math.QA 2011-09-12 0

Module categories over finite pointed tensor categories

We study exact module categories over the representation categories of finite-dimensional quasi-Hopf algebras. As a consequence we classify exact module categories over some families of pointed tensor categories with cyclic group of invertible objets…

Authors: Cesar Galindo, Martin Mombelli

MODULE CA TEGORIES O VER FINITE POINTED TENSOR C A TEGORIES C ´ ESAR GALINDO AND MAR T ´ IN MOMBE LLI Abstra ct. W e study exact mo dule categories ov er the representation categories of fin ite-dimensional quasi-Hopf algebras. As a consequence w e clas sify exact mod u le categor ies ov er some fami lies of pointed tensor categories with cy clic group of invertible ob jets of order p , where p is a prime num b er. Mathematics Subje ct Classific ation (2010): 18D10, 16W30, 19D23. Keywor ds: T ensor c ate gory, mo dule c ate gory, quasi-Hopf algebr a. Introduction F or a giv en tensor categ ory C a mo dule c ate gory ov er C , or a C - mo dule , is the categorificatio n of the n otion of mo dule ov er a ring, it consist of an Ab elian catego ry M together with a b iexact f unctor ⊗ : C × M → M satisfying natural asso ciativit y and unit axioms. A mo dule catego ry M is exact [EO1] if f or any pro jectiv e ob ject P ∈ C and any M ∈ M the ob ject P ⊗ M is again pr o jectiv e. The notion of mo d ule category has b een used w ith profit in the theory of tensor categories, see [DGNO],[ENO1], [ENO2]. Int erestingly , the notion of mo dule categories is related with d iv erse areas of mathematics and mathe- matical ph ysics such as su bfactor theory [Oc], [BEK]; extensions of ve rtex algebras [KO], Calabi-Y au alg ebras [Gi], Hopf algebras [N], affine Hec ke al- gebras [BO] and conformal field theory , see for example [BFS], [CS1], [CS2], [FS1], [FS2 ], [O1 ]. The classification of exact mo d ule catego ries o v er a giv en tensor category w as undertak en by sev eral authors: 1. When C is the semisimple quotien t of U q ( sl 2 ) [Oc], [KO], [EO2], 2. o ve r the category of finite-dimensional S L q (2)-co mo dules [O3 ], 3. o ve r the tensor categ ories of representat ions of finite su p ergroups [EO1], 4. for an y group-theoretical tensor category [O2], 5. o ve r the T am b ara-Y amagami categories [Ga2], [MM ], Date : March 30, 2018. The w ork of M.M. w as supp orted by CON ICET, Secyt (UNC), Mincyt ( C´ ordoba) Argentina. 1 2 GALINDO, MOMBELLI 6. o ve r the Hageerup fusion catego ries [GS], 7. o ve r Rep( H ), where H is a lifting of a quantum linear space [Mo2 ]. In this pap er we a re concerned with the classification of exact mo dule cat- egories o v er s ome families of finite n on-semisimple p oin ted tensor categories that are not equiv alen t to the r epresen tation categories of Hopf algebras. An ob j ect X in a tensor category is in ve rtible if there is another ob j ect Y suc h that X ⊗ Y ≃ 1 ≃ Y ⊗ X . A p ointe d tensor c ate gory is a tensor category suc h that ev ery s imple ob ject is inv ertible. The in v ertible ob jects form a group. P oint ed tensor catego ries with cyclic group of inv er tib le ob jects were studied in [EG1], [EG2], [EG3] and later in [A]. An y finite p oin ted tensor category is equiv alent to the represent ation cat- egory of a fi nite-dimensional quasi-Hopf algebra A . In the case w h en the group of inv ertible elemen ts is a cyclic group G there exists an action of G on Rep( A ) suc h that the equiv arian tization R ep( A ) G is equiv alent to th e represent ation categ ory of a finite-dimensional p oin ted Hopf algebra H , see [A]. The p urp ose of this wo rk is to relate mod ule categories o v er Rep( A ) and mo dule categories o v er Rep( H ) and whenev er is p ossible obtain a clas- sification of exact mo dule cat egories o ver Rep( A ) assu ming th at we kno w the classification for Rep ( H ). Module ca tegories o ver any quasi-Hopf alge - bra are paramete rized by Mo rita equ iv arian t equiv alence classes of comod ule algebras. W e w ould like to establish a corresp ondence as follo ws:    Morita quiv alence classes of H -comodu le algebras suc h that G ⊆ K 0    − − − → ← − − −    Morita equiv alence classes of A -comodu le algebras ( K , Φ λ )    . The conten ts of the pap er are the follo wing. In Section 2 we recall the notion of exact mo dule catego ry , the notion of tensor pro d uct of mo dule catego ries o ver a tens or category . In Section 3 we recal l the notion of G - graded te nsor catego ries, G -actions of tensor categories an d crossed p r o ducts tensor categories. W e also recall the G -equiv ariantiz ation construction of tensor categories and mo d u le categories. Section 4 is devot ed to stud y como dule algebras o ve r quasi-Hopf algebras and ho w they giv e rise to m o dule catego ries. Next, in Section 5 w e s tu dy the equiv arian tization of the r epresen tation category of a qu asi-Hopf algebra and the equiv arian tizatio n of como d ule alge br as. W e describ e the datum that giv es rise to an action in a represen tation catego ry of a comod ule algebra, that we cal l a cr osse d system and w e pro v e that the equiv ariantiza tion of mo dule categories are mo dules o ver a certain crossed p ro du ct como d ule algebra. In Section 6.1 we recall the definition of a family of finite-dimensional basic quasi-Hopf algebras int ro duced by I. Angiono [A] that are denoted b y A ( H, s ), where H is a coradically graded Hopf algebra with cyclic group of group-lik e elemen ts. A particular class of these qu asi-Hopf algebras w ere MODULE CA TEGORIES OVER FINITE POINTED TENSOR C A TEGORIES 3 in tro du ced b y S. Gelaki [Ge] and later used b y Etingof and Gela ki to clas- sify certain families of p oint ed tensor categ ories. There is an action of a group G ⊆ G ( H ) on R ep ( A ( H, s )) suc h that Rep( A ( H, s )) G ≃ Rep ( H ) [A]. F or an y left H -comod ule algebra K suc h th at k G ⊆ K 0 w e con- struct a left A ( H, s )-como dule algebra. W e pr o v e that in the case that | G ( H ) | = p 2 , wh ere p is a p rime n umber, the repr esen tation category of this family of como d u le alge br as is big enough to con tain all mo d ule cate- gories o ve r Rep( A ( H , s )). W e apply this result to classify mo du le categories in th e case wh en H is th e b osonization of a quantum linear space. Ac kno wledgmen ts. W e are ve ry grateful to Iv´ an An giono f or many fru it- ful con v ersations and for p atien tly answering our q u estions on his w ork [A]. W e also thank the referee for his constru ctiv e commen ts. 1. Preliminaries and not a tion Hereafter k will denote an algebraically closed field of characte ristic 0. All vecto r spaces and algebras w ill b e considered ov er k . If H is a Hopf a lgebra and A is an H -como dule algebra via λ : A → H ⊗ k A , w e shall say that a (right ) ideal J is H -costable if λ ( J ) ⊆ H ⊗ k J . W e shall sa y that A is (right) H -simple, if ther e is no non trivial (righ t) ideal H - costable in A . If H is a finite-dimensional Hopf algebra then H 0 ⊆ H 1 ⊆ · · · ⊆ H m = H will denote the coradical filtration. When H 0 ⊆ H is a Hopf subalgebra then the asso ciated graded algebra gr H is a coradically graded Hopf al gebra. If ( A, λ ) is a left H -como du le algebra, the coradical filtration on H induces a filtration on A , giv en by A n = λ − 1 ( H n ⊗ k A ) called the L o ewy filtr ation . 1.1. Finite tensor cate gories and tensor functors. A tensor c ate gory over k is a k -linear Ab elian rigid monoidal catego ry . A finite tensor c ate gory [EO1] is a tensor category suc h that it has a finite num b er of isomorp hism classes of simple ob jects, Hom spaces are fi nite-dimensional k -ve ctor spaces, all ob j ects ha ve finite length, ev ery simple ob ject h as a pro jectiv e cov er and the un it ob ject is simp le. Hereafter all tensor catego ries w ill b e considered ov er k and ev ery fun ctor will b e assu med to b e k -linear. If C , D are tensor categories, the collection ( F , ξ , φ ) : C → D is a tensor functor if F : C → D is a functor, φ : F ( 1 C ) → 1 D is an isomorph ism and for an y X , Y ∈ C the family of natural isomorphisms ζ X,Y : F ( X ) ⊗ F ( Y ) → F ( X ⊗ Y ) satisfies (1.1) ζ X,Y ⊗ Z (id F ( X ) ⊗ ζ Y , Z ) a F ( X ) ,F ( Y ) ,F ( Z ) = F ( a X,Y ,Z ) ζ X ⊗ Y , Z ( ζ X,Y ⊗ id F ( Z ) ) , (1.2) l F ( X ) = F ( l X ) ζ 1 ,X ( φ ⊗ id F ( X ) ) , (1.3) r F ( X ) = F ( r X ) ζ X, 1 (id F ( X ) ⊗ φ ) , 4 GALINDO, M OMBELLI If ( F , ζ ) , ( G, ξ ) : C → D are tensor fun ctors, a natur al tensor tr ansfor- mation γ : F → G is a natural transformation s uc h th at γ X ⊗ Y ζ X,Y = ξ X,Y ( γ X ⊗ γ Y ) for all X , Y ∈ C . 2. M odule ca tegories A (left) mo dule c ate gory o v er a tensor category C is an Ab elian cate - gory M equipp ed with an exac t bifunctor ⊗ : C × M → M , that we will sometimes refer as the action , natural associativit y and unit isomorphisms m X,Y ,M : ( X ⊗ Y ) ⊗ M → X ⊗ ( Y ⊗ M ), ℓ M : 1 ⊗ M → M sub ject to natural asso ciativit y and u nit y axioms. See for example [EO 1]. A mo du le category M is exact , [EO 1], if for an y pro jectiv e ob ject P ∈ C the ob ject P ⊗ M is pro jectiv e in M for all M ∈ M . Sometimes w e shall also sa y that M is a C -mo du le. Righ t mod ule categ ories and b imo dule categories are defined similarly . If M is a left C -mo d u le then M op is the right C -mo dule o ver the opp osite Ab elian cate gory with action M op × C → M op , ( M , X ) 7→ X ∗ ⊗ M and asso ciativit y isomorphisms m op M ,X ,Y = m Y ∗ ,X ∗ ,M for all X , Y ∈ C , M ∈ M . If C , C ′ , E are tensor categ ories, M is a ( C , E )-bim o dule category and N is an ( E , C ′ )-bimo dule category , we shall denote the tensor pro du ct o ve r E b y M ⊠ E N . Th is category is a ( C , C ′ )-bimo dule category . F or more details on the tensor pro d uct of mo d ule catego ries the reader is referred to [ENO3], [Gr]. A modu le fu n ctor b et w een mo d u le categories M and M ′ o v er a tensor catego ry C is a pair ( T , c ), w h ere T : M → M ′ is a fun ctor and c X,M : T ( X ⊗ M ) → X ⊗ T ( M ) is a natural isomorp hism su c h that for any X , Y ∈ C , M ∈ M : (id X ⊗ c Y , M ) c X,Y ⊗ M T ( m X,Y ,M ) = m X,Y ,T ( M ) c X ⊗ Y ,M (2.1) ℓ T ( M ) c 1 ,M = T ( ℓ M ) . (2.2) W e shall u se the notatio n ( T , c ) : M → M ′ . Ther e is a comp osition of mo dule functors: if M ′′ is another mo dule catego ry and ( U, d ) : M ′ → M ′′ is another mo dule fun ctor then the comp osition (2.3) ( U ◦ T , e ) : M → M ′′ , wh ere e X,M = d X,U ( M ) ◦ U ( c X,M ) , is also a mo dule functor. Let M 1 and M 2 b e mo dule categories o v er C . W e d enote by Hom C ( M 1 , M 2 ) the category whose ob jects are mo dule fu n ctors ( F , c ) from M 1 to M 2 . A morphism b et w een ( F , c ) and ( G , d ) ∈ Hom C ( M 1 , M 2 ) is a natural trans- formation α : F → G suc h that for an y X ∈ C , M ∈ M 1 : d X,M α X ⊗ M = (id X ⊗ α M ) c X,M . (2.4) Tw o modu le cate gories M 1 and M 2 o v er C are e quivalent if there exist mo dule fu nctors F : M 1 → M 2 and G : M 2 → M 1 and natural isomor- phisms id M 1 → F ◦ G , id M 2 → G ◦ F that satisfy (2.4). MODULE CA TEGORIES OVER FINITE POINTED TENSOR C A TEGORIES 5 The direct sum of t w o mo dule catego ries M 1 and M 2 o v er a tensor cat- egory C is the k -linear category M 1 × M 2 with co ordinate-wise mo dule structure. A mod ule cat egory is inde c omp osable if it is not equiv alen t to a direct su m of t w o n on trivial mo d ule categories. If ( F , ξ ) : C → C is a tensor fun ctor and ( M , ⊗ , m ) is a mo d ule cat- egory ov er C we shall denote by M F the mo dule cate gory ( M , ⊗ F , m F ) with the same und erlying Ab elian catego ry with action and asso ciativit y isomorphisms d efined by X ⊗ F M = F ( X ) ⊗ M , m F X,Y ,M = m F ( X ) ,F ( Y ) ,M ( ξ − 1 X,Y ⊗ id M ) , for all X , Y ∈ C , M ∈ M . 3. Equ iv ariantiza tion of tens or ca te gories 3.1. Group actions on tensor categories. W e br iefly recall the group actions on tensor categories and the equiv ariantiza tion construction. F or more details the reader is referred to [DGNO]. Let C b e a tensor category and let Aut ⊗ ( C ) b e the monoidal category of tensor au to-equiv alences of C , arrows are tensor natural isomorphisms and tensor pr o duct the comp osition of m onoidal f u nctors. W e shall denote b y Aut ⊗ ( C ) the group of isomorp hisms classes of tensor auto-equiv alences of C , with th e multiplica tion induced by the comp osition, i.e. [ F ][ F ′ ] = [ F ◦ F ′ ]. F or an y group G w e shall d enote b y G the monoidal category where ob jects are elemen ts of G and tensor pro duct is giv en b y the pro duct of G . An action of the group G o ver a C , is a monoidal f u nctor ∗ : G → Aut ⊗ ( C ). In another words for an y σ ∈ G there is a tensor functor ( F σ , ζ σ ) : C → C , and for an y σ, τ ∈ G , there are natur al tensor isomorphism s γ σ ,τ : F σ ◦ F τ → F στ . 3.2. G -graded t e nsor categories. Let G b e a group and C b e a tensor catego ry . W e sh all say that C is G -graded, if there is a decomp osition C = ⊕ σ ∈ G C σ of C in to a direct sum of fu ll Ab elian su b categories, suc h that for all σ, τ ∈ G , the bifunctor ⊗ m ap s C σ × C τ to C στ . Give n a G -graded tensor category C , and a subgroup H ⊂ G , w e sh all denote b y C H the tensor sub category ⊕ h ∈ H C h . 3.3. G -equiv ariantiz ation of tensor categories. Let G b e a group acting on a tensor category C . An e quivariant ob ject in C is a pair ( X , u ) where X ∈ C is an ob ject together with isomorph isms u σ : F σ ( X ) → X satisfying u στ ◦ ( γ σ ,τ ) X = u σ ◦ F σ ( u τ ) , for all σ, τ ∈ G . A G -equiv arian t morph ism φ : ( V , u ) → ( W , u ′ ) b etw een G -equiv arian t ob j ects ( V , f ) and ( W , σ ), is a morphism φ : V → W in M suc h that φ ◦ u σ = u ′ σ ◦ F σ ( φ ) f or all σ ∈ G . 6 GALINDO, M OMBELLI The tensor category of equ iv arian t ob jects is denoted by C G and it is called the e quivariantization of C . The tensor pro duct of C G is defined by ( V , u ) ⊗ ( W , u ′ ) := ( V ⊗ W , ˜ u ) , where ˜ u σ = ( u σ ⊗ u ′ σ ) ζ − 1 σ , for an y σ ∈ G . The unit ob ject is (1 , id 1 ). 3.4. Crossed pro duct tensor categories and G -inv arian t mo dule cat- egories. Giv en an action ∗ : G → Aut ⊗ ( C ) of G on C , the G -crossed pro d uct tensor category , denoted b y C ⋊ G is defined as follo ws. As an Ab elian cat- egory C ⋊ G = L σ ∈ G C σ , where C σ = C as an Abelian categ ory , the tensor pro du ct is [ X, σ ] ⊗ [ Y , τ ] := [ X ⊗ F σ ( Y ) , σ τ ] , X, Y ∈ C , σ, τ ∈ G, and the unit ob j ect is [1 , e ]. See [T a ] f or the associativit y constrain t and a pro of of the p entag on identit y . If C = Rep( A ) is the repr esen tation category of a finite-dimensional quasi- Hopf algebra A th en C ⋊ G is also a represen tation categ ory of a fi n ite- dimensional quasi-Hopf algebra B . This is an immediate consequence of [EO1, Prop. 2.6] since eac h simp le ob ject W ∈ C ⋊ G is isomorph ic to [ V , e ] ⊗ [1 , σ ], where σ ∈ G and V ∈ Rep( A ) is s im p le. Let d : K 0 ( C ) → Z the Perron-F rob enius dimension, then d ([ V , e ] ⊗ [1 , σ ]) = d ( V ) d ([1 , σ ]) = d ( V ) ∈ Z , wh ere d ([1 , σ ]) = 1 b ecause [1 , σ ] is m ultiplicativ ely in v ertible. 3.5. Equiv ariantization of mo dule categories. W e shall explain anal- ogous pro cedures for equiv arian tization in mo dule categories. Equiv arian t mo dule categories app eared in [ENO2 ]. W e shall use the app roac h giv en in [Ga1]. Let G b e a group and C b e a tensor category equipp ed with an action of G . L et M b e a mo du le category o ve r C . F or any g ∈ G we sh all denote b y M σ the mo dule catego ry M F σ . If σ ∈ G , we shall sa y that an endofun ctor T : M → M is σ - invariant if it has a mo d ule stru cture ( T , c ) : M → M σ . If σ, τ ∈ G and T is σ -inv arian t and U is τ -inv ariant then T ◦ U is σ τ - in v ariant. Indeed, let us assume that the f unctors ( T , c ) : M → M σ , ( U, d ) : M → M τ are mo dule functors then ( T ◦ U, b ) : M → M στ is a mo dule functor, where b X,M = (( γ σ ,τ ) X ⊗ id ) c F τ ( X ) ,M T ( d X,M ) , (3.1) for all X ∈ C , M ∈ M . Definition 3.1. Let F ⊆ G b e a subgroup . 1. The monoidal categ ory of σ -equiv ariant functors for some σ ∈ F in M will b e denoted by Aut F C ( M ) . 3. An F - e quiv ariant mo dule category is a mo dule category M equipp ed with a monoidal functor (Φ , µ ) : F → Au t F C ( M ), such that Φ( σ ) is a σ -in v ariant f unctor for any σ ∈ F . MODULE CA TEGORIES OVER FINITE POINTED TENSOR C A TEGORIES 7 In another words, an F -equiv arian t mo dule category is a mo du le catego ry M end o w ed with a family of mo dule fu nctors ( U σ , c σ ) : M → M σ for an y σ ∈ F and a family of natural isomorp hisms µ σ ,τ : ( U σ ◦ U τ , b ) → ( U στ , c στ ) σ, τ ∈ F suc h that (3.2) ( µ σ ,τ ν ) M ◦ U σ ( µ τ ,ν ) M = ( µ στ ,ν ) M ◦ ( µ σ ,τ ) U ν ( M ) , (3.3) c στ X,M ◦ ( µ σ ,τ ) X ⊗ M = (( γ σ ,τ ) X ⊗ ( µ σ ,τ ) M ) ◦ c σ F τ ( X ) ,U τ ( M ) ◦ U σ ( c τ X,M ) , for all σ , τ , ν ∈ F , X ∈ C , M ∈ M . Equation (3.2) follo ws from (1. 1) and (3.3) f ollo ws f rom (2.4 ). Example 3.2. C is a G -equiv ariant mod ule category ov er itself. F or an y g ∈ G set ( U σ , c σ ) = ( F σ , θ σ ) and µ σ ,τ = γ σ ,τ for all σ, τ ∈ G . If M is an F -equiv ariant mo dule category , an e qui v ariant obje ct (see [ENO2, Def. 5.3]) is an ob ject M ∈ M toge ther with isomorph isms { v σ : U σ ( M ) → M : σ ∈ F } suc h that for all σ, τ ∈ F v στ ◦ ( µ σ ,τ ) M = v σ ◦ U σ ( v τ ) . (3.4) The catego ry of F -equiv ariant ob jects is denoted by M F . A m orphism b et wee n tw o F -equiv arian t ob jects ( M , v ), ( M ′ , v ′ ) is a morph ism f : M → M ′ in M su c h that f ◦ v σ = v ′ σ ◦ U σ ( f ) for all σ ∈ F . Lemma 3.3. The c ate gory M F is a C G -mo dule c ate gory. Pr o of. If ( X , u ) ∈ C G and ( M , v ) ∈ M F the action is defin ed by ( X, u ) ⊗ ( M , v ) = ( X ⊗ M , e v ) , where e v σ = ( u σ ⊗ v σ ) c σ X,M for all σ ∈ F . The ob ject ( X ⊗ M , e v ) is equiv arian t due to equation (3.3). Th e asso ciativit y isomorph ism s are the same as in M .  The notion of F -equiv arian t mo du le category is equiv alent to the n otion of C ⋊ F -mo dule cateory . If M is an F -equiv ariant C -mo d u le category for some subgroup F of G , then M is a C ⋊ F -mo dule with action ⊗ : C ⋊ F × M → M giv en by [ X , g ] ⊗ M = X ⊗ U g ( M ), for all X ∈ C , g ∈ F and M ∈ M . T he asso ciativit y isomorphisms are giv en by m [ X,g ] , [ Y ,h ] ,M =  id X ⊗ ( c g Y , U h ( M ) ) − 1 (id F g ( Y ) ⊗ µ − 1 g ,h ( M ))  m X,F g ( Y ) ,U gh ( M ) , for all X , Y ∈ C , g , h ∈ F and M ∈ M . In the next statemen t w e collect sev eral well-kno wn results that are, b y no w, part of the folklore of the su b ject. Prop osition 3.4. L et G b e a finite gr oup acting over a finite tensor c ate gory C . If F ⊂ G is a sub g r oup, and M is an F -e q u ivariant C -mo dule c ate gory, then: 1. If M is an exact ( inde c omp osable) C -mo dule c ate gory then M is an exact (r esp e ctively inde c omp osable) C ⋊ F -mo dule c ate gory. 8 GALINDO, M OMBELLI 2. M F is an exact mo dule c ate gory if and only if M is an exact mo dule c ate gory. 3. Ther e is an e quivalenc e of C G -mo dule c ate gories (3.5) M F ≃ Hom C ⋊ F ( C , M ) ≃ C op ⊠ C ⋊ F M ≃  C ⋊ G ⊠ C ⋊ F M  G . 4. If N is an inde c omp osable (exact) mo dule c ate gory over C G ther e ex- ists a sub gr oup F of G and an F - e quivariant inde c omp osable (exact) mo dule c ate gory M over C such that N ≃ M F . 5. If M 1 , M 2 ar e G -e quivariant C -mo dule c ate gories such that M G 1 ≃ M G 2 as C G -mo dule c ate gories then M 1 ≃ M 2 as C - mo dule c ate- gories. Pr o of. 1. Let P ∈ C ⋊ G b e a pro jectiv e ob ject. Thus, there exists a family of pro jectiv e ob jects P σ ∈ C su c h that P = ⊕ σ ∈ G [ P σ , σ ]. Let M ∈ M , then P ⊗ M = L σ ∈ G P σ ⊗ U σ ( M ), and since M is an exact C -mo dule catego ry P σ ⊗ U σ ( M ) is pro jectiv e for all σ , thus P ⊗ M is pro jectiv e. 2. Under the corresp ond ence d escrib ed in [T a, Thm. 4.1] is enough to sho w that a C ⋊ F -mo dule catego ry M is exact if and only if M is an exact C -mo du le category . The pr o of follo ws f rom p art (1) of this prop osition. 3. An ob ject ( F , c ) ∈ Hom C ⋊ F ( C , M ) is determined uniquely by an ob ject M ∈ M su c h that F ( X ) = X ⊗ M together with an isomorphism v σ = c [ 1 ,σ ] , 1 : U σ ( M ) → M . This corresp on d ence establish an equiv alence M F ≃ Hom C ⋊ F ( C , M ). The equiv alence Hom C ⋊ F ( C , M ) ≃ C op ⊠ C ⋊ F M follo ws from [Gr, Thm . 3.20]. Since C ⋊ G ⊠ C ⋊ F M is a C ⋊ G -mo dule then it is a G -equiv ariant C -mo du le catego ry , thus  C ⋊ G ⊠ C ⋊ F M  G ≃ C op ⊠ C ⋊ G  C ⋊ G ⊠ C ⋊ F M  ≃ C op ⊠ C ⋊ F M ≃ M F . The first equ iv alence is [T a, T h m 4.1]. 4. By [EO 1 , Prop osition 3.9] ev ery indecomp osable exact tensor category o v er a fi nite tensor cate gory is a simple mo dule category in the sense of [Ga1], so the resu lt follo ws b y th e main result of [Ga1 ], and the item (1) of this p rop osition. 5. Since M 1 , M 2 are G -equiv arian t then they are C ⋊ G -mo dule cate- gories. It follo ws from [T a, Thm. 4.1] that this are equiv alen t C ⋊ G -mo du le catego ries. Th is equiv alence induces an equ iv alence of C -mo dule cat egories (see [T a, Ex. 2.5]) .  It follo w s from Prop osition 3.4 (4) that the equiv arian tization constru c- tion of mo d ule categories by a fixed sub group is injectiv e. Moreo ver, if the equiv arian tization of a mod ule catego ry by t wo subgroups giv es the same result then the group s m us t b e co nj ugate. W e shall giv e the precise state- men t in the follo win g. First w e need a definition and a resu lt from the pap er [Ga2]. MODULE CA TEGORIES OVER FINITE POINTED TENSOR C A TEGORIES 9 Definition 3.5. [Ga2, Def. 4.3] Let C b e a G -graded tensor ca tegory . If ( M , ⊗ ) is a C e -mo dule category , then a C -extension of M is a C -mo du le catego ry ( M , ⊙ ) such that ( M , ⊗ ) is obtained b y restriction to C e . Prop osition 3.6. [Ga2, Prop. 4.6] L et C b e a G -gr ade d finite tensor c ate- gory and let F, F ′ ⊂ G b e su b gr oups and ( N , ⊙ ) , ( N ′ , ⊙ ′ ) b e a C F -extension and a C F ′ -extension of the inde c omp osable C e -mo dule c ate gories N and N ′ , r esp e ctively. Then C ⊠ C F ′ N ′ ∼ = C ⊠ C F N as C -mo dules if and only if ther e exists σ ∈ G su c h that F = σ F ′ σ − 1 and C σF ′ ⊠ C F ′ N ′ ∼ = N as C e -mo dule c ate gories. Theorem 3.7. L et G b e a finite g r oup action on a finite tensor c ate gory C and let F , F ′ ⊂ G b e sub gr oups. L et N and N ′ b e an F -e q u ivariant and an F ′ -e quiv ariant mo dule c ate gories r esp e ctively, such tha t N and N ′ ar e inde c omp osable as C -mo dule c ate gories and N F ∼ = N ′ F ′ as C G -mo dule c ate gories. Then F and F ′ ar e c onjugate sub gr oups in G . Pr o of. It follo ws from Prop osition 3.4 (3) that there is an equiv alence of C G -mo dules  C ⋊ G ⊠ C ⋊ F N  G ≃  C ⋊ G ⊠ C ⋊ F ′ N ′  G . Hence b y Prop osition 3.4 (5) there is an equiv alence of C ⋊ G -mo du les C ⋊ G ⊠ C ⋊ F N ≃ C ⋊ G ⊠ C ⋊ F ′ N ′ , th us the result follo ws from Prop osition 3.6.  4. Q uasi-Hopf algebr a s A qu asi-bialgebra [D] is a f our-tuple ( A, ∆ , ε, Φ) where A is an asso ciativ e algebra with u nit, Φ ∈ ( A ⊗ A ⊗ A ) × is calle d the asso ciator , and ∆ : A → A ⊗ A , ε : A → k are algebra homomorph ism s satisfying the id en tities Φ(∆ ⊗ id )(∆( h )) = (id ⊗ ∆)(∆( h ))Φ , (4.1) (id ⊗ ε )(∆( h )) = h ⊗ 1 , ( ε ⊗ id )(∆( h )) = 1 ⊗ h, (4.2) for all h ∈ A . Th e asso ciator Φ has to b e a 3-co cycle, in the sense th at (1 ⊗ Φ)(id ⊗ ∆ ⊗ id )(Φ)(Φ ⊗ 1) = (id ⊗ id ⊗ ∆)(Φ)(∆ ⊗ id ⊗ id )(Φ) , (4.3) (id ⊗ ε ⊗ id )(Φ) = 1 ⊗ 1 ⊗ 1 . (4.4) A is called a quasi-Hopf algebra if, moreo ver, there exists an anti- morp hism S of the algebra A an d elemen ts α, β ∈ A suc h that, for all h ∈ A , w e ha v e: S ( h (1) ) αh (2) = ε ( h ) α and h (1) β S ( h (2) ) = ε ( h ) β , (4.5) Φ 1 β S (Φ 2 ) α Φ 3 = 1 and S (Φ − 1 ) α Φ − 2 β S (Φ − 3 ) = 1 . (4.6) Here w e use th e notation Φ = Φ 1 ⊗ Φ 2 ⊗ Φ 3 , Φ − 1 = Φ − 1 ⊗ Φ − 3 ⊗ Φ − 3 . If A is a quasi-Hopf algebra, w e sh all denote by Rep( A ) the tensor category of finite-dimensional r epresen tations of A . 10 GALINDO, M OMBELLI An inv ertible elemen t J ∈ A ⊗ A is called a t wist i f ( ε ⊗ id )( J ) = 1 = (id ⊗ ε )( J ). I f A is a qu asi-Hopf algebra and J = J 1 ⊗ J 2 ∈ A ⊗ A is a t w ist with inv erse J − 1 = J − 1 ⊗ J − 2 , then w e can defin e a quasi-Hopf alg ebr a on the same algebra A k eeping the counit and an tip o de and replacing the com ultiplication, asso ciator and the elements α and β by ∆ J ( h ) = J ∆( h ) J − 1 , (4.7) Φ J = (1 ⊗ J )(id ⊗ ∆)( J )Φ(∆ ⊗ id )( J − 1 )( J − 1 ⊗ 1) , (4.8) α J = S ( J − 1 ) αJ − 2 , β J = J 1 β S ( J 2 ) . (4.9) W e shall denote this new quasi-Hopf algebra by ( A J , Φ J ). If Φ = 1 then , in this case, we shall denote Φ J = dJ . 4.1. Como dule algebras o ver quasi-Hopf algebras. Let ( A, Φ , α, β , 1) b e a fin ite dimensional quasi-Hopf algebra. Definition 4.1. A left A -comodu le algebra is a family ( K , λ, Φ λ ) suc h that K is an algebra, λ : K → A ⊗K is an algebra map, Φ λ ∈ A ⊗ A ⊗K is an in ve rtible elemen t s u c h that (1 ⊗ Φ λ )(id ⊗ ∆ ⊗ id )(Φ λ )(Φ ⊗ 1) = (id ⊗ id ⊗ λ )(Φ λ )(∆ ⊗ id ⊗ id )(Φ λ ) , (4.10) (id ⊗ ǫ ⊗ id )(Φ λ ) = 1 , (4.11) Φ λ (∆ ⊗ id ) λ ( x ) =  (id ⊗ λ ) λ ( x )  Φ λ , x ∈ K (4.12) W e sh all sa y that a comod ule algebra ( K , λ, Φ λ ) is righ t A -simp le if it h as no non-trivial righ t id eals J ⊆ K suc h that J is costable, that is λ ( J ) ⊆ A ⊗K . R emark 4.2 . The notion of como dule algebra for quasi-Hopf algebras do es not coincide with the notion of como dule algebra for (usual) Hopf algebras. F or quasi-Hopf algebras th e coaction may not b e coassociativ e. If ( K , λ, Φ λ ) is a left A -como dule algebra, the categ ory A K M A consists of ( K , A )-bimo dules M equipp ed w ith a ( K , A )-bimo d u le map δ : M → A ⊗ M suc h that for all m ∈ M Φ λ (∆ ⊗ id ) δ ( m ) = (id ⊗ δ ) δ ( m )Φ , (4.13) ( ε ⊗ id ) δ = id . (4.14) The follo wing result will b e usefu l to present examples of exact mod ule catego ries, it is a consequence of some freeness results on como d u le algebras o v er quasi-Hopf algebras pro ven by H. Henk er. Lemma 4.3. L et ( K , λ, Φ λ ) b e a right A -simple left A -c omo dule algebr a. If M ∈ K M then A ⊗ M ∈ K M is pr oje ctive. MODULE CA TEGORIES OVER FINITE POINTED TENSOR C A TEGORIES 11 Pr o of. The ob ject A ⊗ M is in the category A K M A as follo ws. The left K - action and the right A -action on A ⊗ M are determined by x · ( a ⊗ m ) = x ( − 1) a ⊗ x (0) · m, ( a ⊗ m ) · b = ab ⊗ m, for all x ∈ K , a, b ∈ A and m ∈ M . The coaction is determined by δ : A ⊗ M → A ⊗ A ⊗ M , δ = Φ λ (∆ ⊗ id M ). It follo ws from [He, Lemma 3.6] that A ⊗ M is a pro jectiv e K -mo dule.  4.2. Como dule algebras o ver radically graded quasi-Hopf algebras. Let A b e a quasi-Ho pf alge br a r adic al ly gr ade d , that is there is an algebra grading A = ⊕ m i =0 A [ i ], where I := Rad A = ⊕ i ≥ 1 A [ i ] and I k = ⊕ i ≥ k A [ i ] for any k = 0 . . . m . Here I 0 = A . Since ∆( I ) ⊆ I ⊗ A + A ⊗ I th en ∆( I ) ⊆ P k j =0 I j ⊗ I k − j for any k = 0 . . . m . In this case A [0] is semisimple, A is generated by A [0] and A [1], and the asso ciator Φ is an element in A [0] ⊗ 3 , see [EG1, Lemma 2.1]. If ( K , λ, Φ λ ) is a left A -comod ule algebra, define K i = λ − 1 ( I i ⊗K ) , i = 0 . . . m. This is an algebra filtration, thus w e can consider the asso ciated graded algebra gr K = ⊕ m i =0 K [ i ] , K [ i ] = K i / K i +1 . Lemma 4.4. 1. The ab ove filtr ation satisfies λ ( K i ) ⊆ i X j =0 I j ⊗K i − j . (4.15) 2. Ther e is a left A -c omo dule algebr a structur e (gr K , λ, Φ λ ) satisfying λ (gr K ( n )) ⊆ ⊕ n k =0 A [ k ] ⊗K [ n − k ] . (4.16) 3. ( K [0] , λ , Φ λ ) is a left A [0] -c omo dule algebr a. Pr o of. Item (1) follo ws from the defi n ition of K i and equation (4.12 ). F or eac h n = 0 . . . m there is a linear map λ : gr K → A ⊗ gr K such th at the follo win g diagram commutes K n λ − − − − → P n j =0 I j ⊗K n − j π   y   y K n / K n +1 − − − − →  P n j =0 I j ⊗K n − j  / P n +1 j =0 I j ⊗K n +1 − j   y   y ≃ K [ n ] λ − − − − → ⊕ n k =0 A [ k ] ⊗ k K [ n − k ] . Defining Φ λ as the pro jection of Φ λ to A [0] ⊗ A [0] ⊗K [0] follo ws imm ed iately that (gr K , λ, Φ λ ) is a left A -como du le algebra.  Lemma 4.5. The fol lowing statements ar e e quivalent: 12 GALINDO, M OMBELLI 1. K is a right A -simple left A - c omo dule algebr a. 2. K [0] is a right A [0] -simple left A [0 ] -c omo dule algebr a. 3. gr K is a right A -simple left A -c omo dule algebr a. Pr o of. Assume K [0] is a righ t A [0]-simple. Let J ⊆ A b e a right ideal A -costable. Consider the filtration J = J 0 ⊇ J 1 ⊇ · · · ⊇ J m giv en by J k = λ − 1 ( I k ⊗ J ) for all k = 0 . . . m . S et J ( k ) = J k /J k +1 for any k and J = ⊕ k J ( k ). It follo ws that for an y n = 0 . . . m λ ( J ( n )) ⊆ ⊕ n k =0 A [ k ] ⊗ J ( k ) . (4.17) In p articular J (0) ⊆ K [0] is a righ t ideal A [0]-costable thus J = K [0] or J = 0. In the first case J = A and in the second case J = J 1 . It follo ws from (4.17) that J (1) ⊆ K [0] is a a right ideal A [0]-costable. Hence J = J 2 . Con tinuing this reasoning we obtain that J = 0. Assume n o w that K is a righ t A -simple. Let J ⊆ K [0] b e a righ t A [0]- costable ideal. Denote π : K → K [0] the canonical p ro jection and J = π − 1 ( J ). Clearly J is a righ t A -costable ideal thus J = 0 or J = K , th us J = 0 or J = K [0] resp ectiv ely .  As a consequence we ha ve the follo win g result. Corollary 4.6. L et ( A, Φ) b e a r adic al ly gr ade d quasi-Hopf algebr a and ( K , λ, Φ λ ) b e a left A -c omo dule algebr a such that K [0] = k 1 then A is twist e quivalent to a Hopf algebr a. Pr o of. Since ( K [0] , λ , Φ λ ) is a left A [0]-comod ule algebra then there exists an in ve rtible elemen t J ∈ A ⊗ A suc h that J ⊗ 1 = Φ λ . Equation (4.10) implies that Φ = dJ .  4.3. Mo dule categories o ver quasi-Hopf algebras. F or any como dule algebra ov er a quasi-Hopf algebra A there is asso ciated a mod ule ca tegory o v er Rep( A ). Lemma 4.7. L et A b e a finite-dimensional quasi-Hopf algebr a. 1. If ( K , λ, Φ λ ) is a left A -c omo dule algebr a then the c ate gory K M is a mo dule c ate gory over Rep( A ) . It is exact if K is right A -simple. 2. If M i s an exact mo dule c ate gory over Rep( A ) ther e exists a left A - c omo dule algebr a ( K , λ, Φ λ ) such that M ≃ K M as mo dule c ate gories over Rep( A ) . Pr o of. 1. The acti on ⊗ : Rep( A ) × K M → K M is giv en by the tensor pro du ct o v er th e field k wh ere the action on the tensor pro duct is giv en by λ . Th e asso ciativit y isomorphisms m X,Y ,M : ( X ⊗ Y ) ⊗ M → X ⊗ ( Y ⊗ M ) are giv en b y m X,Y ,M ( x ⊗ y ⊗ m ) = Φ 1 λ · x ⊗ Φ 2 λ · y ⊗ Φ 3 λ · m , for all x ∈ X , y ∈ Y , M ∈ M , X , Y ∈ Rep( A ), M ∈ K M . T o pr ov e that K M is exact, it is enough to verify that A ⊗ M is pro jectiv e for any M ∈ K M but this is Lemma 4.3 . MODULE CA TEGORIES OVER FINITE POINTED TENSOR C A TEGORIES 13 2. T his is a straigh tforward consequence of [EO1, Thm. 3.17], the p ro of of [AM, Prop. 1.19] extends mutatis mutandis to the quasi-Hopf s etting.  Definition 4.8. Tw o left A -como dule algebras ( K , λ, Φ λ ), ( K ′ , λ ′ , Φ ′ λ ) are e quivariantly Morita e qui valent if the corresp ondin g mo dule catego ries are equiv alen t. 4.4. Como dule algebras coming from t wisting. Let ( A, Φ) b e a quasi- Hopf algebra and J ∈ A ⊗ A b e a t wist. Let ( K, λ, Φ λ ) b e a left A -como d ule algebra. Let us denote by ( K J , λ J , e Φ λ ) the follo wing left A J -comod ule alge- bra. As algebras K J = K , the coactio n λ J = λ and e Φ λ = Φ λ ( J − 1 ⊗ 1). The follo wing results are straigh tforwa rd . Lemma 4.9. ( K J , λ J , e Φ λ ) is a left A J -c omo dule algebr a. It is right A -simple if and only if ( K, λ, Φ λ ) is right A -si mple.  Lemma 4.10. L et J ∈ A ⊗ A b e a twist. If ( K, λ, Φ λ ) and ( K ′ , λ ′ , Φ ′ λ ) ar e e quivariantly Morita e quivalent A -c omo dule algebr as then ( K J , λ J , Φ λ ( J − 1 ⊗ 1)) and ( K ′ J , λ ′ J , Φ ′ λ ( J − 1 ⊗ 1)) ar e e quivariant Morita e quiv alent A J -c omo dule al- gebr as.  5. Equ iv ariantiza tion of quasi-Hopf algebras F or a quasi-Hopf algebra A w e shall explain th e notion of a cr osse d system over A and discuss its r elation with the equ iv arian tization of the category Rep( A ). Let A 1 , A 2 b e qu asi-Hopf algebras. A twiste d homomorp hism b et we en A 1 and A 2 is pair ( f , J ) consisting of a h omomorphism of algebras f : A 1 → A 2 and an inv ertible elemen t J ∈ A ⊗ 2 2 suc h that (5.1) Φ 2 (∆ ⊗ id )( J )( J ⊗ 1) = (id ⊗ ∆)( J )(1 ⊗ J )( f ⊗ 3 )(Φ 1 ) , (5.2) ( ε ⊗ id )( J ) = (id ⊗ ε )( J ) = 1 , (5.3) ε ( f ( a )) = ε ( a ) , (5.4) ∆( f ( a )) J = J ( f 2 ⊗ (∆( a ))) , for all a ∈ A. R emark 5.1 . I f ( f , J ) : A 1 → A 2 is a t wisted homomorphism, then J − 1 ∈ A 2 ⊗ A 2 is a t wist and f : A 1 → ( A 2 ) J − 1 is a homomorphism of quasi- bialgebras. W e define th e cate gory End Tw ( A 1 , A 2 ) whose ob jects are twiste d homo- morphism from A 1 to A 2 . A morphism b etw een tw o t wisted homomorphisms ( f , J ) , ( f ′ , J ′ ) : A 1 → A 2 is an element c ∈ A 2 suc h that cf ( a ) = f ′ ( a ) c for an y a ∈ A 1 and ∆( c ) J = J ′ ( c ⊗ c ). The comp osition of a : f → g , b : g → h , is ba : f → h . If ( f , J f ) : A 1 → A 2 and ( g , J g ) : A 2 → A 3 are t wisted homomorphism, w e define the comp osition as the t wisted h omomorpshism ( g ◦ f , J g ( g ⊗ g )( J f )) : A 1 → A 3 . 14 GALINDO, M OMBELLI T o any t wisted homomorph ism ( f , J ) : A 1 → A 2 there is asso ciated a tensor fu nctor ( f ∗ , ξ J ) : Rep( A 2 ) → Rep( A 1 ) , where f ∗ ( V ) = V for all V ∈ Rep( A 2 ), and f ∗ is th e iden tit y o v er arr o ws. The A 1 -action on f ∗ ( V ) is giv en through the morphism f . The monoidal structure is giv en by app lying the elemen t J ∈ A ⊗ 2 2 : ξ J M ,N : f ∗ ( M ) ⊗ f ∗ ( N ) → f ∗ ( M ⊗ N ) , ξ J M ,N ( m ⊗ n ) = J ( m ⊗ n ) , for an y M , N ∈ Rep ( A 2 ), m ∈ M , n ∈ N . Morphisms b et w een twiste d homomorphisms f , f ′ : A 1 → A 2 of quasi-Hopf algebras corresp ond to tensor natural trans formations b et wee n th e asso ciated tensor functors. 5.1. Crossed system ov er a quasi-Hopf algebra. Given a quasi-Hopf algebra A w e shall denote b y Aut Tw ( A ) the (monod ial) sub category of End Tw ( A ) where ob jects are t w isted automorph isms of A , and arr ows are isomorphisms of twisted automorphism s. Let G b e a group, and let A b e a quasi-Hopf algebra. A G -cr osse d system o v er A is a monoidal fun ctor ∗ : G → Aut Tw ( A ) suc h that e ∗ = (id A , 1 ⊗ 1). More exp licitly a G -crossed system consists of the follo win g data: • A t wisted automorph ism ( σ ∗ , J σ ) for eac h σ ∈ G , • an elemen t θ ( σ ,τ ) ∈ A × for eac h σ , τ ∈ G , suc h that for all a ∈ A , σ, τ , ρ ∈ G , ε ( θ ( σ ,τ ) ) = 1 , (5.5) (1 ∗ , J 1 ) = (id , 1 ⊗ 1) , (5.6) θ ( σ ,τ ) ( σ τ ) ∗ ( a ) = σ ∗ ( τ ∗ ( a )) θ ( σ ,τ ) , (5.7) θ ( σ ,τ ) θ ( στ ,ρ ) = σ ∗ ( θ ( τ ,ρ ) ) θ σ ,τ ρ , (5.8) θ (1 ,σ ) = θ ( σ , 1) = 1 , (5.9) ∆( θ ( σ ,τ ) ) J στ = J σ ( σ ∗ ⊗ σ ∗ )( J τ )( θ ( σ ,τ ) ⊗ θ ( σ ,τ ) ) . (5.10) Let A # G b e the ve ctor sp ace A ⊗ k k G with pro duct and copro du ct ( x # σ )( y # τ ) = xσ ∗ ( y ) θ ( σ ,τ ) # σ τ , ∆( x # σ ) = x (1) J 1 σ # σ ⊗ x (2) J 2 σ # σ, for all x, y ∈ A , σ, τ ∈ G . Prop osition 5.2. The for e going op er ations makes the ve ctor sp ac e A # G into a quasi-b i algebr a with asso ciator Φ 1 # e ⊗ Φ 2 # e ⊗ Φ 3 # e , and c ounit ε ( x # σ ) = ε ( x ) for al l x ∈ A, σ ∈ G . Pr o of. It is straigh tforwa rd to see that A # G is an asso ciativ e algebra with unit 1# e . Equ ation (4.1) follo ws from (5.1). Th e map ε is an algebra morphism b y (5.3) and (5.5). Equ ations (4.2) follo w from (5.2), equations (4.3) and (4. 4) follo w b y the definition of the asso ciator. Finally ∆ is an algebra morp hism by (5.4) and (5.10).  MODULE CA TEGORIES OVER FINITE POINTED TENSOR C A TEGORIES 15 5.2. Antipo des of crossed systems. Let G b e a group , ( A, Φ , S, α, β ) b e a quasi-Hopf algebra and ( σ ∗ , θ ( σ ,τ ) , J σ ) σ ,τ ∈ G a G -crossed system o v er A . An an tip o d e for ( σ ∗ , θ ( σ ,τ ) , J σ ) σ ,τ ∈ G is a fun ction υ : G → A × suc h that υ στ ( σ τ ) ∗ ( S ( θ σ ,τ )) = υ τ ( τ − 1 ) ∗ ( υ σ ) θ τ − 1 ,σ − 1 , (5.11) υ − 1 σ S ( x ) υ σ = ( σ − 1 ) ∗ ( S ( σ ∗ ( x ))) , (5.12) υ σ ( σ − 1 ) ∗ (( S ( J 1 σ ) αJ 2 σ )) θ σ − 1 ,σ = α, (5.13) J 1 σ σ ∗ ( β υ σ ( σ − 1 ) ∗ ( S ( J 2 σ ))) θ σ ,σ − 1 = β , (5.14) for all σ ∈ G , wh ere J σ = J 1 σ ⊗ J 2 σ . The next prop osition follo ws by a straigh tforw ard verificatio n. Prop osition 5.3. L et υ : G → A × b e an antip o de for ( σ ∗ , θ ( σ ,τ ) , J σ ) σ ,τ ∈ G . Then ( S, α # e, β # e ) is an antip o de for A # G , wher e S ( x # σ ) = υ σ ( σ − 1 ) ∗ ( S ( x ))# σ − 1 , for al l σ ∈ G, x ∈ A .  5.3. Equiv ariantization and cross ed systems. Let us assume that G is an Ab elian group. In this case a G -crossed system ov er A giv es rise to a G -action on the category Rep( A ). Indeed, for any σ ∈ G we can define the tensor f u nctors ( F σ , ζ σ ) : Rep( A ) → Rep( A ) describ ed as follo ws. F or an y V ∈ Rep( A ), F σ ( V ) = V as ve ctor spaces and the action on F σ ( V ) is giv en by a · v = σ ∗ ( a ) v for all a ∈ A , v ∈ V . F or an y V , W ∈ Rep( A ) the isomorphisms ( ζ σ ) V , W : V ⊗ W → V ⊗ W are give n b y ( ζ σ ) V , W ( v ⊗ w ) = J σ · ( v ⊗ w ) for all v ∈ V , w ∈ W . F or any σ, τ ∈ G the n atural tensor transformation γ σ ,τ : F σ ◦ F τ → F στ , ( γ σ ,τ ) V ( v ) = θ − 1 ( σ ,τ ) v for all V ∈ Rep( A ), v ∈ V . Lemma 5.4. If θ ( σ ,τ ) = θ ( τ ,σ ) for al l σ, τ ∈ G then the tensor functors ( F σ , ζ σ ) describ e d ab ove define a G -action on R ep ( A ) . Pr o of. The conmutativit y of G and equation θ ( σ ,τ ) = θ ( τ ,σ ) for all σ, τ ∈ G imply that the maps γ σ ,τ are morphism s of A -mo dules. The pro of that the tensor fu nctors ( F σ , ζ σ ) defin e a G -action is straigh tforw ard.  Giv en a G -crossed system ( σ ∗ , θ ( σ ,τ ) , J σ ) σ ,τ ∈ G o v er A we consider the cat- egory Rep ( A ) G of G -equiv arian t A -mo du les. Prop osition 5.5. L e t G b e an Ab elian gr oup, A b e a quasi-Hopf algebr a and ( σ ∗ , θ ( σ ,τ ) , J σ ) σ ,τ ∈ G a G -cr osse d system over A such that θ ( σ ,τ ) = θ ( τ ,σ ) for al l σ, τ ∈ G . Then ther e is a tensor e quivalenc e b etwe en Rep( A ) G and Rep( A # G ) . Pr o of. Let ( V , u ) b e a G -equiv arian t ob j ect. Th e linear isomorphisms u σ : F σ ( V ) → V satisfy u σ ( σ ∗ ( a ) · v ) = a · u σ ( v ) , u σ ( u τ ( v )) = u στ ( θ ( σ ,τ ) · v ) (5.15) 16 GALINDO, M OMBELLI for all v ∈ V , a ∈ A , σ, τ ∈ G . Equation (5.15) tog ether with the fact that θ ( σ ,τ ) = θ ( τ ,σ ) for all σ, τ ∈ G imply that there is a wel l-defin ed action of the crossed p ro duct A # G on V determined by (5.16) ( a # σ ) · v = au − 1 σ ( v ) , for all a ∈ A, v ∈ V , σ ∈ G . Morp h isms of G -equiv arian t rep r esen tations are exactly morphisms of A # G -mo dules. Hence we ha ve defined a f unctor F : Rep( A ) G → Rep( A # G ) , whic h clea rly is a tensor fun ctor. Assu me that W ∈ Rep( A # G ). Then, by restriction, W is a representat ion of A . Moreo v er ( W , u ) is a G -equiv ariant ob ject in R ep ( A ), letting u σ : W → W, u σ ( w ) = ( θ − 1 ( σ ,σ − 1 ) # σ − 1 ) · w , for eve ry σ ∈ G . W e ha ve thus a fun ctor G : Rep( A # G ) → Rep( A ) G . It is clear that F and G are inv erse equiv alences of categories.  R emark 5.6 . A v ersion of the ab o v e r esult app ears in [Na, Prop. 3.2]. 5.4. Crossed product of quasi-bialgebras. Definition 5.7. Let ( A, Φ , S , α, β ) b e a qu asi-Hopf algebra, and let G b e a group. W e shall say that A is a G - cr osse d pr o duct if there is a decomp osition A = L σ ∈ G A σ , where: • Φ ∈ A e ⊗ A e ⊗ A e , • A σ A τ ⊆ A στ for all σ, τ ∈ G , • A σ has an inv ertible element for eac h σ ∈ G , • ∆( A σ ) ⊆ A σ ⊗ A σ for eac h σ ∈ G . • S ( A σ ) ⊆ A σ − 1 , for eac h σ ∈ G . • α, β ∈ A e Prop osition 5.8. Ev e ry G -cr osse d pr o duct A i s of the form B # G for some quasi-Hopf algebr a B . Mor e over, ther e exists an antip o de υ : G → B × such that B # G is isomorphic to A as quasi-Hopf algebr as. Pr o of. Let A b e a G -crossed pro du ct. Set B = A e . Since ev ery A σ has an in ve rtible elemen t, w e may c ho ose for eac h σ ∈ G some inv ertible element t σ ∈ A σ , with t e = 1. Then it is clear that A σ = t σ A e = A e t σ , and the set { t σ : σ ∈ G } is a b asis for A as a left (and righ t) A e -mo dule. Note that ε ( t σ ) 6 = 0, b ecause ε is an algebra m ap and t σ is in v ertible. Th us, w e ma y and shall assume that ε ( t σ ) = 1 for eac h σ ∈ G . Let u s define the maps σ ∗ ( a ) = t σ at − 1 σ , for eac h σ G and a ∈ A e , and θ : G × G → A by θ ( σ ,τ ) = t σ t τ t − 1 στ for σ, τ ∈ G. W e ha ve that ∆( t σ ) ∈ A σ ⊗ A σ can b e uniquely expressed as ∆( t σ ) = J σ ( t σ ⊗ t σ ), with J σ ∈ A e ⊗ A e . Sin ce ∆ is an algebra morphism, J σ is MODULE CA TEGORIES OVER FINITE POINTED TENSOR C A TEGORIES 17 in ve rtible, and for th e normalization ε ( t σ ) = 1, ( ε ⊗ id )( J σ ) = (id ⊗ ε )( J σ ) = 1. Then, it is straigh tforward to see that the data ( σ ∗ , θ ( σ ,τ ) , J σ ) σ ,τ ∈ G , defines a G -crossed system o v er the su b-quasi-bialgebra A e ⊆ A , and A e # G is isomorphic to A as quasi-bialgebras. The antip o de S : A → A is ant i-isomorphism of algebras, and the con- dition S ( A σ ) ⊂ A σ − 1 implies that there is a unique fun ction υ : G → A × e suc h that S ( t σ ) = θ σ t σ − 1 for all σ ∈ G . Hence, it is straightforw ard to see that υ is ant ip o de for the crossed system ( σ ∗ , θ ( σ ,τ ) , J σ ) σ ,τ ∈ G , and A e # G is isomorphic to A as quasi-Hopf algebras.  5.5. Tw isted homomorphisms of como dule algebras. Let A b e a qu asi- Hopf algebra. A twiste d homom orphism of left A -como dule algebras ( K , λ, Φ λ ) and ( K ′ , λ ′ , Φ ′ λ ) is pair ( f , J ) consisting of a h omomorphism of algebras f : K → K ′ and an inv ertible element J ∈ A ⊗ K ′ suc h that (5.17) Φ λ ′ (∆ ⊗ id )( J ) = (id ⊗ λ ′ )( J )(1 ⊗ J )(id ⊗ id ⊗ f )(Φ λ ) , (5.18) ( ε ⊗ id )( J ) = 1 , (5.19) λ ′ ( f ( a )) J = J (id ⊗ f )( λ ( a )) , for all a ∈ K . A morphism b etw een t wo t wisted homomorp h isms ( f 1 , J 1 ) , ( f 2 , J 2 ) : K → K ′ is an elemen t c ∈ K ′ suc h that c f 1 ( a ) = f 2 ( a ) c f or an y a ∈ K and λ ′ ( c ) J 1 = J 2 (1 ⊗ c ). T o an y t wisted homomorp hism of como d ule algebras ( f , J ) : K → K ′ there is asso ciated a Rep( A )-mo dule functor ( f ∗ , ξ J ) : Rep( K ′ ) → Rep( K ) , where, for all V ∈ Rep( K 2 ), f ∗ ( V ) = V with actio n giv en b y x · v = f ( x ) v , x ∈ K , v ∈ V . Th e natur al transformation ξ J is giv en b y ξ J X,M : f ∗ ( X ⊗ M ) → X ⊗ f ∗ ( M ) , ξ J X,M ( x ⊗ m ) = J − 1 · ( x ⊗ m ) , for an y X ∈ Rep( A ) , M ∈ Rep( K 2 ), x ∈ X , m ∈ M . Morp h isms b etw een t wisted h omomorphisms f , f ′ : K → K ′ of A -comod ule algebras corresp ond to m o dule natur al tr an s formations b et we en the mo d ule f u nctors. Let A b e a quasi-Hopf algebra and ( K , λ, Φ λ ) b e a left A -comod u le algebra. F or eac h twisted endomorphism ( f , J ) : A → A , w e define a new left A - comod ule algebra ( K f , λ f , Φ f λ ), where K f = K as algebras and λ f ( x ) = ( f ⊗ id ) λ ( x ) , Φ f λ = ( f ⊗ f ⊗ id )(Φ λ )( J − 1 ⊗ 1) , for all x ∈ K . Definition 5.9. Let A b e a quasi-Hopf alge br a and ( K , λ, Φ λ ) b e a left A -comod u le algebra. Given a t wisted endomorphism ( f , J ) of A , a ( f , J )- twiste d endo morphism of K is a t wisted h omomorphism from ( K f , λ f , Φ f λ ) 18 GALINDO, M OMBELLI to ( K , λ, Φ λ ). Explicitly a ( f , J )-t wisted endomorphism is a pair ( f , J ) con- sisting of an algebra end omorphism f : K → K and an in v ertible element J ∈ A ⊗ K , such that: (5.20) ( ε ⊗ id )( J ) = 1 , (5.21) Φ λ (∆ ⊗ id )( J )( J ⊗ 1) = (id ⊗ λ )( J )(1 ⊗ J )( f ⊗ f ⊗ f )(Φ λ ) (5.22) λ ( f ( x )) J = J ( f ⊗ f )( λ ( x )) , for all x ∈ K . Lemma 5.10. L et ( σ ∗ , θ ( σ ,τ ) , J σ ) σ ,τ ∈ G b e a cr osse d system over a quasi-Hopf algebr a A , and ( K , λ, Φ λ ) a left A -c omo dule algebr a. If ( f σ , J σ ) , ( f τ , J τ ) : K → K ar e ( σ ∗ , J σ ) -twiste d and ( τ ∗ , J τ ) -twiste d e ndomorphism, then ( f σ , J σ ) ◦ ( f τ , J τ ) = ( f σ ◦ f τ , J σ ( σ ∗ ⊗ f σ )( J τ )( θ σ ,τ ⊗ 1)) is a (( σ τ ) ∗ , J στ ) -twiste d endomorph ism. M or e over, this c omp osition is asso- ciative, i.e., if ( f σ , J σ ) , ( f τ , J τ ) , ( f ρ , J ρ ) : K → K ar e ( σ ∗ , J σ ) -twiste d, ( τ ∗ , J τ ) - twiste d, and ( ρ ∗ , J ρ ) -twiste d endomo rphism, then [( f σ , J σ ) ◦ ( f τ , J τ )] ◦ ( f ρ , J ρ ) = ( f σ , J σ ) ◦ [( f τ , J τ ) ◦ ( f ρ , J ρ )] Pr o of. If we use the follo w ing notation J σ ◦ J τ = J σ ( σ ∗ ⊗ f σ )( J τ )( θ σ ,τ ⊗ 1) , th us w e need to pro v e: (1) ( ε ⊗ id )( J σ ◦ J τ ) = 1, (2) Φ λ (∆ ⊗ id )( J σ ◦ J τ )( J στ ⊗ 1) = (id ⊗ λ )( J σ ◦ J τ )(1 ⊗ J σ ◦ J τ )(( σ τ ) ∗ ⊗ ( σ τ ) ∗ ⊗ f σ ◦ f τ )(Φ λ ), (3) λ ( f σ ◦ f τ ( x )) J σ ◦ J τ = J σ ◦ J τ ( f ⊗ f )( λ ( x )) for all x ∈ K . (1) The fir s t equation follo ws immediately using ε ⊗ id ( J σ ) = 1, an d ε is an algebra morph ism that comm utes with σ ∗ for all σ ∈ G . (2) F or the second equation, firs t we shall see some equalities: (∆ ⊗ id )( σ ∗ ⊗ f σ )( J τ )( J σ ⊗ 1) = ( J σ ⊗ 1)( σ ∗ ⊗ σ ∗ ⊗ f σ )(∆ ⊗ id )( J τ ) (5.23) (1 ⊗ J σ )( σ ∗ ⊗ σ ∗ ⊗ f σ )[(id ⊗ λ )( J τ )] = (id ⊗ λ )( σ ∗ ⊗ f σ )( J τ )(1 ⊗ J σ ) (5.24) The equation (5.23) follo ws b y axiom (5.4) of J στ , and the equation (5.24) follo ws b y axiom (5.22) of J σ . (5.25) Φ λ (∆ ⊗ id )( J σ ( σ ∗ ⊗ f σ )( J τ ))( J σ ⊗ 1) (5.23) = Φ λ (∆ ⊗ id )( J σ )(∆ ⊗ id )( σ ∗ ⊗ f σ )( J τ )( J σ ⊗ 1) (5.21) = Φ λ (∆ ⊗ id )( J σ )( J σ ⊗ 1)( σ ∗ ⊗ σ ∗ ⊗ f σ )(∆ ⊗ id )( J τ ) = (id ⊗ λ )( J σ )(1 ⊗ J σ )( σ ∗ ⊗ σ ∗ ⊗ f σ )(Φ λ )( σ ∗ ⊗ σ ∗ ⊗ f σ )(∆ ⊗ id )( J τ ) = (id ⊗ λ )( J σ )(1 ⊗ J σ )( σ ∗ ⊗ σ ∗ ⊗ f σ )[Φ λ (∆ ⊗ id )( J τ )] MODULE CA TEGORIES OVER FINITE POINTED TENSOR C A TEGORIES 19 (5.26) Φ λ (∆ ⊗ id )( J σ ( σ ∗ ⊗ f σ )( J τ ))[ J σ ( σ ∗ ⊗ σ ∗ )( J τ ) ⊗ 1] (5.25) = (id ⊗ λ )( J σ )(1 ⊗ J σ )( σ ∗ ⊗ σ ∗ ⊗ f σ )[Φ λ (∆ ⊗ id )( J τ )](( σ ∗ ⊗ σ ∗ )( J τ )) ⊗ 1 = (id ⊗ λ )( J σ )(1 ⊗ J σ )( σ ∗ ⊗ σ ∗ ⊗ f σ )[Φ λ (∆ ⊗ id )( J τ )( J τ ⊗ 1)] (5.21) = (id ⊗ λ )( J σ )(1 ⊗ J σ )( σ ∗ ⊗ σ ∗ ⊗ f σ )[(id ⊗ λ )( J τ )(1 ⊗ J τ )( τ ∗ ⊗ τ ∗ ⊗ f τ )(Φ λ )] = (id ⊗ λ )( J σ )(1 ⊗ J σ )( σ ∗ ⊗ σ ∗ ⊗ f σ )[(id ⊗ λ )( J τ )(1 ⊗ J τ )]( σ ∗ τ ∗ ⊗ σ ∗ τ ∗ ⊗ f σ f τ )(Φ λ ) = (id ⊗ λ )( J σ )(1 ⊗ J σ )( σ ∗ ⊗ σ ∗ ⊗ f σ )[(id ⊗ λ )( J τ )] × [1 ⊗ ( σ ∗ ⊗ f σ )( J τ )]( σ ∗ τ ∗ ⊗ σ ∗ τ ∗ ⊗ f σ f τ )(Φ λ ) (5.24) = (id ⊗ λ )( J σ )(id ⊗ λ )( σ ∗ ⊗ f σ )( J τ )(1 ⊗ J σ ) × [1 ⊗ ( σ ∗ ⊗ f σ )( J τ )]( σ ∗ τ ∗ ⊗ σ ∗ τ ∗ ⊗ f σ f τ )(Φ λ ) = (id ⊗ λ )[( J σ )( σ ∗ ⊗ f σ )( J τ )](1 ⊗ J σ ) × [1 ⊗ ( σ ∗ ⊗ f σ )( J τ )]( σ ∗ τ ∗ ⊗ σ ∗ τ ∗ ⊗ f σ f τ )(Φ λ ) = (id ⊗ λ )( f σ ◦ f τ )(1 ⊗ f σ ◦ f τ )( θ − 1 ( σ ,τ ) ⊗ θ − 1 ( σ ,τ ) ⊗ 1)( σ ∗ τ ∗ ⊗ σ ∗ τ ∗ ⊗ f σ f τ )(Φ λ ) (5.7) = (id ⊗ λ )( J σ ◦ J τ )(1 ⊗ J σ ◦ J τ )(( σ τ ) ∗ ⊗ ( σ τ ) ∗ ⊗ f σ f τ )(Φ λ )( θ − 1 ( σ ,τ ) ⊗ θ − 1 ( σ ,τ ) ⊗ 1) Φ λ (∆ ⊗ id )( J σ ◦ J τ )( J στ ⊗ 1) = Φ λ (∆ ⊗ id )( J σ ( σ ∗ ⊗ f σ )( J τ )( θ σ ,τ ⊗ 1))( J στ ⊗ 1) = Φ λ (∆ ⊗ id )( J σ ( σ ∗ ⊗ f σ )( J τ ))(∆( θ σ ,τ ) J στ ⊗ 1) (5.10) = Φ λ (∆ ⊗ id )( J σ ( σ ∗ ⊗ f σ )( J τ ))[( J σ ( σ ∗ ⊗ σ ∗ )( J τ )( θ ( σ ,τ ) ⊗ θ ( σ ,τ ) )) ⊗ 1] = Φ λ (∆ ⊗ id )( J σ ( σ ∗ ⊗ f σ )( J τ ))[ J σ ( σ ∗ ⊗ σ ∗ )( J τ ) ⊗ 1] × [ θ ( σ ,τ ) ⊗ θ ( σ ,τ ) ⊗ 1] (5.26) = (id ⊗ λ )( J σ ◦ J τ )(1 ⊗ f σ ◦ f τ )(( σ τ ) ∗ ⊗ ( σ τ ) ∗ ⊗ f σ f τ )(Φ λ )( θ − 1 ⊗ θ − 1 ⊗ 1) × [ θ ( σ ,τ ) ⊗ θ ( σ ,τ ) ⊗ 1] = (id ⊗ λ )( J σ ◦ J τ )(1 ⊗ f σ ◦ f τ )(( σ τ ) ∗ ⊗ ( σ τ ) ∗ ⊗ f σ f τ )(Φ λ ) The pro of of the second equation is ov er. (3) No w w e sh all pro ve the third equation: λ ( f σ ◦ f τ ( x )) J σ ◦ J τ = λ ( f σ ◦ f τ ( x )) J σ ( σ ∗ ⊗ f σ )( J τ )( θ σ ,τ ⊗ 1) (5.22) = J σ ( σ ∗ ⊗ f σ ) λ ( f τ ( x ))( σ ∗ ⊗ f σ )( J τ )( θ σ ,τ ⊗ 1) = J σ ( σ ∗ ⊗ f σ )[ λ ( f τ ( x )) J τ ]( θ σ ,τ ⊗ 1) (5.22) = J σ ( σ ∗ ⊗ f σ )[ J τ ( τ ∗ ⊗ f τ ) λ ( x )]( θ σ ,τ ⊗ 1) = J σ ( σ ∗ ⊗ f σ )( J τ )( σ ∗ τ ∗ ⊗ f σ f τ )( λ ( x ))( θ σ ,τ ⊗ 1) (5.7) = J σ ( σ ∗ ⊗ f σ )( J τ )( θ σ ,τ ⊗ 1)(( σ τ ) ∗ ⊗ f σ f τ )( λ ( x )) = ( J σ ◦ J τ )(( σ τ ) ∗ ⊗ f σ f τ ) λ ( x )] 20 GALINDO, M OMBELLI Finally , we shall pro ve the asso ciativit y of ◦ , [ J σ ◦ J τ ] ◦ J ρ = [ J σ ( σ ∗ ⊗ f σ )( J τ )( θ σ ,τ ⊗ 1)] ◦ J ρ = J σ ( σ ∗ ⊗ f σ )( J τ )( θ σ ,τ ⊗ 1)(( σ τ ) ∗ ⊗ ( f σ ◦ f τ ))( J ρ )( θ στ ,ρ ⊗ 1) (5.7) = J σ ( σ ∗ ⊗ f σ )( J τ )( σ ∗ τ ∗ ⊗ ( f σ ◦ f τ ))( J ρ )( θ σ ,τ θ στ ,ρ ⊗ 1) (5.8) = J σ ( σ ∗ ⊗ f σ )( J τ )( σ ∗ τ ∗ ⊗ ( f σ ◦ f τ ))( J ρ )( σ ∗ ( θ τ ,ρ ) θ σ ,τ ρ ⊗ 1) = J σ ( σ ∗ ⊗ f σ )[ J τ ( τ ∗ ⊗ f τ )( J ρ )( θ τ ,ρ ⊗ 1)]( θ σ ,τ ρ ⊗ 1) = J σ ( σ ∗ ⊗ f σ )[ J τ ◦ J ρ ]( θ σ ,τ ρ ⊗ 1) = J σ ◦ [ J τ ◦ J ρ ] .  5.6. Crossed system of como dule algebras. Let A b e a quasi-Hopf al- gebra ( K , λ, Φ λ ) b e a left A -comod u le algebra and ( σ ∗ , θ ( σ ,τ ) , J σ ) σ ,τ ∈ G a G - crossed sys tem o ver A . W e d efi ne the monoidal catego ry Aut Tw G ( K ) of t wisted automorph isms as follo ws . Ob jects in Au t Tw G ( K ) are ( σ ∗ , J σ )-t w isted automorph isms of K for σ ∈ G , the set of arro ws are the isomorphisms of t wisted homomorphisms of A -como du le algebras, the tensor pro duct of ob ject is defin ed by th e com- p osition explained in Lemma 5.10. The un it y ob ject is the id K , and tens or pro du ct of arrows is as in Aut Tw ( A ). Let F ⊂ G b e a subgroup. An F - cr osse d system for a left A -como dule algebra K , compatible w ith the G -crossed system ( σ ∗ , θ ( σ ,τ ) , J σ ) σ ,τ ∈ G is a monoidal fun ctor ( ) : F → Aut Tw G ( K ), th at is, an F -crossed system consists of the follo wing d ata: • A ( σ ∗ , J σ )-t w isted automorphism ( σ , J σ ) for eac h σ ∈ F , • an elemen t θ ( σ ,τ ) ∈ K × for eac h σ, τ ∈ F , suc h that ( 1 , J 1 ) = (id , 1 ⊗ 1) , (5.27) θ ( σ ,τ ) ( σ τ ) ( k ) = σ ( τ ( k )) θ ( σ ,τ ) , (5.28) θ ( σ ,τ ) θ ( στ ,ρ ) = σ ( θ ( τ ,ρ ) ) θ σ ,τ ρ , (5.29) θ (1 ,σ ) = θ ( σ , 1) = 1 , (5.30 ) λ ( θ ( σ ,τ ) ) J στ = J σ  ( σ ∗ ⊗ σ )( J τ )  θ ( σ ,τ ) ⊗ θ ( σ ,τ ) , (5.31) for all k ∈ K , σ , τ , ρ ∈ F . Let K # F b e the vec tor space K ⊗ k k F with pro du ct and coact ion giv en by ( x # σ )( y # τ ) = x σ ( y ) θ ( σ ,τ ) # σ τ , δ ( x # σ ) = x ( − 1) J 1 σ # σ ⊗ x (0) J 2 σ # σ, (5.32) for all x, y ∈ K , σ, τ ∈ F . Prop osition 5.11. The for e going op er ations make the sp ac e K # F into a left A # G -c omo dule algebr a with asso ciator Φ δ = Φ 1 λ #1 ⊗ Φ 2 λ #1 ⊗ Φ 3 λ #1 .  MODULE CA TEGORIES OVER FINITE POINTED TENSOR C A TEGORIES 21 Definition 5.12. Let G b e a group, and F ⊆ G b e a sub group. Let A b e a G -crossed pr o duct quasi-bialgebra, and let ( K , λ, Φ λ ) b e a left A -como dule algebra. W e s hall say that K is an F - cr osse d pr o duct , if there is a decomp o- sition K = L σ ∈ F K σ , suc h that • Φ λ ∈ A e ⊗ A e ⊗ K e , • K σ K τ ⊆ K στ for all σ, τ ∈ F , • K σ has an inv ertible element for eac h σ ∈ F , • λ ( K σ ) ⊆ A σ ⊗ K σ for eac h σ ∈ F . Let A b e a qu asi-Hopf algebra and ( σ ∗ , θ ( σ ,τ ) , J σ ) σ ,τ ∈ G b e a crossed system for the group G . W e h a v e similar r esu lts as for quasi-Hopf algebras. The pro of is analogous to the pro of of Prop osition 5.8. Prop osition 5.13. L et ( L , δ ) b e a F -cr osse d A # G -c omo dule algebr a, for a sub gr oup F ⊆ G . Then ther e is an A -c omo dule algebr a K , and an F -cr osse d system over K c omp atible with the cr osse d system ( σ ∗ , θ ( σ ,τ ) , F σ ) σ ,τ ∈ G , such that K # F and L ar e isomorph ic A # G -c omo dule algebr as. Pr o of. Let L b e a F -crossed A # G -como dule algebra, for a s u bgroup F ⊆ G . Set K = L e . S ince ev ery L σ has an in ve rtible elemen t, we may c ho ose f or eac h σ ∈ F some in v ertible elemen t u σ ∈ L σ , w ith u e = 1. T hen it is clear that L σ = u σ L e = L e u σ , and the set { u σ : σ ∈ F } is a basis for L as a left (and r igh t) L e -mo dule. Let us define the maps σ ( a ) = u σ au − 1 σ , for eac h σ F and a ∈ L e , and θ : G × G → L e b y θ ( σ ,τ ) = u σ u τ u − 1 στ for σ, τ ∈ F . Note that { (1# σ ) ⊗ u τ } σ ∈ G, τ ∈ F is a basis for A # F ⊗ L as a left (and righ t) A ⊗ L e -mo dule. W e hav e that δ ( u σ ) ∈ A # σ ⊗ L σ can b e uniquely expressed as δ ( u σ ) = J σ ((1# σ ) ⊗ u σ ), with J σ ∈ A ⊗ L e , for all σ ∈ F . Then, it is straigh tforwa rd to see th at the d ata ( σ , θ ( σ ,τ ) , J σ ) σ ,τ ∈ F , de- fine an F -crossed system o v er the A -co mo dule alge br a L e , and L e # F is isomorphic to L as A # G como du le algebras.  Let G b e an Ab elian group, F ⊆ G a subgroup, ( σ ∗ , θ ( σ ,τ ) , J σ ) σ ,τ ∈ G b e a crossed system o ver a quasi-Hopf algebra A , and ( σ , θ ( σ ,τ ) , J σ ) σ ,τ ∈ F b e an F -crossed system for a A -como dule algebra K . W e shall fur ther assu me that (5.33) θ ( σ ,τ ) = θ ( τ ,σ ) , θ ( ρ,ν ) = θ ( ν,ρ ) , for all σ, τ ∈ G , ρ, ν ∈ F . W e can consider th e action of G on th e category Rep( A ) describ ed in Lemma 5.4. Prop osition 5.14. Under the ab ove assumptions the fol lowing assertions hold. 1. The Rep( A ) -mo dule c ate gory K M is F -e quivariant. 2. Ther e is an e quivalenc e b etwe en ( K M ) F and K # F M as Rep( A ) G - mo dule c ate gories. 22 GALINDO, M OMBELLI Pr o of. 1. F or an y ρ ∈ F define ( U ρ , c ρ ) : K M → ( K M ) ρ the Rep( A )-mo dule functor giv en as follo ws. F or an y M ∈ K M , U ρ ( M ) = M as ve ctor spaces and the action of K is given by: x · v = ρ ( x ) · v , for all x ∈ K , v ∈ M . F or an y X ∈ Rep( A ), M ∈ K M the maps c ρ X,M : U ρ ( X ⊗ k M ) → F ρ ( X ) ⊗ k U ρ ( M ) are d efi ned by c ρ X,M ( x ⊗ v ) = J − 1 ρ · ( x ⊗ v ), for any x ∈ X, v ∈ M . E q u ation (2.1) f or the pair ( U ρ , c ρ ) follo ws f r om (5.21) . F or an y σ , τ ∈ F define µ σ ,τ : U σ ◦ U τ , → U στ as follo w s. F or any M ∈ K M , m ∈ M µ σ ,τ ( m ) = θ − 1 ( σ ,τ ) · m. It follo ws from equation (5.7) th at µ σ ,τ is a morphism of K -mo dules. Equa- tion (3.2) follo ws fr om (5.7 ) and (3.3) follo ws from (5.10). 2. Let T : ( K M ) F → K # F M b e the mo du le functor defined as follo ws. If ( M , v ) is an F -equiv arian t ob ject th en f or any σ ∈ F we ha v e isomorph isms v σ : U σ ( M ) → M satisfying v στ ( θ − 1 ( σ ,τ ) · m ) = v σ ( v τ ( m ) , v σ ( σ ( x ) · m ) = x · v σ ( m ) , for all σ, τ ∈ F , x ∈ K , m ∈ M . In th is case th er e is a wel l-defin ed action of K # F on M d etermined b y ( x # σ ) · m = x · v − 1 σ ( m ) , (5.34) for all σ ∈ F , x ∈ K , m ∈ M . W e define T ( M ) = M with the ab o v e describ ed action. If ( X , u ) ∈ Rep( A ) G , ( M , v ) ∈ ( K M ) F the action of K # F on X ⊗ M usin g the coaction give n in (5.32) coincides with the action (5.34) using the isomorphism e v describ ed in Lemma 3.3. The pro of that T is an equiv alence is analogo us to the pro of of Pr op osition 5.5.  The category of F -equiv ariant ob j ects in a mo du le category is alw a ys of the form K # F M for some left A -comodu le algebra K and some group F . Prop osition 5.15. L et A b e a finite dimensional quasi-H opf algebr a and G b e a finite Ab elian gr oup and F ⊂ G a sub gr oup. L e t ( σ ∗ , θ ( σ ,τ ) , J σ ) σ ,τ ∈ G b e a G -c r osse d system over A , and M b e an exact F -e quivariant Rep( A ) - mo dule c ate gory. Then ther e is a left A - c omo dule algebr a ( K , λ, Φ λ ) such that K M ∼ = M as Rep( A ) -mo dule c ate gories and ther e is an F -cr osse d system c omp atible with ( σ ∗ , θ ( σ ,τ ) , J σ ) σ ,τ ∈ G such that K # F M ≃ M F as Rep( A ) G - mo dule c ate gories. Pr o of. Let B b e a finite-dimensional quasi-Hopf alg ebra suc h that there is a quasi-Hopf algebra p ro jection π : B → A and an equiv alence Rep( B ) ≃ Rep( A ) ⋊ F of tensor categories, see section 3.4. Since M is F -equiv ariant follo ws from Prop osition 3.4 that M is an exact Rep( B ) mo dule catego ry . Hence there exists a left B -como du le alge br a ( K , λ, Φ λ ) suc h that M ≃ K M as Rep ( B )-mo dules. Let u s recall that the equ iv arian t structur e is giv en b y ( U σ , c σ ) : M → M σ , U σ ( M ) = [ 1 , σ ] ⊗ M , MODULE CA TEGORIES OVER FINITE POINTED TENSOR C A TEGORIES 23 for all σ ∈ F , M ∈ M together with a family of n atural isomorph isms µ σ ,τ : U σ ◦ U τ → U στ for any σ, τ ∈ F . Under the equiv alence Rep( B ) ≃ Rep( A ) ⋊ F the ob ject [ 1 , σ ] corresp ond to a 1-dimensional rep resen tation of B . F or an y σ ∈ F let us d enote by χ σ : B → k the corresp onding c haracter and the algebra map σ : K → K , σ ( k ) = χ σ ( k ( − 1) ) k (0) , for all k ∈ K . Define λ π = ( π ⊗ id ) λ , then ( K , λ π , ( π ⊗ π ⊗ id )(Φ λ )) is a left A -como dule algebra that w e will d enote by K π . The equ iv alence M ≃ K M of Rep( B )- mo dule cate gories indu ces an equiv alence M ≃ K π M of Rep( A )-mo dules. Under this equiv alence the functors U σ : K π M → ( K π M ) σ are giv en as follo ws . F or any M ∈ K π M , U σ ( M ) = M and the action of K on M is giv en b y k · m = σ ( k ) · m, for all k ∈ K , m ∈ M . F or an y σ, τ ∈ F den ote J σ = c σ A, K (1 ⊗ 1) − 1 , θ σ ,τ = ( µ τ ,σ ) K (1) − 1 . T urn s out that the collectio n ( σ , θ ( σ ,τ ) , J σ ) σ ,τ ∈ F is an F -crossed system com- patible with ( σ ∗ , θ ( σ ,τ ) , J σ ) σ ,τ ∈ G for th e A -como du le algebra K π . In deed for an y σ ∈ F the pair ( σ , J σ ) is a ( σ ∗ , J σ )-t w isted automorphism since equation (5.21) follo ws from the fact that c σ satisfies (2.1) and equation (5.22) follo w s since c σ is a K -mo dule morphism. Equation (5.28) follo ws since µ σ ,τ is a morphism of K -mo d ules, equation (5.29) follo ws from (3.2) and equation (5.31) follo ws from (3.3). The equiv alence K π # F M ≃ M F as Rep( A ) G - mo dule categories follo ws from Prop osition 5.14.  6. M odule ca tegories over t he quasi-Hopf algebras A ( H, s ) 6.1. Basic Quasi-Hopf algebras A ( H , s ) . W e recall the defin ition of a the family of basic qu asi-Hopf algebras A ( H , s ) introduced by I. Angiono [A] and used to giv e a classification of p oint ed tensor categories with cyclic group of inv ertible ob jects of order m suc h that 210 ∤ m . Let m ∈ N and H = ⊕ n ≥ 0 H ( n ) b e a fin ite-dimensional radically graded p ointe d Hopf alge br a generated b y a group lik e elemen t χ of ord er m 2 and sk ew primitiv e elemen ts x 1 , ..., x θ satisfying (6.1) χx i χ − 1 = q d i x i , ∆( x i ) = x i ⊗ 1 + χ − b i ⊗ x i , for an y i = 1 , . . . , θ , where q is a p rimitiv e r o ot of 1 of order m 2 , H = B ( V )# k C m 2 , wh er e B ( V ) is the asso ciated Nic hols alge br a of the Y etter- Drinfeld mo dule V ∈ k C m 2 k C m 2 Y D . W e sh all fu rther assume th at B ( V ) h as a b asis { x s 1 1 . . . x s θ θ : 0 ≤ s i ≤ N i } . R emark 6.1 . The ab ov e condition do es not hold for an y Nic hols alg ebra. If V has diagonal b raiding with Cartan matrix of typ e A 3 then B ( V ) is not generated by elemen ts of degree 1. This conditions is satisfied for example for any quantum linear space. 24 GALINDO, M OMBELLI Set σ := χ m and d en ote by { 1 i : i ∈ C m 2 } , { 1 j : j ∈ C m } the families of primitiv e idemp oten ts in k C m 2 and k C m resp ectiv ely . That is 1 i = 1 m 2 m 2 − 1 X k =0 q − k i χ k , 1 j = 1 m m − 1 X l =0 q − mlj σ l . F or an y 0 ≤ s ≤ m − 1 set J s = P m 2 − 1 i,j =0 c ( i, j ) s 1 i ⊗ 1 j , where c ( i, j ) := q j ( i − i ′ ) . Here j ′ denotes the r emainder in the d ivision b y m . The associator Φ s = dJ s is wr itten explicitly as (6.2) Φ s := m − 1 X i,j,k =0 ω s ( i, j, k ) 1 i ⊗ 1 j ⊗ 1 k , where ω s : ( C m ) 3 → k × is the 3-cocycle defined by ω s ( i, j, k ) = q sk ( j + i − ( j + i ) ′ ) . Consider the quasi-Hopf algebra ( H J s , Φ s ) obtained b y t wisting H . Denote Υ( H ) = { 1 ≤ s ≤ m − 1 : b i ≡ sd i mo d( m ) , 1 ≤ i ≤ θ } . F or any s ∈ Υ( H ) the qu asi-Hopf algebra A ( H, s ) is defined as th e subalgebra of H generated b y σ and x 1 , ..., x θ . The algebra A ( H , s ) is a quasi-Hopf subalgebra of H J s with asso ciator Φ s suc h that A ( H , s ) / Rad A ( H , s ) ∼ = k [ C m ]. See [A, Prop. 3.1.1]. F or an y 1 ≤ i ≤ θ we ha ve that ∆ J s ( x i ) = m − 1 X y = 0 q b i y 1 y ⊗ x i + m − 1 X z =0   m − d ′ i − 1 X y = 0 q ( d ′ i − d i ) sz x i 1 y ⊗ 1 z + m − 1 X j = m − d ′ i q ( d ′ i + m − d i ) sz x i 1 y ⊗ 1 z   . R emark 6.2 . Our definition of A ( H , s ) is sligh tly differen t that the one giv en in [A, § 3]. Th is is not a problem since our quasi-Hopf algebras are isomor- phic to the ones defined in lo c. cit. except that the s ma y change. Th e difference comes from the fact that we are u sing ( χ − b i , 1) sk ew-primitive elemen ts instead of (1 , χ b i ) sk ew-primitiv e elements. 6.2. C m -crossed system ov er A ( H , s ) . The cyclic group with m elemen ts will b e denoted b y C m = { 1 , h, h 2 , . . . , h m − 1 } . F or any 0 ≤ i < m set (( h i ) ∗ , J h i ) the t wisted end omorphism of A ( H , s ) giv en by J h i = 1 ⊗ 1 , ( h i ) ∗ ( a ) = χ i ′ aχ − i ′ for all a ∈ A ( H , s ) . F or an y 0 ≤ i, j < m defin e θ ( i,j ) = θ ( h i ,h j ) = σ ( i + j ) − ( i + j ) ′ m . R emark 6.3 . If i + j ≤ m then θ ( i,j ) = 1 and if i + j > m then θ ( i,j ) = σ . In pr inciple the algebra maps ( h i ) ∗ are defin ed in H bu t when restricted to A ( H, s ) they are w ell-defined. MODULE CA TEGORIES OVER FINITE POINTED TENSOR C A TEGORIES 25 These data is a C m -crossed system o ve r A ( H, s ) su c h that the equiv ari- an tization Rep( A ) C m is tensor equiv alen t to Rep( H ). This is cont ained in the next resu lt wh ich giv es an alternativ e pro of for [A, Th m. 4.2.1]. Prop osition 6.4. 1. (( h i ) ∗ , θ ( i,j ) , J h i ) h i ,h j ∈ C m is a C m -cr osse d system over A ( H , s ) . 2. Ther e is an isomorphism of quasi-Hopf algebr as A ( H , s )# C m ≃ H J s . 3. Ther e is a tensor e quivalenc e Rep( A ) C m ≃ Rep( H ) . Pr o of. 1. it follo ws by a straigh tforward compu tation. 2. Define ϕ : A ( H , s )# C m → H J s the linear map give n by ϕ ( a # h i ) = aχ i ′ , for all 0 ≤ i < m , a ∈ A . Let 0 ≤ i, j < m , a, b ∈ A th en ϕ (( a # h i )( b # h j )) = ϕ ( a ( h i ) ∗ ( b ) θ ( i,j ) # h i + j ) = aχ i ′ bχ − i ′ θ ( i,j ) χ ( i + j ) ′ . On the other hand ϕ ( a # h i ) ϕ ( b # h j ) = aχ i ′ bχ j ′ . It is enou gh to pro ve that χ i ′ bχ j ′ = χ i ′ bχ − i ′ θ ( i,j ) χ ( i + j ) ′ for b = x l , 1 ≤ l ≤ θ . If i + j ≤ m then χ i ′ x l χ − i ′ θ ( i,j ) χ ( i + j ) ′ = q d l i x l χ i + j = χ i x l χ j . If i + j = m + k , k > 0 then χ i ′ x l χ − i ′ θ ( i,j ) χ ( i + j ) ′ = q d l i x l σ χ k = q d l i x l χ i + j = χ i x l χ j . It follo ws immediately that ϕ is a coalgebra map and it is injectiv e and b y a dimens ion argument is bijectiv e. 3. It follo ws f rom Pr op osition 5.5.  R emark 6.5 . Ther e is a grading on H compatible with th e isomorphism of Prop osition 6.4 (2). Namely , if σ ∈ G then th e v ector space H σ has basis { x s 1 1 . . . x s θ θ σ } . Defin e H ( i ) = ⊕ m − 1 j =0 H χ mj + i , thus H = ⊕ m − 1 j =0 H ( j ) . It is not difficult to pro ve that with this grading H is a C m -crossed pro duct (see definition 5.7) and this crossed pro duct is compatible w ith the isomorphism of Prop osition 6.4 (2). 6.3. Righ t simple A ( H , s ) -como dule algebras. W e sh all present some families of righ t A ( H, s )-simple left A ( H , s )-como dule alg ebras. This class will b e big enough to classify mod u le categories o v er Rep( A ( H, s )) in some cases. Let ( K, λ ) b e a finite-dimensional left H -como d u le algebra. W e sa y that ( K, λ ) is of typ e 1 if the follo wing assum p tions are satisfied: • Ther e exists a sub group F ⊆ C m 2 and t ∈ N such that K has a basis { y r 1 1 . . . y r t t e f : 0 ≤ r j < N j , f ∈ F , t ≤ θ } suc h that e χ a y l = q ad l y l e χ a , i f χ a ∈ F . 26 GALINDO, M OMBELLI • there is an inclusion ι : K ֒ → H of H -como dules su ch that ι ( e f ) = f , ι ( y l ) = x l , for all f ∈ F , l = 1 . . . t . Observe that in this case w e h a v e that λ ( e f ) = f ⊗ e f , λ ( y l ) = x l ⊗ 1 + χ − b l ⊗ y l . Definition 6.6. W e shall sa y that a Hopf algebra H = B ( V )# k G is of t yp e 1 if (1) B ( V ) h as a basis { x s 1 1 . . . x s θ θ : 0 ≤ s i ≤ N i } , wh ere V is the v ector space generated b y { x 1 , . . . , x θ } , (2) an y right H -simple left H -comodu le algebra ( K, λ ) is equiv arian tly Morita equiv alen t to a como dule algebra of t yp e 1. R emark 6.7 . If H = B ( V )# k Γ is the b osonization of a Nic h ols algebra an d a group algebra a finite group Γ then H is of t yp e 1 when V is a quan tum linear s p ace and Γ is an Ab elian group [Mo2] or wh en V is constructed f r om a rac k and Γ = S 3 , S 4 [GM]. Let ( K, λ ) b e a type 1 left H -comodu le alge br a suc h that K 0 = k F where F ⊆ C m 2 is a s u bgroup su ch that < σ > ⊆ F we s h all denote by λ J s : K → H ⊗ K the map giv en by λ J s ( x ) = J s λ ( x ) J − 1 s , for all x ∈ K. Here J s is identified w ith an element in H ⊗ K via the inclus ion id H ⊗ ι . T he same calculation as in [A, Prop. 3.1.1] prov es that λ J s ( K ) ⊆ H ⊗ K . Define ( K J s , λ J s , Φ s ( J s ⊗ 1)) the left H -como d u le algebra with un derlying algebra K J s , coactio n λ J s and asso ciator Φ s ( J s ⊗ 1). It follo ws from Lema 4.9 that ( K J s , λ J s , Φ s ) is a left H J s -comod ule algebra. Lemma 6.8. The left H - c omo dule algebr as ( K , λ ) and ( K J s , λ J s , Φ s ( J s ⊗ 1)) ar e e qui v ariantly Morita e q uivalent, that is K M , K J s M ar e e quivalent as Rep( H ) -mo dules.  Pr o of. F or an y X ∈ Rep( H ), M ∈ K M and any x ∈ X , m ∈ M defin e c X,M : X ⊗ k M → X ⊗ k M , c X,M ( x ⊗ m ) = J s · ( x ⊗ m ) . It is immediate to pro ve that the iden tit y functor (Id , c ) : K M → K J s M is an equiv alence of mo dule categ ories.  Definition 6.9. Let ( K , λ, Φ λ ) b e a left H J s -comod ule algebra s u c h that the asso ciator Φ λ ∈ A ( H , s ) ⊗ k A ( H, s ) ⊗ k K . Defin e b K = λ − 1 ( A ( H, s ) ⊗ k K ) and denote b λ the restriction of λ to b K . Th en ( b K , b λ, Φ s ) is a left A ( H , s )-como dule algebra. T u r ns out that this pro cedure is the inv erse of the crossed pr o duct. MODULE CA TEGORIES OVER FINITE POINTED TENSOR C A TEGORIES 27 6.4. Actions on mo dule categories ( b K , b λ, Φ s ) M . F or the rest of this sec- tion we shall assume no w that m = p is a p rime num b er . Let ( K, λ ) b e a t yp e 1 left H -como du le algebra suc h that K 0 = k F where F = C d is a cyclic group. There are t wo p ossible cases; wh en < σ > ⊆ F or F = { 1 } . Let us treat the first case. So w e assume that p | d . L et s, l ∈ N b e suc h that d = ps and sl = p . Let u s denote b F = C s = < χ lp > . By hyp othesis th e v ector sp ace K has a d ecomp osition K = ⊕ f ∈ F K f where K f is the v ector space with basis { y r 1 1 . . . y r t t e f : 0 ≤ r j ≤ N j } . F or an y i = 0 . . . s − 1 defin e K ( i ) = M j : χ mj + i ∈ C d K χ mj + i . Observe that b K = K (0) . With this grading K is an b F -crossed pro du ct. Lemma 6.10. Under the ab ove assumptions ( b K , b λ, Φ s ) M is an b F -e quiv ariant Rep( A ( H, s )) -mo dule c ate gory and  ( b K , b λ, Φ s ) M  b F ≃ K M as mo dule c ate- gories over Rep( H ) . Pr o of. It f ollo ws f rom Pr op osition 5.13 and Prop osition 5.14.  No w , let us assume th at F = { 1 } . Let u s endow the s p ace K ⊗ k k C p with the pr o duct d etermined by ( y l ⊗ σ a )( y s ⊗ σ b ) = q pad s y l y s ⊗ σ a + b . The space K ⊗ k k C p is a left H -comod ule algebra with copro d uct determined b y λ ( y l ⊗ σ a ) = x l σ a ⊗ 1 ⊗ σ a + σ a χ − b l ⊗ y l ⊗ σ a . It is clear that ( K ⊗ k k C p ) 0 = k C p . Thus we can consider the left A ( H, s )- comod ule algebra ( K ⊗ k k C p , b λ, Φ s ) . Lemma 6.11. Under the ab ove c onventions the fol lowing ho lds. 1. The mo dule c ate gory ( k C p ,λ, Φ s ) M has a C p -action such that ther e is an e quivalenc e  ( k C p ,λ, Φ s ) M  C p ≃ V ect k as Rep( H ) -mo dules. 2. The mo dule c ate gory ( K ⊗ k k C p , b λ, Φ s ) M has a C p -action such that ther e is an e quivalenc e  ( K ⊗ k k C p , b λ, Φ s ) M  C p ≃ K M as Rep( H ) -mo dules. Pr o of. 1. It f ollo ws from (2) taking K = k . 2. S et M = ( K ⊗ k k C p , b λ, Φ s ) M . F or an y i = 0 , . . . , p − 1 define the fun ctors ( U i , c i ) : M → M σ i as follo w s . F or any M ∈ M U i ( M ) = M with a new action ⊲ : ( K ⊗ k k C p ) ⊗ k M → M of K ⊗ k k C p giv en by y l ⊲ m = q id l y l · m , σ ⊲ m = q ip σ · m, 28 GALINDO, M OMBELLI for all l = 1 , . . . , t , m ∈ M . F or an y X ∈ Rep( A ) , M ∈ M the map c i X,M : U i ( X ⊗ k M ) → F i ( X ) ⊗ k U i ( M ) is the identit y . The isomorph ism µ i,j : U i ◦ U j → U i + j is giv en by the action of σ − ( i + j ) − ( i + j ) ′ p . Altoget her mak es the category ( K ⊗ k k C p , b λ, Φ s ) M a C p -equiv arian t Rep( A )- mo dule category . Let N ∈ K Mo d . Define F ( N ) = ⊕ p − 1 i =0 N i where N i = N as vec tor spaces. Let us define a new actio n of ⇀ : K ⊗ k k C p ⊗ k F ( N ) → F ( N ) as follo ws. If n ∈ N i then σ ⇀ n = q pi n ∈ N i , y l ⇀ n = q d l i y l · n ∈ N ( d l + i ) ′ . Recall that a ′ denotes the r emainder of a in the division b y p . Not e also that for an y i, j = 0 , . . . , p − 1 U i ( N j ) = N i + j . The mo dule F ( N ) is a C p - equiv arian t ob ject in ( K ⊗ k k C p , b λ, Φ s ) M , indeed for an y i = 0 , . . . , p − 1 defin e the isomorp hisms v i : U i ( F ( N )) → F ( N ) as f ollo ws: v i ( n ) = q − i n ∈ N i + j for an y n ∈ N j . This maps are K ⊗ k k C p -mo dule isomorphisms and they sat- isfy equation (3.4). This d efi nes a fun ctor F : K Mo d → ( K ⊗ k k C p , b λ, Φ s ) M that together with the ident ity isomorphisms c X,N : F ( X ⊗ k N ) → X ⊗ k F ( N ) b e- comes a mo d ule fun ctor. If M ∈ ( K ⊗ k k C p , b λ, Φ s ) M then M = ⊕ p − 1 i =0 M i where M i is the eigenspace of the eigen v alue q pi of the action of σ . The sp ace M 0 has a K -action as follo ws . Since M is C p -equiv arian t there are isomorph isms v i : U i ( M ) → M suc h that the restrictions v i | M 0 : M 0 → M i are isomorphisms. If m ∈ M 0 , y l ∈ K then y l · m ∈ M d l , th us w e can define ⇀ : K ⊗ k M 0 → M 0 y l ⇀ m = v − 1 d l ( y l · m ) , for all m ∈ M 0 . Th e map M 7→ M 0 is fun ctorial and d efines an inv erse functor for F .  6.5. Exact mo dule categories o v er Rep( A ( H , s )) . No w we can form ulate the main result of this section. Theorem 6.12. L et H b e a Hopf algebr a of typ e 1 (se e definition 6.6) and let M b e an exact inde c omp osable mo dule c ate gory over Rep ( A ( H , s )) . Then the fol lowing statements hold. (1) ther e exists a right H -simple left H -c omo dule algebr a ( K , λ ) with trivial c oinvariants such that K 0 ⊇ C p and ther e is an e quivalenc e of mo dule c ate gories M ≃ ( b K , b λ, Φ s ) M . (2) If ther e is an e quivalenc e ( c K ′ , b λ ′ , Φ ′ s ) M ≃ ( b K , b λ, Φ s ) M as Rep( A ( H , s )) - mo dules then ( K, λ ) and ( K ′ , λ ′ ) ar e e quivariantly M orita e qui valent H -c omo dule algebr as. Pr o of. 1. By Lemma 4.7 there exists a left A ( H , s )-como du le algebra ( K , λ, Φ) suc h th at M ≃ K M . The catego ry K M is F -equiv arian t for some sub- group F ⊆ C p . Thus it follo ws f rom [AM, Thm 3.3] that there is a righ t MODULE CA TEGORIES OVER FINITE POINTED TENSOR C A TEGORIES 29 H -simple left H -como dule algebra ( S, δ ) with trivial coin v ariants suc h that  K M  F ≃ S M as Rep( H )-mo du les. Hence S 0 = k 1, S 0 = k C p or S 0 = k C p 2 . In an y case, it follo ws from Lemmas 6.10, 6.11 that there is a righ t H - simple left H -como dule algebra ( K, λ ) with trivial coinv ariants suc h that K 0 ⊇ C p and there is an equiv alence S M ≃  ( c K ′ , b λ ′ , Φ ′ s ) M  F . Whence  K M  F ≃  ( c K ′ , b λ ′ , Φ ′ s ) M  F , thus using Prop osition 3.4 (5) we get the result. 2. There exists a subgroup F ⊆ C p suc h that b oth mo d ule categories ( c K ′ , b λ ′ , Φ ′ s ) M , ( b K , b λ, Φ s ) M are F -equiv arian t and there are equiv alences of mo d- ule categories o ver Rep( H J s ) ( K,λ, Φ s ) M ≃  ( b K , b λ, Φ s ) M  F ≃  ( c K ′ , b λ ′ , Φ ′ s ) M  F ≃ ( K ′ ,λ ′ , Φ ′ s ) M . Th us b y Lemma 6.8 follo ws that K M ≃ K ′ M .  6.6. Some classification results. W e apply Theorem 6.12 to obtain the classification of mo dule catego ries o v er Rep( A ( H , s )) where H is the b osoniza- tion of a quantum linear sp ace. Let g 1 , . . . , g θ ∈ C p 2 , χ 1 , . . . , χ θ ∈ d C p 2 b e a datum for a quant um linear space and let V = V ( g 1 , . . . , g θ , χ 1 , . . . , χ θ ) the asso ciated Y etter-Drinfeld mo dule o ve r k C p 2 generated as a vecto r space by x 1 , . . . , x θ . F or more details see [AS ]. The Hopf algebra H = B ( V )# k C p 2 is a t yp e 1 Hopf algebra, see [Mo2 ]. Let us define no w a family of righ t H -simp le left H -comod ule algebras. Let F ⊆ C p 2 b e a sub group and ξ = ( ξ i ) i =1 ...θ , α = ( α ij ) 1 ≤ i

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