Central Exact Sequences of Tensor Categories, Equivariantization and Applications
aa r X i v : . [ m a t h . QA ] D ec CENTRAL EXACT SEQUENCES OF TENSOR CATEGORIES,EQUIVARIANTIZATION AND APPLICATIONS
ALAIN BRUGUI`ERES AND SONIA NATALE
Abstract.
We define equivariantization of tensor categories under tensorgroup scheme actions and give necessary and sufficient conditions for an exactsequence of tensor categories to be an equivariantization under a finite group orfinite group scheme action. We introduce the notion of central exact sequenceof tensor categories and use it in order to present an alternative formulationof some known characterizations of equivariantizations for fusion categories,and to extend these characterizations to equivariantizations of finite tensorcategories under finite group scheme actions. In particular, we obtain a simplecharacterization of equivariantizations under actions of finite abelian groups.As an application, we show that if C is a fusion category and F : C → D is adominant tensor functor of Frobenius-Perron index p , then F is an equivari-antization if p = 2, or if C is weakly integral and p is the smallest prime factorof FPdim C . Introduction
In this paper we pursue the study of exact sequences of tensor categories initiatedin [3]. Exact sequences of tensor categories generalize (strict) exact sequences ofHopf algebras, due do Schneider, and in particular, exact sequences of groups.By a tensor category over a field k , we mean a monoidal rigid category ( C , ⊗ , )endowed with a k -linear abelian structure such that • Hom spaces are finite dimensional and all objects have finite length, • the tensor product ⊗ is k -linear in each variable and the unit object isscalar, that is, End( ) = k .A tensor category is finite if it is k -linearly equivalent to the category of finitedimensional right modules over a finite dimensional k -algebra.We will mostly work with finite tensor categories, with special attention to fusioncategories . A fusion category is a split semisimple finite tensor category ( split semisimple means semisimple with scalar simple objects).A tensor functor is a strong monoidal, k -linear exact functor between tensorcategories over k ; it is faithful. A tensor functor F : C → D is dominant if anyobject Y of D is a subobject of F ( X ) for some object X of C .A tensor functor F : C → D is normal if any object X of C admits a subobject X ′ ⊂ X such that F ( X ′ ) is the largest trivial subobject of F ( X ). An object is trivial if it is isomorphic to n for some natural integer n .We denote by Ker F ⊂ C the full tensor subcategory of objects X of C such that F ( X ) is trivial.Let C ′ , C , C ′′ be tensor categories over k . A sequence of tensor functors Date : September 18, 2018.1991
Mathematics Subject Classification. Dominant functors between finite tensor categories are called surjective in [6]. (1.1) C ′ i −→ C F −→ C ′′ is called an exact sequence of tensor categories if • F is dominant and normal, • i is a full embedding whose essential image is Ker F .An exact sequence of finite tensor categories C ′ −→ C F −→ C ′′ is perfect if the left(or equivalently, the right) adjoint of F is exact. Such is always the case if C ′′ is afusion category. The monadic approach.
Let F : C → C ′′ be a tensor functor between finite tensorcategories. Then F is monadic, that is, it admits a left adjoint L . The endofunctor T = F L of C ′′ is a k -linear Hopf monad on C ′′ , and C is tensor equivalent to thecategory C ′′ T of T -modules in C ′′ . Moreover, F is dominant if and only if T isfaithful, and F is normal if and only if T ( ) is trivial, in which case T is said to benormal.Via this construction, exact sequences of finite tensor categories C ′ i −→ C F −→ C ′′ are classified by k -linear right exact faithful normal Hopf monads on C ′′ [3, Theorem5.8]. Examples.
Any (strictly) exact sequence of Hopf algebras H ′ → H → H ′′ overa field k in the sense of Schneider [16] gives rise to an exact sequence of tensorcategories of finite dimensional comodules:( E S ) comod- H ′ → comod- H → comod- H ′′ , and if H is finite-dimensional we also have an exact sequence of tensor categoriesof finite dimensional modules:mod- H ′′ → mod- H → mod- H ′ . Equivariantization is another source of examples. Let G be a finite group actingon a tensor category D by tensor autoequivalences. Then the equivariantization D G is a tensor category and the forgetful functor D G → D gives rise to a (perfect)exact sequence of tensor categories(1.2) rep G → D G → D , see [3, Section 5.3]. This is extended in Section 3.2 to the case where G is a finitegroup scheme. If D is a fusion category, k is algebraically closed and the order of G is not a multiple of char( k ), then (1.2) is an exact sequence of fusion categories.A tensor functor F : C → D is an equivariantization if there is an action of afinite group scheme G on D and a tensor equivalence C ≃ D G such that the triangleof tensor functors C ≃ / / % % ▲▲▲▲ D G y y rrr D commutes up to a k -linear monoidal isomorphism.An exact sequence of tensor categories C ′ i −→ C F −→ C ′′ is called an equivari-antization exact sequence if it is equivalent to an exact sequence defined by anequivariantization, or equivalently, if F is an equivariantization.A braided exact sequence is an exact sequence of tensor categories where all cat-egories and functors are braided. If C ′ → C → C ′′ is a braided exact sequence, then C ′ is a subcategory of the category T ⊂ C of transparent objects of C (see [1]). Wesay that C ′ → C → C ′′ is a modularization exact sequence if C ′′ is modular, that is, ENTRAL EXACT SEQUENCES OF TENSOR CATEGORIES 3 if all transparent objects of C ′′ are trivial. In that case C ′ = T . Examples of mod-ularization exact sequences of fusion categories arise through the modularizationprocedures introduced in [1, 10]. Equivariantization criteria.
Generalizing a result of [3], we show that an exactsequence of finite tensor categories C ′ → C → C ′′ is an equivariantization exactsequence if and only if the associated normal Hopf monad is exact and cocommuta-tive in the sense of [3]. If C ′ is finite and k is an algebraically closed field such thatchar( k ) does not divide dim C ′ , then the corresponding group scheme is discrete sothat we have an equivariantization in the usual sense.In particular, any perfect braided exact sequence of finite tensor categories is anequivariantization exact sequence.On the other hand, [7, Proposition 2.10](i) affirms that a fusion category C isan equivariantization under the action of a finite group G if there is a full braidedembedding j of the category rep G into into the Drinfeld center Z ( C ) of C , suchthat U j : rep G → C is full, where U denotes the forgetful functor Z ( C ) → C . Seealso [5, Theorem 4.18].In order to unify those two result, we introduce the notion of central exactsequence of tensor categories. An exact sequence of finite tensor categories C ′ i −→ C F −→ C ′′ is central if, denoting by ( A, σ ) its central commutative algebra, the tensor functor i : C ′ → C lifts to a tensor functor ˜ i : C ′ → Z ( C ) such that ˜ i ( A ) = ( A, σ ). Such alift, if it exists, is essentially unique.We show that an exact sequence of finite tensor categories is central if and onlyif its normal Hopf monad is cocommutative. We give two proofs of this result.The first one works in the fusion case. In that situation our characterizationis a reformulation in terms of exact sequences of the characterization of equivari-antizations given in [7, Proposition 2.10], [5, Theorem 4.18]. However our proofis organized differently, and boils down to showing that a central exact sequenceof fusion categories is ‘dominated’ in a canonical way by a modularization exactsequence, thus: C ′ / / = (cid:15) (cid:15) C Z ( C ) ( C ′ ) / / (cid:15) (cid:15) Z ( C ′′ ) (cid:15) (cid:15) C ′ / / C / / C ′′ , where C Z ( C ) ( C ′ ) denotes the centralizer of C ′ viewed as a fusion subcategory of Z ( C ),and that an exact sequence dominated by an equivariantization exact sequence isitself an equivariantization exact sequence.The second one works for finite tensor (not necessarily semisimple) categoriesand actions of finite group schemes, and relies on the construction of the doubleof a Hopf monad in [4]. It is based on the existence of a commutative diagram oftensor categories C ′ / / = (cid:15) (cid:15) Z ( C ) / / (cid:15) (cid:15) Z F ( C ′′ ) (cid:15) (cid:15) C ′ / / C F / / C ′′ , whose first line is an exact sequence of tensor categories, Z F ( C ′′ ) denoting the centerof C ′′ relative to the functor F . ALAIN BRUGUI`ERES AND SONIA NATALE
Application.
The
Frobenius-Perron index of a dominant tensor functor F : C → D between fusion categories is defined in [3] to be the ratioFPind( F ) = FPind( C : D ) = FPdim C FPdim D . It is an algebraic integer by [6, Corollary 8.11]. According to [3, Proposition 4.13],a dominant tensor functor F of Frobenius-Perron index 2 is normal. In this paperwe prove the following refinement of this result: Theorem 6.1.
Let F : C → D be a dominant tensor functor between fusion cate-gories over a field of characteristic . If FPind( C : D ) = 2 , then F is an equivari-antization. This generalizes the fact that that subgroups of index 2 are normal. An analoguein the context of finite dimensional semisimple Hopf algebras was proved in [9,Proposition 2], [13, Corollary 1.4.3]. We also generalize the fact that subgroups ofa finite group whose index is the smallest prime factor of the order of the largergroup are normal. Recall that a fusion category is weakly integral if its Frobenius-Perron dimension is a natural integer.
Theorem 6.2.
Let F : C → D be a dominant tensor functor between fusion cate-gories over a field of characteristic . Assume that FPdim C is a natural integer,and that FPind( C : D ) is the smallest prime number dividing FPdim C . Then F isan equivariantization. In particular under the hypotheses of Theorem 6.2 the functor F is normal. Ananalogue in the context of semisimple Hopf algebras was proved in [9, Proposition2], [13, Corollary 1.4.3]. Observe that Theorem 6.2 gives some positive evidence infavor of the conjecture that every weakly integral fusion category is weakly group-theoretical [7].Regarding the ‘dual’ situation, namely, when C is a weakly integral fusion cate-gory and D ⊆ C is a full fusion subcategory such that the quotient
FPdim C FPdim D is thesmallest prime factor of FPdim C , it may be the case that D is not normal in C ;we give an example of this where p = 2 and C is a Tambara-Yamagami category(see Proposition 6.5). This actually provides examples of simple fusion categoriesof Frobenius-Perron dimension 2 q , where q is an odd prime number. Organization of the text.
In Section 2 we introduce central exact sequences oftensor categories. We also discuss dominant tensor functors on weakly integral fu-sion categories in terms of induced central algebras and show that this class of fusioncategories is closed under extensions; see Corollary 2.13. We give a general criterionfor an exact sequence to be an equivariantization exact sequence in Section 3. Inorder to do so, we generalize the notion of equivariantization to tensor actions ofgroup schemes on tensor categories in Section 3.2. The main results, Theorems 3.5and 3.6, assert that equivariantization exact sequences coincide with central exactsequences, and also with exact sequences whose Hopf monad is exact cocommuta-tive. It is proved in Section 5 using the notion of double of a Hopf monad. We applythis characterization in Section 4 to several special cases: braided exact sequencesof tensor categories, exact sequences of fusion categories, equivariantizations underthe action of abelian groups. Lastly we prove Theorems 6.1 and 6.2 in Section 6.
Conventions and notation.
We retain the conventions and notation of [3].If C is a monoidal category and A is an algebra in C , we denote by C A the categoryof right A -modules in C , and by F A : C 7→ C A the free A -module functor, defined by X X ⊗ A . If C is additive, so is C A . In that case, we say that A is self-trivializing if F A ( A ) ≃ F A ( ) n for some natural integer n . ENTRAL EXACT SEQUENCES OF TENSOR CATEGORIES 5
Let C be a tensor category over a field k , and let X be a object, or a set ofobjects of C . We denote by hX i the smallest full replete tensor subcategory of C containing X . Its objects are the subquotients of finite direct sums of tensorproducts of elements of X and their duals. We denote by the unit object of C .The tensor subcategory h i is the category of trivial objects of C and it is tensorequivalent to vect k .Given an object X of C , we denote by ∨ X and X ∨ the left dual and the right dualof X respectively. An object X of C is called invertible if there exists an object Y of C such that X ⊗ Y ≃ ≃ Y ⊗ X . In that case Y ≃ ∨ X ≃ X ∨ . Invertible objects of C are both simple and scalar. We denote by Pic( C ) the set of isomorphism classesof invertible objects of C ; it is a group for the tensor product, called the Picardgroup of C . We set C pt = h Pic( C ) i ⊂ C .Now assume C is a fusion category. The multiplicity of a simple object X in anobject Y of C is defined as m X ( Y ) = dim Hom C ( X, Y ). We have Y ≃ ⊕ X ∈ Irr( C ) X m X ( Y ) , where Irr( C ) denotes the set of simple objects of C up to isomorphism. An object X of C is invertible if and only if its Frobenius-Perron dimension is 1.2. Central exact sequences of tensor categories
Tensor functors.
Let F : C → D be a tensor functor between tensor cate-gories over k . We denote by Ker F ⊂ C the full subcategory of C of objects c suchthat F ( c ) is trivial in D . The category Ker F is a tensor category over k , and it isendowed with a fibre functor ω F (cid:26) Ker F → vect k x Hom( , F ( x ))By Tannaka reconstruction, this defines a Hopf algebra H = coend( ω F ) = Z x ∈ Ker F ω ( x ) ∨ ⊗ ω ( x )such that Ker F ≃ comod- H .The tensor functor F admits a left adjoint L if and only if it admits a rightadjoint R ; if they exist, the adjoints of F are related by R ( X ) = L ( ∨ X ) ∨ . If C isfinite, then F admits adjoints.If the tensor functor F admits adjoints, we say that F is perfect if its left, orequivalently, its right adjoint is exact. Such is always the case if D is a fusioncategory.The tensor functor F is dominant if for any object d of D , there exists an object c of C such that d is a subobject of F ( c ). It is normal if for any object c of C , thereexists a subobject c ⊂ c such that F ( c ) is the largest trivial subobject of F ( c ).If F admits adjoints L and R , then F is dominant if and only if L , or equivalently R , is faithful, and F is normal if and only if L ( ), or equivalently R ( ), is a trivialobject of C .If F is a normal tensor functor, the Hopf algebra H = coend( ω F ) is called the induced Hopf algebra of F .2.2. Central induced (co)algebras.
Let C and D be finite tensor categories over k and let F : C → D be a dominant tensor functor. Then F admits a left adjoint L : D → C which is faithful and comonoidal. Consequently ˆ C = L ( ) is a coalgebrain C , called the induced coalgebra of F , with coproduct L ( , ) and unit L , where( L , L ) denotes the comonoidal structure of L . We have Hom C ( ˆ C, ) ≃ k . Inaddition, ˆ C is endowed with a canonical half-braiding ˆ σ : ˆ C ⊗ id C → id C ⊗ ˆ C , whichmakes it a cocommutative coalgebra in the center Z ( C ) of C ; see [2] for details of ALAIN BRUGUI`ERES AND SONIA NATALE this construction. The cocommutative coalgebra ( ˆ C, ˆ σ ) is called the induced centralcoalgebra of F . Dually, under the same hypotheses the functor F also admits a right adjoint R ,related to L by R ( X ) = L ( ∨ X ) ∨ . The functor R is faithful and monoidal. As aresult, A = R ( ) = ˆ C ∨ is an algebra in C , called the induced algebra of F , and it isendowed a canonical half-braiding σ : A ⊗ id C → id C ⊗ A , making it a commutativealgebra in Z ( C ). The commutative algebra ( A, σ ), which is the right dual of ( ˆ C, ˆ σ ),is called the induced central algebra of F . See [3, Section 6].The category C A = C ( A,σ ) of right A -modules in C is an abelian k -linear monoidalcategory over k with tensor product induced by ⊗ A and the half-braiding σ , andthe functor F ( A,σ ) : C → C A , F ( A,σ ) ( X ) = X ⊗ A , is strong monoidal and k -linear.If F is dominant and perfect (that is, R is faithful exact), then by [3, Proposition6.1] C A is a tensor category, and there is a tensor equivalence κ : D → C A such thatthe following diagram of tensor functors commutes up to tensor isomorphims: C F / / F A & & ▲▲▲▲▲▲▲▲ D κ (cid:15) (cid:15) C A Note that if F is dominant and D is a fusion category, then R is exact, so C A ≃ D is a fusion category and in that case, A is semisimple. Lemma 2.1.
Let F : C → D be a tensor functor between finite tensor categories,with induced algebra A . The following assertions are equivalent:(i) F is normal;(ii) A belongs to Ker F ;(iii) A is a self-trivializing algebra, that is, A ⊗ A ≃ A n in C A for some integer n .Moreover if these hold, the integer n of assertion (iii) is the dimension of the inducedHopf algebra of F .Proof. Since A = R ( ), we have (i) ⇐⇒ (ii). If A is in Ker F , then F A ( A ) ≃ κF ( A ) ≃ κF ( ) n ≃ F A ( ) n , i.e. A is self-trivializing, so (ii) = ⇒ (iii). Conversely,assume A is self-trivializing, that is, A ⊗ A ≃ A n as right A -modules. By adjunc-tion, Hom D ( F ( A ) , ) ≃ Hom C ( A, R ( )) = Hom C ( A, A ) ≃ Hom C A ( F A ( A ) , F A ( )) ≃ Hom C A ( F A ( ) n , F A ( )) ≃ Hom C ( , A ) n ≃ k n . Thus, there exists an epimorphism s : F ( A ) → n in D . Now s ⊗ F ( A ) is an epimorphism F ( A ) ⊗ F ( A ) → F ( A ) n ,and since those two objects are isomorphic and of finite length, s ⊗ F ( A ) is anisomorphism. Thus ker( s ) ⊗ F ( A ) = 0, so ker( s ) = 0 and s is an isomorphism F ( A ) ∼ −→ n . This shows (iii) = ⇒ (ii).If the assertions of the lemma hold, then F ( A ) = F R ( ) ≃ F ( L ( ) ∨ ) ≃ F ( L ( )) ∨ .Hence ∨ F ( A ) ≃ H ⊗ , and we get n = dim H . (cid:3) Remark . A terminological summary might help: if F : C → D is a tensorfunctor between finite tensor categories, the induced central algebra A = ( A, σ ) of F is a commutative algebra in Z ( C ); the induced algebra A of F is an algebra in C .If F is normal, with induced Hopf algebra H = coend( ω F ), then F ( A ) ≃ H ⊗ and,denoting by T = F L the Hopf monad of F , we have T ( ) ≃ K ⊗ with K = H ∗ .2.3. Exact sequences of tensor categories.
In this section we recall some basicfacts about exact sequences of tensor categories which we will use throughout thispaper, see [3] for details. Let k be a field.An exact sequence of tensor categories over k is a diagram of tensor functors( E ) C ′ i −→ C F −→ C ′′ ENTRAL EXACT SEQUENCES OF TENSOR CATEGORIES 7 between tensor categories C ′ , C , C ′′ over k , such that F is normal and dominant , i ( C ′ ) ⊂ Ker F and i induces a tensor equivalence C ′ → Ker F .The induced Hopf algebra H = coend( ω F ) of F is also called the induced Hopfalgebra of ( E ) , and we have an equivalence of tensor categories C ′′ ≃ comod- H . By[3, Proposition 3.15], the induced Hopf algebra of ( E ) is finite dimensional if andonly if the tensor functor F has a left adjoint, or equivalently, a right adjoint. Inthat case, we say that ( E ) is perfect if F is perfect, that is, R (or F ) is exact.If ( E ) = ( C ′ → C → C ′′ ) and ( E ) = ( C ′ → C → C ′′ ) are two exact sequencesof tensor categories over k , a morphism of exact sequences of tensor categories from ( E ) to ( E ) is a diagram of tensor functors: C ′ / / (cid:15) (cid:15) C / / (cid:15) (cid:15) C ′′ (cid:15) (cid:15) C ′ / / C / / C ′′ which commutes up to tensor isomorphisms. Such a morphism induces a mor-phism of Hopf algebras w : H → H , where H and H denote the induced Hopfcoalgebras of ( E ) and of ( E ), respectively.A morphism of exact sequences of tensor categories is an equivalence of exactsequences of tensor categories if the vertical arrows are equivalences. Lemma 2.3.
Consider a morphism of exact sequences of finite tensor categories ( E ) (cid:15) (cid:15) C ′ i / / U ′ (cid:15) (cid:15) C F / / U (cid:15) (cid:15) C ′′ U ′′ (cid:15) (cid:15) ( E ) C ′ i / / C F / / C ′′ . Denote by H , H the induced Hopf algebras of F and F respectively.(1) The morphism of exact sequences induces a Hopf algebra morphism φ : H → H , and U ′ is dominant ( respectively, an equivalence) if and only if φ is a monomorphism ( respectively, and isomorphism);(2) If U ′ and U ′′ are dominant, then so is U .Proof.
1) We may assume C ′ = comod- H , C ′ = comod- H , U ′ being compatiblewith the forgetful functors. Then U ′ is of the form φ ∗ , for some Hopf algebramorphism φ : H → H , U ′ is an equivalence if and only if φ is an isomorphism,and by [3, Remark 3.12], U ′ is dominant if and only if φ is surjective.2) Let X be an object of C . We are to show that X is a subobject of U ( Y )for some Y ∈ Ob( C ). Now F U = U ′′ F is dominant, so F ( X ) ⊂ F U ( Z ) forsome Z ∈ Ob( C ). Denote by R the right adjoint of F ; being a right adjoint, itpreserves monomorphisms so R F ( X ) ⊂ R F U ( Z ). We have A = R ( ), andwe have an isomorphism RF ≃ A ⊗ ? coming from the fact that the adjunction( F , R ) is a Hopf monoidal adjunction (see [3, Proof of Proposition 6.1], and [2] forthe original statement in terms of Hopf monads).Thus A ⊗ X ⊂ A ⊗ U ( Z ). Since we have ⊂ A , we obtain X ⊂ A ⊗ U ( Z ).On the other hand, A belongs to the essential image of i and U ′ is dominant,so there exists Z ′ ∈ Ob( C ′ ) such that A ⊂ i U ′ ( Z ′ ) ≃ U i ( Z ′ ). Consequently X ⊂ U ( i ( Z ′ ) ⊗ Z ), which shows that U is dominant, as claimed. (cid:3) Central exact sequences of finite tensor categories. If i : C ′ → C isa strong monoidal functor between monoidal categories, a central lifting of i is astrong monoidal functor ˜ i : C ′ → Z ( C ) such that U ˜ i = i , where U denotes the ALAIN BRUGUI`ERES AND SONIA NATALE forgetful functor Z ( C ) → C . Note that if i is full, given a central lifting ˜ i thereexists a unique braiding on C ′ such that ˜ i is braided.An exact sequence of finite tensor categories C ′ i −→ C F −→ C ′′ , with inducedcentral algebra A = ( A, σ ), is called central if the restriction of the forgetful functor U : Z ( C ) → C induces an equivalence of categories h A i → h A i . Theorem 2.4.
Consider an exact sequence of finite tensor categories ( E ) C ′ i −→ C F −→ C ′′ with induced central algebra ( A, σ ) and induced central coalgebra ( ˆ C, ˆ σ ) . The fol-lowing assertions are equivalent:(i) The exact sequence ( E ) is central;(ii) There exists a central lifting ˜ i of i such that ˜ i ( A ) = ( A, σ ) ;(iii) There exists a central lifting ˜ i of i such that ˜ i ( ˆ C ) = ( ˆ C, ˆ σ ) .Moreover, if these assertions hold, the central liftings ˜ i of i appearing in assertions(ii) and (iii) are essentially unique and they coincide. The central lifting ˜ i is called the canonical central lifting of the central exactsequence ( E ) .Proof. Assertions (ii) and (iii) are equivalent because (
A, σ ) is the right dual of( ˆ C, ˆ σ ) and strong monoidal functors preserve duals.Denote by j the tensor functor h ( A, σ ) i → h A i induced by the forgetful functor U . In particular j ( A ) = A .We have (ii) = ⇒ (i) because if ˜ i is a central lifting of i such that ˜ i ( A ) = ( A, σ ),then ˜ i ( h A i ) ⊂ h ( A, σ ) i and by definition of a central lifting, j ˜ i = id h A i . This showsthat j is full and essentially surjective; since on the other hand j is faithful, it istherefore an equivalence, with quasi-inverse ˜ i .We have (i) = ⇒ (ii) because if j is an equivalence, then it admits a quasi-inverse k , which is also a tensor functor. One defines k by picking for each object X of h A i an object k ( X ) in Z ( C ) such that U k ( X ) ≃ X . One may further impose that U k ( X ) = X , and k ( A ) = ( A, σ ). Then ˜ i : h A i → Z ( C ) , X k ( X ) is a centrallifting of i sending A to ( A, σ ), which proves assertion (ii).If ˜ i exists, it is a quasi-inverse of j and as such, it is essentially unique. (cid:3) Example 2.5. If G is a finite group acting on a tensor category C by tensorautoequivalences, then the corresponding exact sequence of tensor categoriesrep G i −→ C G F −→ C is central. Indeed, if (
V, r ) is a representation of G and ( X, ρ ) an object of C G ,one verifies that the trivial isomorphism V ⊗ X ∼ −→ X ⊗ V lifts to an isomorphism σ ( V,r ) , ( X,ρ ) = i ( V, r ) ⊗ ( X, ρ ) ≃ ( X, ρ ) ⊗ i ( V, r ) in C G , and this defines a centrallifting ˜ i : rep G → Z ( C G ) of i , ( V, r ) (( V, r ) , σ ( V,r ) , − ). Moreover, the inducedcentral algebra ( A, σ ) of F is defined by A = k G , with G -action defined by righttranslations, and σ = σ A, − , hence centrality of the exact sequence.If C ′ → C → C ′′ is a central exact sequence of tensor categories, then C ′ issymmetric. More precisely, we have the following lemma. Proposition 2.6.
Consider a central exact sequence ( E ) C ′ i −→ C F −→ C ′′ , with canonical central lifting ˜ i . Then the induced Hopf algebra H of ( E ) is commuta-tive, so that C ′ ≃ comod - H is endowed with a symmetry, and with this symmetry on ENTRAL EXACT SEQUENCES OF TENSOR CATEGORIES 9 C ′ , the tensor functor ˜ i : C ′ → Z ( C ) is braided. If in addition H is split semisimple,then G = Spec H is a discrete finite group and C ≃ rep G .Remark . As a special case of Proposition 2.6, for any finite dimensional Hopfalgebra H the corresponding exact sequencecomod- H → comod- H → vect k is central if and only if H is commutative. Remark . Note that if C ′ → C → C ′′ is a central exact sequence of fusioncategories over an algebraically closed field of characteristic 0, G = Spec H is adiscrete finite group and we have C ′ ≃ rep G . Proof of 2.6.
Let (
A, σ ) be the induced central algebra of ( E ). We may replace ( E )with the equivalent exact sequence Ker F = h A i −→ C −→ C ′′ . Consider the morphism of exact sequences of tensor categories: h A i / / = (cid:15) (cid:15) h A i / / incl. (cid:15) (cid:15) h i incl. (cid:15) (cid:15) h A i / / C / / C ′′ Denote by ( E ) the top exact sequence in this diagram. Its induced central algebrais ( A, σ |h A i ). Moreover it is central, with canonical lifting ˜ i defined by ˜ i ( X ) =( X, s X |h A i ), where ˜ i ( X ) = ( X, s X ) denotes the canonical lifting of ( E ).Thus, it is enough to prove the theorem for ( E ). We may again replace ( E ) bythe equivalent exact sequencecomod- H → comod- H → vect k , where H is the induced Hopf algebra of ( E ), which is the situation of Remark 2.7.In that situation, we have A = ( H, ∆) and the half-braiding σ is defined by(2.1) σ V : H ⊗ V → V ⊗ Hh ⊗ v v (0) ⊗ S ( v (1) ) hv (2) in Sweedler’s notation, for any finite dimensional right H -comodule V (see [3, Ex-ample 6.3]).Centrality of the exact sequence means that we have a central lifting of theidentity of comod- H , which is nothing but a braiding c on comod- H , and in additionthis braiding is required to be such that c A, ? = σ . What we have to prove is that H is commutative and c is the standard symmetry.Let r : H ⊗ H → k be the coquasitriangular structure corresponding to thisbraiding, so that for any pair of right comodules V and W , we have(2.2) c V,W : V ⊗ W → W ⊗ V, c
V,W ( v ⊗ w ) = r ( v (1) , w (1) ) w (0) ⊗ v (0) . Comparing 2.1 and 2.2, we obtain by a straightforward computation that thecondition c A,A = σ A implies r = ε ⊗ ε . That means that the forgetful functorcomod- H → vect k is braided, so H is commutative and c is the standard symme-try. This concludes the proof of the proposition. (cid:3) Normality and centrality criteria.
The following theorem gives a suffi-cient condition for a dominant tensor functor between finite tensor categories to benormal in terms of the induced central algebra.
Theorem 2.9.
Let F : C → D be a dominant tensor functor between finite ten-sor categories C , D , and let ( A, σ ) be its induced central algebra. Assume that A decomposes as a direct sum of invertible objects of C . Then:(i) The functor F is normal, the isomorphism classes of simple direct summandsof A form a group Γ , and we have an exact sequence of tensor categories ( E ) Γ - vect −→ C F −→ D , where Γ - vect denotes the tensor category of finite dimensional Γ -graded vectorspaces.(ii) If in addition ( A, σ ) decomposes as a direct sum of invertible objects in Z ( C ) ,then the exact sequence ( E ) is central.Proof. An invertible object in a tensor category is both simple and scalar. Let R denote the right adjoint of F . For any invertible object g of C , we have byadjunction dim Hom C ( g, A ) = dim Hom D ( F ( g ) , ), because A = R ( ). Now F ( g )is invertible in D , so dim Hom C ( g, A ) = 1 if F ( g ) ≃ , and dim Hom C ( g, A ) = 0otherwise.In other words: (1) the invertible factors of A are exactly the invertible objectsof C which are trivialized by F ; therefore, they form a group for the tensor product;and (2) their multiplicity in A is exactly one.In particular if A is the direct sum of its invertible factors, then A itself istrivialized by F , that is, F is normal. In that case, we have an exact sequence Ker F → C → D . Now Ker F is the tensor subcategory of C generated by A ; it isa pointed tensor category whose invertible objects are the invertible factors of A ,whose isomorphism classes form a group Γ. So Ker F is a pointed tensor category; inaddition, Ker F admits a fiber functor, hence it is tensor equivalent to the categoryΓ- vect. This proves assertion (i).Now assume that ( A, σ ) decomposes as a direct sum of invertible objects of Z ( C ),that is, ( A, σ ) = L ni =1 ( g i , σ i ). Then A = L ni =1 g i , where the g i ’s are invertible in A , so that the first part of the theorem applies. The category h ( A, σ ) i is generatedas a tensor category by the ( g i , σ i ). Let us show that h ( A, σ ) i is additively generatedby the ( g i , σ i ). For this it is enough to show that ( g i , σ i ) ⊗ ( g j , σ j ) is a direct factorof ( A, σ ) for all i, j ∈ { , . . . n } . Now the product µ : A ⊗ A → A embeds g i ⊗ g j into A , and lifts to a morphism in Z ( C ), namely the product of ( A, σ ); consequentlyit embeds ( g i , σ i ) ⊗ ( g j , σ j ) into ( A, σ ). Thus h ( A, σ ) i → h A i is full, which provesthat ( E ) is central. (cid:3) Example: Tambara-Yamagami categories.
A Tambara-Yamagami cate-gory is a fusion category having exactly one non-invertible simple object X , withthe additional condition that X is not a factor of X ⊗ X . These categories, whichare in a sense the simplest non-pointed categories, have been classified in [17].Let T Y be a Tambara-Yamagami category. Denote by Γ the Picard group of
T Y . It is a finite abelian group. Denote by X a non-invertible simple object. Themaximal pointed fusion subcategory of C , denoted by C pt , is tensor equivalent tothe category Γ- vect of finite dimensional Γ-graded vector spaces. The followingproposition characterizes normal tensor functors on C . Proposition 2.10.
Let
T Y be a Tambara-Yamagami category, with Picard group Γ , and let F : C → D be a dominant tensor functor, with induced central algebra ( A, σ ) . Then F is normal if and only if F is a fiber functor, or A belongs to C pt . ENTRAL EXACT SEQUENCES OF TENSOR CATEGORIES 11
In the latter case, we have an exact sequence of tensor categories: G - vect −→ T Y F −→ D , where G is a subgroup of Γ .Proof. The only proper fusion subcategories of C are those contained in C pt . Thisshows the ‘only if’ direction. Conversely, any fiber functor F on C is normal, with Ker F = C . Suppose on the other hand that F is not a fiber functor. Then F isnormal by Theorem 2.9. This finishes the proof of the proposition. (cid:3) It is known that if a Tambara-Yamagami category
T Y admits a fiber functor, sothat
T Y ≃ rep H for some semisimple Hopf algebra H , then H fits into an abelianexact sequence of Hopf algebras k Z → H → k Γ [12]. Hence in this case
T Y fitsinto an exact sequence of fusion categories rep Γ → T Y → rep Z . In particular, T Y is not simple.2.7.
Extensions of weakly integral fusion categories.
Recall that a fusioncategory is weakly integral if its Frobenius-Perron dimension is a natural integer.In this section we discuss dominant tensor functors on weakly integral fusion cate-gories.
Lemma 2.11.
Let F : C → D be a dominant tensor functor between fusion cate-gories, with induced central algebra ( A, σ ) . Then we have:(i) FPind( C : D ) = FPdim A .(ii) If X ∈ Irr( C ) , then m X ( A ) ≤ FPdim X .(iii) F is normal if and only if for all X ∈ Irr( C ) we have m X ( A ) = 0 or m X ( A ) = FPdim X .Proof. By [3, Proposition 4.3], we have FPind( C : D ) = FPdim R ( ). This proves(i) since A = R ( ).By adjunction, we have Hom D ( F ( X ) , ) ≃ Hom C ( X, A ), so that m X ( A ) = m ( F ( X )). This implies (ii) since m ( F ( X )) ≤ FPdim F ( X ) = FPdim X .We next show (iii). The only if part follows from [3, Proposition 6.9]. Conversely,suppose that for all X ∈ Irr( C ) we have m ( F ( X )) = m X ( A ) ∈ { , FPdim X } .Let X ∈ Irr( C ) and assume m ( F ( X )) = 0. Then m ( F ( X )) = FPdim X =FPdim( F ( X )), so F ( X ) is trivial. Thus F is normal, which completes the proof of(iii) and of the lemma. (cid:3) Proposition 2.12.
Let F : C → D be a dominant tensor functor between fusioncategories C and D and let ( A, σ ) be the induced central algebra of F . Then thefollowing assertions are equivalent:(i) C is weakly integral.(ii) D is weakly integral and FPdim A ∈ Z .Proof. (ii) ⇒ (i) results immediately from Lemma 2.11 (i).(i) ⇒ (ii). Notice first that since FPdim A FPdim D = FPdim C and FPdim A isan algebraic integer, it is enough to verify that FPdim D is a natural integer, thatis, D is weakly integral.Recall that D is tensor equivalent to the fusion category C A of right A -modulesin C . Since A is an indecomposable algebra in C , the category A C A of A -bimodulesin C is a fusion category and it satisfies FPdim A C A = FPdim C [6, Corollary 8.14].Therefore A C A is weakly integral. We have a full tensor embedding C A ⊂ A C A .Now, in a weakly integral fusion category the Frobenius-Perron dimensions ofsimple objects are square roots of natural integers [6, Proposition 8.27], and asa result, a full fusion subcategory of a weakly integral fusion category is weaklyintegral. So D ≃ C A is weakly integral, and we are done. (cid:3) In the case where the functor F is normal, we have FPdim A = FPdim Ker F ,and since Ker F admits a fibre functor it is weakly integral. Thus we have: Corollary 2.13.
Let C ′ → C → C ′′ be an exact sequence of fusion categories. Then C is weakly integral if and only if C ′′ is weakly integral. In particular, the class ofweakly integral fusion categories is closed under extensions. (cid:3) Equivariantization revisited
The aim of this section is to state and discuss equivariantization criteria. In orderarrive at a synthetic statement, we have to extend the notion of equivariantizationto actions of finite group schemes. Thanks to this generalization, we can statethat an exact sequence of finite tensor categories is central if and only if its Hopfmonad is normal cocommutative, and that it is an equivariantization exact sequenceif and only if it is perfect and central, which extends a result of [3] concerningdiscrete groups, and also reformulates and extends a result of [7] concerning fusioncategories.3.1.
Cocommutative normal Hopf monads.
Let C be a tensor category. A k -linear right exact normal Hopf monad T on C is cocommutative (see [3]) if for anymorphism x : T ( ) → and any object X of C ( x ⊗ T X ) T ( , X ) = ( T X ⊗ x ) T ( X, ) . Note that, if V is a trivial object, and X is an arbitraty object of C , there is acanonical isomorphism τ V,X : V ⊗ X ∼ −→ X ⊗ V , which is characterized by the factthat for all x : V → , we have ( X ⊗ x ) τ V,X = x ⊗ X . The natural isomorphism τ V, − is a half-braiding, called the trivial half-braiding of V . We have the following characterizations of normal cocommutative Hopf monads.
Lemma 3.1.
Let C be a tensor category and let T be a normal Hopf monad on C ,with induced central coalgebra ( ˆ C, ˆ σ ) . The following assertions are equivalent:(i) T is cocommutative;(ii) T ( X, ) = τ T ,T X T ( , X ) for X in C ;(iii) ˆ σ ( M,r ) = τ T ,M for ( M, r ) in C T , or in short: ˆ σ ‘is the trivial half-braiding’.Proof. Assertion (ii) is just a reformulation of the definition of cocommutativity interms of trivial half-braidings, so (i) ⇐⇒ (ii). Let ( M, r ) be a T -module. Since T is a Hopf monad, we have fusion isomorphismsΦ r ( M,r ) = ( T ⊗ r ) T ( , M ) : T M ∼ −→ T ⊗ M, Φ l ( M,r ) = ( r ⊗ T ) T ( M, ) : T M ∼ −→ M ⊗ T , and by definition ˆ σ = Φ l Φ r − . If (ii) holds, we have Φ l ( M,r ) = τ T ,M Φ r ( M,r ) byfunctoriality of τ , so ˆ σ ( M,r ) = τ T ,M , which shows (ii) = ⇒ (iii). Conversely,applying (iii) to ( M, r ) = (
T X, µ X ) and composing on the right by T ( η X ) gives T ( X, ) = τ T ,T X T ( , X ), so (iii) = ⇒ (ii). (cid:3) From [3, Theorem 5.21 and Theorem 5.24], one deduces immediately
Proposition 3.2.
A dominant tensor functor F : C −→ D between finite tensorcategories is an equivariantization under the action of a finite group G if and onlyif the following two conditions are met:(1) the Hopf monad T of F is normal and cocommutative;(2) the induced Hopf algebra H of F is split semisimple.If these conditions hold, then F is perfect, that is T is exact, and G = Spec( H ) . ENTRAL EXACT SEQUENCES OF TENSOR CATEGORIES 13
This suggests that a (perfect) dominant tensor functor between finite tensorcategories is an equivariantization under the action of a finite group scheme ifand only if its Hopf monad is normal cocommutative; the group scheme beingthe spectrum of the induced Hopf algebra of T - which, in this case, is a finitedimensional commutative Hopf algebra.We will now define group scheme actions in order to give a mathematical meaningto this claim, and then prove it.3.2. Group scheme actions.
Let A be a monoidal category and let M be acategory. An action of A on M is a strong monoidal functor ρ : A →
End( M ),where End( M ) denotes the strict monoidal category of endofunctors of M . Givensuch an action ρ , we say that M is an A -module category, and we usually write ρ ( a, m ) = a ⊙ m . Let M , M ′ be two A -module categories. A functor of A -modulecategories M → M ′ is a pair ( F, F ), where F : M → M ′ is a functor and F is anatural isomorphism F ( a, m ) : a ⊙ F ( m ) ∼ −→ F ( a ⊙ m ), a ∈ A , m ∈ M , such thatthe following diagrams commute: F (( a ⊗ b ) ⊙ m ) ≃ / / F ( a ⊗ b,m ) (cid:15) (cid:15) F ( a ⊙ ( b ⊙ m )) F ( a,b ⊙ m ) (cid:15) (cid:15) a ⊙ F ( b ⊙ m ) a ⊙ F ( b,m ) (cid:15) (cid:15) ( a ⊗ b ) ⊙ F ( m ) ≃ / / a ⊙ ( b ⊙ F ( m )) F ( ⊙ m ) ≃ & & ▼▼▼▼▼▼▼ F ( ,m ) (cid:15) (cid:15) F ( m ) ⊙ F ( m ) ≃ qqqqqqq where the unlabeled isomorphisms come from the monoidal structure of the action.Now assume A is endowed with a strong monoidal functor ω : A → vect k , M is a k -category, and ρ is an action of A on M by k -linear endofunctors. The equivariantization of M under the action ρ is the category M ρ , also denoted by M A , defined as follows. Objects of M ρ are data ( m, α ) where m is an object of M and α = ( α λc ) c ∈ Ob( C ) ,λ ∈ ω ( c ) ∗ is a family of morphisms α λc : c ⊙ m → m satisfyingthe following conditions:(1) functoriality : α λc is linear in λ , and if f : c → c ′ is a morphism in C and λ ∈ ω ( c ′ ) ∗ , then α λc ′ ( f ⊙ m ) = α f ∗ λc , where f ∗ λ = λω ( f );(2) ρ -compatibility: we have commutative diagrams( a ⊗ b ) ⊙ m ≃ (cid:15) (cid:15) α λ ⊗ µa ⊗ b / / m ⊗ m α ω / / ≃ " " ❉❉❉❉❉❉❉❉❉❉ ma ⊙ ( b ⊙ m ) a ⊙ α µb / / a ⊙ m α λa O O m = O O with a , b objects of A and λ ∈ ω ( a ) ∗ , µ ∈ ω ( b ) ∗ .Morphisms in M A from ( m, α ) to ( n, β ) are morphisms f : m → n in M satisfying f α = β ( a ⊙ f ).Note that if G is a discrete group, viewed as a monoidal category G whose objectsare the elements of the group, and equipped with the trivial strict monoidal functor ω : G → k , g k , then a G -action is the same thing as a G -action in the usualsense, and in the case of a k -linear action on a k -category M , M G is isomorphicto the usual equivariantization M G . Proposition 3.3.
Let T be a k -linear faithful exact normal Hopf monad on a tensorcategory C , with induced Hopf algebra H . Let K = H ∗ and L = comod - K . Then there is a natural action of L on C by k -linear endofunctors, defined by V ⊙ X = V (cid:3) K T ( X ) , and C L is canonically isomorphic to C T as a k -linear category.Proof. For simplicity, we identify a finite dimensional vector space E with the trivialobject E ⊗ in C . Then T = H and the comonoidal structure of T defines astructure of H -bicomodule on T ( X ). This enables us to define V ⊙ X = V (cid:3) K T ( X )for V a finite-dimensional right K -comodule and X in C . From the fact that T isa faithful exact Hopf monad, one deduces natural isomorphisms ( V ⊗ W ) ⊙ X ≃ V ⊙ ( W ⊙ X ) and ( k , ε ) ⊙ X ≃ X which make ⊙ an action of L on C . Let A = ( H, ∆)be the trivializing algebra of L . Then A generates L . An object ( m, α ) of C L isentirely determined by α εA : A ⊙ X ≃ T ( X ) → X , which can be interpreted as a T -action on X because we have a canonical isomorphism A ⊙ X ≃ T ( X ). Thisdefines a k -linear isomorphism C L ∼ −→ C T . (cid:3) Note that the action of Proposition 3.3 is not compatible in any clear way withthe tensor product of C . In order to take care of the monoidal structure of C , wenow introduce the notion of L -module tensor category.Denote by ab k the 2-category of abelian k -linear categories having finite di-mensional Hom spaces and objects of finite length, 1-morphisms being k -linear leftexact functors, and 2-morphisms being natural transformations. We equip ab k witha tensor product `a la Deligne, denoted by ⊠ , and characterized by the fact thatgiven three objects M , M ′ , M ′′ in ab k , the category of k -linear left exact functors M ⊠ M ′ → M ′′ is equivalent to the category of functors M × M ′ → M ′′ whichare k -linear left exact in each variable. This tensor product makes ab k a monoidal2-category with unit object vect k , with a symmetry τ M , M ′ : M ⊠ M ′ ∼ −→ M ′ ⊠ M defined by m ⊠ m ′ m ′ ⊠ m .Now if A is a tensor category over k , define an A -module category to be an object M of ab k endowed with a k -linear action of A such that the functor ⊙ : A × C → C is k -linear right exact in the first variable (it is automatically exact in the secondvariable because A is autonomous). Thus we may view ⊙ as a k -linear right exactfunctor A ⊠ C → C .Let L be a finite dimensional cocommutative Hopf algebra. The tensor category L = comod- L is tannakian; it is endowed with a strong monoidal symmetric functor∆ ∗ : L → L ⊠ L , which is coassociative, and the symmetric fiber functor ε ∗ : L → vect k is a counit for ∆ ∗ . Thus L is a bialgebra in the monoidal 2-category ab k .If ( M , ρ ) and ( M ′ , ρ ′ ) are two L -module categories then one defines a new L -module category ( M , ρ ) ⊠ ( M ′ , ρ ′ ) = ( M ⊠ M ′ , ρ ′′ ), where ρ ′′ = ( ρ ⊠ ρ ′ )( L ⊠ τ L , M ⊠ N )(∆ ∗ ⊠ M ⊠ N )and this tensor product defines a monoidal structure on the 2-category of L -modulecategories.An L -module tensor category is a tensor category over k , endowed with • a structure of L -module ρ : L ⊠ C → C of L on C ; • natural isomorphisms α V,X,Y : V ⊙ ( X ⊗ Y ) ∼ −→ ⊗ ( V ⊙ ( X ⊠ Y )) β : V ⊠ ∼ −→ V ⊗ making the tensor product ⊗ C : C ⊠ C → C and the unit functor u : vect k →C , k morphisms of L -module categories.If C is an L -module tensor category, then C L is monoidal. Proposition 3.4.
Let C be a tensor category over k . ENTRAL EXACT SEQUENCES OF TENSOR CATEGORIES 15 (1) Let L be a finite dimensional cocommutative Hopf algebra, L = comod - L .Then a structure of L -module tensor category on C defines a k -linear faithfulexact Hopf monad T = A ⊙ ? on C , where A = ( H, ∆) ;(2) If T is a normal cocommutative k -linear faithful exact Hopf monad on C ,the action of L = comod - L on C defined in Proposition 3.3 makes C a L -module tensor category.Moreover, these construction are essentially mutually inverse. Given a Hopf monadas in Assertion (2) and the corresponding structure of L -module tensor category on C , the canonical isomorphism C L ≃ C T is a tensor isomorphism.Proof. Assume we have a structure of L -module tensor category on C , with action ρ : L →
End( C ), and set T = ρ ( A ) = A ⊙ ?. Then T is a monad on C , thatis, an algebra in End( C ), because A = ( H, ∆) is an algebra in C and ρ is strongmonoidal. Moreover T is k -linear exact, and it is faithful because ρ is right exactand is a subobject of A , so X = ⊙ X is a subobject of A ⊙ X = T ( X ).Moreover the L -module tensor category structure defines isomorphisms T ( X ⊗ Y ) ≃ T ( X ) (cid:3) K T ( Y ) and T ( ) = A ⊙ ≃ K ⊗ , which define a Hopf monad structureon T , which is normal and clearly cocommutative.Conversely, if T is a normal cocommutative k -linear faithful exact Hopf monadon C , consider the action ρ of L on C defined by V ⊙ X = V (cid:3) K T ( X ). Then X ρ ( X ) is k -linear left exact, so the action may be viewed as a k -linear exactfunctor L ⊠ C → C . The structure of Hopf monad, cocommutativity and normalitydefine isomorphisms A ⊙ ( X ⊗ Y ) ≃ ( A ⊙ X ) (cid:3) K ( A ⊙ Y ) and A ⊙ ≃ L , whichgive rise to structures of morphisms of L -module morphisms on the tensor product C ⊠ C → C and the unit functor vect k → C , making ρ a structure of L -module tensorcategory on C . (cid:3) Equivariantization and centrality criteria.
Let G be a finite group schemeover k . A tensor action of G on a tensor category C is a structure of L -moduletensor category on C , where L = comod- k [ G ] = O ( G )-mod, where O ( G ) is the Hopfalgebra of regular functions on G . The equivariantization of C under a tensor actionof a finite group scheme G is the tensor category C G = C L .From the results of the previous section, we deduce: Theorem 3.5.
Let F : C −→ D be a normal dominant tensor functor between finitetensor categories, and let T be its Hopf monad and H its induced Hopf algebra. Thefollowing assertions are equivalent:(i) The tensor functor F is an equivariantization under the tensor action of afinite group scheme on D ;(ii) The normal Hopf monad T is exact and cocommutative.If these assertions hold then the induced Hopf algebra H of F is commutative andthe group scheme of assertion (i) is G = Spec H . Theorem 3.6.
Let C ′ → C → C ′′ be an exact sequence of finite tensor categories,and let T be the associated normal Hopf monad on C ′′ . Then the following assertionsare equivalent:(i) The exact sequence C ′ → C → C ′′ is central;(ii) The normal Hopf monad T is cocommutative. Theorem 3.6 will be proved in Section 5.3.
Corollary 3.7.
Consider a morphism of exact sequences of finite tensor categories ( E ) (cid:15) (cid:15) C ′ / / W (cid:15) (cid:15) C / / U (cid:15) (cid:15) C ′′ V (cid:15) (cid:15) ( E ) C ′ / / C / / C ′′ such that the vertical arrows are dominant tensor functors. Then(1) if ( E ) is central, so is ( E ) .(2) If ( E ) is an equivariantization exact sequence for a finite group scheme G , with G discrete or ( E ) perfect, then ( E ) is an equivariantization exactsequence for a subgroup G ′ ⊂ G acting on C ′′ in a manner compatible with V . Moreover if W is an equivalence, then G ′ = G .Proof. Let T , T be the normal Hopf monads, and H , H the induced Hopf algebrasof the exact sequences ( E ) and ( E ) respectively. Then we may assume that ( E )and ( E ) are of the form comod- H → C ′′ T → C ′′ and comod- H → C ′′ T → C ′′ respectively, and we have a diagram of tensor functors:comod- H / / W (cid:15) (cid:15) C ′′ T U T / / U (cid:15) (cid:15) C ′′ V (cid:15) (cid:15) comod- H / / C ′′ T U T / / C ′′ with U ′ , U , U ′′ dominant, which commutes up to tensor isomorphisms. By trans-port of structure, we may assume that U T U = V U T as tensor functors.Let us assume ( E ) is central, and let us show that ( E ) is central. By Theorem 3.6, T is cocommutative, and we are to prove that T is cocommutative too.If ( X, r ) is an object of C ′′ T then U ( X, r ) = ( V ( X ) , λ ( X, r )), so that we have anatural transformation λ : T V U T → V U T , which by adjunction can be encodedas a natural transformation Λ : T V → V T such that U ( X, r ) = (
V X, V r Λ X ) , for any ( X, r ) in C ′′ T .The transformation Λ is compatible with the monad structures of T and T , and itis comonoidal because U T U = V U T as tensor functors.The tensor functor W is induced by a morphism of Hopf algebras φ : H → H ,which is surjective because W is dominant (see Lemma 2.3). In particular H iscommutative, and the group scheme G = Spec H is a subgroup of the group scheme G = Spec( H ) associated with the central exact sequence ( E ).On the other hand, we have T V ( ) = H ∗ ⊗ and V T ( ) = H ∗ ⊗ , and viathese isomorphisms Λ is the transpose of φ ; therefore Λ is a monomorphism (infact, one can show that Λ is monomorphism, a fact we will not use). Denote by( ˆ C , ˆ σ ) and ( ˆ C, ˆ σ ) the induced central coalgebras of T and T respectively. Let( M, r ) be a T -module. One deduces easily from the comonoidality of Λ that thefollowing diagram commutes: T ⊗ V M ˆ σ U ( M,r ) (cid:15) (cid:15) Λ ⊗ V M / / V T ⊗ V M ≃ / / V ( T ⊗ M ) V ((ˆ σ ) ( M,r ) ) (cid:15) (cid:15) T V ( M ) ≃ f f ▲▲▲▲▲▲▲▲▲▲▲ ≃ x x rrrrrrrrrr Λ M / / V T ( M ) ≃ ♥♥♥♥♥♥♥♥♥♥♥♥ ≃ ' ' PPPPPPPPPPPP
V M ⊗ T V M ⊗ Λ / / V M ⊗ V T ≃ / / V ( M ⊗ T ) ENTRAL EXACT SEQUENCES OF TENSOR CATEGORIES 17 where the slanted arrows are the fusion isomorphisms. Since Λ is a monomorphismand, by assumption, ˆ σ is the trivial half-braiding, we see that ˆ σ U ( M,r ) is the trivialhalf-braiding. The tensor functor U being dominant, ˆ σ is the trivial half-braiding,that is, T is cocommutative.In particular if ( E ) is an equivariantization under the action of a group scheme G , it is central so T is cocommutative. If G is discrete or ( E ) is perfect (hence T is exact), then by Proposition 3.2 or Theorem 3.5 ( E ) is an equivariantizationexact sequence, corresponding with a tensor action of G ⊂ G on C ′′ , which byconstruction is compatible with the tensor action of G on C ′′ via V . If W is anequivalence, φ is an isomorphism so G = G . (cid:3) Equivariantization: special cases
The braided case. A braided exact sequence of tensor categories is an exactsequence of tensor categories C ′ i −→ C F −→ C ′′ such that the tensor categories C , C ′′ and the tensor functor F are braided. Thisimplies that C ′ admits a unique braiding such that i is braided, too. Proposition 4.1.
A braided exact sequence of finite tensor categories is central.In particular, if it is perfect, it is an equivariantization exact sequence.Proof.
Let C ′ i −→ C F −→ C ′′ be a braided exact sequence of tensor categories. Thenthe Hopf monad of F is braided by [3, Proposition 5.29], and it is k -linear, rightexact, and normal, so it is cocommutative by [3, Proposition 5.30]. Therefore, theexact sequence is central by Theorem 3.6 and, if it is perfect, it is equivariantizationexact sequence by Theorem 3.5. (cid:3) The fusion case.
Recall that if B is a braided category with braiding c , and A ⊂ B is a set of objects, or a full subcategory of B , then the centralizer of A in B , denoted by C B ( A ), is the full monoidal subcategory of B of objects b satisfying c b,a c a,b = id a ⊗ b for any object a in A . If B is a fusion braided category, then C B ( A )is a full fusion subcategory of B (see [11]). Proposition 4.2.
Let k be an algebraically closed field of characteristic . Considera central exact sequence of fusion categories ( E ) C ′ −→ C −→ C ′′ with canonical lifting ˜ i : C ′ → Z ( C ) , and set A = ˜ i ( C ′ ) . Let C Z ( C ) ( A ) denote thecentralizer of A in Z ( C ) . Then the following holds:(1) we have a braided exact sequence of fusion categories (in fact a modularizationexact sequence) ( E ) A −→ C Z ( C ) ( A ) −→ Z ( C ′′ ); (2) we have a morphism ( E ) → ( E ) of exact sequences of fusion categories: A ≃ (cid:15) (cid:15) / / C Z ( C ) ( A ) (cid:15) (cid:15) / / Z ( C ′′ ) (cid:15) (cid:15) C ′ / / C / / C ′′ , where the vertical arrows are dominant forgetful functors;(3) there is a finite group G acting on Z ( C ′′ ) and on C ′′ by tensor autoequivalencesin a compatible way, in such a way that E and E are the equivariantizationexact sequences relative to these actions. Remark . This proposition says essentially the same thing as [7, Proposition2.10](i), with a different viewpoint. Indeed [7, Proposition 2.10](i) asserts that if C is a fusion category over C and A ⊂ Z ( C ) is a full tannakian subcategory suchthat the forgetful functor U : Z ( C ) → C induces an equivalence of A with a tensorsubcategory of C , then A encodes a de-equivariantization of C , that is, a tensoraction of a finite group G (such that A ≃ rep G ) on a category D such that C ≃ D G .This can be deduced from Proposition 4.2, as follows. Assume A ⊂ Z ( C ) is asabove. Since A is tannakian, it contains a self-trivializing semisimple commutativealgebra A = ( A, σ ) such that Hom( , A ) = k . The forgetful functor U induces byassumption a full tensor embedding i : A → C . Moreover, we have a tensor functor F A : C → C A and an exact sequence of fusion categories( E ) A i −→ C F A −→ C A . The inclusion
A ⊂ Z ( C ) is a central lifting of i which makes ( E ) a central exactsequence, hence the tensor functor C → C A is an equivariantization. Proof.
Notice first that assertion (3) derives immediately from assertions (1) and(2) and previous results: if (1) and (2) hold, then by Proposition 4.1 ( E ) is anequivariantization exact sequence for the action of a finite group G on Z ( C ′′ ). ByCorollary 3.7, ( E ) is also an equivariantization exact sequence for an action of thesame group G .So the whole point is to construct the exact sequence ( E ) of assertion (1) andthe morphism of exact sequences of assertion (2). This is based on the followinglemma. Let us say that an object X of a braided category B with braiding c is symmetric if c X,X = id X ⊗ X . Lemma 4.4.
Let C be a fusion category, let A = ( A, σ ) be a semisimple, symmetric,self-trivializing commutative algebra in Z ( C ) such that Hom( , A ) = k , and let A = h A i be the fusion subcategory of Z ( C ) generated by A . Then C Z ( C ) ( A ) A = dys Z ( C ) A ≃ Z ( C A ) , where dys Z ( C ) A denotes the category of dyslectic A -modules in Z ( C ) .Proof. Recall that if B is a braided category, with braiding c , and A is a com-mutative algebra in B , then a dyslectic A -module ([14, Definition 2.1]) is a right A -module ( M, r : M ⊗ A → M ) in B satisfying rc A ,M c M, A = r . The category dys B A of dyslectic A -modules is a full monoidal subcategory of B A and it is braided withbraiding induced by c .In the situation of the lemma, Z ( C ) A is a fusion category (because A is semisimpleand Hom( , A ) = k ), and dys Z ( C ) A is a full fusion category of Z ( C ) A .On the other hand, the fact that A is symmetric means that it belongs to C Z ( C ) ( A ), and the category C Z ( C ) ( A ) A is also a full fusion subcategory of Z ( C ) A .Moreover, we have C Z ( C ) ( A ) A ⊂ dys Z ( C ) A because a A -module ( M, r ) such that M belongs to the centralizer of A is dyslectic.Now it follows from [15, Corollary 4.5] that there is a natural equivalence ofbraided tensor categories dys Z ( C ) A ≃ Z ( C A ) which is compatible with the forgetfulfunctors to C .All that remains to do is to show that the full, replete inclusion of C Z ( C ) ( A ) A in dys Z ( C ) A is an equality, which we do by showing that those two fusion cate-gories have the same Frobenius-Perron dimension. Now C A is a fusion category andFPdim C A = FPdim C / FPdim A by Lemma 2.11(i). Since dys Z ( C ) A ≃ Z ( C A ), wehave FPdim dys Z ( C ) A = FPdim Z ( C A ) = (cid:18) FPdim C FPdim A (cid:19) by [6, Proposition 8.12]. On the other hand,FPdim C Z ( C ) ( A ) = FPdim Z ( C )FPdim A by [5, Theorem 3.14], since Z ( C ) is a nondegenerate fusion category. We haveFPdim Z ( C ) = FPdim( C ) again by [6, Proposition 8.12], and since A is self-trivializing, FPdim( A ) = FPdim( A ) = FPdim A , because the forgetful functor Z ( C ) → C preserves Frobenisus-Perron dimensions. SoFPdim C Z ( C ) ( A ) A = FPdim( C ) FPdim( A ) = FPdim dys Z ( C ) A , and we are done. This finishes the proof of the lemma. (cid:3) Now we apply the lemma. Let ˜ F = F A : C Z ( C ) ( A ) → C Z ( C ) ( A ) A ≃ Z ( C A ) be thefunctor ‘free A -module’ X X ⊗ A . Then ˜ F is a braided dominant normal fusionfunctor because A is semisimple and self-trivializing, so we have a braided exactsequence of fusion categories( E ) A → C Z ( C ) ( A ) → Z ( C A ) , which is an equivariantization exact sequence by Proposition 4.1, for an action ofa certain finite group G on Z ( C ). All our constructions are compatible with theforgetful functors, hence we get a morphism of exact sequences of fusion categories( E ) → ( E ): A ≃ (cid:15) (cid:15) / / C Z ( C ) ( A ) (cid:15) (cid:15) / / Z ( C ′′ ) (cid:15) (cid:15) C ′ / / C / / C ′′ The vertical arrow on the right is an equivalence because ( E ) is central and C ′ ≃ h A i ,and the left vertical arrow is dominant because it is the forgetful functor of thecenter. The middle vertical arrow is therefore dominant by virtue of Lemma 2.3, soassertion (2) holds. By Corollary 3.7, ( E ) is an equivariantization exact sequence,for an action of G on C which is compatible with the action of G on Z ( C ) and theforgetful functor Z ( C ) → C . (cid:3) The abelian case.
Let k be a field. We say that a finite abelian group G hasthe Kummer property ( w.r.t. k ) if k contains e distincts e -th roots of 1, where e isthe exponent of G . If such is the case, the group of characters ˆΓ k of G is isomorphicto G . If k is algebraically closed of characteristic 0, all finite abelian groups havethe Kummer property. Proposition 4.5.
Let F : C → D be a dominant tensor functor between finitetensor categories over field k , and denote by A = ( A, σ ) its induced central algebra.The following assertions are equivalent:(i) The functor F is an equivariantization associated with an action of a finiteabelian group G having the Kummer property;(ii) The induced central algebra A of F is a direct sum of invertible objects of Z ( C ) , and the finite abelian group Γ formed by the isomorphism classes ofthese invertible objets has the Kummer property;If these equivalent assertion hold, the groups G of assertion (i) and Γ of assertion(ii) are in duality, that is G = ˆΓ k . Proof. (i) = ⇒ (ii). Assume that F is an equivariantization under a finite abeliangroup G having the Kummer property. We have a tensor action of G on D , and wemay assume that C = D G . The exact sequencerep G i −→ C −→ D is central, that is, i admits a central lifting ˜ i : rep G → Z ( C ) such that ˜ i ( A ) =( A, σ ) = A (see Example 2.5). Now, since G is abelian and has the Kummerproperty, mod- G ≃ Γ- vect is pointed, so A splits as a sum of invertible objects,and so does A = ˜ i ( A ), so (ii) holds.(ii) = ⇒ (i). Assume that A splits as a direct sum of invertible objects of Z ( C ).Then by Theorem 2.9, F is normal and fits into a central exact sequence C ′ −→ C −→ D . Denote by H the induced Hopf algebra of this exact sequence, which is commutativeby Proposition 2.6. The tensor category C ′ is pointed with Picard group Γ becauseit is tensor equivalent to h A i via the canonical lifting ˜ i , and since C ′ = comod- H ,we see that H is cocommutative and split cosemisimple. Thus G = Spec H = ˆΓ k isa discrete abelian group, so H is split semisimple. We conclude by Proposition 3.2that F is an equivariantization under the group G . (cid:3) Equivariantization and the double of a Hopf monad
Relative centers and centralizers.
Let C , D be monoidal categories, andlet F : C → D be a comonoidal functor. Define a half-braiding relative to F to be apair ( d, σ ), where d is an object of D and σ is a natural transformation d ⊗ F → F ⊗ d satisfying: ( F ( c, c ′ ) ⊗ d ) σ c ⊗ c ′ = ( F ( c ) ⊗ σ c ′ )( σ c ⊗ F ( c ′ ))( d ⊗ F ( c, c ′ )) , ( F ⊗ d ) σ = σ ( d ⊗ F ) . Half-braidings relative to F form a category called the center of D relative to F and denoted by Z F ( D ), or Z C ( D ) if the functor F is clear from the context. It ismonoidal, with the tensor product defined by( d, σ ) ⊗ ( d ′ , σ ′ ) = ( d ⊗ d ′ , ( σ ⊗ d ′ )( d ⊗ σ ′ )) , and the forgetful functor U : Z F ( D ) → D is monoidal strict.Now assume F is strong monoidal (in particular, it can be viewed as a comonoidalfunctor). Then we have a strong monoidal functor ˜ F : Z ( C ) → Z F ( D ), defined by˜ F ( c, σ ) = ( F ( c ) , ˜ σ ), where ˜ σ = F ( σ ) up to the structure isomorphisms of F .If F is strong monoidal and has a left adjoint L , then T = F L is a bimonad on D and by adjunction, Z F ( D ) is isomorphic as a monoidal category to the center Z T ( D ) of D relative to the bimonad T defined in [4, Section 5.5].Let ( E ) = (cid:0) C ′ i −→ C F −→ C ′′ (cid:1) be an exact sequence of finite tensor categories overa field k . We will show that, if ( E ) is central, with canonical central lifting ˜ i , thenwe have an exact sequence of tensor categories C ′ ˜ i −→ Z ( C ) ˜ F −→ Z F ( C ′′ )and a morphism of exact sequences of tensor categories C ′ / / = (cid:15) (cid:15) Z ( C ) / / (cid:15) (cid:15) Z F ( C ′′ ) (cid:15) (cid:15) C ′ / / C / / C ′′ . ENTRAL EXACT SEQUENCES OF TENSOR CATEGORIES 21
Quantum double of a Hopf monad.
Let T be a Hopf monad on a rigidmonoidal category C . We say that T is centralizable ([4]) if for all objects X in C the coend Z T ( X ) = Z Y ∈C ∨ T Y ⊗ X ⊗ Y exists. In that case the assignment X → Z T ( X ) defines a Hopf monad Z T on C ,called the centralizer of T . Denoting by j X,Y : ∨ T Y ⊗ X ⊗ Y → Z T ( X ) the universaldinatural transformation (in Y ) associated with the coend Z T ( X ), set ∂ X,Y = (
T X ⊗ j X,Y )(coev
T Y ⊗ X ⊗ Y ) : X ⊗ Y → T Y ⊗ Z T ( X ) . If T is centralizable, we have an isomorphism of tensor categories Z T ( C ) ∼ −→ C Z T ,and so, an isomorphism of tensor categories K : Z U T ( C ) ∼ −→ C Z T .Moreover, if T is centralizable there also exists a canonical comonoidal distribu-tive law Ω : T Z T → Z T T , which is an isomorphism. It is characterized by thefollowing equation(5.1) ( µ X ⊗ Ω Y ) T ( T Y, Z T ( X )) T ( ∂ X,Y ) = ( µ X ⊗ Z T T ( Y )) ∂ T X,T Y T ( X, Y ) . This invertible distributive law serves two purposes: it defines (via its inverse) alift ˜ T of the Hopf monad T to C Z T , and it also defines a structure of a Hopf monad D T = Z T ◦ Ω T on the endofunctor Z T T of C . The Hopf monad D T is called thedouble of T ; it is quasi-triangular, so that C D T is braided, and we have a canonicalbraided isomorphism K ′ : C D T ∼ −→ Z ( C T ).Lastly, we have a commutative diagram of tensor functors Z ( C T ) ˜ U T (cid:15) (cid:15) K ′ ∼ / / C D T U DT / / ˜ U (cid:15) (cid:15) C TU T (cid:15) (cid:15) Z ˜ U T ( C ) ∼ K / / C Z T U ZT / / C If C is a finite tensor category over a field k , and T a k -linear right exact Hopfmonad on C , then C T is a finite tensor category and the forgetful functor U T : C T → C is a tensor functor. Moreover, T is centralizable and Z T is a k -linear Hopfmonad on C , which is right exact (being an inductive limit of right exact functors)so Z T ( C ) ≃ C Z T is a finite tensor category and the forgetful functor Z T ( C ) → C isa tensor functor.5.3. Proof of Theorem 3.6.
Let ( E ) be an exact sequence of finite tensor cate-gories over a field k . Up to equivalence, we may assume that ( E ) is of the form h A i −→ C T U T −→ C , where C is a finite tensor category, T is a k -linear right exact normal Hopf monadon C , and A is the induced algebra of U T .We are to show that the following assertions are equivalent:(i) T is cocommutative;(ii) ( E ) is a central exact sequence.Our proof will rely on the following Lemma 5.1.
Let T be a centralizable Hopf monad on an rigid category C . Thenthe induced central algebra (resp. coalgebra) of U T is the induced algebra (resp.coalgebra) of ˜ U T . Before we prove this lemma, let us show how it enables us to conclude. Considerthe tensor functor ˜ U T : Z ( C ) → Z T ( C ). It is dominant. Indeed U T is dominant byassumption, which means that the unit of T is a monomorphism, and so is the unitof ˜ T because it is a lift of T .Moreover, ˜ U T is normal if and only if T is cocommutative. This can be seen asfollows. Denote by ( ˆ C, ˆ σ ) the induced central coalgebra of U T : C T → C , which isalso the induced coalgebra of ˜ U T by Lemma 5.1. We have ˜ T ( ) = ˜ U T ( ˆ C, ˆ σ ), andalso T ( ) = ˆ C . In particular, ˜ U T is normal if and only if ( ˆ C, ˆ σ ) is trivial in Z T ( C ),that is ˆ C is trivial in C (which is true because we have assumed T is normal) andˆ σ coincides with the trivial half-braiding. The latter condition means that T iscocommutative by Lemma 3.1.Denote by A = ( A, σ ) the induced central algebra of U T , which is the right dualof ( ˆ C, ˆ σ ). It is also the induced algebra of ˜ U T .Now assume T is cocommutative. As we have just seen this means that ˜ U T isnormal and dominant, so we have an exact sequence of tensor categories( E ) h A i −→ Z ( C ) ˜ U T −→ Z T ( C ) . Moreover, we have a morphism of exact sequences of tensor categories h A i / / V (cid:15) (cid:15) Z ( C T ) / / U (cid:15) (cid:15) Z T ( C ) W (cid:15) (cid:15) h A i / / C T / / C where U , V , W denote the forgetful functors. We have U ( A ) = A , so V is anequivalence of categories, that is, ( E ) is central.Conversely, assume ( E ) is central. That means that the forgetful functor in-duces a tensor equivalence h A i → h A i . Since A is self-trivializing, so is A . But byLemma 5.1, A is also the induced algebra of ˜ U T , so this tensor functor is normalby Lemma 2.1; and as we have seen above, this implies that T is cocommutative.Thus, we have shown the equivalence of (i) and (ii). Proof of Lemma 5.1.
The induced (central) algebra being the dual of the induced(central) coalgebra, it is enough to prove the assertion for coalgebras. Let ˜ L denotethe left adjoint of ˜ U . The induced coalgebra ˜ C of ˜ U T is K − ˜ L ( ).The functor ˜ U is monadic, and its monad ˜ T is the lift of T defined by thedistributive law Ω − . This means that we have ˜ L ( c, r ) = ( T ( c ) , T ( r )Ω − c ) for ( c, r )in C Z T . We also have K − ( c, ρ ) = (( c, r ) , s ), where r = ρη Z T T c and s is thehalf-braiding defined by s ( x,r ) = ( r ⊗ ρZ T ( η c )) ∂ c,x for ( x, r ) in C T . As a result,˜ C = (( T , µ ) , Σ), whereΣ ( c,r ) = ( r ⊗ T (( Z T ) )Ω − ) ∂ T ,c . On the other hand, we have ˆ C = ( T ( ) , µ ), and the half-braiding ˜ σ is characterizedby the fact that the following diagram commutes: T ⊗ c ˆ σ ( c,r ) / / c ⊗ T T ⊗ T c T ⊗ r O O T c T ( c, ) / / ≃ ♦♦♦♦♦♦♦♦♦♦♦♦♦♦ T ( ,c ) o o ≃ g g ❖❖❖❖❖❖❖❖❖❖❖❖❖❖ T c ⊗ T r ⊗ T O O ENTRAL EXACT SEQUENCES OF TENSOR CATEGORIES 23 so it is enough to verify: Σ ( c,r ) ( T ⊗ r ) T ( , c ) = ( r ⊗ T ) T ( , c ). Now we haveΣ ( c,r ) ( T ⊗ r ) T ( , c ) = ( r ⊗ T (( Z T ) )Ω − ) ∂ T ,c ( T ⊗ r ) T ( , c )= ( rT ( r ) ⊗ T (( Z T ) )Ω − ) ∂ T ,T c T ( , c ) by functoriality of ∂ = ( rµ c ⊗ T (( Z T ) )Ω − ) ∂ T ,T c T ( , c )= ( rµ c ⊗ T (( Z T ) )) T ( T c, Z T ) T ( ∂ ,c ) by (5.1)= ( rT ( r ) ⊗ T (( Z T ) )) T ( T c, Z T ) T ( ∂ ,c )= ( r ⊗ T ) T ( c, ) T (( r ⊗ ( Z T ) ) ∂ ,c ) by functoriality of T = ( r ⊗ T ) T ( c, ) T ( rη c ) by construction of ( Z T ) = ( r ⊗ T ) T ( c, ) . This concludes the proof of the lemma. (cid:3) Tensor functors of small Frobenius-Perron index
In this section we prove Theorems 6.1 and 6.2.6.1.
Tensor functors of Frobenius-Perron index 2.Theorem 6.1.
Let F : C → D be a dominant tensor functor between fusion cate-gories over a field of characteristic . If FPind( C : D ) = 2 , then F is an equivari-antization associated with an action of Z on D .Proof. Let F : C → D be a dominant tensor functor of Frobenius-Perron index 2between fusion categories. By [3, Proposition 4.13], F is normal and so it inducesan exact sequence of fusion categories(6.1) rep Z → C F → D . Now let A = ( A, σ ) be the induced central algebra of F . We have FPdim A =FPdim A = 2 by Lemma 2.11. Since A contains the unit object, we have A = ⊕ S ,with S invertible. By Proposition 4.5, F is an equivariantization relative to anaction of Z . This concludes the proof of the theorem. (cid:3) Tensor functors of small prime Frobenius-Perron index.Theorem 6.2.
Let F : C → D be a dominant tensor functor between fusion cate-gories over a field of characteristic . Assume that FPdim C is a natural integer,and that FPind( C : D ) is the smallest prime number dividing FPdim C . Then F isan equivariantization associated with an action of Z p on D .Proof. Assume C is a weakly integral fusion category and let p be the smallest primefactor of FPdim C . Consider a dominant tensor functor F : C → D , where D is afusion category, such that FPind( C : D ) = p .Recall that a fusion category is integral if its objects all have integral Frobenius-Perron dimension. If C is not integral, then by [8, Theorem 3.10] it is Z -graded,and so FPdim C is even. Thus p = 2, and so Theorem 6.1 applies and we are done.From now on we assume that C is integral. Then Z ( C ) is also an integral fusioncategory. Let A = ( A, σ ) be the induced central algebra of F . We have FPdim A =FPdim A = FPind( C : D ) = p . Let us decompose A as a direct sum of simpleobjects of Z ( C ):(6.2) A = W ⊕ · · · ⊕ W r . We have r ≥ A is not simple (it contains the unit object), so the Frobenius-Perron dimension of W i is an integer < p for all i .The center Z ( C ) is a non-degenerate braided fusion category. According to[7, Theorem 2.11] (i), (FPdim W i ) divides FPdim Z ( C ) = (FPdim C ) , and so FPdim W i divides FPdim C . We have FPdim W i = 1, because p is by assumptionthe smallest prime divisor of FPdim C , and so W i is invertible in Z ( C ).This implies that A belongs to Z ( C ) pt . By Theorem 2.9, we have an exactsequence(6.3) rep Z p → C F → D . which is central, and by Theorem 3.5, it is an equivariantization exact sequence.This concludes the proof of the theorem. (cid:3) Fusion subcategories of index are not always normal. Let C be afusion category. A full fusion subcategory D ⊂ C is normal in C (see [3]) if theinclusion D ⊂ C extends to an exact sequence of fusion categories
D → C → C ′′ . Inthat case we have FPdim C ′′ = FPdim C FPdim D .If D is a full fusion subcategory of C then the ratio FPdim C FPdim D is an algebraic integer(see [6, Proposition 8.15]). If C is weakly integral, so is D , and therefore FPdim C FPdim D isa natural integer.Is it true that if C is weakly integral and FPdim C FPdim D is the smallest prime numberdividing FPdim C , then D is normal in C ? We show that even for p = 2 such isnot the case, by exhibiting counterexamples in Tambara-Yamagami categories (seeSection 2.6). Proposition 6.3.
Let C be a Tambara-Yamagami category. Then we have FPdim C FPdim C pt = 2 , but C pt is not normal in C if the order of the Picard group of C is not a square.Proof. Let Γ = Pic( C ) and denote by X the simple non-invertible object of C . Wehave X ⊗ X ≃ P g ∈ Γ g , so FPdim X = p | Γ]. We have C pt = h Γ i , andFPdim C FPdim C pt = 2 | Γ || Γ | = 2 . Now assume assume that | Γ | is not a square. Then C is not integral becauseFPdim X is not an integer. We conclude by the following lemma. Lemma 6.4.
Let C be a fusion category and let D ⊂ C be a normal fusion subcat-egory such that
FPdim C FPdim D is prime. Then C is integral.Proof. Consider the exact sequence of fusion categories
D −→ C F −→ C ′′ comingfrom the fact that D is normal in C . We have FPdim C ′′ = p . By [6, Corollary8.30], C ′′ admits a quasi-fiber functor ω : C ′′ → vect k . Then C admits a quasi-fiberfunctor ωK , and therefore C is integral. (cid:3) A contrario the lemma shows that C pt is not normal in C . (cid:3) Proposition 6.5.
Let C be a Tambara-Yamagami category with Picard group ofprime order. Then C is a simple fusion category.Proof. Assume we have an exact sequence of fusion categories
D → C → C ′′ . Since C contains a simple object of Frobenius-Perron dimension √ p , it is not integral.Consequently C ′′ admit no quasi-fiber functor. In particular FPdim( C ′′ ) is neither1 nor a prime number. Therefore FPdim( C ′′ ) = 2 p , and D is trivial. Hence C issimple. (cid:3) ENTRAL EXACT SEQUENCES OF TENSOR CATEGORIES 25
References [1] A. Brugui`eres,
Cat´egories pr´emodulaires, modularisations et invariants des vari´et´es de di-mension
3, Math. Ann. , 215–236 (2000).[2] A. Brugui`eres, S. Lack, and A. Virelizier,
Hopf monads on monoidal categories , Adv. Math.,preprint arXiv:1003.1920 .[3] A. Brugui`eres and S. Natale,
Exact sequences of tensor categories , Int. Math. Res. Not. (24), 5644–5705 (2011).[4] A. Brugui`eres and A. Virelizier,
Quantum double of Hopf monads and categorical centers ,Trans. Am. Math. Soc. , 1225–1279 (2012).[5] V. Drinfeld, S. Gelaki, D. Nikshych and V. Ostrik,
On braided fusion categories I , Sel. Math.New Ser. , 1–119 (2010).[6] P. Etingof, D. Nikshych and V. Ostrik, On fusion categories , Ann. Math. (2) , 581–642(2005).[7] P. Etingof, D. Nikshych and V. Ostrik,
Weakly group-theoretical and solvable fusion categories ,Adv. Math. , 176–205 (2011).[8] S. Gelaki and D. Nikshych,
Nilpotent fusion categories , Adv. Math. , 1053–1071 (2008).[9] T. Kobayashi and A. Masuoka,
A result extended from groups to Hopf algebras , Tsukuba J.Math. , 55–58 (1997).[10] M. M¨uger, Galois theory for braided tensor categories and the modular closure , Adv. Math. , 151–201 (2000).[11] M. M¨uger,
On the structure of modular categories , Proc. London Math. Soc. (3) , 291–308(2003).[12] S. Natale, On group-theoretical Hopf algebras and exact factorizations of finite groups , J.Algebra , (2003) 199–211.[13] S. Natale,
Semisolvability of semisimple Hopf algebras of low dimension , Mem. Amer. Math.Soc. , 123 pp. (2007).[14] B. Pareigis,
On braiding and dyslexia , J. Algebra , 413–425 (1995).[15] P. Schauenburg,
The monoidal center construction and bimodules , J. Pure Appl. Algebra , 325–346 (2001).[16] H.-J. Schneider,
Some remarks on exact sequences of quantum groups , Comm. Alg. (1993),3337-3357.[17] D. Tambara and S. Yamagami, Tensor categories with fusion rules of self-duality for finiteabelian groups , J. Algebra , 692–707 (1998).
Alain Brugui`eres: D´epartement de Math´ematiques. Universit´e Montpellier II.Place Eug`ene Bataillon. 34 095 Montpellier, France
E-mail address : [email protected] URL:
Sonia Natale: Facultad de Matem´atica, Astronom´ıa y F´ısica. Universidad Nacionalde C´ordoba. CIEM – CONICET. Ciudad Universitaria. (5000) C´ordoba, Argentina
E-mail address : [email protected] URL: ∼∼