Rank-finiteness for G-crossed braided fusion categories
Corey Jones, Scott Morrison, Dmitri Nikshych, Eric C. Rowell
aa r X i v : . [ m a t h . QA ] F e b RANK-FINITENESS FOR G-CROSSED BRAIDED FUSIONCATEGORIES
COREY JONES, SCOTT MORRISON, DMITRI NIKSHYCH, ERIC C. ROWELL
Abstract.
We establish rank-finiteness for the class of G -crossed braided fu-sion categories, generalizing the recent result for modular categories and in-cluding the important case of braided fusion categories. This necessitates astudy of slightly degenerate braided fusion categories and their centers, whichare interesting for their own sake. Introduction
The question of whether there are finitely many fusion categories with a fixednumber of isomorphism classes of simple objects (i.e., fixed rank ) was first raisedby Ostrik in [O1], where an affirmative answer was given for rank 2. In [ENO1]the special case of categories with integral Frobenius-Perron dimension (i.e. weaklyintegral categories) was also settled. Around 2003 Wang conjectured that there arealways finitely many modular categories of a given fixed rank, which was explicitlyverified for rank at most 4. A proof of this rank-finiteness conjecture was obtainedrecently [BNRW]. The main goal of this article is to extend rank-finiteness tothe generality of G -crossed braided fusion categories, which includes the importantcase of braided fusion categories, and does not require the existence of a sphericalstructure.The primary obstacle to overcome is the existence of slightly degenerate braidedfusion categories, with symmetric center equivalent the braided fusion categorysVec of super vector spaces. These are interesting in their own right, with themain open question being whether or not every slightly degenerate braided fusioncategory admits a minimal non-degenerate extension. As a step towards answeringthis question we analyze the structure of the Drinfeld center of a slightly degeneratebraided fusion category.As a technical tool, we prove a bound on the rank of invertible ( C − D )-bimodulecategories. In particular, we show that for any invertible C -bimodule category,rank( M ) ≤ rank( C ). In addition, we show that the set of equivalence classes ofinvertible bimodule categories realizing this bound forms a subgroup of BrPic( C ),and discuss some examples. Date : February 19, 2019.ECR is partially supported by NSF grant DMS-1664359. DN was partially supported by theNSA grant H98230-16-1-0008 and the NSF grant DMS-1801198. This paper was initiated whileECR and DN were visiting CJ and SM at the Australian National University, and gratefullyacknowledge the support of that institution. Preliminaries
We work over an algebraically closed field k of characteristic 0. All fusion ca-tegories and their module categories are assumed to be k -linear. For the basics ofthe theory of fusion categories we refer the reader to [EGNO] and [DGNO].By the rank of a fusion category we mean the number of isomorphism classes ofits simple objects.Let Vec and sVec denote the braided fusion categories of vector spaces andsuper vector spaces over k . For any braided fusion category C let Z sym ( C ) denoteits symmetric (or M¨uger) center. Definition 2.1.
A braided fusion category C is called slightly degenerate [DNO]if Z sym ( C ) = sVec. A slightly degenerate ribbon fusion category is called super-modular .The smallest example of a slightly degenerate braided fusion category is sVecitself. Example 2.2.
One can construct a slightly degenerate braided fusion category asfollows. Let ˜ C be a non-degenerate braided fusion category and let sVec ֒ → ˜ C be abraided tensor functor (it is automatically an embedding). Then the centralizer ofthe image of sVec in C is slightly degenerate.Let C be a slightly degenerate braided fusion category. Below we recall somefacts about C from [DNO, BNRW].Let δ denote the simple object generating Z sym ( C ). Then δ ⊗ X ≇ X for eachsimple object X in C (see [Mu1, Lemma 5.4] and [DGNO, Lemma 3.28]). In par-ticular, the rank of a slightly degenerate braided fusion category is even.We say that C is split if C ∼ = C ⊠ sVec, where C is a non-degenerate braidedfusion category. Any pointed slightly degenerate braided fusion category is split,see [ENO3, Proposition 2.6(ii)] or [DGNO, Corollary A.19].The following definition is due to M¨uger [Mu2]. Definition 2.3. A minimal extension of a slightly degenerate braided fusion (re-spectively, super-modular) category C is a braided tensor functor ι : C ֒ → ˜ C , where˜ C is a non-degenerate braided fusion (respectively, modular) category such that thecentralizer of C in ˜ C is the image of sVec.Note that the above functor ι is an embedding by [DMNO, Corollary 3.26].Clearly, every slightly degenerate braided fusion category that admits a minimalextension can obtained via the construction from Example 2.2 and vice versa.An equivalence of minimal extensions is defined in an obvious way. Example 2.4.
The category sVec has 16 inequivalent minimal extensions [DNO,Kt]: 8 Ising categories and 8 pointed categories. The Witt classes of these extensionsform a subgroup of the categorical Witt group isomorphic to Z / Z .It follows that FPdim( ˜ C ) = 2FPdim( C ). By [Mu1, DGNO] this is the minimal possible value of the Frobenius-Perron dimension of a non-degenerate braided fusioncategory containing C . This explains our terminology. Lemma 2.5.
Let D be a fusion category and let D ⊂ D be a fusion subcategorysuch that FPdim ( D ) = 2 FPdim ( D ) . Then D is faithfully Z / Z -graded with thetrivial component D . ANK-FINITENESS FOR G-CROSSED BRAIDED FUSION CATEGORIES 3
Proof.
Let D = D ⊕ D be a decomposition of D into the sum of D and itsdirect complement D . Then D is a D -bimodule subcategory of D . To provethe statement it suffices to check that the tensor product of D maps D × D to D . Let d := FPdim( D ) = FPdim( D ). Let R denote the (virtual) regularobject in C . We can write it as R = R + R , where R is the regular object of D and R is a regular object of the D -module category D [ENO1] such thatFPdim( R ) = FPdim( R ) = d . Then R R = dR = d ( R + R ). On the otherhand, R R = R ( R + R ) = R R + R = dR + R , since a regular object of D is unique up to a scalar multiple. Hence, R = dR ,which implies that the tensor product of any two objects of D is in D . (cid:3) Thus, a minimal extension of a slightly degenerate braided fusion category is thesame thing as a faithful Z / Z -extension which is a non-degenerate braided fusioncategory. 3. Maximal rank bimodule categories
In this section, we show that invertible bimodule categories over a fusion categoryexhibit a rank bound, and that the bimodule categories realizing this bound actuallyform a subgroup of the Brauer-Picard group. We refer the reader to [ENO2] fordefinitions and properties of invertible bimodule categories.
Proposition 3.1.
Let C , D be fusion categories, and M an invertible ( C − D ) -bimodule category. Then rank ( M ) ≤ ( rank ( C ) rank ( D )) . In particular, for aninvertible C − C bimodule category, rank ( M ) ≤ rank ( C ) .Proof. First consider M as a left C -module category. Then the associated full cen-ter provides us with a Lagrangian algebra L ∈ Z ( C ) [D2]. Let F C : Z ( C ) → C be the forgetful functor, and I C its adjoint. Then as an algebra in C , F C ( L ) ∼ = L M ∈ Irr( M ) Hom(
M, M ) , where the internal hom is taken as a left C module cat-egory. Note that each Hom( M, M ) is a separable, connected algebra, and thusdim(Hom C ( , F C ( L )) = rank( M ). But we have a canonical isomorphismHom C ( , F C ( L )) ∼ = Hom Z ( C ) ( I C ( ) , L ) . However, by [ENO2], the bimodule category M induces a canonical braidedequivalence α : Z ( C ) → Z ( D ) such that α ( L ) ∼ = I D ( ), thus we havedim(End Z ( C ) ( I C ( ))) = dim(Hom C ( , F C ( I C ( )))) = rank( C ) , dim(End Z ( C ) ( L )) = dim(End Z ( D ) ( I D ( ))) = rank( D ) . Here we have used that as an object F C ( I ( )) ∼ = L X ∈ Irr( C ) X ⊗ X ∗ . Thereforeby the Cauchy-Schwartz inequality,rank( M ) = dim(Hom Z ( C ) ( I ( ) , L ))= X X ∈ Irr( Z ( C )) dim(Hom Z ( C ) ( I ( ) , X )) dim(Hom Z ( C ) ( L, X )) ≤ dim(End Z ( C ) ( I ( ))) dim(End Z ( C ) ( L )) = (rank( C )rank( D )) . (cid:3) COREY JONES, SCOTT MORRISON, DMITRI NIKSHYCH, ERIC C. ROWELL
Remark 3.2.
Note the bound rank( M ) ≤ rank( C ) requires invertibility. Considerfor example the rank 4 fusion category C = Rep( D ), where D is the group ofsymmetries of the regular pentagon. Then there exists a rank 5 indecomposablebimodule category, namely Rep( Z ), where the (left and right) actions of Rep( D )are induced from the restriction functor (here Z is the subgroup of rotations of D ).The above proposition leads us to the following definition. Definition 3.3.
We say that an invertible C -bimodule category M has maximalrank if rank( M ) = rank( C ). Proposition 3.4.
Let
Ψ : BrPic( C ) → Aut br ( Z ( C )) be the canonical group iso-morphism of [ENO2] . Then M is maximal rank if and only if Ψ( M ) preserves theisomorphism class of the object I ( ) .Proof. Returning to the proof of Proposition 3.1 and identifying D with C thenΨ( M ) = α , and we are interested in the case when the Cauchy-Schwartz inequalityyields equality. But this happens precisely when there exists a scalar λ such thatdim(Hom Z ( C ) ( I ( ) , X )) = λ dim(Hom Z ( C ) ( α ( I ( )) , X )) . But rank( C ) = X X ∈ Irr( Z ( C ) dim(Hom Z ( C ) ( I ( ) , X )) = λ X X ∈ Irr( Z ( C ) dim(Hom Z ( C ) ( α ( I ( )) , X )) = λ rank( C ) . Since the dimension of morphism spaces is non-negative, we see that we musthave λ = 1. Thusdim(Hom Z ( C ) ( I ( ) , X )) = dim(Hom Z ( C ) ( α ( I ( )) , X ))for all X ∈ Irr( Z ( C )) and the conclusion follows. (cid:3) Corollary 3.5.
The maximal rank invertible bimodule categories form a subgroupof
BrPic( C ) . This result seems somewhat surprising, since in general the behavior of the rankof bimodule categories is notoriously difficult to understand under relative tensorproducts.Recall there is a canonical subgroup Out( C ) ≤ BrPic( C ) which consists of equiv-alence classes of invertible bimodule categories which are trivial as a left modulecategory. This implies the right action must be the usual right action twisted byan auto-equivalence of C . More explicitly, let β be a tensor autoequivalence of C and C β the associated bimodule category, which is C as an underlying category andwith actions X ⊲ Y = X ⊗ Y , X ⊳ Y = X ⊗ β ( Y ), and the obvious associators. Theimage of these bimodule categories in BrPic( C ) forms the subgroup Out( C ).Using the correspondence between module categories and Lagrangian algebras,we see that this is precisely the subgroup of BrPic( C ) which preserve I ( ) as an al-gebra object . In particular, Out( C ) forms a subgroup of the maximal rank bimodulecategories. In many cases, this is the whole group. ANK-FINITENESS FOR G-CROSSED BRAIDED FUSION CATEGORIES 5
Proposition 3.6.
For any pointed fusion category C , the group of maximal rankbimodule categories is Out( C ) .Proof. Any pointed fusion category C is monoidally equivalent to Vec( G, ω ) for afinite group G and 3-cocycle ω ∈ Z ( G, C × ). By [O2], the module categories for thisfusion category are classified by subgroups H ≤ G together with a trivialization of ω | H . The rank of the resulting module category is the index [ G : H ]. Thus thereis a unique rank | G | indecomposable module category, where H = { e } , which isVec( G, ω ) acting on itself. The dual category is thus Vec(
G, ω ), hence any invertiblerank | G | bimodule category is of the form Out( C ). (cid:3) There exist maximal rank invertible bimodule categories that are not of the formOut( C ). One such example is constructed by Ostrik in the appendix of [CMS] usingan extension of the Izumi-Xu fusion category. See [CMS, Theorem A.5.1] and [O3,Remark 2.19 and Example 2.20].To find a maximal rank bimodule category not of the form Out( C ), we need notonly a distinct etale algebra structure on I ( ), but we need this algebra structureto be the image of I ( ) under a braided autoequivalence, which makes findinginvertible bimodule categories not of the form Out( C ) difficult in general.To find such examples, we move in a different direction. If C is braided, we cantry to understand invertible module categories over C . Recall from [DN1, Remark2.13] that we can characterize the bimodule categories M ∈
BrPic( C ) which arein the image of the map from Pic( C ) as the one-sided bimodule categories. Bydefinition, these are bimodule categories for which there exists natural isomorphisms d M,X : M ⊳ X ∼ = X ⊲ M satisfying a collection of coherences. It is not hard to seethat these coherences imply the only one-sided invertible bimodule category whichis trivial as a left module category is the trivial bimodule category C . Thus allnontrivial maximal rank invertible module categories are not of the form Out( C )and thus provide interesting examples.We will now provide a characterization of maximal rank invertible module ca-tegories for non-degenerate fusion categories in terms of braided autoequivalences.In [D1], Davydov introduced the notion of a soft monoidal functor, which is simplya monoidal functor which is isomorphic to the identity functor as a linear func-tor. Equivalently, a soft monoidal functor is one which fixes equivalence classes ofobjects.Recall from [ENO2],[DN1, Section 2.9], α -induction provides us with an isomor-phism ∂ : Pic( C ) → Aut br ( C ). The following result is originally due to Kirillov Jr[Kr] (see also [T, Section II.3]) in the case of modular categories. Proposition 3.7. If C is a non-degenerate braided fusion category and M is aninvertible module category, the rank of M is the number of equivalence classes ofsimple objects fixed by ∂ ( M ) . In particular, the image of the group of maximal rankinvertible module categories is the group of soft braided tensor autoequivalences of C .Proof. M induces a braided autoequivalence of Ψ( M ) ∈ Z ( C ), which by [DN1,Lemma 4.4] is Id C ⊠ ∂ , acting on Z ( C ) ∼ = C ⊠ C rev . But I ( ) ∼ = M X ∈ Irr( C ) X ⊠ X ∗ COREY JONES, SCOTT MORRISON, DMITRI NIKSHYCH, ERIC C. ROWELL hence Ψ( M )( I ( )) = M X ∈ Irr( C ) X ⊠ ∂ ( M )( X ∗ ) . Thus rank( M ) = dim(Hom C ⊠ C rev ( I ( ) , Ψ( M )( I ( )))) is precisely the number offixed points of ∂ ( M ) acting on Irr( C ). (cid:3) Davydov [D1] has computed the group of soft braided autoequivalences for thenon-degenerate braided tensor category Z (Vec( G )) for finite groups G . The an-swer is somewhat involved, but he shows it is a certain subgroup of the image ofOut(Vec( G )) ∼ = H ( G, C × ) ⋊ Out( G ) inside Aut br ( Z (Vec( G ))) satisfying a compat-ibility condition with respect to double class functions [D1], Theorem 2.12. He thenpresents several examples which have non-trivial soft braided autoequivalences, thesmallest of which has order 64, though there may certainly be smaller examples.In any case, these provide examples of non-trivial maximal rank invertible modulecategories. 4. Rank finiteness for braided fusion categories
The rank finiteness theorem for modular categories was proved in [BNRW]. Itstates that up to a braided equivalence there exists only finitely many modular ca-tegories of any given rank. Below we extend this result to braided fusion categoriesthat are not necessarily spherical or non-degenerate. The plan is first to establishthis result for non-degenerate and slightly degenerate categories and then pass toequivariantizations.
Corollary 4.1.
Let C = ⊕ a ∈ A C a be a fusion category faithfully graded by a group A . Then rank ( C ) ≤ | A | rank ( C e ) .Proof. The components C a are invertible C e -bimodule categories so this is immedi-ate from Proposition 3.1. (cid:3) Lemma 4.2.
Let C be a fusion category and let G be a finite group acting on G .Then | G | rank ( C ) ≤ rank ( C G ) ≤ | G | rank ( C ) . Proof.
Simple objects of C G are parameterized by pairs consisting of orbits of simpleobjects of C under the action of G and certain irreducible projective representationsof stabilizers. Each orbit has at most | G | elements, so the number of orbits is atleast rank( C ) / | G | . This implies the first inequality.On the other hand, there are at most rank( C ) orbits and each stabilizer has atmost | G | irreducible projective representations, which gives the second inequality. (cid:3) Proposition 4.3.
There are finitely many equivalence classes of non-degeneratebraided fusion categories of any given rank.Proof.
Let N be a positive integer. By [BNRW], it suffices to show that there is apositive integer M such that any non-degenerate braided fusion category C of rank N is a subquotient of a modular category of rank ≤ M . Here by a subquotientwe mean a surjective image of a subcategory. Let ˜ C be the sphericalization of C [ENO1]. It is a degenerate ribbon category (its symmetric center is Rep( Z / Z )with a non-unitary ribbon structure) of rank 2 N . ANK-FINITENESS FOR G-CROSSED BRAIDED FUSION CATEGORIES 7
As ˜ C is a Z / Z -equivariantization of C , its center Z ( ˜ C ) is a Z / Z -graded mod-ular category with the trivial component Z ( ˜ C ) = Z ( C ) Z / Z by [GNN]. UsingCorollary 4.1 and Lemma 4.2 we estimaterank( Z ( ˜ C )) ≤ Z ( ˜ C ) ) = 2 rank( Z ( C ) Z / Z ) ≤ Z ( C )) = 4 N , so one can take M = 4 N . Indeed, C is a quotient of ˜ C and so is a subquotient of Z ( ˜ C ). (cid:3) Let C , C be braided fusion categories with embeddings sVec ֒ → Z sym ( C i ) , i =1 ,
2. Then C ⊠ sVec C has a canonical structure of a braided fusion category [DNO].Namely, it is equivalent to the category of A -modules in B ⊠ B , where A is theregular algebra of the maximal Tannakian subcategory of sVec ⊠ sVec ⊂ C ⊠ C . If C and C are slightly degenerate then so is C ⊠ sVec C . Proposition 4.4.
There are finitely many equivalence classes of slightly degeneratebraided fusion categories of any given rank.Proof.
Let C be a slightly degenerate braided fusion category of rank N . Itscenter Z ( C ) contains a fusion subcategory C ∨ C rev ∼ = C ⊠ sVec C rev of Frobenius-Perron dimension FPdim( C ) = FPdim( Z ( C )). Hence, Z ( C ) is Z / Z -graded byLemma 2.5 and rank( Z ( C )) ≤ C ⊠ sVec C rev ) = 2 × N N by Corollary 4.1. Since C is a fusion subcategory of Z ( C ) the result follows. (cid:3) Remark 4.5.
It was observed in [BGNPRW], following [BRWZ] that if
C ⊂ ˜ C is a minimal modular extension of a super-modular category then rank( C ) ≤ rank( ˜ C ) ≤ C ). This could be used in place of the more general Corollary 4.1in the proof above. Theorem 4.6.
There are finitely many equivalence classes of braided fusion cate-gories of any given rank.Proof.
Let C be a braided fusion category of rank N . Let E ∼ = Rep( G ) be themaximal Tannakian subcategory of Z sym ( C ). Then C = D G , where D is eithernon-degenerate or slightly degenerate braided fusion category. By Lemma 4.2rank( D ) ≤ | G | rank( C ) = | G | N. Now let M be the maximal order of a group with at most N isomorphism classesof irreducible representations ( M exists since the number of such groups is finiteby Landau’s theorem). We have rank( D ) ≤ M N , so there are finitely many choicesfor D , thanks to Lemmas 4.3 and 4.4. There are also finitely many choices for thegroup G and for each such a choice there are finitely many different actions of G on D [ENO1]. Thus, there are finitely many possible C ’s. (cid:3) Corollary 4.7.
There are finitely many equivalence classes of G -crossed braidedfusion categories of any given rank.Proof. Follows immediately from Theorem 4.6 and Lemma 4.2, since any G -crossedbraided fusion category is obtained as a de-equivariantization of a braided fusioncategory [DGNO, Theorem 4.4.]. (cid:3) COREY JONES, SCOTT MORRISON, DMITRI NIKSHYCH, ERIC C. ROWELL The center of a slightly degenerate braided fusion category
Let C be a slightly degenerate braided fusion category. We have Z sym ( C ) ∼ = sVec.Let δ denote the non-trivial invertible object in Z sym ( C ).For any C -module category M let us denote M s := M ⊠ sVec Vec . In particular, C s := C ⊠ sVec Vec is equivalent to the category of A -modules in C ,where A is the regular algebra of sVec. We have M s = M ⊠ C C s . Note thatrank( C s ) = rank( C ). Lemma 5.1. C s is an invertible C -module category of order .Proof. This follows from straightforward equivalences: C s ⊠ C C s = ( C ⊠ sVec Vec) ⊠ C ( C ⊠ sVec Vec) ∼ = C ⊠ sVec (Vec ⊠ sVec Vec) ∼ = C , where we used the obvious fact Vec ⊠ sVec Vec ∼ = sVec. (cid:3) Lemma 5.2.
We have C s ⊠ C M ∼ = M ⊠ C C s for any C -module category M .Proof. Let B ∈ C be an algebra such that M ∼ = C B . Then A ⊗ B ∼ = B ⊗ A asalgebras since A ∈ Z sym ( C ). This yields the statement. (cid:3) Let C , C be slightly degenerate braided fusion categories. Let E ∈ sVec ⊠ sVec ⊂ C ⊠ C be a canonical ´etale algebra. Recall that the braided fusion category C ⊠ sVec C is defined as the category of E -modules in C ⊠ C . There are obvious embeddings C , C ֒ → C ⊠ sVec C .Let M and M be module categories over C and C . Define a C ⊠ sVec C -module category M ⊠ sVec M to be the category of E -modules in M ⊠ M withthe module action given by X ⊙ M = X ⊗ E M, X ∈ C ⊠ sVec C , M ∈ M ⊠ sVec M . Let M be an indecomposable C ⊠ sVec C -module category and let M = M i ∈ I M i , M = M j ∈ J N j be its decompositions into direct sums of indecomposable C -module categories and C -module categories, respectively. Proposition 5.3.
There exist indecomposable C i -module categories L i , i = 1 , , such that M ∼ = L ⊠ sVec L if and only if M i ∩ N j is an indecomposable sVec-module category for some i ∈ I and j ∈ J .Proof. One implication is obvious.Suppose that M i ∩ N j is an indecomposable sVec-module category. There aretwo possible cases.(Case 1) M i ∩ N j ∼ = sVec. Let X ∈ M i ∩ N j be a simple object. Let δ i denotethe non-trivial invertible object in C i , i = 1 ,
2. Then δ i ⊗ X = X . Let us view M as a C ⊠ C -module category and compute the internal Hom:Hom C ⊠ C ( X, X ) ∼ = Hom C ( X, X ) ⊠ Hom C ( X, X ) ⊕ Hom C ( X, δ ⊗ X ) ⊠ Hom C ( δ ⊗ X, X ) ∼ = (cid:0) Hom C ( X, X ) ⊠ Hom C ( X, X )) (cid:1) ⊗ E, ANK-FINITENESS FOR G-CROSSED BRAIDED FUSION CATEGORIES 9 where E = ⊗ ⊕ δ ⊠ δ is the canonical algebra in sVec ⊠ sVec ⊂ C ⊠ C .Therefore, as a C ⊠ sVec C -module category, M ∼ = L ⊠ sVec L , where L i is thecategory of Hom C i ( X, X )-modules in C i , i = 1 , M i ∩ N j ∼ = Vec. In this case the C ⊠ sVec C -module category( C s ⊠ sVec C ) ⊠ C ⊠ sVec C M satisfies the condition of (Case 1) above and, hence, is equivalent to L ⊠ sVec L .Consequently, M ∼ = L s ⊠ sVec L . (cid:3) Remark 5.4.
The pair of module categories L , L in Proposition 5.3 is deter-mined up to a simultaneous substitution of L , L by L s , L s . Example 5.5.
Let C = C = sVec. Then C ⊠ sVec C = sVec andsVec ∼ = sVec ⊠ sVec sVec ∼ = Vec ⊠ sVec Vec , Vec ∼ = Vec ⊠ sVec sVec ∼ = sVec ⊠ sVec Vecas sVec-module categories.
Proposition 5.6.
Let C be a slightly degenerate braided fusion category and let D = D ⊕ D be a minimal extension (see Definition 2.3) of D := C ⊠ sVec C rev . Thereexists an invertible C -module (respectively, C rev -module) category M (respectively, N ) such that D ∼ = M ⊠ sVec N as a C ⊠ sVec C rev -module category.The equivalence classes of module categories M and N are determined up to asimultaneous substitution by M s and N s .Proof. Note that D is a Z / Z -graded extension of D by Lemma 2.5.Let n be the number of C -module components of D . By [DGNO, Corollary 3.6]the number of C -module components of D is equal to the rank of the centralizerof C in D . The latter is C rev . Since the number of C -module components in D = C ⊠ sVec C rev is rank( C ) we conclude that n = 12 rank( C ) . Note that n is also equal to the number of C rev -module components of D .Let ⊕ ni =1 M i (respectively, ⊕ nj =1 N j ) be decompositions of D into direct sumsof indecomposable C -module (respectively, C rev -module) subcategories. In view ofProposition 5.3 it suffices to check that for some i, j the intersection M i ∩ N j is anindecomposable sVec-module category.By Proposition 3.1 we haverank( D ) ≤ rank( D ) = 12 rank( C ) = 2 n . Since D is indecomposable as a D -bimodule category each M i ∩N j , i, j = 1 , . . . , n is non-zero. If any of these intersections has rank 1, then it is sVec-indecomposable.This happens automatically if either rank( M i ) or rank( N j ) is less than 2 n for some i or j (indeed, Irr( M i ) intersects non-trivially with n disjoint sets Irr( N j ) , j =1 , . . . , n ).So let us assume that all intersections M i ∩ N j have rank ≥ M i and N j have rank ≥ n . The latter implies that rank( M i ) = rank( N j ) = 2 n andrank( M i ∩ N j ) = 2 for all i and j since otherwise rank( D ) > n × n = 2 n . Hence,rank( D ) = 2 n = rank( D ), i.e., D is a maximal rank invertible D -bimodulecategory. By Proposition 3.4 the Lagrangian algebras corresponding to D -bimodule cate-gories D and D are isomorphic as objects of Z ( D ). In particular, their forgetfulimages in D are isomorphic: M X ∈ Irr( D ) X ⊗ X ∗ ∼ = M X ∈ Irr( D ) X ⊗ X ∗ . The object on the left does not contain δ since δ acts freely on Irr( D ) by [DGNO,Lemma 3.28]. Hence, the same is true for the object on the right, i.e., δ also actsfreely on Irr( D ). Thus, every M i ∩N j is sVec-indecomposable and D = M ⊠ sVec N by Proposition 5.3.The following equivalences: D ∼ = D ⊠ D D ∼ = ( M ⊠ sVec N ) ⊠ C ⊠ sVec C rev ( M ⊠ sVec N ) ∼ = ( M ⊠ C M ) ⊠ sVec ( N ⊠ C rev N ) , imply that M ⊠ C M is equivalent to C or C s and, hence, M is invertible. Similarly, N is invertible. (cid:3) Corollary 5.7.
Let C be a slightly degenerate braided fusion category. There existan invertible C -module categories M and N such that Z ( C ) ∼ = ( C ⊠ sVec C rev ) ⊕ ( M ⊠ sVec N ) as a C ⊠ sVec C rev -module category. Remark 5.8.
It is possible to show that the above M and N are braided C -modulecategories of order 2, see [DN2]. Remark 5.9.
It will be interesting to see if C is a slightly degenerate braidedfusion category then for such an M as above C ⊕ M has a structure of a minimalextension of C . One expects that there are 16 choices of M in this case, by theresults of [BGNPRW, KLW]. Notice that if ˜ C = C ⊕ N is a minimal extension ofsome slightly degenerate braided fusion category C then Z ( C ) has the form as inCorollary 5.7, as can be seen as follows: Z ( ˜ C ) ∼ = ˜ C ⊠ ˜ C rev contains a Tannakiansubcategory D ∼ = Rep( Z / Z ) as the diagonal of sVec ⊠ sVec. The centralizer of D in Z ( ˜ C ) is ( C ⊠ C rev ) ⊕ ( N ⊠ N rev ), so that the de-equivariantization is( C ⊠ sVec C rev ) ⊕ ( N ⊠ sVec N rev ) ∼ = [ Z ( ˜ C ) Z / Z ] ∼ = Z ( C ) . References [BRWZ] P. Bonderson, E. Rowell, Z. Wang, Q. Zhang, Congruence subgroups and super-modularcategories, Pacific J. Math. (2), 257-270.[BGNPRW] P. Bruillard, C. Galindo, S.-H. Ng, J. Plavnik, E. Rowell, and Z. Wang, Classificationof super-modular categories by rank, preprint. arXiv:1705.05293.[BNRW] P. Bruillard, S.-H. Ng, E. Rowell, and Z. Wang.
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