Constructing dynamical twists over a non-abelian base

CONSTRU CTING D YNAMICAL TWISTS OVER A NON-ABELIAN BASE JUAN MAR T ´ IN MOMBELLI Abstra ct. W e giv e examp les of dynamical twists in fi n ite-dimensional Hopf algebras o ve r an arbitrary Hopf subalgebra. The construction is based on the categorica l approach of dynamical t wists in troduced by Donin and Mudrov [DM1]. Introduction The theory of dynamic al quantum gr oups initiated by G. F elder [F1] is no w adays an activ e branch of mathematics. This theory arose from the no- tion of dynamic al Y ang-Baxter e qu ation , also kno wn as the Gerva is-Neveu- F elder e quation in connection with int egrable mo d els of conformal field the- ories and Liouville theory , s ee for example [F2], [GN], [ABB]. F o r a d etailed review and bibliograph y on the d ynamical Y ang-Baxter equation the reader is referred to [E], [ES], [Fh]. In [B], [BBB], the notion of Drinfeld’s twist for Hopf algebras w as gen- eralized in th e dynamical setting. A dynamic al twist for a Hopf algebra H o v er A , wh er e A is an ab elian subgroup of the group of group lik e ele- men ts in H , is a function J ( λ ) : b A → ( H ⊗ H ) × satisfying certain non-linear equations. When the group A is trivial we reco v er the notio n of Drinfeld’s t w ist. When H is a quasi-triangular Hopf algebra with R -matrix R then R ( λ ) = J − 1 ( λ ) 21 RJ ( λ ) satisfies the dyn amical quan tu m Y ang-Ba xter equa- tion. See [EN1]. In the fin ite-dimensional case, dynamical t wists w ere studied firs t in [EN1] and la ter in [M]. In the first pap er dynamical t w ists ov er an ab elian group for the group alge bra of a finite group are classified. F ollo wing closely th is w ork, in [M], dy n amical twists o ver an ab elian group for an y finite-dimensional Hopf algebra are d escrib ed. In this work w e extend the results of [M ] in the case where A is an ar- bitrary Hopf su balgebra of H . T o this end w e rely on the defin ition and catego rical construction of d y n amical twists introd uced in [DM1] from dy- namical adjoin t fun ctors. Here, th e Hopf s u balgebra A pla ys the same role Date : Octob er 26, 201 8. 2000 Mathematics Subje ct Cl assific ation : 16W30, 18D10, 19D23 keywor ds : Dynamical Y ang-Baxter equation; dynamical t wists; Hopf algebras. 1 2 MOMBELLI as the Levi subalgebra of a r eductiv e Lie alge bra in [DM1 ]. As in [M ], the language of mo dule c ate gories has b een used with p r ofit. The con tents of the p ap er are organized as f ollo ws: in Section 1 w e recall the notion of stabilizers for Hopf algebra actions in tro duced by Y an and Zh u [YZ], and the definition of mo du le categorie s o v er a tensor category . In Section 2 we giv e a b rief accoun t of the results and definitions app eared in [DM1]. W e explain th e definition of dynamical extension of a tensor catego ry , dynamical twist and dyn amical adjoint functors. W e also recall the construction of a dynamical t wist coming from a p air of dynamical functors. Section 3 is dev oted to the construction of dynamical t wists for a fi nite- dimensional Hopf algebra H o v er a Hopf subalgebra A . F ollo wing [EN1] we shall sa y that a dynamic al datum for ( H , A ) is a pair ( K, T ), where • K is a H -simple left H -comod ule algebra with K co H = k , • and T : Rep( A ) → K M is a functor suc h that for any V , W ∈ Rep( A ) there are natural isomorphisms Stab K ( T ( V ) , T ( W )) ≃  Ind H A ( V ⊗ k W ∗ )  ∗ . F or an y d y n amical datum ( K, T ) we shall show that the pair ( T , R ), w here R : Rep( H ) → Rep( A ) is the restriction fu nctor, is a pair of dynamical adjoin t fun ctors. Therefore, applying the to ols exp lained in Section 2, we obtain a dyn amical t wist o v er the base A . This construction generalize the pro cedure app eared in [M] when A is a commutat iv e co comm utativ e Hopf subalgebra. Also, we shall p ro v e that an y dy n amical t wist b ased in A for H comes from a dynamical d atum. Finally w e sho w some explicit examples of dynamical data. 1. Preliminaries and not a tion Throughout this pap er k will denote an arb itrary field. All categories and functors are assumed to b e k -linear. All vecto r spaces and algebras are assume to b e ov er k . If K is an algebra, we sh all denote b y K M the ca tegory of finite-dimensional left K -mo du les. If V is a vec tor space, we shall denote b y h , i : V ∗ ⊗ k V → k the ev aluation map. By H w e shall denote a Hopf algebra w ith counit ε , and an tip o de S . W e shall d enote by Rep( H ) the category of finite-dimensional left H -mo d u les emphasizing the canonical tensor str u cture. If K is an H -como du le algebra with coacti on δ : K → H ⊗ K , an H - c ostable ide al of K is a t w o-sided id eal I of K su ch that δ ( I ) ⊆ H ⊗ I . W e shall sa y that K is H - simple if it h as n o non-trivial H -costable ideal of K . W e sh all d enote H M K the category of left H -comod u les, righ t K -mo dules suc h that the K -mo dule structure is an H -comod ule map. I f P ∈ H M K then End K ( P ) has a n atural left H -comodu le algebra structure via δ : CONSTRUCTING DYNAMICAL TWISTS 3 End K ( P ) → H ⊗ k End K ( P ), T 7→ T ( − 1) ⊗ T (0) , d etermined by (1.1) h α, T ( − 1) i T 0 ( p ) = h α, T ( p (0) ) ( − 1) S − 1 ( p ( − 1) ) i T ( p (0) ) (0) , T ∈ End K ( P ) , p ∈ P , α ∈ H ∗ . See [AM, Lemma 1.26]. Lemma 1.1. L et A ⊆ H b e a H opf sub algebr a and V an A -mo dule. The sp ac e Hom A ( H , V ) has a natur al H -mo dule structur e and ther e ar e natur al H -mo dule isomor phisms  Ind H A V  ∗ ≃ Hom A ( H , V ∗ ) . Pr o of. The H -mo du le structure on Hom A ( H , V ) is as follo ws. If t, h ∈ H , T ∈ Hom A ( H , V ) then ( h · T )( t ) = T ( th ) . It is not difficult to pr o ve that the maps θ :  Ind H A V  ∗ → Hom A ( H , V ∗ ) , e θ : Hom A ( H , V ∗ ) →  Ind H A V  ∗ , giv en by the formulas θ ( α )( h ) = X i α ( S ( h ) ⊗ v i ) v i , e θ ( β )( h ⊗ v ) = h β ( S − 1 ( h )) , v i , are w ell defined isomorphisms , one t he in verse of ea c h other, and they are H -mo du le maps. Here α ∈  Ind H A V  ∗ , β ∈ Hom A ( H , V ∗ ), and ( v i ) , ( v i ) are dual b asis for V and V ∗ , h ∈ H , h ⊗ v ∈ Ind H A V .  1.1. St abilizers for Hopf a lgebra actions. W e r ecall very b riefly th e notion of Hopf algebra stabilizers introdu ced in [YZ], s ee also [AM]. Let K b e a finite-dimensional left H -como du le algebra and V , W t w o left K -mo dules. The Y an-Zhu stabilizer Stab K ( V , W ) is defined as the intersec- tion Stab K ( V , W ) = Hom K ( H ∗ ⊗ V , H ∗ ⊗ W ) ∩ L  H ∗ ⊗ Hom( V , W )  . Here the map L : H ∗ ⊗ Hom( V , W ) → Hom( H ∗ ⊗ V , H ∗ ⊗ W ) is defi ned by L ( γ ⊗ T )( ξ ⊗ v ) = γ ξ ⊗ T ( v ), for ev ery γ , ξ ∈ H ∗ , T ∈ Hom( V , W ), v ∈ V . The K -action on H ∗ ⊗ V is giv en b y k · ( γ ⊗ v ) = k ( − 1) ⇁ γ ⊗ k (0) · v , for all k ∈ K , γ ∈ H ∗ , v ∈ V . Here ⇁ : H ⊗ H ∗ → H ∗ is the act ion defined b y h h ⇁ γ , t i = h γ , S − 1 ( h ) t i , for all h, t ∈ H , γ ∈ H ∗ . Also, we denote Stab K ( V ) = Stab K ( V , V ). 4 MOMBELLI Prop osition 1.2. [AM, Prop. 2.7, Prop. 2.16] The fol lo wing assertions holds. 1. F or any left K -mo dules V , W , U ther e is a natur al c omp osition Stab K ( V , W ) ⊗ k Stab K ( U, V ) → Stab K ( U, W ) making S tab K ( V ) a left H - mo dule algebr a. 2. If K is H -simple then (1.2) dim( K ) d im(Stab K ( V , W )) = dim( V ) dim( W ) dim( H ) . 3. F or any X ∈ Rep( H ) ther e ar e na tur al isom orphisms Hom H ( X, Stab K ( V , W )) ≃ Hom K ( X ⊗ k V , W ) , wher e the action on X ⊗ k V is given by the c o action of K .  The follo wing result concernig Y an -Z h u stabilizers will b e usefu l later. If A ⊆ H is a Hopf su balgebra, and R = K co A ⊆ K is a left A -Hopf Galois extension th en there are H -mo dule algebra isomorph isms Stab K ( V , W ) ≃ Hom A ( H , Hom R ( V , W )) (1.3) for an y left K -mo dules V , W . W orth to mentio n that th e actio n of A on Hom R ( V , W ) is given by ( a · T )( v ) = a [1] · T ( a [2] · v ) , for all a ∈ A, T ∈ Hom R ( V , W ) , v ∈ V . R ecall that the map γ : A → K ⊗ R K , γ ( a ) = a [1] ⊗ a [2] is defin ed b y γ ( a ) = can − 1 ( a ⊗ 1), where can : K ⊗ R K → A ⊗ k K is the canonical map can ( k ⊗ s ) = k ( − 1) ⊗ k (0) s , k , s ∈ K . F or more details see [Sc h, Rmk. 3.4], [AM, Thm. 2.23]. 1.2. Mo dule categories. W e br iefly recal l the d efi nition of modu le c ate- gory and the defin ition introd u ced by Etingof-Ostrik of exact mo dule cate- gories. W e refer to [O1], [O2], [EO]. Let us fix C a fi nite tensor categ ory . A mo dule c ate gory o ver C is a collect ion ( M , ⊗ , m, l ) where M is an Ab elia n category , ⊗ : C × M → M is an exact bifun ctor, asso ciativit y and un it isomorph isms m X,Y ,M : ( X ⊗ Y ) ⊗ M → X ⊗ ( Y ⊗ M ), l M : 1 ⊗ M → M , X , Y ∈ C , M ∈ M , su c h that m X,Y ,Z ⊗ M m X ⊗ Y ,Z ,M = (id X ⊗ m Y , Z,M ) m X,Y ⊗ Z, M ( a X,Y ,Z ⊗ id M ) , (1.4) (id X ⊗ l M ) m X, 1 , M = r X ⊗ id M , (1.5) for all X , Y , Z ∈ C , M ∈ M . Sometimes we shall simply say th at M is a modu le category omitting to men tion ⊗ , m a nd l whenever n o co nfusion arises. In this p ap er w e furth er assu me that all mo dule cate gories hav e finitely man y isomorphism classes of sim p le ob jects. CONSTRUCTING DYNAMICAL TWISTS 5 Let M , M ′ b e t w o mo du le categ ories o v er C . A mo du le functor b et w een M and M ′ is a pair ( F , c ) where F : M → M ′ is a functor and c X,M : F ( X ⊗ M ) → X ⊗ F ( M ) is a family of n atural isomorp hisms such that m ′ X,Y , F ( M ) c X ⊗ Y ,M = (id X ⊗ c Y , M ) c X,Y ⊗ M F ( m X,Y ,M ) , (1.6) l ′ F ( M ) c 1 ,M = F ( l M ) , (1.7) for all X , Y ∈ C , M ∈ M . I f ( G , d ) : M → M ′ is another mo dule f unctor, a morphism b etw een ( F , c ) and ( G , d ) is a natural transformation α : F → G suc h that for any X ∈ C , M ∈ M 1 : d X,M α X ⊗ M = (id X ⊗ α M ) c X,M . (1.8) The m o dule structur e o v er M ⊕ M ′ is defi ned in an obvio us wa y . A mo dule category is inde c omp osable if it is not e quiv alen t t o the direct sum of t w o n on-trivial mo d ule categories. A mo dule cat egory M is exact [EO] if for an y pro jectiv e ob ject P ∈ C and an y M ∈ M the ob ject P ⊗ M is again pro jectiv e. 1.3. Mo dule categories ov er Hopf algebras. Let H b e a fi nite-dimensional Hopf algebra. L et K b e a left H -comod ule algebra. T hen K M is a left mo dule category o v er Rep H via the coaction λ : K → H ⊗ K . That is, ⊗ : Rep H × K M → K M is giv en by X ⊗ V := X ⊗ k V , for X ∈ Rep H and V ∈ K M with action k · ( x ⊗ v ) = k ( − 1) · x ⊗ k (0) · v , for all k ∈ K , x ∈ X , v ∈ V . Moreo ver, any exac t mo dule category is of this form. Theorem 1.3. 1. If K is H - simple left H -c omo dule algebr a then K M is an inde c omp osable exact mo dule c ate gory. 2. If M i s an inde c omp osable exact mo dule c ate gory over Rep( H ) then ther e exists an H -simple left H -c omo dule algebr a K such that M ≃ K M . Pr o of. See [AM, Prop. 1.20, Th. 3.3].  Let S b e another H -simple left H -como d ule algebra. Prop osition 1.4. [AM, Prop. 1.24] The mo dule c ate gories K M , S M over Rep( H ) ar e e quivalent if and only if ther e exists P ∈ H M K such that S ≃ End K ( P K ) as H -mo dule algebr as. M or e over the e qu i valenc e i s given by F : K M → S M , F ( V ) = P ⊗ K V , f or al l V ∈ K M .  2. D ynamical twists constructed from dynamical functors W e recall a construction due to Donin and Mudrov of d ynamical t w ists from dynamical adjoint functors, see [DM1, § 6]. There are some d ifferences in our statemen ts and those app eared in lo c. cit. s in ce we use le ft mod ule catego ries in stead of righ t ones. 6 MOMBELLI 2.1. Dynamical exte nsions of tensor categories. In [DM1] f or an y ten- sor cate gory C an d a mo dule cat egory M o ver C the authors in tro d uced a new tensor catego ry , that we will denote by M ⋉ C . This tensor category is called the dynamic al extension of C ov er M . Ob jects in the category M ⋉ C are fu n ctors F X : M → M , F X ( M ) = X ⊗ M , f or all X ∈ C , M ∈ M . Morphisms are natural transf orm ations. Observe that f or eac h f ∈ Hom C ( X, Y ) there is a n atural transformation η f : F X → F Y , given by ( η f ) M : X ⊗ M → Y ⊗ M , ( η f ) M = f ⊗ id M , for all M ∈ M . W e briefly r ecall the monoidal structure of M ⋉ C . The tensor pro duct is F X ⊗ F Y = F X ⊗ Y , X , Y ∈ C , and the asso ciativit y constrain t is e a X,Y ,Z : ( F X ⊗ F Y ) ⊗ F Z → F X ⊗ ( F Y ⊗ F Z ) , ( e a X,Y ,Z ) M = ( a X,Y ,Z ⊗ id M ) , for all M ∈ M . F or any X ∈ C the left and r igh t unit isomorphisms are giv en by l X : F X ⊗ F 1 → F X , r X : F 1 ⊗ F X → F X , where l X,M = l X ⊗ id M and r X = r X ⊗ id M for all M ∈ M . If η : F X → F Z , φ : F Y → F W are tw o n atural transformation the tensor pro du ct η ⊗ φ : F X ⊗ Y → F Z ⊗ W is giv en by the comp osition ( η ⊗ φ ) M = m − 1 Z W M η W ⊗ M (id X ⊗ φ M ) m X Y M , (2.1) for all M ∈ M . The u nit elemen t is F 1 . R emark 2.1 . Note that for any X, Y , U, V ∈ C and f : X → Y , g : U → V , ( η f ⊗ η g ) M = ( f ⊗ g ) ⊗ id M , (2.2) for all M ∈ M . Prop osition 2.2. If M ≃ N as mo dule c ate gories then ther e is a tensor e quivalenc e M ⋉ C ≃ N ⋉ C . Pr o of. Assume that ( F , c ) : M → N and ( G , d ) : N → M is a pair of equiv- alence of mo dule cate gories. Let θ : Id → F ◦ G b e a n atur al isomorphism of mo du le functors, that is θ satisfies c X, G ( N ) F ( d X,N ) θ X ⊗ N = id X ⊗ θ N , (2.3) for all X ∈ C , N ∈ N . Define Φ : M ⋉ C → N ⋉ C the functor Φ( F X ) = e F X , for any X ∈ C . Here, w e denote e F X : N → N the functor e F X ( N ) = X ⊗ N , for all N ∈ N . If X, Y ∈ C and η : F X → F Y is a natural transform ation then Φ( η ) : e F X → e F Y is giv en by the comp osition CONSTRUCTING DYNAMICAL TWISTS 7 X ⊗ N id X ⊗ θ N − − − − − − → X ⊗F ( G ( N )) c − 1 X, G ( N ) − − − − − → F ( X ⊗G ( N )) − → F ( η G ( N ) ) − − − − − − → F ( Y ⊗G ( N )) F ( d − 1 Y ,N ) − − − − − → F G ( Y ⊗ N ) θ − 1 Y ⊗ N − − − − → Y ⊗ N , (2.4) for all N ∈ N . The tensor structure on Φ is giv en by the iden tit y . Th at is, for any X, Y ∈ C the n atur al isomorphism s ξ : Φ( F X ⊗ F Y ) → Φ( F X ) ⊗ Φ( F Y ) are giv en ξ X,Y ,N : e F X ⊗ Y ( N ) → e F X ⊗ Y ( N ) , ξ X,Y ,N = id X ⊗ Y ⊗ id N , for all N ∈ N . W e hav e to c heck that for all X , Y , Z ∈ C , N ∈ N the follo wing identit y h olds: ( a X Y Z ⊗ id N )( ξ X Y id Z ) N ξ X ⊗ Y ,Z ,N = (id X ⊗ ξ Y Z ) N Φ( a X Y Z ⊗ id N ) . (2.5) The left hand side of (2.5) is equal t o ( a X Y Z ⊗ id N ), the righ t hand side is equal to Φ ( a X Y Z ⊗ id N ), and usin g (2.4 ), is equal to θ − 1 ( X ⊗ ( Y ⊗ Z )) ⊗ N F ( d − 1 X ⊗ ( Y ⊗ Z ) ,N ) ( a X Y Z ⊗ id F ( G ( N )) ) c − 1 ( X ⊗ Y ) ⊗ Z, G ( N ) (id ⊗ θ N ) = ( a X Y Z ⊗ id N ) θ − 1 (( X ⊗ Y ) ⊗ Z ) ⊗ N F ( d − 1 ( X ⊗ Y ) ⊗ Z, N ) c − 1 ( X ⊗ Y ) ⊗ Z, G ( N ) (id ⊗ θ N ) = ( a X Y Z ⊗ id N ) . The last equalit y follo ws by (2.3).  The follo wing defin ition seems to b e well-kno wn. Definition 2.3. A c o cycle in C is a family of isomorphisms J X,Y ∈ Hom C ( X ⊗ Y ) suc h that for all X , Y , Z ∈ C a X Y Z J X ⊗ Y ,Z ( J X,Y ⊗ id Z ) = J X,Y ⊗ Z (id X ⊗ J Y , Z ) a X Y Z , (2.6) J X, 1 = id X ⊗ 1 , J 1 ,X = id 1 ⊗ X . (2.7) If J is a co cycle in C then there is a new monoidal category , C J defined as follo w s. Th e ob jects and morphisms are the same as in C . The tensor pro du ct of C J coincides with th e tensor pro d uct of C on ob jects. If f : X → Y , g : Z → W is a pair of morphisms th en f e ⊗ g = J Y , W ( f ⊗ g ) J − 1 X,Z . Eviden tly if J comm u tes with morp hisms in C then the tensor category C J is equiv alent to C . Let ( M , m, l ) b e a m o dule category o v er C . Th e follo w ing defin ition is due to Donin and Mud ro v, see [DM1 , Definition 5.2]. Definition 2.4. A dynamic al twist for the extension M ⋉ C is a co cycle J in M ⋉ C su c h that J comm u tes w ith morph isms in C , that is J Z,W ( η f ⊗ η g ) = ( η f ⊗ η g ) J X,Y , (2.8) for all f ∈ Hom C ( X, Z ) , g ∈ Hom C ( Y , W ). 8 MOMBELLI More explicitly , a dynamical t wist is a family of isomorphisms J X,Y ,M : ( X ⊗ Y ) ⊗ M → ( X ⊗ Y ) ⊗ M , X, Y ∈ C , M ∈ M such that ( a X Y Z ⊗ id M ) J X ⊗ Y ,Z ,M m − 1 X ⊗ Y ,Z ,M J X,Y ,Z ⊗ M m X ⊗ Y ,Z ,M = = J X,Y ⊗ Z, M m − 1 X,Y ⊗ Z, M (id X ⊗ J Y , Z,M ) m X,Y ⊗ Z ,M ( a X Y Z ⊗ id M ) , (2.9) ( l X ⊗ id M ) J X, 1 , M = ( l X ⊗ id M ) , J 1 ,X, M ( r X ⊗ id M ) = ( r X ⊗ id M ) , (2.10) for all X , Y , Z ∈ C , M ∈ M . Equation (2.8) implies that J Z,W,M m − 1 Z,W,M ( f ⊗ ( g ⊗ id M )) m X,Y ,M = m − 1 Z,W,M ( f ⊗ ( g ⊗ id M )) m X,Y ,M J X,Y ,M , for all morphism s f : X → Z , g : Y → W in C , and all M ∈ M . 2.2. Mo dule categories coming from dynamical t wists. If J is a dy- namical t wist for the dynamical extension M ⋉ C w e will d enote b y M ( J ) the catego ry M with the follo wing mo d ule category structure; th e actio n is the same as in M and the asso ciativit y isomorphisms are b m X,Y ,M : ( X ⊗ Y ) ⊗ M → X ⊗ ( Y ⊗ M ) , b m X,Y ,M = m X,Y ,M J − 1 X,Y ,M , for all X , Y ∈ C , M ∈ M . Prop osition 2.5. L et J b e a dynamic al twist f or the dynamic al extension M ⋉ C . 1. M ( J ) is a mo dule c ate gory over C . If M is inde c omp osable then so is M ( J ) . 2. Ther e is a tenso r e q uivalenc e M ( J ) ⋉ C ≃ ( M ⋉ C ) J . Pr o of. Straigh tforw ard.  The idea of u sing the modu le category la nguage in the study of dynam- ical twists is due to Ostrik, see [O1]. In lo c. cit. the author relates the classification of mo d u le categories o ver Rep( G ), G a finite g roup, with t he results obtained by Etingof and Niksh yc h on dynamical t wists o ver th e group algebra of the group G [EN1]. This idea was used with p r ofit in [M]. 2.3. Dynamical t wists and dynamical adjoint functors. Let C b e a tensor catego ry . Co cycles in C are in b ijectiv e corresp ondence with natur al asso ciativ e operations in the space Hom C . If J is a co cycle then ⊛ : Hom C ( V , U ) ⊗ Hom C ( V ′ , U ′ ) → Hom C ( V ⊗ V ′ , U ⊗ U ′ ) defined by φ ⊛ ψ = ( φ ⊗ ψ ) J − 1 V U is an asso ciativ e op eration for all V , U, V ′ , U ′ ∈ C . CONSTRUCTING DYNAMICAL TWISTS 9 The follo wing result is analogous to [DM1, L emma 6.1]. Lemma 2.6. Assume th at th er e is an asso ciative op er ation ⊛ : Hom C ( V , U ) ⊗ k Hom C ( V ′ , U ′ ) → Hom C ( V ⊗ V ′ , U ⊗ U ′ ) for al l V , U, V ′ , U ′ ∈ C such that ( φ ⊛ ψ ) = ( φ ⊗ ψ )(id ⊛ id ) , (2.11) φ ⊛ ξ = φ ⊗ ξ , ξ ⊛ φ = ξ ⊗ φ (2.12) for al l morphisms φ, ψ , α, β in C and ξ ∈ Hom C ( V , 1 ) . Assume also that for any U, V ∈ C I U V = id U ⊛ id V is invertible. Then J U V = I − 1 U V , U, V ∈ C , is a c o cyc le in C . Pr o of. The pro o f is en tirely similar to the pro of of [DM1, Lemma 6.1].  Let C , C ′ b e tw o tensor categories. Let ( M , m, l ) b e a mo dule catego ry o ver C and ( M ′ , m ′ , l ′ ) b e a mo dule category ov er C ′ . Let ( R , b ) : C → C ′ b e a tensor functor. The follo wing defi n ition is [DM1 , Def. 6.2] for r igh t mo dule categories. Definition 2.7. A functor T : M ′ → M is s aid to b e a d ynamic al adjoint to R if there exists a family of natural isomorph ism s ξ X,M ,N : Hom M ( X ⊗ T ( M ) , T ( N )) ≃ − − → Hom M ′ ( R ( X ) ⊗ M , N ) , for all X ∈ C , M , N ∈ M ′ . W e further assu me that for any M ∈ M ′ ξ 1 ,M ,M ( l T ( M ) ) = l ′ M . (2.13) R emark 2.8 . F or an y M , M ′ , N , N ′ ∈ M ′ and X , Y ∈ C and morph isms f : N → N ′ , g : M : → M ′ , α : X → Y , the naturalit y of ξ implies that f ◦ ξ X,M ,N ( β 1 ) = ξ X,M ,N ′ ( T ( f ) β 1 ) , (2.14) ξ X,M ′ ,N ( β 2 ) ◦ (id R ( X ) ⊗ g ) = ξ X,M ,N ( β 2 (id X ⊗ T ( g ))) , (2.15) ξ Y , M ,N ( β 3 )( R ( α ) ⊗ id M ) = ξ X,M ,N ( β 3 ( α ⊗ id T ( M ) )) , (2.16) for any β 1 ∈ Hom M ( X ⊗ T ( M ) , T ( N )) , β 2 ∈ Hom M ( X ⊗ T ( M ′ ) , T ( N )) , β 3 ∈ Hom M ( Y ⊗ T ( M ) , T ( N )) . The category M ′ is a mo dule category ov er C via R . Th e action is giv en b y e ⊗ : C × M ′ → M ′ , X e ⊗ M = R ( X ) ⊗ M , for all X ∈ C , M ∈ M ′ . The asso ciativit y isomo rphisms are e m X,Y ,M = m ′ R ( X ) ,R ( Y ) ,M ( b X Y ⊗ id M ) for all X , Y ∈ C , M ∈ M ′ . 10 MOMBELLI F or any p air of dynamical adj oint fu nctors ( R, T ) we will rep eat the con- struction giv en in [DM1] of a dynamical twist for the extension M ′ ⋉ C . F or this we will defi n e an asso ciativ e operation in Hom M ′ ⋉ C . In some sense, the dynamical t wist constructed from the pair ( R, T ) me a- sur es ho w far is the functor T fr om b eing a m o dule fu nctor. Let X , Y , U, V ∈ C and φ : F X → F Y , ψ : F U → F V b e morphisms in M ′ ⋉ C . So for eac h M ∈ M ′ w e ha ve that φ M : R ( X ) ⊗ M → R ( Y ) ⊗ M , ψ M : R ( U ) ⊗ M → R ( V ) ⊗ M are m orphisms in M ′ . Set e φ M = ξ − 1 X,M ,R ( Y ) ⊗ M ( φ M ), and e ψ M = ξ − 1 U,M ,R ( V ) ⊗ M ( ψ M ). Th us we define ( φ ⊛ ψ ) M as th e image b y ξ of the comp osition ( X ⊗ U ) ⊗ T ( M ) m X,U,T ( M ) − − − − − − − − → X ⊗ ( U ⊗ T ( M )) id X ⊗ e ψ M − − − − − − − → X ⊗ T ( R ( V ) ⊗ M ) → e φ R ( V ) ⊗ M − − − − − − − → T ( R ( Y ) ⊗ ( R ( V ) ⊗ M )) T ( e m − 1 Y V M ) − − − − − − − − → T ( R ( Y ⊗ V ) ⊗ M ) . That is, for all M ∈ M ′ , ( φ ⊛ ψ ) M equals to ξ X ⊗ U,M, R ( Y ⊗ V ) ⊗ M  T ( e m − 1 Y V M ) e φ R ( V ) ⊗ M (id X ⊗ e ψ M ) m X,U,T ( M )  . Lemma 2.9. F or any X, Y , U, V ∈ C and mo rphisms f : X → Y , g ∈ U → V , φ : F X → F Y , ψ : F U → F V ( η f ⊛ η g ) = (id Y ⊛ id V )( η f ⊗ η g ) . (2.17) φ ⊛ ψ = ( φ ⊗ ψ )(id X ⊛ id U ) . (2.18) Pr o of. Using (2.16) w e hav e that ( η f ⊛ η g ) equals to = ξ X ⊗ U,M, R ( Y ⊗ V ) ⊗ M  T ( e m − 1 Y V M ) ξ − 1 Y , M ,R ( Y ) ⊗ M (id )( f ⊗ id T ( M ) ) (id X ⊗ ξ − 1 V ,M ,R ( V ) ⊗ M (id )( g ⊗ id T ( M ) )) m X,U,T ( M )  = ξ X ⊗ U,M, R ( Y ⊗ V ) ⊗ M  T ( e m − 1 Y V M ) ξ − 1 Y , M ,R ( Y ) ⊗ M (id )(id Y ⊗ ξ − 1 V ,M ,R ( V ) ⊗ M (id )) ( f ⊗ ( g ⊗ id T ( M ) )) m X,U,T ( M )  = ξ X ⊗ U,M, R ( Y ⊗ V ) ⊗ M  T ( e m − 1 Y V M ) ξ − 1 Y , M ,R ( Y ) ⊗ M (id )(id Y ⊗ ξ − 1 V ,M ,R ( V ) ⊗ M (id )) m Y , V ,T ( M ) (( f ⊗ g ) ⊗ id T ( M ) )  = ξ X ⊗ U,M, R ( Y ⊗ V ) ⊗ M  T ( e m − 1 Y V M ) ξ − 1 Y , M ,R ( Y ) ⊗ M (id )(id Y ⊗ ξ − 1 V ,M ,R ( V ) ⊗ M (id )) m Y , V ,T ( M )  ( R ( f ⊗ g ) ⊗ id M ) = (id Y ⊛ id V )( η f ⊗ η g ) . The third equality follo ws from the naturalit y of m and the four th equalit y , again, follo ws from (2.16). Thus we hav e pro v ed (2.17). CONSTRUCTING DYNAMICAL TWISTS 11 Note that for any φ : F X → F Y , ψ : F U → F V , equations (2.14) and (2.15) implies that for any M ∈ M ′ e φ R ( V ) ⊗ M (id X ⊗ e ψ M ) equ als to ξ − 1 X,R ( U ) ⊗ M , R ( Y ) ⊗ ( R ( V ) ⊗ M )  φ R ( V ) ⊗ M (id X ⊗ ψ M )   id X ⊗ ξ − 1 U,M ,R ( U ) ⊗ M (id )  . Also, u sing (2.14) we get that ξ − 1 X,R ( U ) ⊗ M , R ( Y ) ⊗ ( R ( V ) ⊗ M )  φ R ( V ) ⊗ M (id X ⊗ ψ M )  is equal to T ( φ R ( V ) ⊗ M ((id X ⊗ ψ M )) ξ − 1 X,R ( U ) ⊗ M , R ( X ) ⊗ ( R ( U ) ⊗ M ) (id ) . Th us, for any M ∈ M ′ , ξ − 1 X ⊗ U,M, R ( Y ⊗ V ) ⊗ M  ( φ ⊛ ψ ) M  is equal to T ( e m − 1 Y , V ,M φ R ( V ) ⊗ M ((id X ⊗ ψ M )) ξ − 1 X,R ( U ) ⊗ M , R ( X ) ⊗ ( R ( U ) ⊗ M ) (id )  id X ⊗ ξ − 1 U,M ,R ( U ) ⊗ M (id )  F ollo ws from (2.1) that T ( e m − 1 Y , V ,M φ R ( V ) ⊗ M ((id X ⊗ ψ M )) = T (( φ ⊗ ψ ) M e m − 1 X,U,M ) , hence ( φ ⊛ ψ ) M equals to ξ X ⊗ U ,M ,R ( Y ⊗ V ) ⊗ M  T (( φ ⊗ ψ ) M e m − 1 X,U,M ) ξ − 1 X,R ( U ) ⊗ M , R ( X ) ⊗ ( R ( U ) ⊗ M ) (id )  id X ⊗ ξ − 1 U,M ,R ( U ) ⊗ M (id )  Using again (2.14) we get that ( φ ⊛ ψ ) M equals to ( φ ⊗ ψ ) M ξ X ⊗ U,M, R ( Y ⊗ V ) ⊗ M  T ( e m − 1 X,U,M ) ξ − 1 X,R ( U ) ⊗ M , R ( X ) ⊗ ( R ( U ) ⊗ M ) (id )  id X ⊗ ξ − 1 U,M ,R ( U ) ⊗ M (id )  , and by definition this is equal to ( φ ⊗ ψ ) M (id X ⊛ id U ) M .  Definition 2.10. F or an y X, Y ∈ C set (2.19) I X,Y = id X ⊛ id Y . Lemma 2.11. F or any X ∈ C , I X, 1 = id X ⊗ id 1 , I 1 ,X = id 1 ⊗ id X . Pr o of. By defin ition, for all M ∈ M ′ , ξ − 1 X ⊗ 1 ,M ,R ( X ⊗ 1 ) ⊗ M ((id X ⊛ id 1 ) M ) equals to T ( e m − 1 X, 1 , M ) ξ − 1 X,R ( 1 ) ⊗ M , R ( X ) ⊗ ( 1 ⊗ M ) (id )(id X ⊗ ξ − 1 1 ,M , 1 ⊗ M (id )) m X, 1 , M (2.20) Using (1.5) and (2.14) for the map id R ( X ) ⊗ l M w e get that (2.20) is equal to T ( b X, 1 r ′− 1 R ( X ) ⊗ id M ) ξ − 1 X, 1 ⊗ M ,R ( X ) ⊗ M (id R ( X ) ⊗ l ′ M )(id X ⊗ ξ − 1 1 ,M , 1 ⊗ M (id )) m X, 1 , M . (2.21) 12 MOMBELLI F rom (2.14) and (2.13) we get that ξ − 1 1 ,M , 1 ⊗ M (id ) = T ( l ′− 1 M ) ξ − 1 1 ,M ,M ( l ′ M ) = T ( l ′− 1 M ) l T ( M ) . No w, u sing (1.5 ) follo ws that (2.21) equals to T ( b X, 1 r ′− 1 R ( X ) ⊗ id M ) ξ − 1 X, 1 ⊗ M ,R ( X ) ⊗ M (id R ( X ) ⊗ l ′ M )(id X ⊗ T ( l ′− 1 M ))( r X ⊗ id M ) . (2.22) F rom (2.15) and (2.15) we get that ξ − 1 X, 1 ⊗ M ,R ( X ) ⊗ M (id R ( X ) ⊗ l ′ M )(id X ⊗ T ( l ′− 1 M )) = ξ − 1 X,M ,R ( X ) ⊗ M (id ) , ξ − 1 X,M ,R ( X ) ⊗ M (id )( r X ⊗ id M ) = ξ − 1 X ⊗ 1 ,M ,R ( X ) ⊗ M ( R ( r X ) ⊗ id M ) th us (2.22) is equal to T ( b X, 1 r ′− 1 R ( X ) ⊗ id M ) ξ − 1 X,M ,R ( X ) ⊗ M (id )( r X ⊗ id M ) = = T ( b X, 1 r ′− 1 R ( X ) ⊗ id M ) ξ − 1 X ⊗ 1 ,M ,R ( X ) ⊗ M ( R ( r X ) ⊗ id M ) . Finally , (id X ⊛ id 1 ) M is equal to = ξ X ⊗ 1 ,M ,R ( X ⊗ 1 ) ⊗ M  T ( b X, 1 r ′− 1 R ( X ) ⊗ id M ) ξ − 1 X ⊗ 1 ,M ,R ( X ) ⊗ M ( R ( r X ) ⊗ id M )  = ( b X, 1 r ′− 1 R ( X ) ⊗ id M ) ξ X ⊗ 1 ,M ,R ( X ) ⊗ M  ξ − 1 X ⊗ 1 ,M ,R ( X ) ⊗ M ( R ( r X ) ⊗ id M )  = ( b X, 1 r ′− 1 R ( X ) ⊗ id M )( R ( r X ) ⊗ id M ) = id R ( X ) ⊗ id 1 ⊗ id M . The equalit y I 1 ,X = id 1 ⊗ id X follo ws in a s im ilar wa y .  Lemma 2.12. F or any U, V , W , X, Y , Z ∈ C , φ : F X → F Y , ψ : F U → F V , χ : F Z → F W ( φ ⊛ ψ ) ⊛ χ = φ ⊛ ( ψ ⊛ χ ) . Pr o of. The pro o f is done by a tedious but str aigh tforward compu tation.  The follo wing result is a re-statemen t of [DM1 , P r op. 6.3]. Theorem 2.13. L et us assume that for any X ∈ C , M ∈ M ′ the map ξ − 1 X,M ,R ( X ) ⊗ M (id ) is an isomorphism. Then for any X , Y ∈ C the maps I X,Y ar e invertible and J X,Y = I − 1 X,Y is a dyna mic al twist for the extension M ′ ⋉ C . Pr o of. Once w e prov e that I X,Y are in vertible for any X , Y ∈ C , the pro of that I − 1 X,Y is a dynamical twist for the extension M ′ ⋉ C f ollo ws immediately from Lemmas 2.6, 2.9, 2.11 and 2.12. Let X ∈ C , M , N ∈ M ′ and β : X ⊗ T ( M ) → T ( N ) b e an inv ertible map with inv erse γ : T ( N ) → X ⊗ T ( M ). There exists an f : R ( X ) ⊗ M → N such that β = ξ − 1 X,M ,N ( f ). Using (2.14) we obtain that β = T ( f ) ξ − 1 X,M ,R ( X ) ⊗ M (id ), hence f is in vertible. Th erefore ξ X,M ,N  β  is inv ertible. In another w ords, ξ maps isomorphisms to isomorph isms. CONSTRUCTING DYNAMICAL TWISTS 13 Let X , Y ∈ C , M ∈ M ′ . By definition ( I X,Y ) M = ξ X ⊗ Y ,M , R ( X ⊗ Y ) ⊗ M  T ( e m − 1 X,Y ,M ) θ m X,Y ,T ( M )  , where θ = ξ − 1 X,R ( Y ) ⊗ M , R ( X ) ⊗ ( R ( Y ) ⊗ M ) (id ) (id X ⊗ ξ − 1 Y , M ,R ( Y ) ⊗ M (id )) . By as- sumption θ is inv ertible, thus T ( e m − 1 X,Y ,M ) θ m X,Y ,T ( M ) is inv ertible, h ence I X,Y is inv ertible.  R emark 2.14 . F or any X , Y ∈ C I X,Y = id X ⊗ Y if and only if the func- tor ( T , c ) : M ′ → M is a m o dule functor, w here c X,M : T ( R ( X ) ⊗ M ) → X ⊗ T ( M ) is defined by c X,M = ( ξ − 1 X,M ,R ( X ) ⊗ M (id )) − 1 for a n y X ∈ C , M ∈ M ′ . Lemma 2.15. L et T b e a dynam ic al adjoint to R and let J b e the dynamic al twist asso ciate d. Then the functor ( T , c ) : ( M ′ ) ( J ) → M is a mo dule functor, wher e c X,M = ( ξ − 1 X,M ,R ( X ) ⊗ M (id )) − 1 for any X ∈ C , M ∈ M ′ .  3. D ynamical twists o ver Hopf algebras In this sectio n we shall fo cus our atten tion to the computation of dynam- ical t wists for a dynamical extension of the category Rep( H ), wh ere H is a finite-dimensional Hopf algebras, and w e shall giv e explici t examples. Hereafter w e sh all denote by H a fi nite-dimensional Hopf algebra. Let S b e a left H -como dule algebra. Definition 3.1. A dynamic al twist with b ase S for H is an inv ertible ele men t J ∈ H ⊗ k H ⊗ k S such that (3.1) J 1 s ( − 2) ⊗ J 2 s ( − 1) ⊗ J 3 s (0) = s ( − 2) J 1 ⊗ s ( − 1) J 2 ⊗ s (0) J 3 for all s ∈ S, (3.2) j 1 (1) J 1 ⊗ j 1 (2) J 2 ⊗ j 2 J 3 ( − 1) ⊗ j 3 J 3 (0) = j 1 ⊗ j 2 (1) J 1 ⊗ j 2 (2) J 2 ⊗ j 3 J 3 , (3.3) h ε, J 1 i J 2 ⊗ J 3 = 1 H ⊗ 1 K = h ε, J 2 i J 1 ⊗ J 3 . Here we use the notation J = J 1 ⊗ J 2 ⊗ J 3 = j 1 ⊗ j 2 ⊗ j 3 a voiding th e sum- mation symb ol. R emark 3.2 . Definition 3.1 coincides with the defin ition of dynamical t wist o ver an Ab elian group giv en in [EN1], [EN2]. Definition 3.3. T w o dyn amical t wists f or H o ver S , J and J ′ , are said to b e gauge e qui valent if there exists an inv ertible elemen t t ∈ H ⊗ S such that h ε, t 1 i t 2 = 1 , t 1 (1) J 1 ⊗ t 1 (2) J 2 ⊗ t 2 J 3 = j 1 t 1 ⊗ j 2 T 1 t 2 ( − 1) ⊗ j 3 T 2 t 2 (0) . Here J = J 1 ⊗ J 2 ⊗ J 3 , J ′ = j 1 ⊗ j 2 ⊗ j 3 , t = t 1 ⊗ t 2 = T 1 ⊗ T 2 . The follo wing Lemma is a straigh tforward consequence of the definitions. 14 MOMBELLI Lemma 3.4. L et J b e a dynamic al twist with b ase S for H . F or any X, Y ∈ Rep( H ) , M ∈ S M define J X,Y ,M : X ⊗ k Y ⊗ k M → X ⊗ k Y ⊗ k M by J X,Y ,M ( x ⊗ y ⊗ m ) = J 1 · x ⊗ J 2 · y ⊗ J 3 · m, for al l x ∈ x, y ∈ Y , m ∈ M . Then J is a dynamic al twist fo r the dynamic al extension S M ⋉ Rep( H ) . Mor e over, any dynamic al twist for the extension S M ⋉ Rep( H ) c omes fr om a dynamic al twist with b ase S over H .  Lemma 3.5. If J and J ′ ar e gauge e quivalent dynamic al twists for H with b ase S then S M ( J ) ≃ S M ( J ′ ) as mo dule c ate gories over Rep( H ) . Pr o of. Let ( F , c ) : S M ( J ) → S M ( J ′ ) b e the fu nctor defined as follo ws. F or an y M ∈ S M and X ∈ Rep( H ), F ( M ) = M and c X,M : X ⊗ k M → X ⊗ k M , c X,M ( x ⊗ m ) = t 1 · x ⊗ t 2 · m , for an y x ∈ X, m ∈ M . It is easy to ve rify that ( F , c ) giv es an equ iv alence of mo d ule catego ries.  3.1. Dynamical t wists and Hopf Galois extensions. Let S b e a left H -como dule algebra. F or an y dynamical twist with base S for H there is asso ciated a H -Galois extension with coin v arian ts S . Set B = H ∗ ⊗ k S . The copro du ct of H ∗ endo ws B with a right H ∗ cop - comod ule structure, that is δ : B → B ⊗ k H ∗ , δ ( α ⊗ s ) = α (2) ⊗ s ⊗ α (1) , for all α ∈ H ∗ , s ∈ S . Clearly B co H ∗ cop = S . If J ∈ H ⊗ k H ⊗ k S w e end o wed B with the follo wing pro duct: ( α ⊗ k )( β ⊗ s ) = ( J 1 ⇀ α )( J 2 k ( − 1) ⇀ β ) ⊗ J 3 k (0) s, (3.4) for all α, β ∈ H ∗ , k , s ∈ S . Prop osition 3.6. Assu me that J ∈ H ⊗ k H ⊗ k S is a dynamic al twist with b ase S , th en B ⊃ S is a H ∗ cop -Hopf Ga lois e xtension. Pr o of. First w e prov e that B is an asso ciativ e algebra with the pr o duct describ ed in (3.4). Let α, β , γ ∈ H ∗ , k , s, r ∈ S , then  ( α ⊗ k )( β ⊗ s )  ( γ ⊗ r ) =  ( J 1 ⇀ α )( J 2 k ( − 1) ⇀ β ) ⊗ J 3 k (0) s  ( γ ⊗ r ) = ( j 1 (1) J 1 ⇀ α )( j 1 (2) J 2 k ( − 2) ⇀ β )( j 2 J 3 ( − 1) k ( − 1) s ( − 1) ⇀ γ ) ⊗ j 3 J 3 (0) k (0) s (0) r . On the other h and ( α ⊗ k )  ( β ⊗ s )( γ ⊗ r )  = ( α ⊗ k )  ( J 1 ⇀ β )( J 2 s ( − 1) ⇀ γ )  ⊗ J 3 s (0) r = ( j 1 ⇀ α )( j 2 (1) k ( − 1) J 1 ⇀ β )( j 2 (2) k ( − 2) J 2 s ( − 1) ⇀ γ ) ⊗ j 3 k (0) J 3 s (0) r = ( j 1 ⇀ α )( j 2 (1) J 1 k ( − 1) ⇀ β )( j 2 (2) J 2 k ( − 2) s ( − 1) ⇀ γ ) ⊗ j 3 J 3 k (0) s (0) r . The last equalit y follo ws by (3.1). F rom (3.2) follo ws that  ( α ⊗ k )( β ⊗ s )  ( γ ⊗ r ) = ( α ⊗ k )  ( β ⊗ s )( γ ⊗ r )  . CONSTRUCTING DYNAMICAL TWISTS 15 The pro of that B is a H ∗ cop -comod ule algebra is straigh tforw ard . Let can : B ⊗ S B → B ⊗ k H ∗ cop b e the canonical m ap; that is can ( a ⊗ b ) = ab (0) ⊗ b (1) , for all a, b ∈ B . In this case, can (( α ⊗ k ) ⊗ ( β ⊗ s )) = ( J 1 ⇀ α )( J 2 k ( − 1) ⇀ β (2) ) ⊗ J 3 k (0) s ⊗ β (1) , for all α, β ∈ H ∗ , k , s ∈ S . It is easy to see th at the in v er s e of can is giv en b y can − 1 : B ⊗ k H ∗ cop → B ⊗ S B , can − 1 ( γ ⊗ r ⊗ β ) = J − 1 ⇀ ( γ S ( β (2) )) ⊗ 1 ⊗ ( J − 2 ⇀ β (1) ) ⊗ J − 3 r , for all γ , β ∈ H ∗ , r ∈ S . T h us B ⊃ S is a H ∗ cop -Galois extension.  3.2. Dynamical t wists coming from dynamical datum. In this subs ection w e shall giv e a metho d for constru cting dynamical t wists with base A , where A ⊂ H is a Hopf subalgebra. This construction is based on the same id eas con tained in [EN1], [M] without assum in g commutat ivit y nor co comm utativit y of the base of the dyn amical t wist. The follo wing definition generalizes [EN1, Def. 4.5], see also [M, Def. 3.8]. Definition 3.7. A dynamic al datum for H o v er A is a pair ( K, T ) where K is a left H -como du le algebra H -simple, with trivial coinv ariants, T : Rep( A ) → K M is a functor suc h that there are natur al H -mo d u le isomorphisms ω V W : Stab K ( T ( V ) , T ( W )) ≃ − − →  Ind H A ( V ⊗ k W ∗ )  ∗ , (3.5) for any V , W ∈ Rep( A ). W e shall further assume that ω V V (1)( h ⊗ v ⊗ f ) = h ε, h ih f , v i , (3.6) for all h ∈ H , v ∈ V , f ∈ V ∗ . Tw o dynamical data ( K, T ) and ( S, U ) are e quivalent if and only if there exists an element P ∈ H M K suc h that S ≃ End K ( P K ) as H -como d ule algebras, and there exists a f amily of natural K -mo du le isomorphisms φ V : P ⊗ K T ( V ) ≃ − − → U ( V ) , for all V ∈ Rep( A ). R emark 3.8 . If ( K, T ) is a dy n amical datum, then for any V ∈ Rep( A ) dim A (dim T ( V )) 2 = dim K (dim V ) 2 . (3.7) These form ula follo ws straightfo rw ard fr om the definition of d ynamical da- tum and form ula (1.2) . Denote by R : Rep( H ) → Rep( A ) the restriction fu nctor. Prop osition 3.9. If ( K, T ) is a dynamic al datum then T is a dynamic al adjoint to R . 16 MOMBELLI Pr o of. Category K M is a mo dule catego ry ov er Rep( H ) as explained in section 1.3. The catego ry Rep( A ) is a mo dule catego ry o v er itself. Let V , W ∈ K M an d X ∈ Rep( H ). Then Hom K ( X ⊗ k T ( V ) , T ( W )) ≃ Hom H ( X, Stab K ( T ( V ) , T ( W ))) ≃ Hom H ( X,  Ind H A ( V ⊗ k W ∗ )) ≃ Hom H (Ind H A ( V ⊗ k W ∗ ) , X ∗ ) ≃ ≃ Hom A ( V ⊗ k W ∗ , R ( X ∗ )) ≃ Hom A ( R ( X ) , W ⊗ k V ∗ ) ≃ ≃ Hom A ( R ( X ) ⊗ k V , W ) . The fi rst isomorphism follo ws from Prop osition 1.2 (3), the second b e- cause ( K, T ) is a dynamical datum and the fourth isomorphism is F rob enius recipro cit y . Let us denote by ξ : Hom K ( X ⊗ k T ( V ) , T ( W )) → Hom A ( R ( X ) ⊗ k V , W ) the composition of the abov e isomorphism s. Is clear that ξ satisfies (2.1 3) since we requested that the isomorphism s ω V W satisfy (3.6).  Definition 3.10. F or an y dynamical datum ( K, T ) w e shall denote by J T the asso ciated d ynamical t wists for th e Hopf algebra H with base A accord- ing to Theorem 2.13 . In the follo wing we shall pro v e that the construction of th e d ynamical t w ist d o es not dep end on the equiv alence class of the dynamical datum. Prop osition 3.11. L et ( K, T ) and ( S, U ) b e two e quivalent dynamic al data over A . Then J T is g auge e quiv alent to J S . F or the pro of w e will need first some tec hnical results. F rom the h yp oth- esis we know that th ere exists P ∈ H M K suc h that S ≃ En d K ( P K ) as H -mo du le algebras and natur al isomorphisms φ V : P ⊗ K T ( V ) ≃ − − − → U ( V ) , for all V ∈ Rep( A ) . F or any F or an y X ∈ Rep( H ), V ∈ Rep( A ) let us denote b y ξ : Hom K ( X ⊗ k T ( V ) , T ( W )) → Hom A ( R ( X ) ⊗ k V , W ) , ζ : Hom S ( X ⊗ k U ( V ) , U ( W )) → Hom A ( R ( X ) ⊗ k V , W ) , the f amily of natural iso morphisms constructed in th e pro of of Prop osit ion 3.9. F or an y X ∈ Rep( H ), V ∈ Rep( A ), M ∈ K M let us d efine θ X,M : X ⊗ k ( P ⊗ K M ) → P ⊗ K ( X ⊗ k M ) as follo ws. F or any x ∈ X, p ∈ P, m ∈ M θ X,M ( x ⊗ ( p ⊗ m )) = p (0) ⊗ S − 1 ( p ( − 1) ) · x ⊗ m. Let us also defi ne σ X,V : X ⊗ k U ( V ) → U ( X ⊗ k V ) CONSTRUCTING DYNAMICAL TWISTS 17 as the comp osition σ X,V = φ X ⊗ k V  id P ⊗ ξ − 1 X,V ,X ⊗ k V (id )  θ X,T ( V ) (id X ⊗ φ − 1 V ) . Clearly , σ X,V and θ X,M are isomorp h isms. Lemma 3.12. F or any X, Y ∈ Rep( H ) , V , W ∈ Rep( A ) , M , N ∈ K M and any morphisms f : X → Y , β : V → W , g : M → N θ X ⊗ k Y , M = θ X,Y ⊗ k M (id X ⊗ θ X,M ) , (3.8) (id P ⊗ id X ⊗ g ) θ X,M = θ X,N (id X ⊗ id P ⊗ g ) , (3.9) (id P ⊗ f ⊗ id M ) θ X,M = θ Y , M ( f ⊗ id P ⊗ id M ) , (3.10) σ Y , V ( f ⊗ id V ) = U ( f ⊗ id V ) σ X,V , (3.11) σ X,W (id X ⊗ β ) = U ( id X ⊗ β ) σ X,V . (3.12) Pr o of. Equations (3.8 ), (3.9) and (3.10) are straigh tforw ard. By definition σ Y , V ( f ⊗ id U ( V ) ) = φ Y ⊗ k V  id P ⊗ ξ − 1 Y , V ,Y ⊗ k V (id )  θ Y , T ( V ) (id Y ⊗ φ − 1 V )( f ⊗ id V ) = φ Y ⊗ k V  id P ⊗ ξ − 1 Y , V ,Y ⊗ k V (id )  (id P ⊗ f ⊗ id T ( V ) ) θ X,T ( V ) (id Y ⊗ φ − 1 V ) = φ Y ⊗ k V  id P ⊗ ξ − 1 Y , V ,Y ⊗ k V ( f ⊗ id V )  θ X,T ( V ) (id Y ⊗ φ − 1 V ) = φ Y ⊗ k V  id P ⊗ T ( f ⊗ id V ) ξ − 1 Y , V ,Y ⊗ k V (id )  θ X,T ( V ) (id Y ⊗ φ − 1 V ) = U (id X ⊗ β ) σ X,V . The s econd equalit y f ollo ws by the naturalit y of φ and (3.10), the third equalit y f ollo ws by (2.16), the f ourth by (2.14) and the fifth again by the naturalit y of φ . Equation 3.12 follo ws in a similar wa y .  F or an y X ∈ Rep( H ), V ∈ Rep( A ) set t X,V : X ⊗ k V → X ⊗ k V the isomorphism of A -mo dules defined as t X,V = ζ X,V ,X ⊗ k V ( σ X,V ) . Lemma 3.13. The maps t X,V ar e natur al isomorph isms. In p articular ther e exists an invertible e lement t = t 1 ⊗ t 2 ∈ H ⊗ k A such that for any x ∈ X, v ∈ V , t X,V ( x ⊗ v ) = t 1 · x ⊗ t 2 · v . Pr o of. Let X , Y ∈ Rep( H ) and let f : X → Y b e an y m orphism. t Y , V ( f ⊗ id V ) = ζ Y , V ,Y ⊗ k V ( σ Y , V ) ( f ⊗ id V ) = ζ X,V ,Y ⊗ k V ( σ Y , V ( f ⊗ id U ( V ) )) = ζ X,V ,Y ⊗ k V ( U ( f ⊗ id V ) σ X,V ) = ( f ⊗ id V ) ζ X,V ,X ⊗ k V ( σ X,V ) = ( f ⊗ id V ) t X,V . The second equalit y f ollo ws from (2.16), the third by (3.11) and the fourth one by (2.14). The naturalit y of t in the s econd v ariable follo ws in an analogous wa y usin g (3.12).  18 MOMBELLI Pr o of of Pr op osition 3.11. Let X , Y ∈ Rep( H ) and V ∈ Rep( A ) . Let I X,Y ,V , e I X,Y ,V : ( X ⊗ k Y ) ⊗ k V → ( X ⊗ k Y ) ⊗ k V b e the isomorphisms defined as I X,Y ,V ( x ⊗ y ⊗ v ) = J − 1 K · x ⊗ J − 2 K · y ⊗ J − 3 K · v , e I X,Y ,V ( x ⊗ y ⊗ v ) = J − 1 S · x ⊗ J − 2 S · y ⊗ J − 3 S · v , for an y x ∈ X , y ∈ Y , v ∈ V . In another wo rds, the family of natural isomorphisms I and e I are giv en by I X,Y ,V = ξ X ⊗ k Y , V , ( X ⊗ k Y ) ⊗ k V  ξ − 1 X,Y ⊗ k V ,X ⊗ k ( Y ⊗ k V ) (id ) (id X ⊗ ξ − 1 Y , V ,Y ⊗ k V (id ))  and e I X,Y ,V = ζ X ⊗ k Y , V , ( X ⊗ k Y ) ⊗ k V  ζ − 1 X,Y ⊗ k V ,X ⊗ k ( Y ⊗ k V ) (id ) (id X ⊗ ζ − 1 Y , V ,Y ⊗ k V (id ))  . W e shall p ro v e th at (3.13) I X,Y ,V t X ⊗ k Y , V = t X,Y ⊗ k V (id X ⊗ t Y , V ) e I X,Y ,V . Using 2.14 we obtain that ζ − 1 X ⊗ k Y , V , ( X ⊗ k Y ) ⊗ k V  I X,Y ,V t X ⊗ k Y , V  = U ( I X,Y ,V ) ζ − 1 X ⊗ k Y , V , ( X ⊗ k Y ) ⊗ k V  t X ⊗ k Y , V  = U ( I X,Y ,V ) σ X ⊗ k Y , V . No w, using the naturalit y of φ and the natur ality of ξ (2.14) w e obtain that U ( I X,Y ,V ) σ X ⊗ k Y , V is equal to φ ( X ⊗ k Y ) ⊗ k V  id P ⊗ ξ − 1 X ⊗ k Y , V , ( X ⊗ k Y ) ⊗ k V ( T ( I X,Y ,V ))  θ X ⊗ k Y , T ( V ) (id X ⊗ k Y ⊗ φ − 1 V ) . By definition of the isomorphism I X,Y ,V and (3.8) this last expression equals to φ ( X ⊗ k Y ) ⊗ k V  id P ⊗ ξ − 1 X,Y ⊗ k V ,X ⊗ k ( Y ⊗ k V ) (id )(id X ⊗ ξ − 1 Y , V ,Y ⊗ k V (id ))  θ X,Y ⊗ k T ( V ) (id X ⊗ θ X,T ( V ) ) (id X ⊗ k Y ⊗ φ − 1 V ) , and using (3 .9) equ als to φ ( X ⊗ k Y ) ⊗ k V  id P ⊗ ξ − 1 X,Y ⊗ k V ,X ⊗ k ( Y ⊗ k V ) (id )  θ X,T ( Y ⊗ k V )  id X ⊗ k P ⊗ ξ − 1 Y , V ,Y ⊗ k V (id )  (id X ⊗ θ X,T ( V ) ) (id X ⊗ k Y ⊗ φ − 1 V ) , whic h is equal to σ X,Y ⊗ k V (id X ⊗ σ Y , V ) . CONSTRUCTING DYNAMICAL TWISTS 19 F rom (2.14) follo ws that ζ − 1 X ⊗ k Y , V , ( X ⊗ k Y ) ⊗ k V  t X,Y ⊗ k V (id X ⊗ t Y , V ) e I X,Y ,V  equals to = T ( t X,Y ⊗ k V ) T (id X ⊗ t Y , V ) ζ − 1 X,Y ⊗ k V ,X ⊗ k ( Y ⊗ k V ) (id )(id X ⊗ ζ − 1 Y , V ,Y ⊗ k V (id )) = T ( t X,Y ⊗ k V ) ζ − 1 X,Y ⊗ k V ,X ⊗ k ( Y ⊗ k V ) (id X ⊗ t Y , V )(id X ⊗ ζ − 1 Y , V ,Y ⊗ k V (id )) = T ( t X,Y ⊗ k V ) ζ − 1 X,Y ⊗ k V ,X ⊗ k ( Y ⊗ k V ) (id )  id X ⊗ T ( t Y , V ) ζ − 1 Y , V ,Y ⊗ k V (id )  = ζ − 1 X,Y ⊗ k V ,X ⊗ k ( Y ⊗ k V ) ( t X,Y ⊗ k V )  id X ⊗ ζ − 1 Y , V ,Y ⊗ k V ( t Y , V )  = σ X,Y ⊗ k V (id X ⊗ σ Y , V ) . The second equalit y f ollo ws from (2.14), the third by (2.15) and the fourth again f rom (2.14 ). T hus, we ha v e pro v en equation (3.13). F ollo ws imme- diately from (3 .13) that the elemen t t − 1 is a ga uge equiv alence for J K and J S .  There is a recipro cate construction, that is, for any dynamical t wist o v er A we can asso ciate a dynamical datum as follo ws. Let J b e a dynamical twist for the dynamical exte nsion A M ⋉ Rep( H ). By Prop osition 2.5 the categ ory A M ( J ) is an exact indecomp osable mo dule catego ry o v er Rep( H ), therefore by T heorem 1.3 there exists an H -simple H -como dule algebra K suc h that K co H and A M ( J ) ≃ K M as mo du le cate- gories o v er Rep( H ). Let us d enote by T : A M ( J ) → K M such equiv alence. Prop osition 3.14. The p air ( K, T ) as ab ove is a dynamic al datum. Pr o of. The pro of is en tirely analo gous to the pro of of [M, P r op. 3.18]. F or completeness we write the p r o of. Let V , W ∈ Rep( A ), X ∈ Rep( H ) then Hom H ( X, Stab K ( T ( V ) , T ( W ))) ≃ Hom K ( X ⊗ k T ( V ) , T ( W )) ≃ ≃ Hom K ( T ( X ⊗ k V ) , T ( W )) ≃ Hom A ( X ⊗ k V , W ) ≃ Hom A ( R ( X ) , W ⊗ k V ∗ ) ≃ Hom A ( V ⊗ k W ∗ , R ( X ∗ )) ≃ Hom H (Ind H A ( V ⊗ k W ∗ ) , X ∗ ) ≃ ≃ Hom H ( X, Ind H A ( V ⊗ k W ∗ ) ∗ ) . The fi rst isomorp hism is a consequence of Pr op osition 1.2 (3), the sixth isomorphism is F rob enius recipro cit y . Thus, the statemen t follo ws from Y oneda’s Lemma.  The constru ction of the d ynamical datum fr om a d ynamical twist is not canonical b u t it does n ot dep end o n the gauge equiv alence class of the dy- namical twist. Prop osition 3.15. L et J , J ′ two gauge e quivalent dynamic al twists and let ( K , T ) and ( S, U ) the dyna mic al data asso ciate d as in Pr op osition 3.14. Then ( K , T ) is e quivalent to ( S, U ) . 20 MOMBELLI Pr o of. By constr u ction the functors T : A M ( J ) → K M and U : A M ( J ′ ) → S M are equiv alences of mo du le categories o ver Rep( H ). By Lema 3.5 the catego ries A M ( J ) and A M ( J ′ ) are equiv alen t. L et G : K M → A M ( J ) b e the in v erse of T . The fun ctor U ◦ G : K M → S M is an equiv alence of mo dule categories, th us Prop osition 1.4 implies that there exists an ob ject P ∈ H M K suc h that S ≃ En d K ( P K ) as H -mo dule algebras and natural isomorphisms U ( G ( M ) ) ≃ P ⊗ K M for all M ∈ K M . In particular there are natural isomorph isms U ( V ) ≃ U ( G ( T ( V ))) ≃ P ⊗ K T ( V ) , for all V ∈ Rep( A ).  R emark 3.16 . It would b e int eresting to kno w, for a fixed Hopf algebra H , whic h mod ule categories are equiv alen t to A M ( J ) for some Hopf subalgebra A and a dynamical t wist J w ith base A . 3.3. Some examples. W e shall giv e concrete examples of dynamical datum and explicit computations of the corresp on d ing dyn amical t wist. 3.3.1. K is an A -Galois extension. Let us assume that K is an A -Galois extension w ith tr ivial coinv arian ts. Let us denote by γ : A → K ⊗ k K the map γ ( a ) = can − 1 ( a ⊗ 1) = a [1] ⊗ a [2] , for all a ∈ A . Let us assume that T : Rep( A ) → K M is a functor such th at for any V , W ∈ Rep( A ) there are A -mo du le isomorphisms T ( W ) ⊗ k T ( V ) ∗ ≃ − − → W ⊗ k V ∗ . The A -mo du le structure on T ( W ) ⊗ k T ( V ) ∗ is give n as follo ws. If w ∈ T ( W ), f ∈ T ( V ) ∗ and a ∈ A , th en a · ( w ⊗ f ) = a [1] · w ⊗ f · a [2] , where ( f · k )( v ) = f ( k · v ), for an y k ∈ K , v ∈ V . This is a we ll defin ed action, see [AM, Lemma 2.21]. Lemma 3.17. Under the ab ove c onditions ( K, T ) is a dyna mic al datum. Pr o of. F or an y V , W ∈ Rep( A ) we ha v e t hat Stab K ( T ( V ) , T ( W )) ≃ Hom A ( H , Hom k ( T ( V ) , T ( W ))) ≃ Hom A ( H , T ( W ) ⊗ k T ( V ) ∗ ) ≃ Hom A ( H , W ⊗ k V ∗ ) ≃  Ind H A V ⊗ k W ∗  ∗ . The fi rst isomorph ism follo ws from (1.3), and the last is Lemma 1.1.  CONSTRUCTING DYNAMICAL TWISTS 21 3.3.2. Dynamic al tw ists over A ( G, χ , g ) . Let G b e a fi nite group, g ∈ Z ( G ) and χ : G → C × a c haracter such that n = | g | = | χ ( g ) | and χ n = 1. Let us denote b y H = A ( G, χ, g ) the alge bra generated by x, g sub ject to the relati ons: x n = 0, xh = χ ( h ) hx for all h ∈ G . The alge bra A ( G, χ , g ) has a Hopf a lgebra stru cture as follo w s: ∆( x ) = 1 ⊗ x + x ⊗ g , ∆( f ) = f ⊗ f , ε ( x ) = 0 , ε ( f ) = 1 , for all f ∈ G . These Hopf al gebras are a sp ecial class of monomial Hopf algebras. S ee [CYZ]. Let λ ∈ C × and let F ⊆ G b e a subgroup su ch that g ∈ F . In this case A ( F , χ, g ) ⊆ A ( G, χ, g ) is a Hopf subalgebra. Let us denote by A ( F , λ ) the algebra generated by elemen ts y , e h : h ∈ F sub ject to th e relations y n = λ 1 , e h e f = e hf , y e h = χ ( h ) e h y , for all h, f ∈ F . Lemma 3.18. L et us denote δ : A ( F , λ ) → A ( G, χ, g ) ⊗ A ( F , λ ) , the map given by δ ( y ) = g − 1 ⊗ y − xg − 1 ⊗ 1 , δ ( e h ) = h ⊗ e h , for al l h ∈ F . Then A ( F , λ ) is a left A ( G, χ, g ) -c omo dule algebr a with trivial c oinvariants. M or e over, A ( F, λ ) is a A ( F , χ, g ) -Galois extension. Pr o of. Straigh tforw ard.  Let B ⊆ F b e a subgroup suc h that B T < g > = { 1 } . Let A b e the group algebra of the group B . Let T : Rep( A ) → A ( F,λ ) M b e the functor defined as f ollo ws. F or an y left A -mo dule V , T ( V ) = L n − 1 i =0 C v i ⊗ V . Th e action of A ( F , λ ) on T ( V ) is the follo wing. F or an y v ∈ V , i = 0 . . . n − 1, h ∈ F y · ( v i ⊗ v ) = µ v i − 1 ⊗ v , y · ( v 0 ⊗ v ) = µ v n − 1 ⊗ v g · ( v i ⊗ v ) = χ i ( g ) v i ⊗ v , e h · ( v i ⊗ v ) = χ i ( h ) v i ⊗ h · v . Here µ n = λ is a fixed n -th ro ot of λ . Prop osition 3.19. The p air ( A ( F , λ ) , T ) is a dynamic al datum for H over A . Pr o of. Let V , W ∈ Rep( A ). Since A ( F , λ ) is A ( F, χ, g )-Galois, us in g (1.3), w e obtain that Stab A ( F,λ ) ( T ( V ) , T ( W )) ≃ Hom A ( F, χ,g ) ( A ( G, χ, g ) , Hom ( T ( V ) , T ( W )) . Also, by Lemma 1.1, we ha v e t hat  Ind A ( G,χ,g ) C B V ⊗ W ∗  ∗ ≃ Hom C B ( A ( G, χ, g ) , Hom ( V , W )) . Th us, the pro of will end if we p ro v e that Hom C B ( A ( G, χ, g ) , Hom ( V , W )) is isomorphic to Hom A ( F, χ,g ) ( A ( G, χ, g ) , Hom ( T ( V ) , T ( W )). 22 MOMBELLI Let G = S l F c l , F = S n − 1 j = 0 B g j b e right coset d ecomp ositions of G and F . Then the algebra A ( G, χ, g ) has a basis consisting of elemen ts { B g j x i c l } j,i,l . Let u s defin e the maps φ : Hom C B ( H , Hom ( V , W )) − → Hom A ( F, χ,g ) ( H , Hom ( T ( V ) , T ( W )) and ψ : Hom A ( F, χ,g ) ( H , Hom ( T ( V ) , T ( W )) − → Hom C B ( H , Hom ( V , W )) defined as follo ws . If ξ ∈ Hom C B ( H , Hom ( V , W )) then φ ( ξ )( c l )( v k ⊗ v ) = n − 1 X s =0 w s ⊗ ξ ( b s x k c l )( v ) , ψ ( α )( g j x i c l )( v ) = ( p j ⊗ id )  α ( c l )( v i ⊗ v )  for any v ∈ V , k = 0 . . . n − 1. Here p j : L n − 1 i =0 C w i → C , p j ( w i ) = δ ij . It is immediate to ve rify that these t wo maps are well defined and they are one the in v erse of eac h other.  Ac knowledgmen ts. 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