If k =
2, then we get that 2 m ′ ≡ ( mod r + ) , and thus m ′ = r + . As m ′ ∣ k , we see thateither r = m ′ =
1, or r = m ′ =
2. The latter case is one of the excluded cases. Forthe former case, it is known that the simple current auto-equivalence acts trivially [11, Theorem1.2].The same argument used in [13, Lemma A.2] shows that the set of simple current auto-equivalences, and the set { b ∈ Z r + mm ′′ n ′ m ′ ∶ b ≡ ( mod n ′ m ′ )} have the same size. We give a bijection n ′ m ′ ⋅ { c ∈ Z × r + mm ′′ ∶ c ≡ ( mod n ′ m ′ )} → { b ∈ Z r + mm ′′ n ′ m ′ ∶ b ≡ ( mod n ′ m ′ )} by sending ( N, c ) ↦ c + N n ′ m ′ . As the simple current auto-equivalences are all distinct (except for C( sl , ) ( Z ) ), and thenumber of them is equal to the upper bound of auto-equivalences extending the identity onthe subcategory C( sl r + , k ) adRep ( Z m ′ ) , we therefore have that every auto-equivalences extendingthe identity on the subcategory C( sl r + , k ) adRep ( Z m ′ ) is isomorphic to the group of simple currentauto-equivalences, which is { a ∈ Z r + mm ′′ ∶ + a rkm ′ is coprime to r + mm ′′ } . (cid:3) A-priori there should be no reason that the upper bound on the number of auto-equivalenceswe construct should be tight. We suspect that something deep going on here that deserves tobe investigated.As a corollary, we can determine which auto-equivalence which extend the identity are braided.
Corollary 3.11.
The group of braided auto-equivalences of C( sl r + , k ) ( Z m ) extending theidentity on the subcategory C( sl r + , k ) ad Rep ( Z m ′ ) is isomorphic to the group Z p + t , where ● p is the number of distinct odd primes dividing r + mm ′′ but not km ′ , and ● t = ⎧⎪⎪⎨⎪⎪⎩ if r + mm ′′ is odd, or if km ′ ≡ ( mod 4 ) , or if both km ′ is odd, and r + mm ′′ ≡ ( mod 4 ) otherwise,unless when r = , k = , and m = , in which case the group is trivial.Proof. This corollary follows as we know that a simple current auto-equivalence is braided pre-cisely when a rkm ′ − a ≡ ( mod 2 r + mm ′′ ) . We can then copy the analysis of [26, Theorem 1] toget the group isomorphism. (cid:3) Now that we completely understand the braided auto-equivalences of C( sl r + , k ) ( Z m ) whichextend the identity on the subcategory C( sl r + , k ) adRep ( Z m ′ ) , we can use a torsor argument tofairly easily leverage this information to understand the auto-equivalences extending the charge-conjugation auto-equivalence on the subcategory C( sl r + , k ) adRep ( Z m ′ ) which fix the distinguishedobject Ω. This completes the proof of Theorem 3.2, the main result of this section. Proof of Theorem 3.2.
All that remains to be done is to show that there exists a braided auto-equivalence of C( sl r + , k ) ( Z m ) which restricts to give the charge-conjugation auto-equivalenceof C( sl r + , k ) adRep ( Z m ′ ) . This follows from the fact that charge conjugation exists for C( sl r + , k ) ,and it preserves the Rep ( Z m ) subcategory. Therefore it descends to the category C( sl r + , k ) ( Z m ) . (cid:3) Candidates for exceptional auto-equivalences
In the previous section we were able to completely determine all non-exceptional braidedauto-equivalences of the categories C( sl r + , k ) ( Z m ) . That is, we could determine all braidedauto-equivalences which fixed the distinguished object Ω. For this section we will focus ondetermining the braided auto-equivalences which move Ω. This section will be combinatorial innature, making use of the rich combinatorics of the categories C( sl r + , k ) . Let us outline thearguments of this section.Our main tool to determine when the category C( sl r + , k ) ( Z m ) has an exceptional auto-equivalence will be the following Lemma, which gives very restrictive necessary conditions. Lemma 4.1.
The category C( sl r + , k ) ( Z m ) has a braided exceptional auto-equivalence only ifthere exists a object X ∈ C( sl r + , k ) such that ● X ∉ [ Λ + Λ r ] , and ● the orbit of X under the action of Z m is closed under charge-conjugation, ● we have [ r ] r,k [ r + ] r,k = dim ( X )∣ Stab Z m ( X )∣ ≥ dim ( X )∣ Stab Z r + ( X )∣ , and ● the twist of X is equal to the twist of Λ + Λ r .Proof. Suppose C( sl r + , k ) ( Z m ) has a braided exceptional auto-equivalence, then by definitionthere is an object ( X, χ X ) ∈ C( sl r + , k ) ( Z m ) such that Ω is mapped to ( X, χ X ) under theexceptional auto-equivalence, and ( X, χ X ) is not in the orbit of Ω under simple currents. YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 27 As Ω is self-dual, we have that ( X, χ X ) is self-dual, and hence the orbit of X under Z m isclosed under conjugation.To obtain the dimension bound for X , we note thatdim ( Ω ) = dim (( X, χ X )) . From this we can obtain the inequalitydim ( Ω ) = dim ( X )∣ Stab Z m ( X )∣ ≥ dim ( X )∣ Stab Z r + ( X )∣ . The dimension of the object Ω is [ r ] r,k [ r + ] r,k , hence we have the result.To get the condition on the twist of X , note that a braided auto-equivalence of C( sl r + , k ) ( Z m ) will preserve twists by [13, Lemma 2.2]. The twist of an object ( X, χ X ) ∈ C( sl r + , k ) ( Z m ) isequal to the twist of X ∈ C( sl r + , k ) . The condition is then immediate. (cid:3) The key restriction here is the existence of an object X ∈ C( sl r + , k ) with [ r ] r,k [ r + ] r,k ≥ dim ( X )∣ Stab Z r + ( X )∣ . If Stab Z r + ( X ) is non-trivial, then we can use Lemma 2.11 to bound the dimension of X below bythe dimension of a simpler object in C( sl r + , k ) , say for example 2Λ . We can then use the hookformula to write the dimension of this simpler object as a product of quantum integers. For our2Λ example we would have the dimension is [ r ] r,k [ r + ] r,k [ r + ] r,k [ ] r,k [ ] r,k . This then gives us an inequalityof quantum integers that must be obeyed for there to exist an exceptional auto-equivalence. Bysuitably bounding this inequality we can then obtain strong restrictions on the rank and levelof the category. With this approach we are able to show that there are only a finite number ofcases where the inequality may hold. From here we can then directly search for X where thecondition [ r ] r,k [ r + ] r,k = dim ( X )∣ Stab Z m ( X )∣ holds. This yields a very small number of candidates for exceptional auto-equivalences.When Stab Z r + ( X ) is trivial, we search for objects X ∈ C( sl r + , k ) which satisfy [ r ] r,k [ r + ] r,k = dim ( X ) . Here there are many candidates for X . In particular, any object in [ Λ + Λ r ] will satisfy thiscondition. However, when paired with the condition that X ∉ [ Λ + Λ r ] , we can again reducethe list of candidates down to a finite list via similar techniques as before. This case is a bitmore fiddly than the case with non-trivial stabaliser group, as now we have to carefully avoidthe objects in the orbit of Λ + Λ r , however the technical details remain the same.In order to suitably bound the inequalities of quantum integers, we have to assume that k ≥ r + k < r + k ≥ r + X ∈ C( sl r + , k ) such that X ∉ [ Λ + Λ r ] andsuch that [ r ] r,k [ r + ] r,k = dim ( X )∣ Stab Z m ( X )∣ . From this finite list we then search for objects which satisfy the remaining conditions of Lemma 4.1to obtain an even smaller list.Finally, we computer search the fusion rings of these remaining candidates, looking for fusionring automorphisms which preserve the twists of the simples. This yeilds the main theorem ofthis section.
Theorem 4.2.
Let r ≥ and k ≥ and m a divisor of r + satisfying m ∣ k ( r + ) if r is even,and m ∣ k ( r + ) if r is odd. Then except for the cases ● C( sl , ) ( Z ) , ● C( sl , ) ( Z ) , ● C( sl , ) ( Z ) , ● C( sl , ) ( Z ) , ● C( sl , ) ( Z ) , ● C( sl , ) ( Z ) , ● C( sl , ) ( Z ) , and ● C( sl , ) ( Z ) every braided auto-equivalence of C( sl r + , k ) ( Z m ) is non-exceptional.For the first four cases, we have that there are two possibilities for the group of braided auto-equivalences: EqBr (C( sl , ) ( Z ) ) ∈ { Z , S } , EqBr (C( sl , ) ( Z ) ) ∈ { D , S } EqBr (C( sl , ) ( Z ) ) ∈ { D , S } , EqBr (C( sl , ) ( Z ) ) ∈ { D , A } . For the remaining four cases, we have that
EqBr (C( sl , ) ( Z ) ) = EqBr (C( sl , ) ( Z ) ) , EqBr (C( sl , ) ( Z ) ) = EqBr (C( sl , ) ( Z ) ) × Z , EqBr (C( sl , ) ( Z ) ) = EqBr (C( sl , ) ( Z ) ) × Z , and EqBr (C( sl , ) ( Z ) ) = EqBr (C( sl , ) ( Z ) ) . With the high-level arguments and end goal in mind. Let us proceed with the fine details of thearguments. Let C( sl r + , k ) ( Z m ) be a category with an exceptional braided auto-equivalence.Then by Lemma 4.1 we get an object X ∈ C( sl r + , k ) satisfying the conditions of the lemma. Wewill have to split into several cases, depending on the stabaliser group of X ∈ C( sl r + , k ) , and onthe size of k compared to r + Case:
Stab Z r + ( X ) = r + . Let us first deal with the case where X has full stabaliser subgroup, i.e. Stab Z r + ( X ) = Z r + .As ∣ Stab Z r + ( X )∣ divides k , we necessarily have k ≥ r + ( X ) ≥ dim ( ) , dim ( X ) ≥ dim ( ) , and dim ( X ) ≥ dim ( ) . Hence we get the inequalities [ r ] r,k [ r + ] r,k ≥ dim ( ) r + , [ r ] r,k [ r + ] r,k ≥ dim ( ) r + , and [ r ] r,k [ r + ] r,k ≥ dim ( ) r + . Let us focus on this first inequality for now. Expanding this inequality with the hook formula,and simplifying gives(4.1) ( r + )[ ] r,k [ ] r,k ≥ [ r + ] r,k . YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 29 The left hand side we can bound above by ( r + ) ⋅
12, giving that the above inequality canonly hold if the weaker inequality(4.2) 12 ( r + ) ≥ [ r + ] r,k holds. As k ≥ r +
1, we have that r + ≤ ( + r + + r ) ≤ ( + r + k ) , so we can apply Lemma 2.5to obtain the bound [ r + ] r,k ≥ r + . This gives us the even weaker inequality48 ( r + ) ≥ ( r + ) which only holds if r ≤ r ≤
47 we still have an infinite number of k where the initial inequality may hold.Let us return to the inequality from Equation (4.2). For each fixed r ≤
47, the left hand sideis constant, while the right hand side is an increasing function of k . Therefore if we can finda smallest k for which this inequality breaks, then we know it will also break for all larger k .This leaves us with a finite list of k for which the initial inequality from Equation (4.1) canhold. Finally, we check each of these finite potential solutions against Equation (4.1) to obtainan even smaller list of potential candidates.We find the following finite list of potential solutions for r ≥ r Potential k
12 13 ≤ k ≤ ≤ k ≤ ≤ k ≤ ≤ k ≤ ≤ k ≤ ≤ k ≤ ≤ k ≤ ≤ k ≤ ≤ r ≤ r + ≤ k ≤ ≤ r ≤ r + ≤ k ≤ r ≥ ∅ Remark 4.3.
We will repeatedly use the above trick in order to leverage an inequality of quantumintegers, into an explicit list of r and k where the inequality holds. To summarise, we begin withan inequality left ≤ right of quantum integers. We then use the bound from Lemma 2.5 to boundthe left equation below, and the bound from Lemma 2.4 to bound the right equation above. Thesebounds remove the quantum integers, and the resulting inequality gives an upper bound on r . Wenow return to the equation left ≤ right, but this time only bound the right hand side above, by afunction of r . We plug each of our finite r into this inequality, giving a new inequality whichstates that a product of quantum integers is less than some constant. As quantum integers are anincreasing function of k (once r is fixed), we can find the smallest k which breaks the inequality,which tells us it also breaks for all larger k . At this point we may find that no k breaks theinequality. When this happens we have to throw away the r , and find a different inequality ofquantum integers to deal with that particular r . This leaves us with a finite number of r wherethe inequality may hold, and for some subset of these r , a finite list of k where the inequality mayhold. To further the finite list of k , we test each possible solution against the initial inequalityof quantum integers. For r ≤
11 there is no k where Equation (4.2) breaks. To deal with the case of r ≤
11 let usnow consider the inequality dim ( X ) ≥ dim ( ) . Expanding this out using the hook formula gives the inequality [ ] r,k [ ] r,k [ ] r,k [ ] r,k [ ] r,k [ r ] r,k ( r + ) ≥ [ r + ] r,k [ r + ] r,k [ r + ] r,k [ r + ] r,k [ r + ] r,k . Playing the game from Remark 4.3 we find this inequality breaks for all k ≥ r and k such that C( sl r + , k ) ( Z m ) can have an exceptionalauto-equivalence. This is still an unreasonable number of cases to computer search through. Forexample C( sl , ) has on the order of 10 simple objects. To refine our finite list of potentialsolutions further we run each solution of r and k through the inequality [ r ] r,k [ r + ] r,k ≥ dim ( ) r + . This yields the following list of r and k , such that C( sl r + , k ) may have an object X withStab Z r + ( X ) = Z r + , and with dim ( X ) Stab Z r + ( X ) ≤ [ r ] r,k [ r + ] r,k . r Potential k ≤ k ≤
402 3 ≤ k ≤
403 4 ≤ k ≤
404 5 ≤ k ≤
155 6 ≤ k ≤
86 7 r ≥ ∅ From this small finite list, we can computer search to find all objects X ∈ C( sl r + , k ) such that dim ( X ) r + = [ r ] r,k [ r + ] r,k . This yields the following result Lemma 4.4.
Let r ≥ , and k ≥ r + . There exists a object X of C( sl r + , k ) with dim ( X ) r + =[ r ] r,k [ r + ] r,k if and only if (1) r = and k = , in which case X = , or (2) r = and k = , in which case X = + , or (3) r = and k = , in which case X = Λ + Λ + Λ + Λ . k ≥ r + and ∣ Stab Z r + ( X )∣ ∉ { , r + } . Let us now consider the case where k ≥ r +
1, and ∣ Stab Z r + ( X )∣ ∉ { , r + } . We can immediatelyassume that r ≥
3, as if r ∈ { , } , then there are no possibilities for ∣ Stab Z r + ( X )∣ which mustdivide r + ∣ Stab Z r + ( X )∣ ∉ { , r + } and r ≥ ≤ r + ∣ Stab Z r + ( X )∣ ≤ r −
1. Hence, usingLemma 2.11, along with Lemma 2.8 we see thatdim ( X ) ≥ dim ( r + d ) ≥ dim ( ) . We expand this inequality as ∣ Stab Z r + ( X )∣[ ] r,k [ ] r,k [ ] r,k ≥ [ r + ] r,k [ r + ] r,k [ r + ] r,k . As ∣ Stab Z r + ( X )∣ ≤ r + , we can bound the left hand side above to get the weaker inequality(4.3) r + [ ] r,k [ ] r,k [ ] r,k ≥ [ r + ] r,k [ r + ] r,k [ r + ] r,k . As r ≥
3, we have that r + , r + , r + ≤ ( + r + + r ) ≤ ( + r + k ) . YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 31 Thus we can apply Lemma 2.5 to get the lower bounds [ r + ] r,k ≥ r + , [ r + ] r,k ≥ r + , and [ r + ] r,k ≥ r + . With these bounds, we can use the methods described in Remark 4.3 to obtain a finite list ofsolutions. We can ignore the r = r = r + ∣ Stab Z r + ( X )∣ must be either 1 or r + r and k , such that C( sl r + , k ) may have an object X withStab Z r + ( X ) ∉ { , r + } , and with dim ( X ) Stab Z r + ( X ) ≤ [ r ] r,k [ r + ] r,k . r Potential k ≤ k ≤
95 6 ≤ k ≤ r ≥ ∅ From this small finite list, we can computer search to find all objects X ∈ C( sl r + , k ) such that dim ( X ) Stab Z m ( X ) = [ r ] r,k [ r + ] r,k . This yields the following result Lemma 4.5.
Let r ≥ , k ≥ r + , and m a divisor of r + such that m ∣ k ( r + ) if r iseven, or such that m ∣ k ( r + ) if r is odd. There exists an object of X ∈ C( sl r + , k ) with ∣ Stab Z m ( X )∣ ∉ { , r + } and dim ( X )∣ Stab Z m ( X )∣ = [ r ] r,k [ r + ] r,k if and only if (1) r = , k = , and m = , in which case X ∈ [ ] , or (2) r = , k = , and m = , in which case X ∈ [ ] . k ≥ r + and d = . We now deal with the case where the object X has trivial stabilizer group.The difficulty here lies in the fact that many objects close to the corners of the Weyl chamberhave trivial stabilizer subgroup, and are of small dimension. In fact, the object 2Λ (nearlyalways) has trivial stabilizer group, and has dimension smaller that [ r ] r,k [ r + ] r,k . To makematters worse, we have that the Lemma 2.11 which was the powerhouse of the previous cases,is useless in this case, as it simply says that dim ( X ) ≥
1. Thus we will need to be smarter withour bounds in this case.For this entire subsection we will assume that k ≥
6, as for k ≤
5, there are only finitely many r to consider.Let us write X as X = r ∑ i = λ i Λ i ∈ C( sl r + , k ) . We have that Stab Z r + ( X ) = { } , and dim ( X ) = [ r ] r,k [ r + ] r,k . Our goal is to show that bar asmall number of exceptions, X must lie in [ Λ + Λ r ] .We have break into cases depending on the structure of the coefficients λ i . This is becausesome X will have dimension larger than [ r ] r,k [ r + ] r,k , while others will be smaller. Our twomain cases will be if all λ i ≤ k −
3, and if there is some λ i ≥ k − Case: All λ i ≤ k − If each λ i ≤ k −
3, then Lemma 2.9 gives the bounddim ( X ) ≥ dim ( ) . The dimension of 3Λ is [ r + ] r,k [ r + ] r,k [ r + ] r,k [ ] r,k [ ] r,k . Hence, we get the inequality [ ] r,k [ ] r,k [ r ] r,k ≥ [ r + ] r,k [ r + ] r,k . As k ≥ r + r ≥
2, we have that r + , r + ≤ ( + r + + r ) ≤ ( + r + k ) , hence we canuse Lemma 2.5 with the method described in Remark 4.3 to obtain that the inequality can only hold if r ≤ k ≤
9. From this finite list, we can now directly search for simple objects X ∈ C( sl r + , k ) with dim ( X ) = [ r ] r,k [ r + ] r,k , and with each λ i ≤ k −
3. This gives the followingLemma.
Lemma 4.6.
Suppose that r + ≤ k , and suppose X = ∑ ri = λ i Λ i ∈ C( sl r + , k ) is a simple objectwith each λ i ≤ k − . Then X has dimension equal to [ r ] r,k [ r + ] r,k if and only if (1) r = , k = , and X ∈ [ ] , or if (2) r = , k = , and X ∈ [ ] , or if (3) r = , k = and X ∈ [ ] . Case: there is some λ i ≥ k − For this case we have to introduce sub-cases, as we have to carefully dance around the objectΛ + Λ r . By applying a simple current symmetry, we can assume that i =
0, and thus ∑ ri = λ i ≤ X has to be of one of the following forms (excluding X = Λ + Λ r ): ● X = Λ i for 1 ≤ i ≤ r , ● X = Λ i + Λ j with 2 ≤ i, j ≤ r − ● X = Λ + Λ j or X = Λ r + Λ j with 3 ≤ j ≤ r − ● X = , X = r , X = Λ + Λ , X = Λ + Λ r − , X = Λ r + Λ , or X = Λ r + Λ r − .In each of these cases, we will show that if X has dimension [ r ] r,k [ r + ] r,k , then either X ∈ [ Λ + Λ r ] , or there are severe restrictions on r and k . Sub-case: X = Λ i for ≤ i ≤ r We first note that we can write Λ + Λ r =
12 2Λ +
12 Λ r , thus by Lemma 2.2 we have dim ( Λ + Λ r ) ≥ dim ( ) , and thus by Lemma 2.8 dim ( Λ + Λ r ) isstrictly larger than dim ( Λ ) . Hence X can not be one ofΛ , or Λ r . If X = Λ or X = Λ r − , then we have the following Lemma which shows that r = k = i with 1 ≤ i ≤ r , the object Λ in the case r = k = Lemma 4.7.
Suppose k ≥ r + . Then we have dim ( Λ + Λ r ) = dim ( Λ ) if and only if r = and k = .Proof. Suppose we had dim ( Λ + Λ r ) = dim ( Λ ) . We can explicitly write this equation as [ r ] r,k [ r + ] r,k = [ r + ] r,k [ r ] r,k [ ] r,k . We can rearrange this to obtain [ r + ] r,k = [ ] r,k [ r + ] r,k = [ r + ] r,k + [ r + ] r,k , hence [ r + ] r,k =
0. This can only occur when k = (cid:3) YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 33 Next up is when X = Λ or X = Λ r − . This case is a bit fiddly, as dim ( Λ ) can be larger, orsmaller than [ r ] r,k [ r + ] r,k depending on r and k . We will show for all r ≥ ( Λ ) isstrictly larger than [ r ] r,k [ r + ] r,k , and also for all r ≤ ( Λ ) is strictly smaller than [ r ] r,k [ r + ] r,k . At r =
8, we get a crossover, which occurs at k = Lemma 4.8.
Assume that k ≥ r + . ● If r ≤ and ( r, k ) ≠ ( , ) , or r = and k ≤ , then dim ( Λ ) < [ r ] r,k [ r + ] r,k . ● If r ≥ , or r = and k ≥ , or ( r, k ) = ( , ) , then we have dim ( Λ ) > [ r ] r,k [ r + ] r,k . ● If r = and k = , then we have dim ( Λ ) = [ r ] r,k [ r + ] r,k . Proof.
Let us begin with the first statement. Aiming for a contradiction suppose thatdim ( Λ ) ≥ [ r ] r,k [ r + ] r,k . Expanding this inequality and simplifying gives [ r + ] r,k [ r − ] r,k ≥ [ ] r,k [ ] r,k [ r + ] r,k . We now use the method described in Remark 4.3 to obtain the solutions.Playing the same game with the inequalitydim ( Λ ) ≤ [ r ] r,k [ r + ] r,k finishes the proof. (cid:3) Now for all 3 ≤ i ≤ r −
2, we have from Lemma 2.8 thatdim ( Λ i ) ≥ dim ( Λ ) . Thus the above Lemma rules out the possibility of X = Λ i when r ≥
9, or r = k ≥
15. Thuswe have a finite number of possibilities left for r and X . Either r ≤ X = Λ , or r ∈ { , } and X = Λ or X = Λ . If r ≤ X = Λ , or if r = X = Λ , then the above Lemma alsorules out the possibility of dim ( X ) = [ r ] r,k [ r + ] r,k except for when r = k =
15. Thus allthat remains to be considered is the case r ∈ { , } and X = Λ . Lemma 4.9.
Suppose r ∈ { , } and k ≥ r + . Then dim ( Λ ) > [ r ] r,k [ r + ] r,k . Proof.
Apply the method from Remark 4.3 to the inequality. (cid:3)
We summerise these results with the following Lemma.
Lemma 4.10.
Suppose that k ≥ r + , and let ≤ i ≤ r . Then dim ( Λ i ) = [ r ] r,k [ r + ] r,k if andonly if r = and k = , in which case i ∈ { , } . Sub-case: X = Λ i + Λ j for ≤ i, j ≤ r − . Now we suppose X is of the form X = Λ i + Λ j with 2 ≤ i, j ≤ r −
1. We can write X = ( i ) + ( j ) . Hence dim ( X ) ≥ min { dim ( i ) , dim ( j )} by Lemma 2.2. As 2 ≤ i, j, ≤ r −
1, then we can use Lemma 2.8 to see thatmin { dim ( i ) , dim ( j )} ≥ dim ( ) . This implies the inequality [ r + ] r,k ≤ [ ] r,k [ ] r,k . The standard trick from Remark 4.3 show that this inequality only holds if r = r = k ∈ { , , } . If r =
2, then there are no X of the imposed form, and if r =
3, then X = , andwe can directly check that dim ( X ) = [ r ] r,k [ r + ] r,k only when k = Lemma 4.11.
Suppose that r + ≤ k , and let ≤ i, j, ≤ r − . Then dim ( Λ i + Λ j ) = [ r ] r,k [ r + ] r,k if and only if r = and k = , in which case i = j = . Sub-case: X = Λ + Λ j or X = Λ r + Λ j with ≤ j ≤ r − . Now we consider the case of X = Λ i + Λ j if both i ∈ { , r } and 3 ≤ j ≤ r −
2. By applying chargeconjugation, we can assume that X is of the form X = Λ + Λ j . Working in the extended Weyl chamber, we can write X = ( ) + (
32 Λ j ) . Hence dim ( X ) ≥ min { dim ( ) , dim ( Λ j )} . As k ≥ r + r ≥ ≤ ≤ k − ≤ j ≤ r − ( X ) ≥ min { dim ( ) , dim ( Λ )} . This gives us that one of the inequalities [ r ] r,k [ r + ] r,k ≥ ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ [ r + ] r,k [ r + ] r,k [ r + ] r,k [ ] r,k [ ] r,k [ r + ] r,k [ r ] r,k [ r − ] r,k [ ] r,k [ ] r,k . holds. With the assumption that r ≥ r ≥ k where the inequality is satisfied,and when r = ≤ k ≤
15. A computer search of these seven remaining casesreveals that Λ + Λ j never has dimension [ r ] r,k [ r + ] r,k , for any of these k .When r ≤ X with the desiredform, and with dim ( X ) = [ r ] r,k [ r + ] r,k . Let us sketch the r = r =
7, there are three possibilities for the object X . They areΛ + Λ , Λ + Λ , and Λ + Λ . The dimensions of these objects are, respectively [ ] r,k [ ] r,k [ ] r,k [ ] r,k [ ] r,k [ ] r,k , [ ] r,k [ ] r,k [ ] r,k [ ] r,k [ ] r,k [ ] r,k [ ] r,k [ ] r,k , and [ ] r,k [ ] r,k [ ] r,k [ ] r,k [ ] r,k [ ] r,k [ ] r,k [ ] r,k [ ] r,k [ ] . As we require that the object X has dimension [ r ] r,k [ r + ] r,k = [ ] ,k [ ] ,k we get an equalityof quantum integers. Let us single out Λ + Λ as the rest follow near identically. For this case,we have the equality [ ] ,k [ ] ,k = [ ] ,k [ ] ,k [ ] ,k [ ] ,k [ ] ,k [ ] ,k , YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 35 which we can simplify to [ ] ,k [ ] ,k = [ ] ,k [ ] ,k . We can treat this as an inequality [ ] ,k [ ] ,k ≤ [ ] ,k [ ] ,k and use the technique from Remark 4.3 to see that the inequality holds for no k ≥ r + + Λ j to have dimension [ r ] r,k [ r + ] r,k . r Λ + Λ j Equality2 ≤ r ≤ + Λ [ ] r,k = [ ] r,k + Λ [ ] r,k [ ] r,k = [ ] r,k [ ] r,k Λ + Λ [ ] r,k [ ] r,k = [ ] r,k [ ] r,k + Λ [ ] r,k [ ] r,k = [ ] r,k [ ] r,k Λ + Λ [ ] r,k [ ] r,k = [ ] r,k [ ] r,k Λ + Λ [ ] r,k [ ] r,k = [ ] r,k [ ] r,k .We find none of these equalities hold for k ≥ r +
1, hence there are no objects of the formΛ + Λ j with dimension equal to [ r ] r,k [ r + ] r,k .Summerising we have the following Lemma. Lemma 4.12.
Suppose that r ≤ k + , and let ≤ j ≤ r − . Then Λ + Λ j and Λ r + Λ j never havedimension equal to [ r ] r,k [ r + ] r,k . Sub-case: X = , X = r , X = Λ + Λ , X = Λ + Λ r − , X = Λ r + Λ , or X = Λ r + Λ r − . Finally we have the objects X = , X = r , X = Λ + Λ , X = Λ + Λ r − , X = Λ r + Λ , and X = Λ r + Λ r − to consider. By charge conjugation, it suffices to consider X = , X = Λ + Λ , and X = Λ + Λ r − . The first is easily ruled out as we can writeΛ + Λ r =
12 2Λ +
12 2Λ r . Thus Lemma 2.2 gives that dim ( Λ + Λ r ) is always strictly bigger than dim ( ) except when r =
1, in which case X = = Λ + Λ r .When X = Λ + Λ , we get the equality [ r ] r,k [ r + ] r,k = [ r ] r,k [ r + ] r,k [ r + ] r,k [ ] r,k , and hence [ ] r,k = [ r + ] r,k . This equation only holds when either r =
2, or k =
3. Both thesecases we can disregard, as when r =
2, we have X = Λ + Λ = Λ + Λ r , and as k ≥ r +
1, we onlyconsider k = r ∈ { , } . When r = k = X = Λ ∈ [ Λ + Λ r ] .When X = Λ + Λ r − , we get the equality [ r ] r,k [ r + ] r,k = [ r − ] r,k [ r + ] r,k [ r + ] r,k [ ] r,k , which we can simplify to [ ] r,k [ r ] r,k = [ r − ] r,k [ r + ] r,k . Using the techniques form Remark 4.3 we see that r ≤
3. If r =
3, then X = Λ + Λ which hasalready been dealt with. If r =
2, then X = which has also already been dealt with.Summerising we have the following. Lemma 4.13.
Let k ≥ r + , and suppose X ∈ { , r , Λ + Λ , Λ + Λ r − , Λ r + Λ , Λ r + Λ r − } with dim ( X ) = [ r ] r,k [ r + ] r,k . Then X ∈ [ Λ + Λ r ] . We have now covered all possibilities for an object X = ∑ ri = λ i Λ i . Taking the union of resultsin this subsection gives the following. Lemma 4.14.
Suppose that k ≥ r + , and let X be a simple object of C( sl r + , k ) with dim ( X ) =[ r ] r,k [ r + ] r,k . Then either X ∈ [ Λ + Λ r ] , or ( r, k ) = ( , ) , and X ∈ [ ] or ( r, k ) = ( , ) , and X ∈ [ ] or ( r, k ) = ( , ) , and X ∈ [ ] or ( r, k ) = ( , ) , and X ∈ [ ] or ( r, k ) = ( , ) , and X ∈ [ Λ ] . All together, we have the following key result.
Lemma 4.15.
Let r ≥ , k ≥ r + , and m a divisor of r + such that m ∣ k ( r + ) if r iseven, or such that m ∣ k ( r + ) if r is odd. Then there exists an object X ∈ C( sl r + , k ) with dim ( X ) Stab Z m ( X ) = [ r ] r,k [ r + ] r,k if and only if X ∈ [ Λ + Λ r ] , or in one of the following cases: r k m X + { } [ ] { } [ ] { } [ ] { } [ ] { } Λ + Λ + Λ + Λ { } [ ] { } [ Λ ] k < r + . We now have to consider the case where the level is small compared to the rank i.e. k < r +
1. Using level-rank duality we can reduce this argument to the k ≥ r + C( sl r + , k ) and C( sl k , r + ) .Given an object X = r ∑ i = λ i Λ i ∈ C( sl r + , k ) YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 37 we can form the r × k Young tableaux T ( X ) ∶= ( r ∑ i = λ i , r ∑ i = λ i , ⋯ , λ r ) . Taking the transpose of this tableaux gives a k × r tableaux. Initially this presents a spannerfor a level rank-duality connection, as the objects of C( sl k , r + ) are identified by ( k − ) ×( r + ) tableauxes. Thus level-rank duality at first glance appears to give a connection between C( sl r + , k ) and C( sl k + , r ) . However this connection is superficial at best, and only shows theranks of the two categories are equal. Instead we will restrict our attention to objects X ∈C( sl r + , k ) with λ ≠
0. With this restriction, the the tableaux T ( X ) can be considered as a r × ( k − ) tableaux, and thus the transpose can be identified with an object of C( sl k , r + ) . Wewrite X T for this transposed object of C( sl k , r + ) . Explicitly we have that X T = k ∑ (cid:96) = ˆ λ (cid:96) Λ (cid:96) , where ˆ λ (cid:96) ∶= RRRRRRRRRRR ≤ j ≤ r ∶ j − ∑ i = λ i = k − (cid:96) RRRRRRRRRRR . Using the hook formula, along with the fact that [ n ] r,k = [ + r + k − n ] k,r we see thatdim ( X ) = dim ( X T ) . In order to apply level-rank duality arguments to study the exceptional auto-equivalences of C( sl r + , k ) ( Z m ) , we need to understand how the the stabaliser group Stab Z m ( X ) is affectedby level-rank duality. There is a subtlety here in that m doesn’t necessarily divide k , and sotalking about Stab Z m ( X T ) doesn’t make sense. We solve this problem, and resolve the subtletyin the following lemma. Lemma 4.16.
Let m a divisor of r + , such that m ∣ k ( r + ) if r is even, or such that m ∣ k ( r + ) if r is odd, and set m ′ = gcd ( k, m ) . Then we have isomorphisms Stab Z m ( X ) ≅ Stab Z m ′ ( X ) ≅ Stab Z m ′ ( X T ) . Proof.
We will first show that Stab Z m ( X ) ≅ Stab Z m ′ ( X ) . Suppose that Stab Z m ( X ) ≅ Z d , then we have that X = r + d ∑ i = λ i d ∑ j = Λ i r + d + j . This implies that d divides k , and therefore Stab Z m ′ ( X ) ≅ Z d .For the second isomorphism we want to show thatStab Z m ′ ( X ) ≅ Stab Z m ′ ( X T ) . Suppose Stab Z m ′ ( X ) ≅ Z d . Then λ i = λ i + r + d for all i ∈ Z r . This implies that for any j ∈ Z r we have j − + r + d ∑ i = j λ i = kd . To prove the claim of the lemma, we have to show thatˆ λ (cid:96) = ˆ λ (cid:96) + kd , for all (cid:96) ∈ Z k .Suppose we have 0 ≤ j ≤ r contributes to ˆ λ (cid:96) . That is j − ∑ i = λ i = k − (cid:96). Then we have that j − r + d − ∑ i = λ i = j − ∑ (cid:96) = λ i − j − ∑ i = j − r + d λ i = k − (cid:96) − kd , and so j − r + d contributes to ˆ λ(cid:96) + kd . This implies thatˆ λ (cid:96) = ˆ λ (cid:96) + kd and hence we have the result. (cid:3) Now suppose k < r +
1, and let m a divisor of r + m ∣ k ( r + ) . Let X be an objectof C( sl r + , k ) with Stab Z m ( X ) = Z d for some d , with dim ( X ) = [ r ] r,k [ r + ] r,k d . By hitting X witha suitable simple current, we can assume that λ ≠
0, and thus we can apply level-rank dualityto get an object X T ∈ C( sl k , r + ) withdim ( X T ) = [ r ] r,k [ r + ] r,k ∣ Stab Z m ′ ( X T )∣ . Furthermore, we note that if X T ∈ [ Λ + Λ k ] , then X ∈ { Λ + Λ r , ( k − ) Λ + Λ , Λ r − + ( k − ) Λ r } ⊂[ Λ + Λ r ] . The following Lemma then follows from Lemma 4.15. Lemma 4.17.
Let r ≥ , ≤ k < r + , and m a divisor of r + such that m ∣ k ( r + ) if r iseven, or such that m ∣ k ( r + ) if r is odd. Then there exists an object X ∈ C( sl r + , k ) with dim ( X ) Stab Z m ( X ) = [ r ] r,k [ r + ] r,k if and only if X ∈ [ Λ + Λ r ] , or in one of the following cases: r k m X { , } [ ] { , , } [ Λ ] { , } [ ] { , , } [ Λ ] { , } [ Λ ] [ Λ + Λ ]
14 9 { , , } [ ]
15 2 { , } [ Λ ] Together with Lemma 4.15, we now have a complete list of objects X ∈ C( sl r + , k ) with dim ( X ) Stab Z m ( X ) = [ r ] r,k [ r + ] r,k . From this small list of objects, we can directly search for objectsthat satisfy the remaining conditions of Lemma 4.1. Namely, objects X with the same twist asΛ + Λ r and such that X ∗ ∈ [ X ] . This yields the following lemma. Lemma 4.18.
Let r ≥ , k ≥ , and m a divisor of r + such that m ∣ k ( r + ) if r is even,or such that m ∣ k ( r + ) if r is odd. There may exist an exceptional auto-equivalence of C( sl r + , k ) ( Z m ) only in the following cases: YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 39 C Image of Ω C( sl , ) ( Z ) ( , χ n ) : n ∈ Z C( sl , ) ( Z ) ( + , χ n ) : n ∈ Z C( sl , ) ( Z ) ( , χ n ) : n ∈ Z C( sl , ) ( Z ) ( + , χ n ) : n ∈ Z C( sl , ) ( Z ) ( , χ n ) : n ∈ Z C( sl , ) ( Z ) ( Λ + Λ + Λ + Λ , χ n ) : n ∈ Z C( sl , ) ( Z ) ( , χ n ) : n ∈ Z C( sl , ) ( Z ) ( , χ n ) : n ∈ Z C( sl , ) ( Z ) ( Λ + Λ , χ n ) : n ∈ Z C( sl , ) ( Z ) ( Λ , χ n ) : n ∈ Z C( sl , ) ( Z ) ( Λ , χ n ) : n ∈ Z Now that we have this extremely small finite list of candidates for exceptional braided auto-equivalences of C( sl r + , k ) ( Z m ) , we can explicitly search the fusion rings of these candidates tosee if the exceptional braided auto-equivalence exists at the level of the fusion ring. Additionally,we check that the fusion ring automorphisms preserve the twists of the simple objects. We obtainthe explicit data for these categories from the results of [19]. From the results of the previoussection, we know precisely the non-exceptional braided auto-equivalences of all of the categories C( sl r + , k ) ( Z m ) . This information helps us in two ways. First, we know that for all cases exceptfor the finite exceptions in the above list, that all braided auto-equivalences are non-exceptional,hence we now fully understand their braided auto-equivalence groups. Second, we also knowthe braided auto-equivalences which fix the object Ω of the finite number of exceptions in theabove list. Via compositional arguments, this allows us to rule out many potential exceptionalauto-equivalences of these categories. This allows us to essentially determine the group structureof the braided auto-equivalence groups, up to the exceptional auto-equivalences existing. Lemma 4.19.
We have that
EqBr (C( sl , ) ( Z ) ) ∈ { Z , S } , EqBr (C( sl , ) ( Z ) ) ∈ { D , S } EqBr (C( sl , ) ( Z ) ) ∈ { D } , EqBr (C( sl , ) ( Z ) ) ∈ { D , S } , EqBr (C( sl , ) ( Z ) ) ∈ { D , A } . Proof.
From the results of [19] we have the fusion rings and twists of each of the five abovecategories. Mathematica files containing these fusion rings and twists can be found attached tothe arXiv submission of this paper. We compute the group of fusion ring automorphisms whichpreserve the twists. We will refer to these groups as the braided fusion ring symmetries.For C( sl , ) ( Z ) we see that every braided fusion ring symmetry is non-exceptional, thusEqBr (C( sl , ) ( Z ) ) = D by Theorem 3.2. For the case of C( sl , ) ( Z ) we have that the braided fusion rings symmetries form agroup isomorphic to S , with generators ( , χ ) ↔ ( , χ ) , and Ω ↔ ( , χ ) . From Theorem 3.2 we know that the first generator is realised as a braided auto-equivalence of C( sl , ) ( Z ) . Thus EqBr (C( sl , ) ( Z ) ) is an intermediate subgroup of Z and S . Thereare only two such intermediate subgroups which are the end points.For the case of C( sl , ) ( Z ) we have that the braided fusion rings symmetries form a groupisomorphic to Z × S , with generators ( , ) ↔ ( , ) , ( + , χ ) ↔ ( + , χ ) , and ( + , χ ) ↦ ( + , χ ) ↦ ( + , χ ) , and Ω ↦ ( + , χ ) ↦ ( + , χ ) ↦ ( + , χ ) . From Theorem 3.2 we know that the first two generators are realised as braided auto-equivalences of C( sl , ) ( Z ) , and form a group isomorphic to D . Further, this theoremtells us that any braided auto-equivalence which fixes Ω must be in the subgroup generated bythe first two generators. Thus EqBr (C( sl , ) ( Z ) ) is an intermediate subgroup of D ⊂ Z × S with the property that if Ω is fixed by an auto-equivalence, then this auto-equivalence lives inthe D subgroup. With this knowledge, we can study the intermediate subgroups of D ⊂ Z × S to see that there are only two such subgroups with this property. These are D with the firsttwo generators, and S with the first two generators, and the new generatorΩ ↔ ( + , χ ) , ( + , χ ) ↔ ( + , χ ) . Thus EqBr (C( sl , ) ( Z ) ) is isomorphic to either D or S .The remaining two cases fall to the same argument. For C( sl , ) ( Z ) the group of braidedfusion ring symmetries is Z × S . We have that the generators ( , χ ) ↔ ( , χ ) , ( + + , χ i ) ↦ ( + + , χ i + ) and ( + Λ , ) ↔ ( Λ + , ) , ( + + , χ ) ↔ ( + + , χ ) form a D subgroup of EqBr (C( sl , ) ( Z ) ) , and that any braided auto-equivalence whichfixes Ω must be in this subgroup. Analysing the subgroup structure between D and Z × S shows that at most there can be one more generatorΩ ↔ ( , χ ) , ( + Λ , ) ↦ ( + + , χ ) ↦ ( Λ + , ) ↦ ( + + , χ ) , in the braided auto-equivalence group, which would form a group isomorphic to S . ThusEqBr (C( sl , ) ( Z ) ) is either isomorphic to D or S .Finally for the case C( sl , ) ( Z ) we have that the group of braided fusion symmetries is S . We have that the generators ( Λ + Λ + Λ + Λ , χ i ) ↦ ( Λ + Λ + Λ + Λ , χ i + ) and ( Λ + Λ + Λ + Λ , χ i ) ↦ ( Λ + Λ + Λ + Λ , χ − i ) YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 41 form a D subgroup of EqBr (C( sl , ) ( Z ) ) , and that any braided auto-equivalence whichfixes Ω must be in this subgroup. Analysing the subgroup structure between D and S showsthat at most there can be one more generatorΩ ↦ ( Λ + Λ + Λ + Λ , χ ) ↦ ( Λ + Λ + Λ + Λ , χ ) , ( Λ + Λ + Λ + Λ , χ ) ↦ ( Λ + Λ + Λ + Λ , χ ) ↦ ( Λ + Λ + Λ + Λ , χ ) in the braided auto-equivalence group, which would form a group isomorphic to A . ThusEqBr (C( sl , ) ( Z ) ) is either isomorphic to D or A . (cid:3) To finish off the proof of the main theorem of this section, we need to deal with the remainingfour cases. These can be dealt with easily by a level-rank duality argument.
Proposition 4.20.
We have the following braided equivalences: C( sl , ) ( Z ) ≃ (C( sl , ) ( Z ) ) rev ⊠ Vec ( Z , { , e πi }) , C( sl , ) ( Z ) ≃ (C( sl , ) ( Z ) ) rev ⊠ Vec ( Z , { , e πi , e πi , e πi , , e πi , e πi , e πi }) , C( sl , ) ( Z ) ≃ (C( sl , ) ( Z ) ) rev ⊠ Vec ( Z , { , e πi , e πi }) , and C( sl , ) ( Z ) ≃ (C( sl , ) ( Z ) ) rev ⊠ Vec ( Z , { , e πi }) . Proof.
Let us do the computation for C( sl , ) ( Z ) , as the other cases follow in a similarfashion.From Lemma 2.19 we know that C( sl , ) ( Z ) has a subcategory braided equivalent to (C( sl , ) ad ) Rep ( Z ) . Via level-rank duality [47], we have a braided equivalence C( sl , ) ad → (C( sl , ) ad ) rev . Thus together we have a braided subcategory of C( sl , ) ( Z ) equivalent to ((C( sl , ) ad ) rev ) Rep ( Z ) .Taking the reverse braiding on the category commutes with taking the de-equivariantization (asRep ( Z ) rev = Rep ( Z ) ), thus we have a subcategory braided equivalent to (C( sl , ) ( Z ) ) rev .As (C( sl , ) ( Z ) ) rev is modular, Mugers theorem [43, Theorem 4.2 ] give us that the category C( sl , ) ( Z ) factors as C( sl , ) ( Z ) ≃ (C( sl , ) ( Z ) ) rev ⊠ D , where D is a modular category with global dimension 2. To identify D we study the object ([ ] , ) ∈ C( sl , ) ( Z ) . The twist of this object is e πi , and the order of the object is 2.Thus ([ ] , ) generates a modular subcategory equivalent to Vec ( Z , e πi ) . As the category C( sl , ) ( Z ) has no non-trivial invertible objects, we must have that ([ ] , ) generates D ,which completes the claim. (cid:3) Corollary 4.21.
We have
EqBr (C( sl , ) ( Z ) ) = EqBr (C( sl , ) ( Z ) ) , EqBr (C( sl , ) ( Z ) ) = EqBr (C( sl , ) ( Z ) ) × Z , EqBr (C( sl , ) ( Z ) ) = EqBr (C( sl , ) ( Z ) ) × Z , and EqBr (C( sl , ) ( Z ) ) = EqBr (C( sl , ) ( Z ) ) . Proof.
We use the canonical embeddingEqBr (C) ×
EqBr (D) →
EqBr (C ⊠ D) . By analysing the fusion rings and twists of the the Deligne products from the previous proposi-tion, we see that this embedding is an isomorphism. (cid:3) Realisation of the exceptionals
In the previous section we identified a finite list of the categories C( sl r + , k ) ( Z m ) whichmay have an exceptional braided auto-equivalence, and furthermore, gave upper bounds for thenumber of such auto-equivalences that may exist. In this section, our goal is to construct all theexceptional braided auto-equivalences of these finite number of categories. That is, we want tocompute the braided auto-equivalence groups of the categories ● C( sl , ) ( Z ) , ● C( sl , ) ( Z ) , ● C( sl , ) ( Z ) , and ● C( sl , ) ( Z ) .While it is not immediate from the above list, there are two situations in play here. Thefirst is for the category C( sl , ) ( Z ) , where the exceptional auto-equivalences come from thecoincidence of categories C( so , ) ≃ (C( sl , ) ( Z ) ) rev ⊠ C( so , ) . The S worth of braided exceptional auto-equivalences of C( sl , ) ( Z ) is then naturally seendue to the triality of the Dynkin diagram D . This connection was initially discovered in [42]. Lemma 5.1. [42, Theorem 4.3]
We have
EqBr (C( sl , ) ( Z ) ) = S . The far more interesting situation occurs with the other three categories. Studying the di-mensions of these three examples, one notices that they are all defined over small quadraticfields. For C( sl , ) ( Z ) the dimensions live in Q [√ ] , for C( sl , ) ( Z ) the dimensions livein Q [√ ] , and for C( sl , ) ( Z ) the dimensions live in Q [√ ] . Such behaviour with the di-mensions does not hold for general C( sl r + , k ) ( Z m ) , and suggests that these dimensions mayhave something to do with the potential exceptional auto-equivalences of these categories.A large class of categories with objects living in quadratic fields are the quadratic categories,where the simple objects consist of the group of invertibles, and an object ρ , along with theorbit of ρ under the action of the invertibles. The natural suspicion to draw, is that the threecategories C( sl , ) ( Z ) , C( sl , ) ( Z ) , and C( sl , ) ( Z ) should in some way be connectedto quadratic categories. The naive guess, that these three categories are quadratic categories onthe nose, is immediately thwarted by fact that the dimensions in these examples take on morethan two values. Further, quadratic categories are almost never modular, where as our threeexamples are. However, this lack of modularity suggests the next place to look for a connection.Taking the Drinfeld centre of a quadratic category gives a modular category whose dimensionslie in the same field as the quadratic category. While in the quadratic category, the dimensionsof the simples can only have two possible values, the dimensions of the simples in the centrecan be any integer combination of 1 and the dimension of the non-invertible of the quadratic.This provides strong evidence that the three categories C( sl , ) ( Z ) , C( sl , ) ( Z ) , and C( sl , ) ( Z ) may be Drinfeld centers of quadratic categories. YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 43 Now that we have an idea of what to look for, we can make educated guesses as to the identityof the quadratic categories. For example, the dimensions of C( sl , ) ( Z ) are1 , + √ , + √ , and 3 + √ . If C( sl , ) ( Z ) were the Drinfeld centre of a quadratic category, then a natural guess for thedimension of the non-invertible would be 3 + √
3, as all the above dimensions can be constructedas integer combinations of 1 and 3 + √
3. There is a known quadratic category with an objectof this dimension, which is a near-group category with group of invertibles Z = { , g, g } , anda single non-invertible with fusion ρ ⊗ ρ ≅ ⊕ g ⊕ g ⊕ ρ. The global dimension of this category is 24 + √
3, so the global dimension of its Drinfeld centreis 1008 + √
3. Where as the global dimension of C( sl , ) ( Z ) is 336 + √
3. These globaldimensions are off by a factor of three, which suggests a Z factor is involved. From all this weconjecture that there is a quadratic category C , , with fusion as above such that Z(C , , ) ≃ C( sl , ) ( Z ) ⊠ Vec ( Z , { , e iπ , e iπ }) . Using similar reasoning we conjecture the existence of a fusion category C , , with invertibles Z × Z = { , e, m, em } and non-invertibles { ρ, mρ } with fusion ρ ⊗ ρ ≅ ⊕ e ⊕ ρ ⊕ mρ, such that Z(C , , ) ≃ C( sl , ) ( Z ) ⊠ Vec ( Z × Z , { , e iπ , − , e iπ }) , and a fusion category C , , with invertibles Z × Z = { , e, m, em } and non-invertibles { ρ, eρ, mρ, emρ } with fusion ρ ⊗ ρ ≅ ⊕ ρ ⊕ eρ ⊕ mρ ⊕ emρ, such that Z(C , , ) ≃ C( sl , ) ( Z ) ⊠ Vec ( Z × Z , { , − , − , − }) . We make all this precise with the following Theorem.
Theorem 5.2.
There exist unitary fusion categories C , , , C , , , and C , , with combinatoricsas above, such that Z(C , , ) ≃ C( sl , ) ( Z ) ⊠ Vec ( Z , { , e iπ , e iπ })Z(C , , ) ≃ C( sl , ) ( Z ) ⊠ Vec ( Z × Z , { , e iπ , − , e iπ }) , and Z(C , , ) ≃ C( sl , ) ( Z ) ⊠ Vec ( Z × Z , { , − , − , − }) . Proof.
Let us begin with the case C( sl , ) ( Z ) . It is well known that there is a conformalinclusion SU ( ) ⊂ ( E ) which induces a commutative algebra object A ∈ C( sl , ) . Thecategory Mod (C( sl , ) , A ) is then a unitary fusion category, as described in [6, Figure 4]. Thecategory Mod (C( sl , ) , A ) is Z -graded. Let C , , be the trivially graded subcategory of thisgrading.By [2, Corollary 4.8] we have that Z( Mod (C( sl , ) , A )) ≃ C( sl , ) ⊠ ( Mod (C( sl , ) , A ) ) rev . For this particular case, we have that Mod (C( sl , ) , A ) ≃ C( e , ) ≃ Vec ( Z , { , e iπ , e iπ }) .Now the results of [28] allow us to compute the Drinfeld centre of the subcategory C , , interms of the Drinfeld centre of Mod (C( sl , ) , A ) . This gives that Z(C , , ) ≃ C( sl , ) ( Z ) ⊠ Vec ( Z , { , e iπ , e iπ }) as desired.The case of C( sl , ) ( Z ) is almost identical, except now we use the conformal inclusion SU ( ) ⊂ Spin ( ) . The structure of the associated algebra object, and category of modules,can be found in [8, Section 2.6].The case of C( sl , ) ( Z ) should follow in the same manner, where now we work with theconformal inclusion SU ( ) ⊂ Spin ( ) . However, the author was unable to find a suitabledescription of the category of modules in this case. Instead we have [52, Theorem 3.2] whichproves the precise statement in this case. They show that C , , is the even part of the 3 Z × Z subfactor. (cid:3) With these alternate identifications of the categories C( sl , ) ( Z ) , C( sl , ) ( Z ) , and C( sl , ) ( Z ) identified, we now have the tools to construct their exceptional braided auto-equivalences, and hence determine their braided auto-equivalence groups. We are able to com-plete this for the cases C( sl , ) ( Z ) and C( sl , ) ( Z ) in this paper. Let us begin with C( sl , ) ( Z ) . Lemma 5.3.
We have
EqBr (C( sl , ) ( Z ) ) = S Proof.
From Theorem 4.2 we have that EqBr (C( sl , ) ( Z ) ) is either D or S . By analysingthe fusion rings and twists of Z(C , , ) ≃ C( sl , ) ( Z ) ⊠ Vec ( Z , { , e iπ , e iπ }) we see that EqBr (Z(C , , )) = EqBr (C( sl , ) ( Z ) ) × Z . It is proven in [34, Section 10.6] that Out (C , , ) = D . From [40] we have an embeddingOut (C , , ) → EqBr (Z(C , , )) . Thus EqBr (C( sl , ) ( Z ) ) × Z has a subgroup isomorphic to D . This is only possible ifEqBr (C( sl , ) ( Z ) ) = S . (cid:3) We now deal with the case of C( sl , ) ( Z ) . This case has been examined in the literaturepreviously [52, 29]. Lemma 5.4.
We have
EqBr (C( sl , ) ( Z ) ) = A . Proof.
From Theorem 4.2 we have that EqBr (C( sl , ) ( Z ) ) is either D or A . By analysingthe fusion rings and twists of Z(C , , ) ≃ C( sl , ) ( Z ) ⊠ Vec ( Z × Z , { , − , − , − }) . we see that EqBr (Z(C , , )) = EqBr (C( sl , ) ( Z ) ) × S . It is proven in [35, Theorem 9.4] that Out (C , , ) = A . Thus EqBr (C( sl , ) ( Z ) ) × S has asubgroup isomorphic to A . This is only possible if EqBr (C( sl , ) ( Z ) ) = A . (cid:3) Unfortunately we are unable to determine the braided auto-equivalence group of the category C( sl , ) ( Z ) in this paper. This is because an explicit construction of the quadratic category C , , has yet to be given. One way to construct this category is via the Cuntz algebra method,where it will be realised as endomorphisms on the C ∗ -algebra O ⋊ Z . The large multiplicityspaces of the quadratic category C , , means that this method required solving for roughly 1700 YPE II QUANTUM SUBGROUPS OF sl N . I : SYMMETRIES OF LOCAL MODULES 45 complex variables in 20000 polynomial equations. This makes the problem too complex, evenfor modern computer algebra programs. It appears that a new approach is needed to deal withthe category C( sl , ) ( Z ) . References [1] Jocelyne Bion-Nadal. An example of a subfactor of the hyperfinite II factor whose principal graph invariantis the Coxeter graph E . In Current topics in operator algebras (Nara, 1990) , pages 104–113. World Sci.Publ., River Edge, NJ, 1991.
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Cain Edie-Michell, Vanderbilt University, Nashville, USA
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