Clifford theory for tensor categories

CLIFF ORD THEOR Y F OR TENSOR CA TEGORIES C ´ ESAR GALINDO Abstract. A g raded te nsor category o ver a group G will be call ed a strongly G -graded tensor catego ry if ev ery homogeneous component has at least one multiplicativ ely in vertible ob ject. Our m ain result is a description of the mo d- ule categories o ver a st rongly G - gr ade d tensor category as induced from module categories ov er tensor sub categories associated with the subgroups of G . 1. Introduction The clas s ical Clifford theo ry is an imp ortant co llection o f results relating repre- sentation of a g roup to the represe n tation of its nor mal subgr oups. The principa l results can be gener alized using strongly graded rings, as in [7]. The goal of this pap er is to describ e a categor ical analogue of the Clifford theor y for tensor ca te- gories. Throughout this article we work ov er a field k . By a tenso r categ ory ( C , ⊗ , α, 1) we understand a k -linear a belian ca tegory C , endow ed with a k -bilinear exa ct bifunctor ⊗ : C × C → C , an ob ject 1 ∈ C , and an ass ociativity constr ain t α V ,W,Z : ( V ⊗ W ) ⊗ Z → V ⊗ ( W ⊗ Z ), such that Mac Lane’s p en tagon axiom holds [5], V ⊗ 1 = 1 ⊗ V = V , α V , 1 ,W = id V ⊗ W for all V , W ∈ C and dim k End C (1) = 1. An in teres ting and active pro blem is the classification of mo dule catego ries ov er a tensor ca tegory . See [2], [10], [1 9], [20], [21]. A left mo dule categor y over a tensor ca tegory C , or a left C -module catego ry , is a k -linear a belian ca tegory M equipp e d with a n exact bifuntor ⊗ : C × M → M and natural iso mo rphisms α X,Y ,M : ( X ⊗ Y ) ⊗ M → X ⊗ ( Y ⊗ M ), X , Y , Z ∈ C , M ∈ M , satisfying natural axioms. Definition 1. 1. Let C b e a tenso r ca teg ory , and let M b e a C -module ca tegory . A C - s ubmodule categor y of M is a Serr e sub category N ⊆ M of M suc h that N is a C -module ca teg ory with resp ect to ⊗ . A C -mo dule ca tegory will b e called simple if it doe s not contain any non-trivial C - s ubmodule categ o ry . R emark 1.2 . A r igid tensor catego ry ov er an algebraica lly closed field is called fi- nite, if it is equiv alen t as an ab elian catego ry to the catego ry of finite repres en tation of a finite dimensional alg ebra, s ee [11]. In this case the rig h t definition of mo d- ule ca teg ory is that of an exact mo dule c ate gory , see lo c. cit. F or exact mo dule categorie s ov er finite tensor categ ories, the notion of simple mo dule catego ry is equiv alen t to that o f indecomp osable mo dule catego ry . In particular a semisimple mo dule categor y over a fusion category is simple if and o nly if it is indecomp o sable. Date : Nov ember 21, 2018. 1991 Mathematics Subje ct Classific ation. 16W30, 18D10. 1 2 C ´ ESAR GALINDO Let C and D b e tensor categor ies. A C - D -bimo dule category is a k -linear ab elian category M , endow ed with a structure of left C -module catego ry and r igh t D - mo dule catego ry , such tha t the “ac tio ns” commu te up to natura l isomorphisms in a coherent wa y . See Sectio n 2 for details on the definitions of C - module categ ory , C - bimo dule catego ry , C -mo dule functor, C - linea r na tur al transformation and their comp osition. F or a right C -module categor y M and a left C - module categor y N , the tensor pro duct categ ory of k -linear mo dule categ ories M ⊠ C N w as defined in [27]; howev er, t ypica lly M ⊠ C N is not an abe lian category . If M is a D - C -bimo dule categ o ry the category M ⊠ C N has a co heren t left D -action. Let G b e a group and C be a tensor category . W e shall say that C is G -graded, if there is a decomp osition C = ⊕ x ∈ G C x of C into a direct s um of full ab elian subcateg ories, such that fo r all σ, x ∈ G , the bifunctor ⊗ maps C σ × C x to C σx . See [12]. Recall that a gr aded ring A = ⊕ σ ∈ G A σ is called strong ly gr a ded, if A x A y = A xy for all x, y ∈ G . If we denote by C σ · C τ ⊆ C στ the full k -linear s ubcategory of C στ whose ob jects are dir ect sums of ob jects of the form V σ ⊗ W τ , for V σ ∈ C σ , W τ ∈ C τ , σ , τ ∈ G , the definition of str ongly gr ade d tensor c ate gory is the following: Definition 1.3. Let C = ⊕ σ ∈ G C σ be a graded tensor categ ory ov er a group G . W e shall say that C is stro ngly gr ade d if the inclusion functor C σ · C τ ֒ → C στ is an equiv alence of k -linear categ ories for a ll σ , τ ∈ G . R emark 1.4 . Note that C σ · C τ is only the ful l k -line ar sub c ate gory o f C στ , and not the full a belian sub category generated by C σ · C τ . F or example the T ambara-Y ama g ami categorie s T Y ( A, χ, ǫ ) (see [2 9]) are Z 2 -graded fusion categor ies not str ongly gr a ded. In fa ct, the simple ob jects of C 0 are in vertible a nd C 1 only hav e one simple. Then the ob jects of the category C 1 · C 1 has the form ( X ⊗ X ) ⊕ n , and the full k -linear sub c ategory C 1 · C 1 is not eq uiv alent to C 0 , if C 0 has mo r e than one simple ob ject. Note that the abelia n sub categor y of C 0 generated by C 1 · C 1 is equiv alent to C 0 . Also note that for every tensor categor y ( C , ⊗ , I ), the k -linear categor y C · C is equiv alen t to C , since V ∼ = I ⊗ V ∈ C · C , for ev ery V ∈ O bj ( C ). By Lemma 3.1, a g raded tenso r categ o ry ov er a group G is a str ongly G -g raded tensor categ ory , if and only if e very homo geneous compo nent has at lea st one in- vertible ob ject. Let C b e a stro ngly G -gra ded tenso r categor y . Given a C e -mo dule category M , we shall denote by Ω C e ( M ) the set of equiv alences class es of simple C e -submo dule categor ies of M . B y Corollar y 4.3, the group G a cts o n Ω C e ( M ) by G × Ω C e ( M ) → Ω C e ( M ) , ( g , [ X ]) 7→ [ C g ⊠ C e X ] . Our main result is: Theorem 1.5 (Clifford Theorem for mo dule categor ies) . L et C b e a str ongly G - gr ade d tensor c ate gory and let M b e a simple ab elia n C -m o dule c ate gory. Then: (1) The action of G on Ω C e ( M ) is tr ansitive, (2) L et N b e a s imple ab elian C e -submo dule s ub c ate gory of M . L et H = st([ N ]) b e the stabilizer sub gr oup of [ N ] ∈ Ω C e ( M ) , and let also M N = X h ∈ H C h ⊗N . CLIFFOR D THEOR Y F OR TENSOR CA TEGORIES 3 Then M N is a simple C H -mo dule c ate gory and M ∼ = C ⊠ C H M N as C - mo dule c ate gori es. An imp ortant family of examples of str ongly graded tensor ca tegories are the crossed pro duct tensor categ ories, see [18], [28]. L e t C be a tenso r categor y and let G b e a gr o up. W e shall deno te b y G the monoida l category , where the ob jects a re the elements o f G , arr o ws a re identities and tenso r pro duct the pro duct of G . Let Aut ⊗ ( C ) be the monoida l category where ob jects are tensor auto- equiv alence s of C , ar rows are tensor natura l isomo r phisms and tens o r pro duct the c o mposition of functors . An actio n of the group G over a monoidal c a tegory C , is a mono idal functor ∗ : G → Aut ⊗ ( C ). Given an action ∗ : G → Aut ⊗ ( C ) of G on C , the G -cro ssed pr oduct tensor category , deno ted by C ⋊ G is defined as follo ws. As an ab elian catego r y C ⋊ G = L σ ∈ G C σ , where C σ = C as an ab elian category , the tens o r pro duct is [ X , σ ] ⊗ [ Y , τ ] := [ X ⊗ σ ∗ ( Y ) , στ ] , X , Y ∈ C , σ , τ ∈ G, and the unit ob ject is [1 , e ]. See [28] for the asso ciativit y constra in t a nd a pr oof o f the p en tago n identit y . The categ ory C ⋊ G is G -gr aded by C ⋊ G = M σ ∈ G ( C ⋊ G ) σ , where ( C ⋊ G ) σ = C σ , and the ob jects [1 , σ ] ∈ ( C ⋊ G ) σ are inv ertible, with in verse [1 , σ − 1 ] ∈ ( C ⋊ G ) σ − 1 . Another useful construction of a tens o r ca tegory starting fr o m a G -action ov er a tensor c a tegory C , is the G -equiv ariantization of C , denoted b y C G . This constr uc- tion has b een used for example in [3 ], [1 4], [1 7], [18], [28]. The category C is a C ⋊ G -mo dule categ ory with action [ V , σ ] ⊗ W = V ⊗ σ ∗ ( W ), see [18], [28]. Mor eo ver, the tenso r category of C ⋊ G -linear endofunctors of C denoted b y F C ⋊ G ( C , C ), is monoidally e q uiv alent to the G -equiv ariantization C G of C , see [18]. With help o f this equiv alence can b e describ e the mo dule c a tegories ov er C G , using the description of the mo dule categ o ries ov er the strong ly G -graded tensor categor y C ⋊ G , see [13] for the fusion ca tegory cas e . The paper is organized as follows: Sectio n 2 co ns ists mainly o f definitions and prop erties of mo dule a nd bimo dule categ ories ov er tensor categ ories and the tensor pro duct of mo dule catego r ies, that will b e need in the sequel. In Section 3 we int ro duce mo dule catego ries gra ded ov er a G -set and give a structure theorem for them. In Section 4 the main theo rem is pr o ved. In Sec tio n 5 we descr ibe the simple mo dule categor ie s o ver C ⋊ G a nd the simple mo dule ca teg ories over C G if G is finite. 2. Preliminaries A k -linear categ o ry o r a c ategory additive over k , is a catego ry in which the sets of arrows betw een t wo ob jects a r e k -vector spaces, the comp ositions are k -bilinea r op erations, finite dir ect sums exist and there is a z ero ob ject. A k -linear functor C → D b e t ween k -linear categ ories, is and additive functor k -linear on the s pa ces of morphisms. The notion of k -bilinear bifunctor C × C ′ → D is the obvious. Definition 2.1. [19, Definition 6.] Let C b e a monoidal category . A left C -mo dule category over C , is a c ategory M together with a bifuntor ⊗ : C × M → M and 4 C ´ ESAR GALINDO natural isomorphisms m X,Y ,M : ( X ⊗ Y ) ⊗ M → X ⊗ ( Y ⊗ M ) , such that ( α X,Y ,Z ⊗ M ) m X,Y ⊗ Z,M ( X ⊗ m Y ,Z,M ) = m X ⊗ Y ,Z,M m X,Y ,Z ⊗ M , 1 ⊗ M = M , for all X , Y , Z ∈ C , M ∈ M . A mo dule categ ory M ov er a tensor categ ories C always will b e ab elian, and the bifunctor ⊗ : C × M → M biexact. A rig h t mo dule categ ory is defined in a similar wa y . R emark 2.2 . F or a catego ry M , the ca tegory of k -linear exact e ndofunctors F ( M , M ) is a k -linear ab elian strict mono ida l categ o ry , where the kerner of mor phis m τ : F → G in F ( M , M ), is the functor K : M → M , defined by K ( M ) = ker( τ M ), and with the comp osition of functors as tenso r pr oduct, and . F or a tensor catego ry C , a structure o f C -mo dule category ( M , ⊗ , m ) on M is the same as an exact monoidal functor ( F, ζ ) : C → F ( M , M ). The bijection is g iv en by the equation V ⊗ M = F ( V )( M ), identifying ( ζ V ,W ) M : ( F ( V ) ◦ F ( W ))( M ) → F ( V ⊗ W )( M ) with m − 1 X,Y ,M : V ⊗ ( W ⊗ M ) → ( V ⊗ W ) ⊗ M . Example 2.3. Let ( A, m, e ) be a n asso ciative algebra in C . L et C A be the categor y of rig h t A -mo dules in C . This is an abelia n left C -mo dule category with action V ⊗ ( M , η ) = ( V ⊗ M , (id V ⊗ η ) α V ,M ,A ) and ass o ciativity constra in t α X,Y ,M , for X , Y ∈ C , M ∈ C A . See [19, sec. 3.1]. Example 2. 4 . W e shall denote b y V ec f the categor y of finite dimensional vector spaces ov er k . This is a semisimple tensor catego ry with only one simple o b ject. F or every k -linear a belian c a tegory M , there is a n unique V ec f -mo dule category structure with a ction k ⊕ n ⊗ X := X ⊕ n . See [24, Lemma 2.2.2 ]. Example 2.5 . L e t H b e a Hopf algebra and let B ⊆ A b e a left faithfully flat H - Galois extension. Let M B and M H be the catego ries of right B -mo dules and right H -como dules, resp ectively . Reca ll that categ ory of r igh t Hopf ( H , A )-mo dules M H A is b y definition the categor y ( M H ) A of r igh t A -mo dules over M H . By Schneider’s structure theorem [26], the functor M B → ( M H ) A , M 7→ M ⊗ B A , is a categ o ry equiv alence w ith inv erse M 7→ M co H . So M B has a M H -mo dule category structure as in Exa mple 2.3. F or tw o C -mo dules categ ories M and N , a C - linear functor or module functor ( F, φ ) : M → N consis ts o f a n exa c t functor F : M → N and natura l is omorphisms φ X,M : F ( X ⊗ M ) → X ⊗ F ( M ) , such that ( X ⊗ φ Y ,M ) φ X,Y ⊗ M F ( m X,Y ,M ) = m X,Y ,F ( M ) φ X ⊗ Y ,M , for all X , Y ∈ C , M ∈ M . If M , N a re k -linea r a b elian categories, then F V e c f ( M , N ) is the ca teg ory o f k -linear exact functors, so F V e c f ( M , N ) = F ( M , N ). CLIFFOR D THEOR Y F OR TENSOR CA TEGORIES 5 A C -linear natura l trans formation b et ween C -linear functors ( F, φ ) , ( F ′ , φ ′ ) : M → N , is a k -linear na tur al trans fo rmation σ : F → F ′ such that φ ′ X,M σ X ⊗ M = ( X ⊗ σ M ) φ X,M , for all X ∈ C , M ∈ M . W e s hall denote the ca tegory of C - linear functors a nd C -linear natural transfor- mations b et ween C -mo dules categories M , N b y F C ( M , N ). Definition 2.6. Let C b e a tensor catego r y and let M b e a C -module categor y . A C - s ubmodule categ ory o f M is a Serre sub categor y N ⊆ M of M , such that is a C - mo dule catego ry with re s pect to ⊗ . A C -mo dule ca tegory will b e called simple if it doe s not contain any non-trivial C - s ubmodule categ o ry . F or C - linear functors ( G, ψ ) : D → M and ( F, φ ) : M → N , the comp osition is a C -linear functor ( F ◦ G, θ ) : D → N , where θ X,L = φ X,G ( L ) F ( ψ X,L ) , for X ∈ C , L ∈ D . So w e have a bifunctor F C ( M , N ) × F C ( D , M ) → F C ( D , N ) (( F, φ ) , ( G , ψ )) → ( F , φ ) ◦ ( G, ψ ) . 2.1. Strict m odul e categories. A monoidal ca tegory is called strict if its asso ciativity constraint is the identit y . In the same wa y we say that a module categor y ( M , ⊗ , α ) ov er a str ict mo no idal category ( C , ⊗ , 1) is strict , if α is the identit y . The main res ult of this subsection establishes tha t every monoidal ca tegory C is monoidally equiv alen t to a stric t mo noidal ca tegory C ′ , such that every mo dule category ov er C ′ is equiv alent to a strict one. Lemma 2.7. L et C b e a m onoidal c ate gory. Then F C ( C , C ) ∼ = C , wher e C is a left C - m o dule c ate gory with the tens or pr o duct and the isomorphism of asso ciativi ty. Mor e over, C op ∼ = F C ( C , C ) as monoidal c ate gorie s (wher e C op = C as c ate gories, and tensor pr o duct V ⊗ op W = W ⊗ V ). Pr o of. W e define the functor d ( − ) : C → F C ( C , C ) as follo ws: given V ∈ C , the functor ( b V , α − , − V ) : C → C , W 7→ W ⊗ V , α X,Y ,V : b V ( X ⊗ Y ) → X ⊗ b V ( Y ) is a C -module functor . If φ : V → V ′ is a mor phism in C , we define the natural transformatio n b φ : b V → c V ′ , as b φ W = id W ⊗ φ : b V ( W ) = W ⊗ V → c V ′ ( W ) = W ⊗ V ′ . The natura l iso morphism α − ,W,V : b V ◦ c W → \ V ⊗ op W , gives a str ucture o f monoidal functor to d ( − ). Let ( F , ψ ) : C → C b e a mo dule functor. Then we hav e a natura l isomorphism σ X = ψ X, 1 : F ( X ) = F ( X ⊗ 1 ) → X ⊗ F (1 ) = [ F (1)( X ) , such that α X,Y ,F (1) σ X ⊗ Y = α X,Y ,F (1) ψ X ⊗ Y , 1 = id X ⊗ ψ Y , 1 ◦ ψ X,Y = id X ⊗ σ Y ◦ ψ X,Y . 6 C ´ ESAR GALINDO That is, σ X is a na tural isomor phism mo dule b et ween ( F , ψ ) and ( F (1) , α − , − ,F (1) ). So the functor is essentially sur jectiv e. Let φ : b V → c V ′ be a C - linear natura l morphism. Then α X, 1 ,V φ X = id X ⊗ φ 1 α X, 1 ,V ′ , so φ X = id X ⊗ φ 1 , and the monoida l functor d ( − ) is faithful a nd full. Hence, by [16, Theorem 1, p. 91] a nd [22, Pro position 4.4.2], the functor is an equiv alence of monoidal ca tegories.  Prop osition 2.8. L et C b e a monoidal c ate gory, then ther e is a strict monoidal c ate gory C , such that every mo dule c ate gory over C is e quivalent to a strict C -mo dule c ate gory and C is monoidal ly e quivalent to C . Pr o of. Let C = F C ( C , C ) op . By Lemma 2.7, C is monoidally equiv alent to C . Let ( M , ⊗ , m ) b e a le ft C - module categ ory . The ca tegory F C ( C , M ) is a strict left C -mo dule categ o ry with the comp osition of C -mo dule functor s. Conversely , if M ′ is a C -module catego ry , then M ′ is a mo dule categor y o ver C , using the tenso r equiv alence c ( − ) : C → F C ( C , C ). In a simila r w ay to the pro of of the Le mma 2.7, the functor M → F C ( C , M ) M 7→ ( c M , m − , − ,M ) , is an equiv alence of C -mo dule catego ries. So every mo dule catego ry ov er C is equiv a- lent to a strict one.  2.2. T ensor pro duct of mo dule categories. Definition 2 . 9. [27, pp. 518] Le t ( M , m ) a nd ( N , n ) be r igh t and left C -module categorie s res p ectively . A C -bilinear functor ( F , ζ ) : M × N → D is a bifunctor F : M × N → D , together with natural isomor phisms ζ M ,X,N : F ( M ⊗ X , N ) → F ( M , X ⊗ N ) , such that F ( m M ,X,Y , N ) ζ M ,X ⊗ Y ,N F ( M , n X,Y ,N ) = ζ M ⊗ X,Y ,N ζ M ,X,Y ⊗ N , for all M ∈ M , N ∈ N , X , Y ∈ C . A natural tra ns formation ω : ( F , ζ ) → ( F ′ , ζ ′ ) b et ween C -bilinear functors , is a natural transforma tion ω M ,N : F ( M , N ) → F ′ ( M , N ) such tha t ω M ,X ⊗ N ζ M ,X,N = α ′ M ,X,N ω M ⊗ X,N , for all M ∈ M , N ∈ N , X ∈ C . Example 2. 1 0. Let C b e a tensor catego ry and let D be a tenso r sub categor y of C . Let ( M , m ) be a C - module categ ory and let N b e a D - module subca tegory of the D -module ca tegory M . Then the functor C × N → M , ( V , N ) → V ⊗ M , has a canonical D -bilinear s tructure. Her e, C is a D -mo dule categor y in the obvious wa y , and the D -bilineal is o morphism is given by m . W e shall denote by Bil( M , N ; D ) the category of C -bilinea r functors. In [27] a k -linear categor y (not neces sarily ab e lian) M ⊠ C N is constructed by gene r ators and relatio ns , together with a C -bilinear functor T : M × N → M ⊠ C N , that induces an equiv alence of k -linear categories F ( M ⊠ C N , D ) → Bil( M , N ; D ), for every k -linear catego ry D . CLIFFOR D THEOR Y F OR TENSOR CA TEGORIES 7 The ob jects of M ⊠ C N ar e finite sums of symbols [ X , Y ], for ob jects X ∈ M , Y ∈ N . Morphisms ar e sums of compo sitions of symbols [ f , g ] : [ X , Y ] → [ X ′ , Y ′ ] , for f : X → X ′ , g : Y → Y ′ , symbols α X,V ,Y : [ X ⊗ V , Y ] → [ X , V ⊗ Y ] , for X ∈ M , V ∈ C , N ∈ N , and symbo ls for the formal inv erse of α X,V ,Y . The generator morphisms s atisfy the following r elations: (i) Linea rit y: [ f + f ′ , g ] = [ f , g ] + [ f ′ , g ] , [ f , g + g ′ ] = [ f , g ] + [ f , g ′ ] , [ af , g ] = [ f , ag ] = a [ f , g ] , for all mor phisms f , f ′ : M → M ′ in M , g , g ′ : N → N ′ in N , and a ∈ k . (ii) F unctor ialit y: [ f f ′ , g g ′ ] = [ f ′ , g ′ ][ f , g ] , [id M , id N ] = id [ M ,N ] , for all f : M → M ′ , f ′ : M ′ → M ′′ in M , and g : N → N ′ , g ′ : N ′ → N ′′ in N . (iii) Natur a lit y: α M ′ ,V ′ ,N ′ [ f ⊗ u, g ] = [ f , u ⊗ g ] α M ,V ,N , for morphisms f : M → M ′ in M , u : V → V ′ in C , and g : N → N ′ in N . (iv) Coher e nc e : [ α M ,V ,W , id N ] α M ,V ⊗ W,N [id M , α V ,W,N ] = α M ⊗ V ,Y ,N α M ,X,Y ⊗ N , for all M ∈ M , N ∈ N , V , W ∈ C . Let M , N b e k -linear catego ries, then the categ ory M ⊠ N := M ⊠ V e c f N , is the tensor pr oduct of k -linear tensor catego ries; see [5, Definition 1.1.1 5]. If M a nd N are semisimple catego r ies, this is the Delig ne ’s tenso r pro duct of ab elian categ ories [9]. Definition 2.11. [27, pp. 517] Let C 1 and C 2 be tensor categor ie s. A C 1 - C 2 - bimo dule categ ory is a k -linear ab elian catego ry M , equipp ed with exact bifunctors ⊗ : C 1 × M → M , ⊗ : M × C 2 → M , and naturals is o morphisms α X,Y ,M : ( X ⊗ Y ) ⊗ M → X ⊗ ( Y ⊗ M ) , α X,M ,S : ( X ⊗ M ) ⊗ S → X ⊗ ( M ⊗ S ) , α M ,S,T : ( M ⊗ S ) ⊗ T → M ⊗ ( S ⊗ T ) , for all X , Y ∈ C 1 , M ∈ M , S, T ∈ C 2 , such tha t M is a left C 1 -mo dule categ ory with α X,Y ,M , it is a right C 2 -mo dule categor y with α M ,S,T , and id X ⊗ α Y ,M ,S α X,Y ⊗ M ,S α X,Y ,M ⊗ id S = α X,Y ,M ⊗ Z α X ⊗ Y ,M ,S , id X ⊗ α M ,S,T α X,M ⊗ S,T α X,M ,S ⊗ id T = α X,M ,S ⊗ T α X ⊗ M ,S,T . If M is a ( C 1 , C 2 )-bimo dule categor y and N is a right C 2 -bimo dule ca tegory , then the catego ry M ⊠ C 2 N ha s a structure of left C 1 -mo dule categor y . The actio n of an ob ject X ∈ C 1 ov er a n ob ject [ M , N ] ∈ M ⊠ C 2 N is given by X ⊗ [ M , N ] = [ X ⊗ M , N ] . 8 C ´ ESAR GALINDO The action o ver the morphisms α M ,Y ,N is given by id X ⊗ α M ,Y ,N = α X ⊗ M ,X,Y , N ◦ [ α − 1 X,M ,Y , N ], and the a ssocia tivit y is [ α X,Y ,M , N ] : [( X ⊗ Y ) ⊗ M , N ] → [ X ⊗ ( Y ⊗ M ) , N ] . Prop osition 2.12. L et C b e a t ensor c ate gory. L et M 1 , M 2 b e C -bimo dule c ate- gories, and let M 3 b e a right C -mo dule c ate gory. Then (1) C ⊠ C M 3 ∼ = M 3 , as left C -mo dule c ate gories. (2) ( M 1 ⊠ C M 2 ) ⊠ C M 3 ∼ = M 1 ⊠ C ( M 2 ⊠ C M 3 ) , as left C -mo dule c ate gories. (3) if M = ⊕ n i M i , N = ⊕ m j N j , as right and left C - mo dule c ate gories, then M ⊠ C N = ⊕ i,j M i ⊠ C N j , as k -line ar c ate gories. Pr o of. By P r opos ition 2.8, we can supp ose that all mo dule categor ie s are str ict. (1) The functor F : M → C ⊠ C M M 7→ [1 , M ], is a ca teg ory equiv alence. In effect, using the isomor phism α 1 ,X,M , we ca n s ee that F is essentially sur jectiv e, and every mor phism b et ween [1 , M ] and [1 , N ] is o f the form [1 , f ], for f : M → N . Then F is faithful a nd full. Moreov er, with the na tural iso morphism η V ,M = α 1 ,V ,M : F ( V ⊗ N ) → V ⊗ F ( N ), the pa ir ( F, η ) is a C -linear functor, since η V ⊗ W,M = α 1 ,V ⊗ W,M = α V ,W,M ◦ α 1 ,V ,W ⊗ M =id V ⊗ α 1 ,W ,M ◦ η V ,W ⊗ M =id V ⊗ η W ,M ◦ η V ,W ⊗ M . (2) F o r every o b ject M 1 ∈ M 1 , the functor λ M 1 : M 2 × M 3 → ( M 1 ⊠ C M 2 ) ⊠ C M 3 , where λ M 1 ( M 2 , M 3 ) = [[ M 1 , M 2 ] , M 3 ] , λ M 1 ( f , g ) = [[id M 1 , f ] , g ] , with the natural transformation η 1 M 2 ,V ,M 3 := α [ M 1 ,M 2 ] ,V ,M 3 , is a C -bilinear func- tor. So we have a family of functors λ M 1 : M 2 ⊠ C M 3 → ( M 1 ⊠ C M 2 ) ⊠ C M 3 , λ M 1 ([ M 2 , M 3 ]) = [[ M 1 , M 2 ] , M 3 ]. No w, the functor M 1 × ( M 2 ⊠ C M 3 ) → ( M 1 ⊠ C M 2 ) ⊠ C M 3 , ( M 1 , [ M 2 , M 3 ]) 7→ λ M 1 ([ M 2 , M 3 ]) , with the na tur al tra nsformation η 2 M 1 ,V , [ M 2 ,M 3 ] = α M 1 ,V , [ M 2 ,M 3 ] , is a C -bilinear func- tor. So we hav e a functor π : M 1 ⊠ C ( M 2 ⊠ C M 3 ) → ( M 1 ⊠ C M 2 ) ⊠ C M 3 , [ M 1 , [ M 2 , M 3 ]] 7→ [[ M 1 , M 2 ] , M 3 ]. The functor π is e s sen tially surjective and π ([ f , [ g , h ]]) = [[ f , g ] , h ] π ([id M 1 α M 2 ,V ,M 3 ]) = α [ M 1 ,M 2 ] ,V ,M 3 π ( α M 1 ,V , [ M 2 ,M 3 ] ) = [ α M 1 ,V ,M 2 , id M 3 ] . So π is faithful and full, hence by [16, Theor em 1, pp. 91], the functor π is a category equiv alence. Finally , note that the functor π is C -linear. (3) Its follows directly b y the c onstruction of M ⊠ C N .  Let C b e a G -graded tenso r categor y . Note that if H ⊆ G is a subgro up o f G , then the ca tegory C H = ⊕ τ ∈ H C τ is a tens o r subca tegory of C . W e shall say that an o b ject U ∈ C is invertible if the functor U ⊗ ( − ) : C → C , V 7→ U ⊗ V is a categor y equiv alence or, equiv alently , if ther e is an o b ject U ∗ ∈ C , such that U ∗ ⊗ U ∼ = U ⊗ U ∗ ∼ = 1. CLIFFOR D THEOR Y F OR TENSOR CA TEGORIES 9 Prop osition 2.13. L et C b e a G -gr ade d c ate gory and let H ⊆ G b e a su b gr oup of G . Su pp ose that the every c ate gory C σ has at le ast one invertible obje ct, for every σ ∈ G . L et M b e a mo dule c ate gory over C H = ⊕ h ∈ H C h . Then the k -line ar c ate gory C ⊠ C H M is an ab elian c ate gory. Mor e ove r, sinc e C is a C - C H -bimo dule c ate gory then C ⊠ C H M is a left mo dule c ate gory over the tensor c ate gory C . Pr o of. W e shall supp ose that the tenso r catego ry C is strict. Let Σ = { e, σ 1 , . . . } b e a set of r epresen tatives of the cos ets G/H . Since C = L σ ∈ Σ C σH as right C H -mo dule categorie s, C ⊠ C H M = L σ ∈ Σ C σH ⊠ C H M , as k -linear ca tegories, by Prop osition 2.12. F or every coset σH in G , let U σ ∈ C σ be an inv ertible ob ject. The functor U σ : C H → C σH , V 7→ U σ ⊗ V is a catego ry equiv alence with a quasi- in verse U ∗ σ : C σH → C H , W → U ∗ σ ⊗ W . Then we ca n assume, up to iso morphisms, that every ob ject o f C σH is of the form U σ ⊗ V , wher e V ∈ C H . Let L i [ V i , M i ] ∈ C σH ⊠ C H M . F o r every V i there exist V ′ i such that V i ∼ = U σ ⊗ V ′ i . Then L i [ V i , M i ] ∼ = [ U σ , L i V ′ i ⊗ M i ], i.e. , we can a ssume, up to isomor phisms, that every ob ject o f C σH ⊠ C H M is o f the form [ U σ , M ]. If U σ ⊗ V ∼ = U σ then V ∼ = 1; so every mor phism [ U σ , M ] → [ U σ , M ′ ] is of the form [id U σ , f ], wher e f : M → M ′ . Then the functor : M → C σH ⊠ C H M , f 7→ [id U σ , f ] is an equiv alence o f k -linear categor ies. W e define the ab elian structur e ov er C σH ⊠ C H M as the induced by this equiv alence. F or the s econd part, note that C ⊠ C H M = M σ ∈ Σ C σH ⊠ C H M as ab elian categor y , so we need to pr o ve that if (2.1) 0 → [ U σ , S ] → [ U σ , T ] → [ U σ , W ] → 0 is an exact s e quence in C σH ⊠ C H M , then the sequence (2.2) 0 → [ X ⊗ U σ , S ] → [ X ⊗ U σ , T ] → [ X ⊗ U σ , W ] → 0 is ex a ct for all X ∈ C . Since C = L σ ∈ G C σ we can supp ose that X ∈ C τ , then [ X ⊗ U σ , S ] , [ X ⊗ U σ , T ] , [ X ⊗ U σ , W ] ∈ C τ σH ⊠ C H M . Let U τ σ ∈ C τ σ with inverse ob ject U ∗ τ σ ∈ C ( τ σ ) − 1 , so w e hav e the following commutativ e dia gram 0 → [ X U σ , S ] [ X U σ , T ] [ X U σ , W ] → 0 0 → [ U τ σ ( U ∗ τ σ X U σ ) , S ] [ U τ σ ( U ∗ τ σ X U σ ) , T ] [ U τ σ ( U ∗ τ σ X U σ ) , W ] → 0 0 → [ U τ σ , ( U ∗ τ σ X U σ ) S ] [ U τ σ , ( U ∗ τ σ X U σ ) T ] [ U τ σ , ( U ∗ τ σ X U σ ) W ] → 0 ❄ ✲ [id ,f ] ❄ ✲ [id ,g ] ❄ ❄ α U τ σ ,U ∗ τ σ X U σ ,S ✲ [id ,f ] ❄ α U τ σ ,U ∗ τ σ X U σ ,T ✲ [id ,g ] ❄ α U τ σ ,U ∗ τ σ X U σ ,W ✲ [id , id U ∗ τ σ X U σ f ] ✲ [id , id U ∗ τ σ X U σ g ] 10 C ´ ESAR GALINDO where tensor symbo ls b et ween ob jects and mor phis m hav e b een omitted a s a space- saving measur e. Then the sequence (2.2) is exact if and only if the sequence (2.3) 0 → [ U τ σ , ( U ∗ τ σ X U σ ) S ] → [ U τ σ , ( U ∗ τ σ X U σ ) T ] → [ U τ σ , ( U ∗ τ σ X U σ ) W ] → 0 is exact. By definition the sequence (2.3) is exact if and only if the sequence 0 → ( U ∗ τ σ X U σ ) S → ( U ∗ τ σ X U σ ) T → ( U ∗ τ σ X U σ ) W → 0 in M is e x act, but since M is a C H -mo dule ca tegory it is exac t.  3. Strongl y graded tensor ca tegories Recall from Definition 1.3 that the G -g r aded categ o ry C is called s trongly g raded if the inclusio n functor C σ · C τ → C στ is a catego ry equiv alence for all σ, τ ∈ G . Lemma 3.1. L et C b e a ten sor c ate gory. Then C is str ongly gr ade d over G if and only if the c ate gory C σ has at le ast one multiplic atively invertible element, for al l σ ∈ G . Mor e over, in this c ase the Gr othendi e ck ring of C is a G -cr osse d pr o duct. Pr o of. If C is strongly gra ded by definition there there exist ob jects V 1 , . . . , V n ∈ C σ , W 1 , . . . , W t ∈ C σ − 1 , such that 1 ∼ = L i,j V i ⊗ W j , then End C ( L i,j V i ⊗ W j ) ∼ = End C (1) ∼ = k , so n = 1 , t = 1. That is, there exist o b jects V ∈ C σ , W ∈ C σ − 1 , suc h that V ⊗ W ∼ = 1. Conv ersely , suppo se that C σ has at leas t an inv ertible ob ject for all σ ∈ G . Let U σ ∈ C σ be an in vertible ob ject with dual ob ject U ∗ σ ∈ C σ − 1 , s o V ∼ = U σ ⊗ ( U ∗ σ ⊗ V ) for every V ∈ C στ . Then the inc lus ion functor is essentially surjective, and therefor e it is an eq uiv alence. Recall that by definition a gra ded r ing A = ⊕ σ ∈ G A σ is a cro ssed pro duct over G if for all σ ∈ G the abe lia n g roup A σ has at leas t an inv ertible element. T hus, by the firs t part o f the lemma, the Grothendieck ring of C is a G -cro ssed pr o duct if C is stro ng ly g raded.  Example 3.2. Let V ec G ω be the semisimple category o f finite dimensional G -gr aded vector spaces, with constraint of as sociativ ity ω ( a, b, c )id abc for all a, b , c ∈ G , where ω ∈ Z 3 ( G, k ∗ ) is a 3-co cycle. Then V ec G ω is a stro ng ly G -gr a ded tenso r category . Example 3 .3. Let C ⋊ G a c rossed pro duct tensor catego ry . As we saw in the Int ro duction, the category C ⋊ G is a strong G -graded tensor category . No w if we take a norma lized 3-co cycle β ∈ Z 3 ( G, k ∗ ) and we define a new as socia tor α β [ U,σ ] , [ V , τ ] , [ W ,ρ ] = β ( σ , τ , ρ ) α [ U,σ ] , [ V , τ ] , [ W ,ρ ] , then the new tenso r categor y is strongly G -graded to o. 3.1. Mo dule categories graded o v er a G -set. Definition 3. 4. Let C = ⊕ σ ∈ G C σ be a gra ded tenso r c ategory a nd let X b e a left G -set. A left X -gr ade d C -mo dule c ate gory is a left C -module category M endow ed with a deco mposition M = ⊕ x ∈ X M x , int o a direct sum of full ab elian sub categories , such that for all σ ∈ G , x ∈ X , the bifunctor ⊗ maps C σ × M x to M σx . An X -gr ade d C -mo dule fun ctor F : M → N is a C -mo dule functor such that F ( M x ) is mapped to N x , for a ll x ∈ X . CLIFFOR D THEOR Y F OR TENSOR CA TEGORIES 11 Definition 3.5. A left X -gra ded C - submodule c a tegory of M is serr e sub category N of M such that N is an X -gra ded C - mo dule categ ory with re s pect to ⊗ , and the gra ding N x ⊆ M x , x ∈ X . An X -gra ded C -module catego ry will b e called simple if it contains no nontrivial X -graded C - s ubmodule categ o ry . Lemma 3.6. L et C b e a G -gr ade d t ensor c ate gory and let H ⊆ G a sub gr oup of G . If N is a left C H -mo dule c ate gory, then the c ate gory C ⊠ C H N is a G/ H -gr ade d C - m o dule c ate gory with gr ading ( C ⊠ C H N ) σH = ( ⊕ τ ∈ σH C τ ) ⊠ C H N . Pr o of. Let Σ = { e, σ 1 , . . . } b e a set o f representativ es of the cosets of G mo dulo H . By Prop osition 2.12, C ⊠ C H N = L σ ∈ Σ C σH ⊠ C H N as k -linear categories, and b y the definition of the action of C , the mo dule c a tegory C ⊠ C H N is G/H -gr aded.  Prop osition 3. 7. L et C b e a st r ongly G -gr ade d tensor c ate gory, and let ( A, m, e ) b e an algebr a in C H . Then C ⊠ C H ( C H ) A ∼ = C A as G/H - gr ade d C -mo dule c ate gories. Pr o of. Let Σ = { e, σ 1 , . . . } a set of repr esen tatives of the co sets of G mo dulo H . The C -module category C A has a canonical G/H -g rading: if ( M , ρ ) is an A -mo dule then ( M , ρ ) = M σ ∈ Σ ( M σH , ρ σH ) , where M σH = L h ∈ H M σh , ρ σH = L h ∈ H ρ σh . Let us consider the cano nic a l C -linear functor F : C ⊠ C H ( C H ) A → C A , [ V , ( M , ρ )] 7→ ( V ⊗ M , id V ⊗ ρ ) . W e shall fir st show that F is a ca tegory equiv alence. Let U σ ∈ C σH be a n inv ertible ob ject for every coset o f H on G . Let ( M , ρ ) ∈ C A be a ho mo geneous A -mo dule of degree σ − 1 H . Then the A -mo dule ( U σ ⊗ M , id U σ ⊗ ρ ) is also an A - mo dule in C H and F ([ U σ − 1 , ( U σ ⊗ M , id U σ ⊗ ρ )]) ∼ = ( M , ρ ) ∈ C A . So F is an essentially surjective functor . W e c an s upp ose, up to iso morphisms, that every ob ject of C σH ⊠ C H ( C H ) A is of the form [ U σ , ( M , ρ )]. Then F ([ U g , ( M , ρ )]) = ( U g ⊗ M , id U g ⊗ ρ ). N ow it is clear that the functor F is faithful and full, so by [1 6, Theo rem 1, p. 91] the functor F is a catego ry equiv alence.  Theorem 3.8. L et C b e a str ongly gr ade d t en sor c ate gory over a gr oup G and let X b e a t ra nsitive G -set. L et M and N b e non zer o X -gr ade d mo dules c ate gori es. Then (1) M ∼ = C ⊠ C H M x as X -gr ade d C -mo dule c ate gories, wher e, for al l x ∈ X , H = s t( x ) is the stabilizer sub gr oup of x ∈ X . (2) Ther e is a bije ctive c orr esp ondenc e b etwe en isomorphisms classes of X - gr ade d C -mo dule functors ( F, η ) : M → N and C H -mo dule funct ors ( T , ρ ) : M x → N x . Pr o of. (1) Cho ose x ∈ X , a nd denote H = s t( x ). In a similar wa y to the pr o of of Prop osition 3.7, the cano nic a l functor µ : C ⊠ C H M x → M [ V , M ] → V ⊗ M , is a catego ry equiv alence a nd it resp ects the g rading. 12 C ´ ESAR GALINDO The pro of of par t (1) of the theor em is completed by showing that the functor µ is a C -mo dule functor. Indeed, by Prop osition 2.8 we can a s sume that the mo dule categorie s are strict, hence µ ( V ⊗ [ W, M x ]) = µ ([ V ⊗ W, M x ]) = ( V ⊗ W ) ⊗ M x = V ⊗ ( W ⊗ M x ) = V ⊗ µ ([ W, M x ]) , i.e. , µ is a C - mo dule functor. (2) By the first part we c a n supp ose N = C ⊠ H N x . Let ( F , µ ) : N x → M x be a C H -mo dule functor, the functor I ( F ) : C × N x → M ( S, N ) 7→ S ⊗ F ( N ) with the na tural transformatio n id S ⊗ µ V ,N : I ( F )( S, V ⊗ N ) → I ( S ⊗ V , N ) is a C H -bilinear functor, so we have a functor I ( F ) : C ⊠ C H N x → M [ S, N ] 7→ V ⊗ F ( N ) α S,V ,N 7→ id S ⊗ µ V ,S , and this is a n X -graded C - mo dule functor in the o bvious wa y . Let ( F = ⊕ s ∈ X F s , η ) : C ⊠ C H N x → M b e an X -gra ded C -module functor. Consider the natur a l iso morphism σ [ V ,N ] := η V , [1 ,N ] : F ([ V , N ]) → V ⊗ F x ([1 , N ]) = I ( F x )([ V , N ]) , σ X ⊗ [ V ,N ] = η X ⊗ V , [1 , N ] = id X ⊗ η V , [1 ,N ] ◦ η X, [ V ,N ] = id X ⊗ σ [ V ,N ] ◦ η X, [ V ,N ] . So σ is a natural isomor phism of mo dule functor s.  Corollary 3.9. L et C b e a st r ongly G -gr ade d tensor c ate gory. Then ther e is a bije ct ive c orr esp ondenc e b etwe en mo dule c ate gories over C e and G -gr ade d C -mo dule c ate gori es. Pr o of. It is a particular c a se of Theorem 3.8, with X = G .  Prop osition 3.1 0. F or every σ, τ ∈ G , the c anonic al functor f σ,τ : C σ ⊠ C e C τ → C στ , f σ,τ ([ X, Y ]) = X ⊗ Y , is an e quivalenc e of C e -bimo dule c ate gori es. Pr o of. Let us co nsider the g raded C -mo dule ca tegory C ( τ ), where C = C ( τ ) a s C - mo dule catego ries, but with grading ( C ( τ )) g = C τ g , for τ ∈ G . Since C ( τ ) e = C τ , b y Theorem 3.8, the canonica l functor µ ( C ( τ ) e ) : C ⊠ C e C τ → C ( τ ), [ X , Y ] 7→ X ⊗ Y is an equiv alence o f G -graded C -module categ o ries. So the r estriction µ ( C ( τ )) σ : C σ ⊠ C e C τ → C ( τ ) σ = C τ σ is a C e -mo dule categor y equiv alence. But by definition µ ( C ( τ )) σ = f σ,τ . It is clear tha t f σ,τ is a C e -bimo dule categor y functor, so the pro of is finished.  CLIFFOR D THEOR Y F OR TENSOR CA TEGORIES 13 4. Clifford Theor y In this se ction we shal l su pp ose that C is a str ongly gr ad e d tensor c ate gory over a gr oup G . W e s hall denote b y Ω C e the s et of equiv alences clas ses of s imple C e -mo dule cat- egories. G iven a C e -mo dule categor y M , we sha ll denote b y Ω C e ( M ) the set o f equiv alences classes of simple C e -submo dule categor ies of M . Lemma 4.1. L et M b e a C e -mo dule c ate gory. Then for al l σ ∈ G , the c ate gory C σ ⊠ C e M is a simple C e -mo dule c ate gory if and only if M is. Pr o of. If N is a prop e r C e -submo dule categor y of M , then the catego ry C σ ⊠ C e N is a C e -submo dule categor y of C σ ⊠ C e M , so the C e -mo dule categor y C σ ⊠ C e M is not simple. By Pro p osition 3.10, we hav e that M ∼ = C g − 1 ⊠ C e ( C g ⊠ C e M ), so if C g ⊠ C e M is not simple, then M is not simple neither.  By Lemma 4.1 and Pro position 3.10, the group G a cts o n Ω C e by G × Ω C e → Ω C e , ( g , [ X ]) 7→ [ C g ⊠ C e X ] . Let M be a C -mo dule catego ry , and let N ⊆ M be a Se r re sub category . W e s hall denote by C σ ⊗N the Serr e sub category g iv en b y Ob ( C σ ⊗N ) = { sub quotients of V ⊗ N : V ∈ C σ , N ∈ N } . (Recall that a sub quotient ob ject is a sub ob ject of a quotient ob ject.) Prop osition 4. 2. L et M b e a C -mo dule c ate gory and let N b e a C e -submo dule c ate gory of M . Then C σ ⊠ C e N ∼ = C σ ⊗N , as C e -mo dule c ate gories, for al l σ ∈ G . Pr o of. Define a G -g raded C -module ca tegory by g r - N = L σ ∈ G C σ ⊗N , with action ⊗ : C σ × C g ⊗N → C σg ⊗N V σ × T 7→ V σ ⊗ T . Since C e ⊗N = N as C e -mo dule categor y , by Theor em 3.8 the canonical functor µ ( N ) : C ⊠ C e N → g r − N is a categor y eq uiv alence of G -g raded C -mo dule catego ries and the restr iction µ σ : C σ ⊠ C e N → C σ ⊗N is a C e -mo dule catego ry equiv alence.  Corollary 4. 3. L et M b e a C -mo dule c ate gory. The action of G on Ω C e induc es an action of G on Ω C e ( M ) . Pr o of. Let N be a simple C e -submo dule catego ry o f M . By Pro position 4.2 the functor µ σ : C σ ⊠ C e N → C σ ⊗N [ V , N ] 7→ V ⊗ N , is a C e -mo dule category eq uiv alence, so C σ ⊠ C e N is equiv alent to a C e -submo dule category of M .  Let M b e a n ab elian categ ory and let N , N ′ be Ser r e subca teg ories of M , we shall deno te N + N ′ the Ser re subc a tegory of M where O b ( N + N ′ ) = { sub quotient s o f N ⊕ N ′ : N ∈ N , N ′ ∈ N ′ } . It will b e calle d the sum categ ory of N and N ′ . 14 C ´ ESAR GALINDO Pr o of of the The or em 1.5. (1) Let N b e a simple ab elian C e -submo dule category of M , the canonical functor µ : C ⊠ C e N → M [ V , N ] 7→ V ⊗ N , is a C -mo dule functor and µ = ⊕ σ ∈ G µ σ , where µ σ = µ | C σ . By Pr opos itio n 4 .2 each µ σ is a C e -mo dule categor y equiv alence with C σ ⊗N . Since M is simple, every o b ject M ∈ M is isomo r phic to some sub quotient of µ ( X ) for some ob ject X ∈ C ⊠ C e N . Then M = P σ ∈ G C σ ⊗N a nd ea c h C σ ⊗N is an ab elian simple C e -submo dule categor y . Let S, S ′ be simple ab elian C e -submo dule ca tegories of M . Then there e x ist σ , τ ∈ G such that C σ ⊠ C e N ∼ = S , C τ ⊠ C e N ∼ = S ′ , and by Prop osition 3.10, S ′ ∼ = C τ σ − 1 ⊠ C e S . So the action is tra nsitiv e. (2) Let H = st([ N ]) b e the stabilizer subgroup of [ N ] ∈ Ω C e ( M ) and let M N = X h ∈ H C h ⊗N . Since H a cts tra nsitiv ely on Ω C e ( M N ), the C H -mo dule ca tegory M N is simple. Let Σ = { e, σ 1 , . . . } b e a s et o f re pr esen tatives of the cosets of G mo dulo H . The map φ : G/H → Ω C H ( M ), φ ( σ H ) = [ C σ ⊗M N ] is an isomorphism of G -sets. Then M has a s tructure of G/ H - graded C -module category , wher e M = ⊕ σ ∈ Σ C σ ⊗ M N . By Pro p osition 3.8, M ∼ = C ⊠ C H M N as C -module ca tegories.  R emark 4.4 . Nikshyc h a nd Gelak i noted the exis tence of a g rading by a transitive G -set for every indecomp osable mo dule category ov er a G - g raded fusion catego ry [15, P ropo sition 5 .1]. Using the Theorem 3.8 and [1 5, Pro position 5.1], we can do an alter nativ e pro of of the main theor em in the case o f s trongly graded fusion categorie s. 5. Simple module ca tegories over crossed product tensor ca tegories and G -equiv ariant of tenso r ca tegories 5.1. G -equiv arian tization of tensor categories. Le t G b e a gro up a cting on a category (not necess a rily by tensor equiv alences) C , ∗ : G → Aut( C ) , so we ha ve the following da ta • functors σ ∗ : C → C , fo r each σ ∈ G , • isomo r phism φ ( σ, τ ) : ( σ τ ) ∗ → σ ∗ ◦ τ ∗ , for a ll σ, τ ∈ G . The category of G -inv ariant ob jects in C , deno ted by C G , is the ca teg ory defined as follows: an ob ject in C G is a pair ( V , f ), where V is an ob ject of M and f is a family of iso morphisms f σ : σ ∗ ( V ) → V , σ ∈ G , such that, for all σ , τ ∈ G , (5.1) φ ( σ , τ ) f στ = f σ σ ∗ ( f τ ) . A G -equiv ariant morphism φ : ( V , f ) → ( W , g ) b et ween G -equiv ariant ob jects ( V , f ) and ( W, g ), is a morphism u : V → W in C such that g σ ◦ σ ∗ ( u ) = u ◦ f σ , for all σ ∈ G . If the catego ry C is a tenso r ca teg ory , and the action is by tensor auto equiv a lences ∗ : G → Aut ⊗ ( C ) , then w e hav e a natural isomor phism • ψ ( σ ) V ,W : σ ∗ ( V ) ⊗ σ ∗ ( W ) → σ ∗ ( V ⊗ W ), for all σ ∈ G , V , W ∈ C . CLIFFOR D THEOR Y F OR TENSOR CA TEGORIES 15 th us C G has a tensor pro duct defined by ( V , f ) ⊗ ( W , g ) := ( V ⊗ W, h ) , where h σ = u σ v σ ψ ( σ ) − 1 V ,W , and unit ob ject (1 , id 1 ). Example 5.1. The como dule category of a co cen tral cleft exact sequence of Hopf algebras. Let G be a group and le t (5.2) k → A → H π → k G → k be a co central cleft exa ct se q uence of Hopf alg ebras, i.e. , the pr o jection π : H → k G admits a k G -colinear sectio n j : k G → H , inv ertible with resp ect to conv olution pro duct. Since the sequenc e is cleft, the Hopf algebr a H has the str ucture of a bicrossed pro duct H ∼ = A τ # σ k G with resp ect a certain compatible datum ( · , ρ, σ, τ ), where · : A ⊗ k G → A is a w eak action, σ : k G ⊗ k G → A is inv ertible co cycle, ρ : k G → k G ⊗ A is a weak coaction, τ : k G → A ⊗ A is a dual co cycle, sub jects to compatibility conditio ns in [1, Theorem 2 .2 0]. The pro jection in (5.2), is called co cent ra l if π ( h 1 ) ⊗ h 2 = π ( h 2 ) ⊗ h 1 , this is equiv alen t to the w eak c oaction ρ to b e trivia l, see [17, Lemma 3.3]. Lemma 5.2. L et H ∼ = A τ # σ k G b e a bicr osse d pr o duct with t r ivial c o action. Then the gr oup G acts over the c ate gory of right A -mo dules A M , and H M ∼ = ( A M ) G as tensor c ate gories, wer e H M is the c ate gory of right H -mo dules . Pr o of. See [17, Lemma 3.3].  R emark 5.3 . Let H b e a semisimple Hopf a lg ebra ov er C . By [15, Pro of of Theor e m 3.8], the fusion categ ory H M of finite dimensional como dules is G -g raded (not necessary stro ngly graded) if a nd only if ther e is a co cen tra l exac t sequence of Hopf alge br as a s in (5.2). In this case, the fusion ca tegory H M is weakly Mor ita equiv alen t to a G -cro ssed tenso r category A M ⋊ G . Tha t is, H M ∼ = F A M ⋊ G ( N , N ), for some indeco mposable A M ⋊ G -mo dule catego r y N . 5.2. The obstruction to a G -action o ver a tens or category. Let C be a tensor category , we shall denote by Aut ⊗ ( C ) the gr o up of tensor a uto-equiv alences, it is the s et o f isomor phisms classes of auto-equiv alences of C , with the m ultiplication induced by the comp osition: [ F ][ F ′ ] = [ F ◦ F ′ ]. Every G -action over a tensor category induces a group homomor phism ψ : G → Aut ⊗ ( C ). W e shall say that a homomorphis m ψ : G → Aut ⊗ ( C ) is realizable if there is some G -action such the induced gr oup homomo rphism c o incides with ψ . The go al of this subsection is show that for every homomorphism ψ : G → Aut ⊗ ( C ), there is an a ssocia ted e le ment in a 3rd coho mology gr oup whic h is zero if and only if ψ is realizable. Moreov er, every realizatio n is in corresp ondence (non natural) with a n element of a 2nd cohomolog y group. 5.2.1. Cate goric al-gr oups. A categorica l-group G is a mono idal c a tegory where ev - ery ob ject, a nd every arr o w is inv ertible, see [4] for a complete r eference. A tr ivial example of a categor ical-group is the discrete categor ic a l-group G , as- so ciated to a g r oup G . The ob jects o f G are the elemen ts of G , the arr o ws are only the identities, and the tenso r pro duct is the multiplication of G . 16 C ´ ESAR GALINDO Complete in v ariants of a categor ical-group G with r espect to monoidal equiv a- lences are π 0 ( G ) , π 1 ( G ) , α ∈ H 3 ( π 0 ( G ) , π 1 ( G )) , where π 0 ( G ) is the g roup of isomor phism cla sses of ob jects, π 1 ( G ) is the a belian group of automor phisms of the unit ob ject. The gro up π 1 ( G ) is a π 0 ( G )-mo dule in the natura l wa y , a nd α is a third cohomolo gy class g iv en by the asso ciator. Complete inv a r ian ts of a mo noidal functor F : G → G ′ betw een categor ical- groups, with r espect to monoidal isomor phisms ar e π 0 ( F ) : π 0 ( G ) → π 0 ( G ′ ) , π 1 ( F ) : π 1 ( G ) → π 1 ( G ′ ) , θ ( F ) : π 0 ( G ) × π 0 ( G ) → π 1 ( G ′ ) where π 0 ( F ) is a morphism of g roups, π 1 ( F ) is a morphism of π 0 ( G )-mo dules and θ ( F ) is a class in C 2 ( π 0 ( G ) , π 1 ( G ′ )) /B 2 ( π 0 ( G ) , π 1 ( G ′ )), such that δ ( θ ( F )) = π 1 ( G ′ ) ∗ ( φ ( G )) − π 0 ( G ′ ) ∗ ( φ ( G ′ )) , where π 0 ( F ) ∗ : C ∗ ( π 0 ( G ′ ) , π 1 ( G ′ )) → C ∗ ( π 0 ( G ) , π 1 ( G ′ )) , π 1 ( F ) ∗ : C ∗ ( π 0 ( G ) , π ( G )) → C ∗ ( π 0 ( G ) , π 1 ( G ′ )) , are the maps of co chain complexes induced by the g roup morphisms π 0 ( F ) and π 1 ( F ). The next r esult follows from the last discussion o r s ee [4 ]. Prop osition 5. 4. L et G b e a c ate goric al gr oup and let f : G → π 0 ( G ) b e a morphism of gr oups. Then ther e is a monoidal functor F : G → G , su ch that f = π 0 ( F ) if and only if the c oho molo gy class of f ∗ ( φ ) is zer o. If f ∗ ( φ ) is zer o, t he classes of e quivalenc e of monoidal fu n ctors F : G → G ar e in one t o one c orr esp ondenc e with H 2 ( G, π 1 (( G ))) . Pr o of. The monoidal category G has inv a riant s π 0 ( G ) = G and π 1 ( G ) = 0. Then, the pro of follows fro m the discussions of this subsection, or se e [4].  5.2.2. The obstruction to a G - action over a tensor c ate gory and cyclic actions. Let Aut ⊗ ( C ) be the monoidal c a tegory of tensor auto-equiv alences of a tensor c ate- gory C , where ar r o ws are tensor natural iso morphisms a nd tensor pro duct given by comp osition of functors. Then Aut ⊗ ( C ) is a c a tegorical-gr oup. The in v ariants asso ciated to Aut ⊗ ( C ) (see Subsection 5 .2 .1) are Then π 0 (Aut ⊗ ( C )) = Aut ⊗ ( C ), and π 1 (Aut ⊗ ( C ) ) = Aut ⊗ (id C ), the gr oup of monoidal natural isomor- phisms of the iden tity functor. Theorem 5. 5. L et C b e a tensor c ate gory and let G b e a gr oup. Conside r the data ( Aut ⊗ ( C ) , A ut ⊗ ( id C ) , [ a ]) asso ciate d to the c ate goric al-gr oup Aut ⊗ ( C ) . Then • a gr oup homomorphism f : G → Aut ⊗ ( C ) is r e ali ze d as a G -action over C if and only if 0 = [ f ∗ ( a )] ∈ H 3 ( G, Aut ⊗ ( id C )) . • If the gr oup homomorp hism f : G → Aut ⊗ ( C ) is r e ali zable, then the set of r e ali zations of f is in 1-1 c orr esp ondenc e with Z 2 ( G, Aut ⊗ ( id C )) , and the set of e quivalenc es classes of r e alizations of f is in 1-1 c orr esp ondenc e with H 2 ( G, Aut ⊗ ( id C )) . Pr o of. The Theor em is a particular ca se o f the Prop osition 5.4.  CLIFFOR D THEOR Y F OR TENSOR CA TEGORIES 17 Recall that if A is a module for the c y clic g roup C m of order m , then: H n ( C m ; A ) = ( { a ∈ A : N a = 0 } / ( σ − 1) A, if n = 1 , 3 , 5 , . . . A C m / N A, if n = 2 , 4 , 6 , . . . , (5.3) where N = 1 + σ + σ 2 + · · · + σ m − 1 , see [31, Theo rem 6.2 .2]. Given an elemen t a ∈ A C m the asso ciated 2-co cycle can b e constructed as follows. γ a ( σ i , σ j ) = ( 1 , if i + j < m, a i + j − m , if i + j ≥ m. (5.4) Let F : C → C b e a monoidal equiv alence, such tha t there is a monoidal natural isomorphism α : F m → id C . By Theorems 5.5 and (5.3), the induced homomo r- phism ψ : C m → Aut ⊗ ( C ) is realizable if a nd only if id F ⊗ α ⊗ id F − 1 = α . In this case, tw o natural isomo rphisms α 1 , α 2 : F m → id C realize eq uiv alent C m -actions if and only if there is a monoidal natura l isomo rphism θ : F 1 → F 2 such that θ m F 1 = F 2 . Corollary 5.6. L et C b e a t ensor c ate gory and let C m b e cyclic gr oup of or der m . Then the set of C m -actions over C ar e in 1-1 c orr esp ondenc e with p airs ( F , α ) , wher e F : C → C is a monoidal e quivalenc e, α : F m → id C is a monoidal natur al isomorphi sm such id F ⊗ α = α ⊗ id F . Two p airs ( F 1 , α 1 ) and ( F 2 , α 2 ) induc e e quivalent C m -actions if and only if ther e is a monoidal natur al isomorphism θ : F 1 → F 2 such that θ m F 1 = F 2 . The description of the 2-co cycle asso ciated to a C m -inv ariant element (5.4), is as follows: the C m -action ψ : C m → Aut ⊗ ( C ) asso ciated to a pair ( F , α ) is ψ (1) = id C , ψ ( σ i ) = F i , i = 1 , . . . m − 1, and the mo noidal natural isomo r phisms φ α ( σ i , σ j ) : F i ◦ F j → F i + j φ α ( σ i , σ j ) = ( id C , if i + j < m, id F ⊗ α i + j − m = α i + j − m ⊗ id F , if i + j ≥ m. (5.5) 5.2.3. The bigalois gr oup of a H op f algebr a. Le t H b e a Hopf a lg ebra. A r igh t H - Galois ob ject is a non-zer o r igh t H -como dule algebra A such that the linear ma p defined by c an : A ⊗ A → A ⊗ H, a ⊗ b 7→ ab (0) ⊗ b (1) is bijective. A fiber functor F : H M → V ec k is an exa ct and faithful monoidal functor that commutes with colimits. Ulbrich defined in [30] a fib er functor F A asso ciated with each H -Galois ob ject A , in the for m F A ( V ) = A  H V , where A  H V is the cotensor pro duct over H of the right H -co module A and the left H -co module V . He show ed in lo c. cit. that this defines a categ o ry equiv alence b etw een H -Galois o b jects a nd fiber functors ov er H M . Similarly , a left H -Galo is ob ject is a non-zero le ft H -como dule a lgebra A such that the linear map can : A ⊗ A → H ⊗ A, a ⊗ b 7→ a ( − 1) ⊗ a (0) b is bijective. Let H a nd Q b e Hopf algebra s. An H - Q -big alois ob ject is an algebr a A which is an H - Q -bicomodule a lgebra and b oth a left H -Galo is o b ject a nd a rig h t Q -Ga lois ob ject. Let A b e an H -Galo is ob ject. Schauen burg shows in [23, Theor em 3.5] that there is a Hopf a lgebra L ( A, H ) such that A is a L ( A, H )- H -bigalo is ob ject. 18 C ´ ESAR GALINDO The Ho pf a lgebra L ( A, H ) is the T a nnakian-Krein reco nstruction fr om the fiber functor a s socia ted to A . By [23, Corollary 5.7 ], the following categor ies are equiv- alent: • The monoidal catego ry BiGa l ( H ) , where ob jects a r e H -bigalo is ob ject, morphism are mor phism of A –bicomo dules algebr a s, a nd tensor pro duct A  H B , the cotensor pro duct ov er H . • The monoidal catego r y Aut ⊗ ( H M ) . Schauen burg defined the g roup BiGal( H ) a s the set of is omorphism classes of H -bigalois ob jects with multiplication induced by the cotenso r pro duct. This group coincides with Aut ⊗ ( H M ). Is ea sy to s ee that for the Hopf a lgebra k G of a group G , BiGal( k G ) = Aut ( G ) ⋊ H 2 ( G, k ∗ ). Ho wev er, it is difficult to find an explicit description in gener al. The group BiGal( H ) ha s b een calculated for some Hopf algebras , for example: T aft algebras [25], monoida l non-se misimple Hopf alg e bras [6], the algebra of function ov er a finite gr oup coprime to 6 [8]. 5.2.4. The ab elian gr oup Aut ⊗ ( id C ) for Hopf algebr as. Prop osition 5.7. L et H b e a H opf algebr a. Then Aut ⊗ ( id H M ) ∼ = G ( H ) ∩ Z ( H ) the gr oup of c entr al gr oup-likes of H . Pr o of. The maps H ⊗ k ( M ⊗ k N ) → ( H ⊗ H N ) ⊗ k ( H ⊗ H N ) , h ⊗ m ⊗ n 7→ ( h (1) ⊗ m ) ⊗ ( h (2) ⊗ n ), and H ⊗ k k → k , h ⊗ 1 7→ ǫ ( h ), induce natura l H -mo dule morphisms F M ,N : H ⊗ H ( M ⊗ N ) → ( H ⊗ H M ) ⊗ ( H ⊗ H N ) F 0 : H ⊗ H k → k . The identit y monoidal functor is naturally isomor phic to ( · H · ⊗ H ( − ) , F , F 0 ), a nd it is well-know that every H -bimo dule endomorphism is of the for m ψ c : H → H , h 7→ ch , for s ome c ∈ Z ( H ). The na tural transfor ma tion asso ciated to ψ c is monoidal if and only if ψ c is a bimo dule co algebra ma p, i.e. , if c is a g r oup-like.  F or the gr oup alge bra k G , we have Aut ⊗ (id kG M ) ∼ = Z ( G ) the center of G , a nd for a Hopf alg ebra C G , where G is a finite group, we hav e Aut ⊗ (id k G M ) ∼ = G/ [ G, G ]. Let C be a complex fusion categ ory , i .e. , a semisimple tensor catego ry with finitely ma ny is omorphisms classes of s imple ob jects. In [1 5] it is s ho wn that every fusion catego ry is naturally gr aded by a g roup U ( C ) called the universal grading group of C . The group U ( C ) only dep ends o f the Grothendieck r ing o f C . In [15, Prop osition 3.9 ] it is shown that if C is a fusion categ ory and G = U ( C ) is the univ ersa l gra ding g roup of C , then Aut ⊗ (id C ) ∼ = d G ab the gro up o f characters of the maxima l ab elian quotient of G . Corollary 5.8. L et H b e a semisimple almost-c o c ommutative Hopf algebr a. Then U ( H M ) ∼ = Z ( H ) ∩ G ( H ) . Pr o of. Since H is almost-co mmutative the Grothendiek ring is co mmutative, hence the universal gra ding gro up is ab elian. By P ropo sition 5.7 and [15, P ropo sition 3.9] U ( H M ) ∼ = Z ( H ) ∩ G ( H ).  CLIFFOR D THEOR Y F OR TENSOR CA TEGORIES 19 5.3. G -inv arian t actions on mo dul e categorie s . Let C b e a tensor categ ory and let ( σ, ψ ) : C → C b e a monoidal functor . If ( M , ⊗ , α ) is a r igh t C - module category , the t wisted C -module categ ory ( M σ , ⊗ σ , α σ ) is defined by: M = M σ as category , with M ⊗ σ V = M ⊗ σ ( V ), and α σ M ,V ,W = id M ⊗ ψ V ,W ◦ α M ,V ,W . Definition 5.9. Le t C be a tensor c ategory , M a left C -mo dule ca tegory and σ : C → C a monoidal functor. W e sha ll say that the functor ( T , η ) : M → M σ is a σ -equiv a rian t functor o f M if is a C -mo dule functor . Given an actio n o f a gr oup G over C , the mo dule category M is ca lled G -inv ariant if there is a σ -equiv ariant functor for each σ ∈ G . Let σ, τ : C → C b e monoidal functor s. Let also ( T , η ) : M → M σ a σ -equiv ariant functor a nd ( T ′ , η ′ ) : M → M τ a τ -in v ariant functor. W e define their comp osition by ( T ′ T , T ′ ( η )( η ′ ( T × T ))) : M → M . This gives a σ ◦ τ -equiv ariant functor o f M . Given a G -action ov er a monoidal categ ory C and a G -in v ariant mo dule cat- egory M , we denote b y Aut G C ( M ) the following mo no idal catego r y: o b jects are σ ∗ -equiv ariant functor s, for all σ ∈ G , mo r phisms are natural iso mo rphisms of mo dule functors, the tensor pro duct is co mposition o f C -mo dule functor s and the unit ob ject is the identit y functor o f M . Definition 5 . 10. Let ( σ ∗ , φ ( σ , τ ) , ψ ( σ )) : G → Aut ⊗ ( C ) b e an action of G ov er a tensor categor y C , and let M be a G -inv ariant C - mo dule category . A G -inv ar ian t functor over M is a monoidal functor ( σ ∗ , φ, ψ ) : G → Aut G C ( M ), such that σ ∗ is a σ ∗ -inv ariant functor, for all σ ∈ G . R emark 5.11 . (1) A C -module category M with a G -inv a rian t functor is called a G -e quivariant C -mo dule c ate gory in [1 3, definition 5.2]. (2) Let C be a G -inv aria n t monoida l categor y . The monoidal categor y Aut G C ( M ) is a graded ca tegorical-gr oup and the gr oup Aut G C ( M ) has a natur al gr oup epimor- phism π : Aut G C ( M ) → G . So, if a gr oup homo mo rphism ψ : G → Aut G C ( M ) is realizable, then π ψ = id G . Suc h g r oup homomo rphisms will be called split . (3) Let ψ : G → Aut G C ( M ) b e a s plit g roup homomo rphism. If a ∈ H 3 (Aut G C ( M ) , H ) is the 3- c ocycle a s socia ted to the categorica l-group Aut G C ( M ) , then like in Theor em 5.5, ψ is r ealizable if a nd only if the 3-co cycle ψ ∗ ( a ) is a 3- cob oundary , and the set of realizations o f ψ is in corres pondence with the elements of a 2nd cohomology group. The following result app ears in [2 8, Sec . 2]. Prop osition 5.1 2. L et C ⋊ G b e a cr osse d pr o duct tens or c ate gory. Then ther e is a bije ct ive c orr esp ondenc e b etwe en stru ctur es of C ⋊ G -mo dule c ate gory and G - invariant functors over a C -mo dule c ate gory M . Pr o of. Let M b e a C ⋊ G -mo dule catego ry . E ac h ob ject [1 , σ ], σ ∈ G , defines an equiv alence σ ∗ : M → M , M 7→ [1 , σ ] ⊗ M . With φ ( σ, τ ) M = α (1 ,σ ) , (1 ,τ ) ,M the constraint of asso ciativity , this defines a monoidal functor G → Aut( M ). The categ ory M is a C - module ca teg ory with V ⊗ M = [ V , e ] ⊗ V a nd since [1 , σ ] ⊗ [ V , e ] = [ σ ∗ ( V ) , e ] ⊗ [1 , σ ] we have a natural isomorphis m ψ ( σ ) V ,M : σ ∗ ( V ) ⊗ σ ( M ) → σ ( V ⊗ M ), by ψ ( σ ) V ,M = α − 1 (1 ,σ ) , ( V ,e ) ,M ◦ α ( σ ∗ ( V ) , e ) , (1 ,σ ) ,M . This defines a G -inv ariant functor. 20 C ´ ESAR GALINDO Conv ersely , if G → Aut G C ( M ) is a G -inv ariant functor, w e hav e natural isomor- phisms φ ( σ, τ ) M : σ ∗ τ ∗ ( M ) → σ τ ∗ ( M ), ψ ( σ ) V ,M : σ ( V ) ⊗ σ ( M ) → σ ( V ⊗ M ). Then, we ma y define the action on M b y ( V , σ ) ⊗ M := V ⊗ σ ∗ ( M ) , and constra int of asso ciativity α ( V ,σ ) , ( W ,τ ) ,M = id V ⊗ σ ∗ ( W ) ⊗ φ ( σ, τ ) M ◦ α V ,σ ∗ ( W ) ,σ ∗ ( τ ∗ ( M )) ◦ id V ⊗ ψ ( σ ) − 1 W ,M .  Suppo se that the group G is finite and the tensor categ ory C is a fusion categor y ov er an alg ebraically close d field of characteristic zer o. Then the module categor ie s ov er C ⋊ G and C G are in bijective corr espondence by [1 8, Prop osition 3.2]. If M is C ⋊ G -mo dule categ ory then, by P ropo s ition 5.1 2, there is a G -a ction on M , a nd the catego ry M G is a C G -mo dule categor y with ( V , f ) ⊗ ( M , g ) := ( V ⊗ M , h ) , where h σ = g σ h σ ψ ( σ ) − 1 V ,M . F or a k -linear mono ida l categor y and G finite where char( k ) . | G | , Theo rem [28, Theorem 4.1 ] says that e v ery C G -mo dule categor y is of the for m M G for a C ⋊ G - mo dule categor y . The following result app ears in [1 3] for fusio n categor ies and finite groups. Theorem 5.13. Simple m o dule c ate gories over C ⋊ G ar e in bije ctive c orr esp ondenc e with the fol lowing data: • a sub gr oup H ⊆ G , • a simple H -invariant C mo dule c ate gory M , • a monoidal functor H → Aut H C ( M ) . If the gr oup G is finite then the mo dule c ate gories over C G ar e in bije ction with the same data. Pr o of. By Theorem 1.5, if N is a n simple C ⋊ G -mo dule categ ory , then it is isomor - phic to C ⊠ C ⋊ H M for some subg roup H ⊆ G a nd a simple C ⋊ H -mo dule category M , such tha t M is H - inv ariant. In particula r it follows that the restriction of M to C is simple. Now the corres p ondence follows from Prop osition 5.1 2. If the g roup G is finite, then the corr espondence follows fro m [28, Theorem 4.1] or [18, Pr opositio n 3.2].  Suppo se that G is a finite group a nd H ∼ = A τ # σ k G is a bicrossed pro duct with trivial coaction. Then the mo dule ca tegories ov er H M are of the form N G , for some G -equiv ariant A M -mo dule c a tegory N . Mor eo ver, the module category is simple if and only if N is s imple. Example 5.14 . Let N ≥ 2 be an integer a nd let q ∈ C b e a primitive N -th r oot of unit y . The T aft algebr a T ( q ) is the C -a lgebra pr esen ted by g enerators g and x with relations g N = 1, x N = 0 and g x = q xg . The algebra T ( q ) car ries a Hopf algebr a structure, determined b y ∆ g = g ⊗ g , ∆ x = x ⊗ 1 + g ⊗ x. Then ε ( g ) = 1 , ε ( x ) = 0 , S ( g ) = g − 1 , and S ( x ) = − g − 1 x . It is known that CLIFFOR D THEOR Y F OR TENSOR CA TEGORIES 21 (1) T ( q ) is a p oin ted non-semisimple Hopf a lgebra, (2) the gr oup of group-like elements of T ( q ) is G ( T ( q )) = h g i ≃ Z / ( N ), (3) T ( q ) ≃ T ( q ) ∗ , (4) T ( q ) ≃ T ( q ′ ) if a nd only if q = q ′ . Prop osition 5.15. L et G b e a gr oup, then the set of G -actions on the tensor c ate gory T ( q ) M of T ( q ) -c omo dules is in 1-1 c orr esp ondenc e with the set of gr oup homomorph ism fr om G to C ∗ ⋉ C , wher e C ∗ acts on C by C ∗ × C → C , ( s, t ) 7→ st . Pr o of. By Pr opositio n 5.7, the ab e lia n group Aut ⊗ (id C ) is triv ial, and by [2 5, The- orem 5], Aut ⊗ ( T ( q ) M ) = BiGa l( T ( q )) ∼ = C ∗ ⋊ C . Then by The o rem 5.5, the set of isomorphism classe s of G -a ctions is given by the set of group homo morphism from G to C ∗ ⋉ C .  If G = Z / ( N ) then, by Prop osition 5.15 the p ossible G -actions are par ameterized by pairs ( r , µ ), where r is a no n-trivial N -th r o ot of the unit and µ ∈ C . W e shall denote by A ( α,γ ) the T ( q )-bigalois ob ject as socia ted to the pair ( r , µ ) ∈ C ∗ ⋉ C ∼ = BiGal( T ( q )). See [25, Theor em 5 ]. The T ( q ) M -mo dule ca tegories of rank one ar e in corresp ondence with fib er func- tors on T ( q ) M , a nd these are in turn in 1- 1 corresp ondence with T ( q )-Galois ob jects. By Theorem 2 in lo c. cit. , every T ( q )-Galois ob ject is isomor phic to A (1 ,β ) , β ∈ C , and tw o T ( q )-Galois ob jects A (1 ,β ) , A (1 ,µ ) are isomorphic if and only β = µ . By Theorem 5.13, if there is a semisimple module catego ry of ra nk one ov er C = T ( q ) M ⋊ Z / ( N ), it must be a T ( q ) M -mo dule category Z / ( N )-in v ariant. Suppo se that A (1 ,β ) is Z / ( N )-inv a rian t. Since A ( r,µ )  T ( q ) A (1 ,β ) ∼ = A ( r,µ + β ) , we hav e that µ = 0 . Then if the action is asso ciated to a pair ( r , µ ) whe r e µ 6 = 0, the category C does not a dmit any fib er functor, i.e. , it is not the ca tegory of co modules of a Hopf a lgebra. How ever, since every simple ob ject is invertible, the Perron-F rob enius dimension of the simple ob jects is one. So, by [11, Prop osition 2 .7], the tensor category T ( q ) M ⋊ Z / ( N ) is equiv alent to the ca tegory of representations of a qua si-Hopf algebra. Note tha t the tensor categor y ( T ( q ) M ) G has at least one fiber functor, for every group and e very group action. In fact, since the forgetful functor U : T ( q ) M G → T ( q ) M is monoidal, then the comp osition with the fib er functor of T ( q ) M gives a fiber functor on ( T ( q ) M ) G . Ac kno wledgem en t. The author thanks Pa vel Etingo f, Gas t´ on Garc ´ ıa, Mart ´ ın Mombelli, Sonia Natale, and Dmitri Nikshych for use ful discussions and a dvice. References [1] BibliographyN. Andruskiewitsch and J. D evoto, ‘Extensions of Hopf algebras’, St. Petersbg. Math. J. 7 (1996), 17-52. [2] BibliographyN. Andruskiewitsc h and J.M. Mombelli, ‘On mo dule categories ov er finite- dimensional Hopf algebras’, J. Algebr a 314 (2007), 383–418. [3] BibliographyS. Ar khipov and D. Gaitsgory, ‘ Anoth er realization of the category of mo dules o ver the small quan tum group’, A dv. Math. 17 3 (2003), 114–143. [4] BibliographyJ. Baez and A. D. Lauda, ‘Higher-dimensional algebra V: 2-groups’, The or. and Appl. Cat. 12 N o. 14 (2004), 423–491. [5] BibliographyB. Bak alo v and A. Kirri lo v Jr., ‘Lectures on T ensor categories and mo dular functors’, AM S, (2001). 22 C ´ ESAR GALINDO [6] BibliographyJ. Bichon, ‘ Galois and biGalois ob jects ov er monomial non-semis i mple Hopf Algebras’, J. Pur e Appl. Algebr a 5 (2006) , 653–480. [7] Bibliograph yE. C. Dade, ‘C omp ounding Clifford’s theory’, Ann. Math. , 91 (1970), 236–290 . [8] BibliographyA. Da vydo v, ‘Twi sted aut omorphisms of gr oup algebras’ Pr eprint arXiv:0708.275 8 . [9] BibliographyP . Deli gne, ‘Cat ´ egories tannakiennes, i n : The Gr othendie ck F est sc hrift ’, V ol. II, Progr. Math., 87 , Bi rkhuser, Boston, MA, (1990), 111-195. [10] BibliographyP . Etingof and V. Ostrik, ‘Mo dule categories ov er r epresen tations of S L q (2) and graphs’, Math. R es. Lett. 11 (2004) , 103–114. [11] BibliographyP . Etingof and V. Ostrik, ‘Finite tensor categories’, Mosc. Math. J. 4 (2004), 627–654, 782–783. [12] BibliographyP . Etingof, D. Ni ksh yc h and V. Ostrik, ‘ On fusion categories’, Ann. Math. 162 (2005), 581–642. [13] BibliographyP . Etingof, D. Nikshyc h and V. O s trik, ‘W eakly group-theoretical and solv able fusion categories’, Pr eprint arXiv:0809.30 31 . [14] BibliographyE. F renkel and E. Witten, ‘Geometric endoscop y and mirr or s ymmetry’, Com- mun. Numb er The ory Phys. 2, No. 1,(2008), 113–283 [15] BibliographyS. Gelaki and D. Nikshyc h, ‘N ilpotent fusi on cat egories’, A dv. M ath. , 217 (2008), 1053-1071. [16] BibliographyS. Mac Lane, ‘Categories for the W orking Mathematician’, 2nd edition. Springer V erlar g (1997). [17] BibliographyS. Natale, ‘Hopf algebra extensions of gr oup algebras and T ambara-Y amagami categories’ Pr eprint arXiv:0805.317 2 v1 , (2008). [18] BibliographyD. Nikshyc h, ‘Non group-theoretical semisimple Hopf algebras from group ac- tions on fusi on categories’, Sel. math., New ser. (2008) 145–161. [19] BibliographyV. Os trik, ‘M odule categories, weak Hopf al geb ras and mo dular inv ari an ts’, T ra nsform. Gr oups , 8 (2003), 177-206. [20] BibliographyV. Ostrik, ‘ M odule categories ov er the D rinfeld double of a finite group’, Int. Math. R es. Not. no. 27, (2003), 1507-1520. [21] BibliographyV. Os trik, ‘M odule categories ov er representat ions of S L q (2) in the non- semisimple case’, Ge om. F unct. A nal. (2008) 17, No. 6, 2005–2017. [22] BibliographyN. Saa ve dra Riv ano, ‘Cat´ egories T annakiennes’, L e ctur e notes in mathematics 265 . Spri nge r V er l ag, 1972. [23] BibliographyP . Sc hauen burg, ‘Hopf bi- Galois extensions’, Comm. Algebr a 24 (1996), 3797– 3825. [24] BibliographyP . Scha uenburg, ‘T annak a duality for arbitrary Hopf algebras’, Algebr a Beri c hte 66 , V erlag Reinhard Fische r, M nc hen (1992). [25] BibliographyP . Schaue nburg, ‘Bigalois ob j ect s ov er the T aft algebras’, Isr ael J. Math. 115 (2000), 101–123. [26] BibliographyH.-J. Schneider, ‘Principal Homogeneus spaces for arbi trary Hopf algebras’, Is- r ael J. of Math. 72 (1990) , 167–195. [27] BibliographyD. T ambara, ‘A Duality for Mo dules o ver Monoidal Categories of Represen ta- tions of Semisimpl e Hopf A lgebras’, J. A lgebr a 296 (2006), 301–322. [28] BibliographyD. T am bara, ‘Inv ariants and semi-direct products for finite group actions on tensor categories’, J. Math. So c. Jap an 53 (2001), 429–456. [29] BibliographyD. T ambara and S. Y amagami, ‘T ensor cate gories with fusion rules of self- dualit y for finite ab elian groups’, J. A lgebr a 209 (1998), 692–707. [30] BibliographyUlbric h, K -H, ‘Fib er functor of finite dimensional comodules’, Manuscripta Math. 65 (1989), 39–46. [31] BibliographyCharles A. Wibel, ‘ An int ro duction to homological algebra’, Cambridge studies in advanc e d mathematics 3 8 , (1994). Dep ar t a mento de M atem ´ aticas Pontificia Univ ersid ad Ja veriana Bogot ´ a, Colomb ia. E-mail addr ess : cesarneyit@gmail .com, cesar.galindo@ gmail.com