Monoidal categories of comodules for coquasi Hopf algebras and Radfords formula

MONOIDAL CA TEGORIES O F COMO DULES FOR COQUASI HOPF AL GEBRAS AND RADF ORD’S F ORMULA W AL TER FERRER SAN TOS AND IGNACIO LOPEZ FRANCO Abstra ct. W e study the basi c monoidal prop erties of the category of Hopf mod u les for a coquasi Hopf algebra. In particular w e discuss the so called fundamental theorem th at establishes a monoidal eq uiv alence b etw een the ca t- egory of co mo dules and the categ ory of Hopf modules. W e present a categorical proof of Radford’s S 4 form ula for the case of a ﬁnite d imensional coq uasi Hopf algebra, by establishing a monoidal isomorphism betw een certain double du al functors. De dic ate d to I. Shestakov on th e o c c asion of his 60th birthda y 1. Introduction The main purp ose o f this p ap er is to study th e monoidal category of Hopf mo dules for a co quasi Hopf algebra. As a consequence w e obtain a p ro of of Radford’s S 4 form ula v alid for ﬁnite dimensional co quasi Hopf alge bras. Ins pired in [8] we sh o w that this formula is in timately relat ed to the existence of certain natural transformation relating the left and the righ t double dual fun ctors for the cate gory of right H –comod u les. Th is natur al transformation comes from the application of the structure th eorem for Hopf mo dules, to ∗ H viewe d as a righ t H –Hopf mo dule. Co quasi Hopf algebras are the dual notion of the quasi Hopf algebras deﬁned in [7]. The m ain diﬀerence w ith Hopf algebras is that for co quasi Hopf algebras the role of the m ultiplicativ e a nd com u ltiplicativ e structures is n ot longer inter- c hangeable. In a coquasi Hopf algebra th e m ultiplicativ e structure is no longer one dimensional, bu t tw o dimens ional; this is expressed in th e f act that the m ul- tiplication is not longer associativ e but only u p to isomorphism, p ro vided b y a functional φ . The an tip o de is also deﬁned as a tw o dimensional structure, the extra d imension p ro vided b y tw o fun ctionals α, β . See b elo w. The ca tegory of Hopf mo d ules in the con text of (co )quasi Hopf alge bras h as b een considered by diﬀeren t authors and it w as initially studied in [10, 22]. In the case of Hopf algebras, Radford’s formula for S 4 w as ﬁ rst p ro v ed in full generalit y in [20], with predecessors in [17] and [24]. A more recen t pro of, app ears in [23]. There are many generalizatio ns of the form ula fr om the case of Hopf algebras to other situations, e.g. : braided Hopf algebras, bF algebras – braided and classical–, quasi Hopf algebras, we ak Hopf algebras, and Hopf algebras o v er rings. The follo wing is a partial list of references for some pro ofs of th ese generalizat ions: [2], [6], [9], [10], [13], [14] and [19]. Close r to the sp ir it of our pap er an analog ue of Radford’s formula for ﬁ nite tensor categ ories app ears in [8 ]. The ﬁrst auth or w ould like to thank, Csic-UDELAR , Conicyt-MEC. The second author ackno wledges t he supp ort of a Internal Graduate Stud entship form T rinity College, Cambridge. 1 2 W AL TER FER R ER SANTOS AND IGNACIO LOPEZ FRA NCO In this In tro duction, after a general description of the pap er, we brieﬂy recall the deﬁn ition an d the ﬁrst b asic prop erties of a co quasi bialgebra and of a co quasi Hopf alg ebra. W e deﬁne the notion of monoidal morp hism that is the adequate notion of morphism b et wee n co quasi bialgebras. L ater in this In tro du ction we establish the basic notations that will b e used along th e pap er. In Section 2, that is th e technical core of th e pap er, we present the b asic prop erties of the monoidal categories u sed later. W e work with the categ ories of comod ules and Hopf mo dules for co quasi bialgebras and reca ll the prop erties of the monoidal str u ctures induced by the cotensor pro du ct o ve r H and by th e tensor pro du ct o ve r k . W e lo ok at some b asic monoidal fu nctors associated to monoidal morphisms giv en b y corestrictio n of scala rs and its adjoints giv en by co indu ction. W e also consider other monoidal fun ctors – e.g. , the left adjoin t como dule and righ t adjoin t comodu le fun ctors– that will b e u sed later. In Section 3 we recall that in the case of co qu asi Hopf algebras w ith in ve rtible an tip o de the tensor categ ories of ﬁn ite dimensional comod ules –or bicomo du les– are rigid, i.e. , eac h ob j ect has a left and a right dual. W e u se this rigidit y in order to describ e explicitly –for ﬁnite dimensional Hopf algebras– the monoidal structure of the an tip o de. In Section 4 w e presen t a p ro of of the v ersion of the fu ndamenta l theorem on Hopf mo du les f or co quasi Hopf algebras that we need later in the pap er. By applying some general results on Hopf mo d u les o ve r autonomous pseudomonoids to our con text we pro ve that the fr ee righ t Hopf mo d ule fu nctor is a m on oidal equiv alence from the category of como dules int o the category of Hopf mo d ules. In Section 5, w e apply the fundamental theorem on Hopf mod ules to ∗ H and obtain the F r ob enius isomorph ism that is a morp hism in the category of Hopf mo dules b et ween H and ∗ H . Along the w a y w e iden tify the one dimen s ional ob ject of coin tegrals in this con text. In Section 6 , usin g the categorical mac hinery constructed ab o v e an d in the same vein than in [8], we prov e the existence of a natur al monoidal isomorphism b et we en the double duals on the left and on the righ t of a ﬁnite dimensional left H –mod ule. T his isomorphism will yield Radford’s form ula. It is not obvi ous a priori th at the form u la obtained for ﬁ nite dimensional co- quasi Hopf algebras is r elated to the classical Radford’s formula for Hopf algebras. Th us, in Sectio n 7, w e apply the previously develo p ed tec hniqu es to the case that H is a classical Hopf algebra in order to deduce the original Radford’s formula for S 4 –see [20]–. In Section 8 we p resen t in an Ap p endix the categorical backg round needed to pro v e some of the basic monoidal prop erties of the cotensor pro duct. W e recall a few basic deﬁnitions and results ab out density of functors and completi ons of catego ries under certain classes of colimits. 1.1. Basic deﬁnitions. Next w e summarize the basic deﬁn itions that w e need. Recall that the category of coa lgebras and morphisms of coalgebras has a monoidal stru cture su c h that the forgetful functor into the category of vect or spaces is monoidal. In other words, th e tensor p ro duct o ve r k of t wo coalgebras is a coalgebra and k is a coal gebra, in a canonical wa y . T his is a consequ en ce of the fact of that category of v ector spaces is braided, and in fact symmetric, w ith the usual switch sw : V ⊗ W → W ⊗ V . MONOIDAL CA TEGORIES OF COM ODULES 3 Assume ( C , ∆ , ε ) is a coalg ebra. The maps ∆ : C → C ⊗ C and ε : C → k are the com ultiplication and the counit resp ectiv ely . W e will u se Swe edler’s notatio n as in tro du ced in [25 ], and write ∆( c ) = P c 1 ⊗ c 2 . W e use the n otatio n ∆ 2 for the morphism ∆ 2 ( c ) = P c 1 ⊗ c 2 ⊗ c 3 . Moreo v er, the con v olution p ro duct will b e denoted b y the sym b ol ⋆ . Deﬁnition 1. A c o quasi bialgebr a s tructure on the coalgebra ( C, ∆ , ε ) is a triple ( p, u, φ ) wh ere p : C ⊗ C → C – the pr o duct – and u : k → C – t he unit – are coalgebra morphisms, and φ : C ⊗ C ⊗ C → k – the asso ciator – is a con v olution–in v ertible functional, satisfying the follo wing axioms. p ( u ⊗ id) = id = p (id ⊗ u ) (1) X ( c 1 d 1 ) e 1 φ ( c 2 ⊗ d 2 ⊗ e 2 ) = X φ ( c 1 ⊗ d 1 ⊗ e 1 ) c 2 ( d 2 e 2 ) (2) X φ ( c 1 d 1 ⊗ e 1 ⊗ f 1 ) φ ( c 2 ⊗ d 2 ⊗ e 2 f 2 ) = = X φ ( c 1 ⊗ d 1 ⊗ e 1 ) φ ( c 2 ⊗ d 2 e 2 ⊗ f 1 ) φ ( d 3 ⊗ e 3 ⊗ f 2 ) (3) φ ( c ⊗ 1 ⊗ d ) = ε ( c ) ε ( d ) (4) The qu adruple ( C , p, u, φ ) is called a c o quasi bialgebr a . Along this pap er coalgebras and co quasi bialgebras will b e den oted with th e letters, C , D , etc. The d ual concept of a quasi bialgebra was originally deﬁned in [7]. In the ab o v e equations w e h a v e w ritten 1 ∈ C for the image under u of the u nit of k , and p ( c, d ) = cd . Mo reo v er, when multiplying three elements of C we used paren thesis in ord er to establish the wa y w e p erformed th e op erations. The equ ation (2) can b e in terpreted in a precise w a y as a natur ality condition on φ . Eq u ations (3) and (4) can b e written as the equ alities φ ( p ⊗ id ⊗ id) ⋆ φ (id ⊗ id ⊗ p ) = ( φ ⊗ ε ) ⋆ φ (id ⊗ p ⊗ id ) ⋆ ( ε ⊗ φ ) and φ (id ⊗ u ⊗ id) = ε ⊗ ε v alid in the con v olution algebras ( C ⊗ C ⊗ C ) ∨ and ( C ⊗ C ) ∨ resp ectiv ely . Applying equation (3) to the case of c = d = 1, w e obtain that φ (1 ⊗ e ⊗ f ) = φ (1 ⊗ e 1 ⊗ f 1 ) φ (1 ⊗ e 2 ⊗ f 2 ). Hence ρ : C ⊗ C → k , ρ ( e ⊗ f ) = φ (1 ⊗ e ⊗ f ) is con v olution inv ertible and ρ ⋆ ρ = ρ . Then ρ = ε ⊗ ε and for later use we record b elo w this and other similar consequence of the axioms of a co qu asi Hopf a lgebra. φ (1 ⊗ c ⊗ d ) = ε ( c ) ε ( d ) φ ( c ⊗ d ⊗ 1) = ε ( c ) ε ( d ) (5) Observ ation 1. If ( C, p, u, φ ) is a coquasi bialgebra, th en C cop has a structure of a co quasi bialgebra with unit u , m ultiplication p sw and asso ciator φ (id ⊗ s w)(sw ⊗ id)(id ⊗ sw ). W e shall d enote this co quasi bialge bra b y C ◦ . In the literature C ◦ is denoted by C copop . Next w e deﬁn e the concept of monoid al mor phism b et w een co qu asi bialgebras. Monoidal morphisms are to coqu asi b ialgebras what bialgebra morphisms are to bialgebras. The monoidal morp hisms are the adequate kind of morphisms f or our category as they pr eserv e m ultiplication and unit u p to coherent isomorph isms. Although w e will only need this concept of morphism, for th e sak e of clariﬁcatio n we also giv e th e deﬁnition of lax monoidal morphism as in this general case the role of the in v ertible s calar ρ app earing in the deﬁnition b elo w is more transparen t. 4 W AL TER FER R ER SANTOS AND IGNACIO LOPEZ FRA NCO Deﬁnition 2. Let C and D b e co quasi bialgebras and f : C → D b e a m orphism of coalgebras. A lax monoidal structur e on f is a functional χ : C ⊗ C → k an d a s calar ρ ∈ k satisfying ( χ ⊗ p ( f ⊗ f ))∆ C ⊗ C = ( f p ⊗ χ )∆ C ⊗ C u = f u (6) ( φ D ( f ⊗ f ⊗ f )) ⋆ ( χ ⊗ ε ) ⋆ ( χ ( p ⊗ id)) = ( ε ⊗ χ ) ⋆ ( χ (id ⊗ p )) ⋆ φ C (7) ρχ ( u ⊗ id) = ε = ρχ (id ⊗ u ) . (8) The lax monoidal str u cture ( χ, ρ ) is called a monoidal structur e wh en χ is in- v ertible, – notice that ρ is alw a ys inv ertib le–. A morphism of coalgebras b e- t w een co quasi b ialgebras equipp ed w ith a (lax) monoidal s tr ucture is called a (lax) monoidal mor phism . A monoidal morphism b etw een t w o coqu asi bialgebras is a morp hism of coal- gebras equipp ed with a monoidal structure. The abov e terminology on monoidal structures comes from cate gory th eory . In fact, the concept of monoidal morphism as deﬁned ab ov e is a sp ecial instance of a th e concept of monoidal 1-cell b et we en pseudomonoids. In p articular, in the case that the monoidal structure is χ = ε ⊗ ε and ρ = 1 equations (6), (7 ) and (8) simply sa y that f preserves th e pro du ct, the unit an d that φ D ( f ⊗ f ⊗ f ) = φ C . Hence, it is clear in particular that a map of coquasi bialgebras that preserves the pr o duct, the copro du ct an d the associator, is a monoidal m orphism. The deﬁn ition ab o v e can b e generalized to the concept of a monoidal ( C , D )– bicomo dule, where C and D are co quasi bialgebras. In that case, f : C → D is a monoidal morph ism if and only if the b icomod ule f + (see Deﬁnition 4) is a monoidal bicomo dule (see Theorem 2). In this pap er we will not co v er these general asp ects of the theory . Observ ation 2. If f : C → D and g : D → E are monoidal morp hisms with monoidal structures ( χ f , ρ f ) and ( χ g , ρ g ) r esp ectiv ely , then gf has canonical monoidal structur e, namely , ( χ g ( f ⊗ f ) ⋆ χ f , ρ f ρ g ). Also, the iden tit y morphism id : C → C is equipp ed with a monoidal structure given by ( ε ⊗ ε, 1). In P rop osition 3 we sho w that the an tip o d e S –see the deﬁnition b elo w – of a ﬁnite dimensional coquasi Hopf alge bra H is a monoidal morp hism from H copo p to H . Deﬁnition 3. An antip o de for the co quasi bialgebra H is a triple ( S, α, β ) where S : H cop → H is a coalg ebra morphism and the functionals α, β : H → k satisfy the follo wing equations. X S ( h 1 ) α ( h 2 ) h 3 = α ( h )1 X h 1 β ( h 2 ) S ( h 3 ) = β ( h )1 (9) X φ − 1 ( h 1 ⊗ S h 3 ⊗ h 5 ) β ( h 2 ) α ( h 4 ) = ε ( h ) (10) X φ ( S h 1 ⊗ h 3 ⊗ S h 5 ) α ( h 2 ) β ( h 4 ) = ε ( h ) (11) A c o quasi Hopf algebr a is a co quasi b ialgebra equipp ed with an an tip o d e. Along this pap er co qu asi Hopf algebras will b e denoted as H . Observ ation 3. If ( S, α, β ) is an an tip o de for the co quasi bialgebra H th en ( S, β , α ) is an antipo de f or the co quasi bialgebra H ◦ considered in Observ ation 1. MONOIDAL CA TEGORIES OF COM ODULES 5 Observ ation 4. F or future use w e record th e follo w ing fact. If a ∈ H is a group lik e elemen t then S ( a ) = a − 1 . Indeed, from the equalit y φ ( S a ⊗ a ⊗ S a ) α ( a ) β ( a ) = 1, we deduce that α ( a ) 6 = 0. Then, from the equalit y S ( a ) α ( a ) a = α ( a )1 we deduce that S ( a ) a = 1. T he equalit y aS ( a ) = 1 can b e pr o v ed in a similar mann er . Moreo ver, if b ∈ H satisﬁes that ab = ba = 1, then S a = ( ba ) S a . Reasso ciating the prod uct, if w e call γ : H → k the fun ctional γ ( x ) = φ ( x ⊗ a ⊗ S a ), w e hav e that: S a = ( ba ) S a = γ − 1 ⇀ b ↼ γ . Then b = γ ⇀ S a ↼ γ − 1 and s in ce S a is a group like element, b = γ ⇀ S a ↼ γ − 1 = S a . In the particular case of the group lik e elemen t 1, we ha v e that S (1) = 1. The deﬁnition of co q u asi bialgebra (or rather its du al concept of quasi bialge- bra) w as introdu ced by Drin f el’d in [7]. Th e cr u cial observ ation w as that in order to guarantee the corresp ond ing mo dule category to b e monoidal, the asso ciativit y of the copro d uct w as only n ecessary u p to conju gation. The concept of antipo de (called by m an y authors a quasi antipo d e) and of qu asi Hopf algebra as deﬁned in [7], is n eeded in order guaran tee the existence of duals in the corresp onding catego ries of ﬁnite dimensional ob jects. Along this pap er we study th e basic prop erties of the categories of mo d ules and comod ules for a co quasi Hopf algebra. Th ese categories hav e b een considered by man y authors –see for example [15] and more sp eciﬁcally [10] and [22]. Our main in terest la ys in the case that the co qu asi Hopf alge bra is ﬁn ite di- mensional as a vec tor space. In this case , it is known (see [4 ] and [22 ]) that the an tip o de S is a bijectiv e linear transformation. The comp osition in v erse of S will b e denoted as S . W e ﬁnish this Introdu ction b y d escribing some of the notations w e u se. W e denote the u sual dualit y functor in the category of ve ctor spaces as V 7→ V ∨ and th e usu al ev aluation and coev aluation maps as e and c. Let C = ( C , ⊗ , k , Φ , l , r ) b e a monoidal cate gory with monoidal structure ⊗ : C × C → C , unit ob ject ob ject k , asso ciativit y co nstraint w ith comp onen ts Φ M ,N , L : ( M ⊗ N ) ⊗ L → M ⊗ ( N ⊗ L ) and left and r igh t un it constrain ts l and r . W e denote as C rev the tensor category ( C , ⊗ rev = ⊗ sw , k , b Φ , b r , b l ) wh er e ⊗ rev ( M ⊗ N ) = N ⊗ M , b Φ M ,N , L = Φ − 1 L,N ,M , b r = l and b l = r . Let C and D b e monoidal cat egories and T : C → D a fu n ctor. A monoidal structure on T is a natural isomorphism ⊗ ( T × T ) ⇒ T ⊗ : C × C → D and an isomorphism k → T ( k ) satisfying a certain n atur al list of coherence axio ms (see [12] for details). A monoidal functor is a fun ctor equipp ed with a monoidal structure. W e assume that the reader is familiar with the b asic concepts concerning rigidit y for tensor categories as pr esen ted for example in [12] or [15]. Recall that monoidal functors pr eserv e duals. In other wo rds , if T : C → D is a m onoidal functor and M ∈ C is a left r igid ob ject in C , th en T ( M ) is also left rigid and there is a canonical natural isomorphism η : T ( ∗ M ) → ∗ T ( M ). This isomorphism is the unique arro w su ch that mak es the diagram b elo w commutativ e T ( ∗ M ) ⊗ T ( M ) η ⊗ i d   a / / T ( ∗ M ⊗ M ) T (ev M ) / / T ( k ) b   ∗ T ( M ) ⊗ T ( M ) ev T ( M ) / / k 6 W AL TER FER R ER SANTOS AND IGNACIO LOPEZ FRA NCO where th e m aps a and b are the maps giv en by the monoidal stru ctur e of T . Also, follo wing th e standard usage we write C M , M D and C M D for the cat- egories of left C –como d ules, righ t D –como dules and ( C , D )–bicomo d ules, where C and D are coalgebras. F or the coactions asso ciated to the ob jects in these catego ries w e also us e Sweedler’s notation. In these situati ons, when w e add the subscript f to the symb ols, w e mean to say that w e are restricting our atten tion to the sub categ ories of ﬁnite–dimensional ob j ects, e.g. , C M D f is the full sub categ ory of C M D whose ob jects are ﬁnite d imensional k –spaces. 2. The ca tegories of bicomodules and of Hopf modules 2.1. The cotensor pro duct. W e start by b rieﬂy reviewing some of the basic prop erties of the cotensor pr o duct. Give n coalgebras C, D and E , one can deﬁne the c otensor pr o duct functor C M D × D M E → C M D as follo ws. If M and N are ob jects of C M D and D M E resp ectiv ely , its cot ensor pro duct o v er D , d enoted b y M  D N is the equalizer of the follo wing diagram M ⊗ N (( ε ⊗ id ⊗ id) χ M ) ⊗ id / / id ⊗ ((id ⊗ id ⊗ ε ) χ N ) / / M ⊗ D ⊗ N endo w ed with the b icomod ule structur e ind u ced b y the left coaction of M and the righ t coact ion of N . If F is another coalgebra, there is a natural isomorphism b et we en the t w o ob- vious functors C M D × D M E × E M F → C M F , with comp onen ts L  D ( M  E N ) ∼ = ( L  D M )  E N ind uced by the universal p rop erty of the equ alizers. Also, the func- tors C  C − an d −  D D : C M D → C M D are canonically isomorphic to the identi t y functor. All these data satisfy coherence conditions ; in cat egorical terminology w e sa y th at the categ ories of bicomo dules form a bic ate gory [1]. F rom th e ab o v e, it is clea r that the cotensor pro du ct p ro vides a monoidal structure to C M C , with unit ob ject ( C , ∆ 2 ). Observ ation 5. T h e cote nsor p ro duct fu nctor C M D × D M E → C M D preserve s ﬁltered col imits in eac h v ariable. T his is b ecause ﬁn ite limits comm ute with ﬁnite colimits. Next we consider corestrictio n f u nctors. Deﬁnition 4. If f : C → D is a morph ism of coalgebras, w e shall denote by f + = C f ∈ C M D the ob ject obtained from the r egular bicomo dule C b y correstriction with f on the right, i.e. , th e coactio n in f + = C f is give n by x 7→ P x 1 ⊗ x 2 ⊗ f ( x 3 ). Similarly , w e shall denote b y f + = f C ∈ D M C the ob ject obtained form C by correstriction on the left. T aking cotensor pr o ducts with bicomo dules that are indu ced b y morphisms of coalge bras h as conv enient prop erties. Observ ation 6. Su pp ose that f , C and D are as ab ov e and that A is an arbitrary coalge bra, (1) F or an y bicomo du le M ∈ A M C , with coactio n χ M , the cotensor pr o duct M  C f + = M  C C f ∈ A M D is canonicall y isomorphic with the bicomo d- ule –sometimes called also M f – with un d erlying space M and coact ion (id A ⊗ id M ⊗ f ) χ M : M → A ⊗ M ⊗ D . MONOIDAL CA TEGORIES OF COM ODULES 7 In a completely analog ous w a y , if N ∈ C M A , then the cote nsor pro duct f +  C N = f C  C N ∈ D M A is canonically isomorphic to the bicomo dule –sometimes called f N – w ith underlying space N and coactio n ( f ⊗ id N ⊗ id C ) χ N : N → D ⊗ N ⊗ A . Hence, −  C f + = −  C C f and f +  C − = f C  C − are the f unctors M 7→ M f : A M C → A M D and N 7→ f N : C M A → D M A giv en by correstriction with f . (2) Giv en a morp hism of coa lgebras f : C → D , it is clea r th at the functor considered ab o v e −  C f + = −  C C f : A M C → A M D is left adjoin t to the so called coinduction functor −  D f + = −  D f C : A M D → A M C . Similarly , the other co rrestriction functor f +  C − = f C  C − : C M A → D M A is left adjoin t to the so called coindu ction f unctor f +  D − = C f  C − : D M A → C M A (3) Assume that we ha v e t wo morphisms of coalgebras f : C → D and g : D → E . In that situatio n we hav e canonical isomorphisms b et w een ( g f ) + ∼ = f +  D g + and ( g f ) + ∼ = g +  D f + . Deﬁnition 5. Assume that the co algebra D has a group like elemen t that we call 1 ∈ D . W e app ly the ab o v e construction to the morphism of coalgebras u : k → D . In this case w e abbreviate ( − ) 0 = −  k u + = ( − ) u : A M → A M D . Similarly w e call 0 ( − ) = u +  k − = u ( − ) : M A → D M A . Observ ation 7. (1) The und erlying space for M 0 is M and the explicit for- m ula for the asso ciated coaction is χ 0 ( m ) = P m − 1 ⊗ m 0 ⊗ 1 if ( M , χ ) ∈ A M . As w e observ ed b efore this functor is left adjoin t to −  k u + : A M D → A M . It is clear that if M ∈ A M D , then N  k u + = N co D . In other w ords, the functor ( − ) 0 that pr o duces from a left A –como dule the ( A, D )– bicomo dule with trivial r igh t structure is left adjoin t to the ﬁxed p oin t functor. (2) The u nderlying space to 0 M is the same th an M and the coact ion is is 0 χ ( m ) = P 1 ⊗ m 0 ⊗ m 1 . Th is functor is left adjoint to u +  k − : D M A → M A , that is the functor that take s the left coinv ariants, i.e., sends M 7→ co D M . In ot her w ords, the f unctor ( − ) 0 that p ro duces fr om a r igh t A –co mo du le th e ( D , A )–bicomodu le with the trivial left structur e is left adjoin t to the left ﬁxed p oin t fun ctor. Theorem 1. L et g , h : C → D b e two morphisms of c o algebr as, and c onsider the fol lowing structur es. (1) F u nctionals γ : C → k satisfying ( γ ⊗ g )∆ = ( h ⊗ γ )∆ . (2) Morphisms of bic omo dules θ : g + → h + . (3) Natur al tr ansformatio ns Θ : ( −  C g + ) ⇒ ( −  C h + ) : M C → M D . Each structur e of typ e (1) i nduc es a structur e of typ e (2) by θ = (id C ⊗ γ )∆ and e ach structur e of typ e (2) i nduc es a structur e of typ e (3) by Θ = −  C θ . Mor e over, if C is ﬁnite d imensional these c orr esp ondenc es ar e bije ctions, with inverses given by γ = εθ and θ = Θ C . The identity and the c omp osition of the natur al tr ansfor- mations in (3) c orr esp ond to the identity and the c omp osition of the morphisms in (2) and to the c ounit ε and the c onvolution pr o duct of the functionals in (1) . In p articular, the natur al tr ansforma tion Θ is invertible iﬀ the asso ciate d morph ism θ is invertible i ﬀ the c orr e sp onding functional γ is c onvolution– invertible. 8 W AL TER FER R ER SANTOS AND IGNACIO LOPEZ FRA NCO Pr o of. That ea c h structur e (1) induces a stru cture (2) and eac h str ucture in (2) induces a structur e (3) is easily v eriﬁed. Next w e pro v e that the ab o v e describ ed maps are indeed a bijection b et w een (1) and (2). The m ap θ s atisﬁes (id ⊗ θ ⊗ g )∆ 2 = (id ⊗ id ⊗ h )∆ 2 θ . (12) Comp osing the ab o v e equalit y with id ⊗ id ⊗ ε we deduce that ∆ θ = (id ⊗ θ )∆. No w, if we comp ose Equation (12) with ε ⊗ ε ⊗ id we obtain ( γ ⊗ g )∆ = hθ , with γ = εθ . A direct calculation sho ws that ( h ⊗ γ )∆ = ( h ⊗ ε )(id ⊗ θ )∆ = ( h ⊗ ε )∆ θ = hθ . Hence, γ = εθ satisﬁes condition (1). Clearly , the corresp ondences giv en ab o ve b et we en elemen ts γ and θ are inv erses of eac h other. If we assume that C is ﬁ nite dimens ional, the b ijection b et ween the s tr uctures in (2) and (3 ) is a consequence of Observ ation 22.  F or a fu nctional γ as in Theorem 1.3, we will denote as γ + : g + → h + the asso ciated morphism of comodules and as Γ th e corresp onding natural transfor- mation. Observ ation 8. In the case th at γ is con volutio n inv ertible, th e condition (3) that relates g and h in T heorem 1, can b e written as g = γ − 1 ⋆ h ⋆ γ or as an y of the equalitie s b elo w v alid for all c ∈ C : g ( c ↼ γ ) = h ( γ ⇀ c ) , g ( c ) = h ( γ ⇀ c ↼ γ − 1 ) (13) 2.2. The tensor product o v er k . When w e consider coalge bras that hav e th e additional structure of a co qu asi bialgebra, the corresp ond ing cat egories of co- mo dules and of bicomodu les ha v e –b esides the monoidal structure giv en b y the cotensor pro duct– another monoidal structure. This monoidal structure is based up on the tensor pro duct ov er the base ﬁeld k with asso ciativit y constr aint de- ﬁned in terms of the corresp onding functional φ . F or example if C and D are co quasi b ialgebras with asso ciators φ C and φ D resp ectiv ely , if L, M , N ∈ C M D the associativit y constraint is the map Φ L,M ,N : ( L ⊗ M ) ⊗ N → L ⊗ ( M ⊗ N ) giv en b y the formula Φ(( l ⊗ m ) ⊗ n ) = X φ C ( l − 1 ⊗ m − 1 ⊗ n − 1 ) l 0 ⊗ ( m 0 ⊗ n 0 ) φ − 1 D ( l 1 ⊗ m 1 ⊗ n 1 ) (14 ) Here we view M ⊗ N as an ob ject in C M D with the usual structure: χ M ⊗ N ( m ⊗ n ) = P m − 1 n − 1 ⊗ m 0 ⊗ n 0 ⊗ m 1 n 1 ∈ C ⊗ M ⊗ N ⊗ D . Notice that the ab ov e form ula for the associativit y constrain t can b e written using the standard ac tions asso ciated to coactions as follo ws: Φ L,M ,N (( l ⊗ m ) ⊗ n ) = φ − 1 D ⇀ l ⊗ m ⊗ n ↼ φ C ∈ L ⊗ ( M ⊗ N ) . The unit constrain ts M ⊗ k ∼ = M and k ⊗ M ∼ = M are the same than in the catego ry of k –v ector spaces. In case that the categories are C M or M C , th e constrain ts are deﬁned s imilarly but us in g only the action b y φ − 1 on the left for M C and of φ on the righ t for C M . The monoidal ca tegories M C and C M can also b e deﬁned as k M C and C M k resp ectiv ely . MONOIDAL CA TEGORIES OF COM ODULES 9 Observ ation 9. If C is a coquasi bialgebra with unit u and m u ltiplication p , the triple ( C , p, u ) ∈ C M C is an asso ciativ e algebra. This can b e p ro v ed directly using equation (2), whic h can b e rewritten as p (id ⊗ p )Φ C,C, C = p ( p ⊗ id) : ( C ⊗ C ) ⊗ C → C . Deﬁnition 6. The cate gory of righ t C –mo du les within C M C will b e denoted as C M C C and called the category of Hopf mo dules. S imilarly w e deﬁ ne the categ ory C C M C . Notice that the unit ob ject k of the monoidal structure ⊗ is canonically a Hopf mo dule with action giv en b y the counit ε . The category of Hopf mo dules in this conte xt w as ﬁrst considered in [10] and [22]. Observ ation 10. a) The  C monoidal structure of C M C lifts to a monoidal structure on C M C C in suc h a wa y that the forgetful fun ctor C M C C → C M C is monoidal. Indeed, if M , N , L, R are in C M C , one easily can d eﬁne –using the universal prop erty of equalizers– a n atur al m orphism of bicomod ules ( M  C N ) ⊗ ( L  C R ) → ( M ⊗ L )  C ( N ⊗ R ) relating b oth monoidal s tructures on C M C . If L = R = C , w e obtain a map ( M  C N ) ⊗ C → ( M ⊗ C )  C ( N ⊗ C ) that comp osed with th e righ t C –actions on M and N — a M : M ⊗ C → M and a N : N ⊗ C → N —endo ws M  C N with the s tructure of a Hopf mo d ule ( M  C N ) ⊗ C → ( M ⊗ C )  C ( N ⊗ C ) a M ⊗ a N − − − − − → M  C N . (15) Clearly the un it ob ject C of  C in C M C is also a u nit ob ject in C M C C . b) If f : k → C and g : C → C are morphisms of coalgebras –in p articular this means that f (1) ∈ C is a group lik e elemen t– then f + ⊗ ( M  C g + ) ∼ = M  C p ( f ⊗ g ) + and ( M  C g + ) ⊗ f + ∼ = M  C p ( g ⊗ f ) + . 2.3. Monoidal functors induced by monoidal morphisms. Theorem 2. L et f : C → D b e a c o algebr a morphism, and c onsider the fol lowing structur es. (1) Monoidal structur es on f , (2) Monoidal structur es on the functor ( −  C f + ) : M C → M D , (3) Monoidal structur es on the functor ( f +  C − ) : C M → D M . Each structur e (1) induc es structur es (2) and (3). Mor e over, if C is ﬁnite- dimensional ther e is a b i je ction b etwe en the thr e e typ es of structur es. Pr o of. First, w e co nsider the relationship of structur es of t yp e (1 ) with structures of type (2). If χ : C ⊗ C → k , ρ ∈ k is a m on oidal structure on f : C → D , then the transformation with comp onent s Θ M ,N : M ⊗ N → M ⊗ N give n b y Θ M ,N ( m ⊗ n ) = χ ⇀ ( m ⊗ n ) = P χ ( m 1 ⊗ n 1 ) m 0 ⊗ n 0 together with the isomorphism k → k giv en by m ultiplication b y ρ is a monoidal structure as in (2). Indeed, for example th e cond ition that the m ap Θ M ,N : M f ⊗ N f → ( M ⊗ N ) f is a morphism of H –comodu les –recall the notatio ns of Observ ation 6– is equiv alen t with condition (6) in Deﬁnition 2. A structure as in (3) is obtained in a similar w a y , using χ − 1 and ρ − 1 . 10 W AL TER FER R ER SANTOS AND IGNACIO LOPEZ FRA NCO If w e no w assume that C is ﬁnite-dimensional we can proceed b ackw ard s in order to go from (2) to (1). Let Θ b e a natural transformation as depicte d b elo w. M C × M C ( −  C f + ) × ( −  C f + ) / / ⊗         Θ M D × M D ⊗   M C ( −  C f + ) / / M D (16) Since all the functors in this diagram preserv e ﬁ ltered co limits, Θ is determined b y its restriction to the categories of ﬁnite-dimensional como dules. So we can substitute th e categories of como d ules in diagram (16) by the corresp onding cat- egories of ﬁnite-dimensional como dules. In the app endix –Section 8– w e prov e that for a ﬁnite d imensional coa lgebra C , comp osition w ith the tensor pro duct functor ⊗ k : M C f × M C f → M C ⊗ C f induces an equiv alence Lex[ M C ⊗ C f , M D f ] ≃ Lex[ M C f , M C f ; M D f ]. The category Lex[ M C f , M C f ; M D f ] app earing on th e right hand side of the equiv alence is the catego ry of functors from M C f ⊗ M C f to M D f whic h are left exact in eac h v ariable–see also the App endix for the d eﬁnition of M C f ⊗ M C f –. Using this fact, w e d ed uce that natural transformations as in dia- gram (16) are in bijection with natural transformations as in the diagram b elo w : M C ⊗ C f −  C ⊗ C ( f ⊗ f ) + / / −  C ⊗ C p +         Θ ′ M D ⊗ D f −  D ⊗ D p +   M C f −  C f + / / M D f . Also these t yp e of natural transformations are in bij ective corresp onden ce w ith bicomo dule morph isms ( p ( f ⊗ f )) + → ( f p ) + . These b icomod ule morphism s corre- sp ond bijectiv ely to functionals χ : C ⊗ C → k satisfying P χ ( c 1 ⊗ c ′ 1 ) f ( c 2 ) f ( c ′ 2 ) = P f ( c 1 c ′ 1 ) χ ( c 2 ⊗ c ′ 2 ). Th e inv ertibilit y of Θ is equiv alent to the in v ertibilit y of χ . Similarly , an isomorphism Σ : k → k  C f + is just an in v ertible scalar ρ su c h that ρf (1) = ρ 1. The axioms of a monoidal stru cture for Θ , Σ translate to the axioms of a m on oidal structure on f for χ, ρ . The relationship b et ween the str u ctures (1) and (3) is as follo ws. A monoidal structure χ : C ⊗ C → k , ρ ∈ k on f induces a monoidal stru cture on ( f +  C − ) giv en b y the D -como dule morphism m ⊗ n 7→ P χ − 1 ( m − 1 ⊗ n − 1 ) m 0 ⊗ n 0 : ( f +  C M ) ⊗ ( f +  C N ) → f +  C ( M ⊗ N ) and b y λ 7→ ρ − 1 λ : k → f +  C k . The pro of of the con v erse, i.e. that when C is ﬁnite-dimensional every m onoidal structure on ( f +  C ) is of this f orm for a uniqu e ( χ, ρ ) is similar to the one pre- sen ted ab o v e for th e case of right como dules.  Corollary 1. In the situation ab ove the functors u +  k − = 0 ( − ) : M D → C M D , ( − ) 0 = −  k u + : C M → C M D , ar e monoida l. Pr o of. It follo ws immediately from the fact th at u : k → C preserv es the pro duct, the unit and the associator. Explicitly , u has a m on oidal s tr ucture pro vided b y χ = id : k → k and ρ = 1 ∈ k . In this case, equalities (6) and (8) are trivial while condition (7) reads as φ (1 ⊗ 1 ⊗ 1) = 1, whic h is true by (5).  MONOIDAL CA TEGORIES OF COM ODULES 11 2.4. Some useful mono idal functors on comodule categories. In this sub - section w e describ e the functors w e use along this w ork. Deﬁnition 7. If C and D are coalgebras we deﬁne the fu n ctors ( − ) ◦ : C M D → D cop M C cop ( − ) r : ( M C f ) op → C M f ( − ) ℓ : ( C M f ) op → M C f The fun ctor ( − ) ◦ is the iden tit y on arro ws, and if M ∈ C M D with coaction χ ( m ) = P m − 1 ⊗ m 0 ⊗ m 1 , then M ◦ has M as und erlying space and coactio n χ ◦ ( m ) = P m 1 ⊗ m 0 ⊗ m − 1 . In the case wh en C, D are co quasi bialgebras, ( − ) ◦ has a canonical structure of a monoidal functor ( C M D ) rev → D ◦ M C ◦ giv en b y the usu al symmetry of ve ctor spaces sw : M ◦ ⊗ N ◦ ∼ = ( N ⊗ M ) ◦ and th e identit y k → k ◦ . The functor ( − ) r is deﬁned as f ollo ws . If M ∈ M C f , the underlying space of M r is M ∨ , the linear dual of M . If c and e denote the standard coev aluation and ev aluation, the coa ction for M r is: M ∨ id ⊗ c − − − → M ∨ ⊗ M ⊗ M ∨ id ⊗ χ ⊗ id − − − − − → M ∨ ⊗ M ⊗ C ⊗ M ∨ e ⊗ id ⊗ id − − − − − → C ⊗ M ∨ . (17 ) On arrows, ( − ) r is giv en b y the usu al (linear) du alit y fu nctor. W e call M r the right adjoint of M . When C is a coquasi b ialgebra, ( − ) r has the follo wing canonical structure of a monoidal fu nctor ( M C f ) op → C M f . Th e unit constraint is th e canonical isomorphism k ∼ = k ∨ ; if M , N ∈ M C f , then th e transform ation M r ⊗ N r → ( M ⊗ N ) r is giv en b y the canonical arro ws M ∨ ⊗ N ∨ → ( M ⊗ N ) ∨ , wh ich are isomorphisms b y dimension considerations. W e should remark that here w e are not thin king M ∨ as a categorical dual of the v ector space M but rather as the in tern al hom V ect ( M , k ). Th is is the reason why ( − ) r do es not reverse the order of the tensor p ro ducts. The d eﬁnition of ( − ) ℓ is analogous, if N ∈ C M f , then: N ∨ c ⊗ id − − − → N ∨ ⊗ N ⊗ N ∨ id ⊗ χ ⊗ id − − − − − → N ∨ ⊗ C ⊗ N ⊗ N ∨ id ⊗ id ⊗ e − − − − − → C ⊗ N ∨ . (18) If N ∈ C M f , we call N ℓ the left adjoint of N . When C is a co quasi bialgebra we ha v e a monoidal functor ( − ) ℓ : ( C M f ) op → M C f . F or future reference w e record the follo wing results that can b e prov ed directly . Lemma 1. Observe that ( − ) r and ( − ) ℓ ar e inverse monoidal e quiv alenc es and that ( − ) r ℓ = ( − ) ℓr . The monoid al isomorphisms M r ℓ ∼ = M ∼ = M ℓr ar e just the c anonic al line ar i somorphisms M ∼ = M ∨∨ . Lemma 2. F or any mo rphism of c o algebr as f : C → D , the diagr ams in Figur e 2 c ommute. If mor e over f is a monoidal morphism, the diagr ams c ommute as diagr ams of monoidal functors. Pr o of. Recall that if f has a monoidal stru cture χ : C ⊗ C → k , ρ ∈ k , the monoidal structures on ( −  C f + ) and ( f +  C − ) are indu ced by χ, ρ and χ − 1 , ρ − 1 , resp ectiv ely . Also, it is easy to sh ow that the monoidal structures on ( −  C f + ) op and ( f +  C − ) op are induced b y χ − 1 , ρ − 1 and χ, ρ resp ectiv ely . The v eriﬁcation of the Lemma is direct.  12 W AL TER FER R ER SANTOS AND IGNACIO LOPEZ FRA NCO ( M C f ) op ( −  C f + ) op / / ( − ) r   ( M D f ) op ( − ) r   C M f f +  C − / / D M f ( C M f ) op ( f +  C − ) op / / ( − ) ℓ   ( D M f ) op ( − ) ℓ   M C f −  C f + / / M D f ( M C ) rev ( −  D f + ) rev / / ( − ) ◦   ( M D ) rev ( − ) ◦   C ◦ M f cop +  C cop − / / D ◦ M Figure 1. Diagrams in Lemma 2 Lemma 3. F or any c o quasi bialgebr a C the fol lowing diagr am of monoidal func- tors c ommutes. ( C M f ) oprev (( − ) ◦ ) op / / (( − ) ℓ ) rev   ( M C ◦ f ) op ( − ) r   ( M C f ) rev ( − ) ◦ / / C ◦ M f Lemma 4. The fol lowing two monoidal functors ar e monoida l ly isomorphic to the identity fu nc tor via the c anonic al maps M 7→ M ∨∨ M C f ( − ) r − − − → C M op f ( − ) ◦ − − − → ( M C ◦ f ) oprev ( − ) r − − − → C ◦ M rev f ( − ) ◦ − − − → M C f C M f ( − ) ℓ − − − → ( M C f ) op ( − ) ◦ − − − → ( C ◦ M f ) oprev ( − ) ℓ − − − → ( M C ◦ f ) rev ( − ) ◦ − − − → M C f 3. Dual ity In the case that the map S is in v ertible –for example if the co quasi Hopf algebra H is ﬁnite dimensional– the monoidal categories H M f , M H f and H M H f are rigid. In the case of H M H , e.g. , w e need to construct for ev ery ob ject M a left and a righ t dual –denoted as ∗ M and M ∗ resp ectiv ely– tog ether with the corresp onding ev aluation and co ev aluation maps. If ( M , χ ) ∈ H M H f and M ∨ is the du al of the underlying ve ctor sp ace, M ∗ = ( M ∨ , χ ∗ ) where χ ∗ is the comp osition M ∨ c ⊗ id − − − → M ∨ ⊗ M ⊗ M ∨ id ⊗ χ ⊗ id − − − − − → M ∨ ⊗ H ⊗ M ⊗ H ⊗ M ∨ sw ⊗ id ⊗ sw − − − − − − → H ⊗ M ∨ ⊗ M ⊗ M ∨ ⊗ H id ⊗ id ⊗ e ⊗ id − − − − − − − → H ⊗ M ∨ ⊗ H S ⊗ i d ⊗ S − − − − − → H ⊗ M ∨ ⊗ H . (19) The ev aluation and co ev aluation morph isms are given by ev ℓ : ∗ M ⊗ M id ⊗ χ − − − → ∗ M ⊗ H ⊗ M ⊗ H id ⊗ β S ⊗ i d ⊗ α − − − − − − − − → ∗ M ⊗ M e − → k (20) and co ev ℓ : k c − → M ⊗ ∗ M χ ⊗ id − − − → H ⊗ M ⊗ H ⊗ ∗ M α S ⊗ i d ⊗ β ⊗ id − − − − − − − − → M ⊗ ∗ M (21) MONOIDAL CA TEGORIES OF COM ODULES 13 It is not hard to c hec k u sing (9) that ev ℓ and co ev ℓ are morphism s in H M H . Moreo ver, the map s ev ℓ : ∗ M ⊗ M → k ∈ H M H and co ev ℓ : k → M ⊗ ∗ M ∈ H M H satisfy id M = M coev ℓ ⊗ id − − − − − → ( M ⊗ ∗ M ) ⊗ M Φ M , ∗ M ,M − − − − − − → M ⊗ ( ∗ M ⊗ M ) id ⊗ ev ℓ − − − − → M (22) and id ∗ M = ∗ M id ⊗ coev ℓ − − − − − → ∗ M ⊗ ( M ⊗ ∗ M ) Φ − 1 ∗ M ,M , ∗ M − − − − − − − → ( ∗ M ⊗ M ) ⊗ ∗ M ev ℓ ⊗ id − − − − → ∗ M . (23) These equations are direct consequen ces of (10) and (11). Analogously , ∗ M = ( M ∨ , ∗ χ ), where ∗ χ is the comp osition M ∨ id ⊗ c − − − → M ∨ ⊗ M ⊗ M ∨ id ⊗ χ ⊗ id − − − − − → M ∨ ⊗ H ⊗ M ⊗ H ⊗ M ∨ sw ⊗ id ⊗ sw − − − − − − → H ⊗ M ∨ ⊗ M ⊗ M ∨ ⊗ H id ⊗ e ⊗ id ⊗ id − − − − − − − → H ⊗ M ∨ ⊗ H S ⊗ i d ⊗ S − − − − − → H ⊗ M ∨ ⊗ H . (24) The corresp onding right ev aluation and coev aluation morphisms are: ev r : M ⊗ M ∗ χ ⊗ id − − − → H ⊗ M ⊗ H ⊗ M ∗ β ⊗ id ⊗ α S ⊗ i d − − − − − − − − → M ⊗ M ∗ e − → k (25) and co ev r : k c − → M ∗ ⊗ M id ⊗ χ − − − → M ∗ ⊗ H ⊗ M ⊗ H id ⊗ α ⊗ id ⊗ β S − − − − − − − − → M ⊗ M ∗ (26) As b efore one easily v eriﬁes that ev r and co ev r are m orphisms in H M H and also that th ey d eﬁne a righ t dualit y . Observ ation 11. In explicit terms the como du le structures for the duals are giv en b y the follo wing form ulæ. If ( M , χ ) ∈ H M H and f ∈ ∗ M and m ∈ M , then ∗ χ ( f ) = P f − 1 ⊗ f 0 ⊗ f 1 ∈ H ⊗ ∗ M ⊗ H if and only if: X f 0 ( m ) f − 1 ⊗ f 1 = X f ( m 0 ) S ( m − 1 ) ⊗ S ( m 1 ) . Similarly if ( M , χ ) ∈ H M H and f ∈ M ∗ and m ∈ M , then χ ∗ ( f ) = P f − 1 ⊗ f 0 ⊗ f 1 ∈ H ⊗ M ∗ ⊗ H if and on ly if: X f 0 ( m ) f − 1 ⊗ f 1 = X f ( m 0 ) S ( m − 1 ) ⊗ S ( m 1 ) . Lemma 5. F or the c ate gory M H , the duality functors c an b e expr esse d in terms of the fu nctors in De ﬁnition 7, in the fol lowing way. ∗ ( − ) : ( M H f ) op ( − ) r − − − → H M ( − ) ◦ − − − → M H cop f −  H cop S + − − − − − − − → M H f ( − ) ∗ : ( M H f ) op (( − ) ◦ ) op − − − − − → ( H cop M f ) op ( − ) ℓ − − − → M H cop f −  H cop S + − − − − − − − → M H f Theorem 3. If H is a ﬁnite dimensional c o quasi H opf algebr a, then its antip o de has a c anonic al structur e of a monoidal mo rphisms of c o quasi bialgebr as S : H ◦ → H . Mor e over, this structur e is given by the functional χ S : H ⊗ H → k χ S ( x ⊗ y ) = X φ − 1 ( S ( y 3 ) ⊗ S ( x 3 ) ⊗ x 5 ) α ( x 4 ) φ ( S ( y 2 ) S ( x 2 ) ⊗ x 6 ⊗ y 5 ) α ( y 4 ) β ( x 8 y 7 ) φ ( S ( y 1 ) S ( x 1 ) ⊗ ( x 7 y 6 ) ⊗ S ( x 9 y 8 )) . and c orr esp onds to the usual mo noidal structur e of the left dual functor ∗ ( − ) . 14 W AL TER FER R ER SANTOS AND IGNACIO LOPEZ FRA NCO Pr o of. By general categorical principles, th e left dual functor has a canonica l monoidal structure ∗ ( − ) : ( M H ) oprev → M H . This can b e explici tly computed in terms of the coquasi Hopf algebra structure of H . On the other hand, w e know the monoidal structures of the equiv alences ( − ) r and ( − ) ◦ , hence we can explicitly compute the monoidal structure of ( −  H cop S + ). The latter is giv en by a monoidal structure on the coquasi bialgebra morphism S : H ◦ → H (see Th eorem 2), and in fact it is giv en by the f u nctional χ S ab o v e.  Note that the formula for χ S ab o v e is wr itten in fun ction of the comultiplica tion of H not of the domain of S : H cop . The functional in the theorem ab o v e app eared in [3], and in the d ual case of quasi Hopf algebras in [7]. Theorem 4. L et H b e a ﬁnite-dimensional c o quasi H opf algebr a and c onsider the monoidal structur e on S intr o duc e d in Pr op osition 3 ab ove. The c anonic al line ar isomorphisms M ∼ = M ∨∨ pr ovide the c omp onents for monoidal natur al isomorph isms ∗∗ ( − ) ∼ = −  H S 2 + : M H f → M H f ( − ) ∗∗ ∼ = −  H S 2 + : M H f → M H f . Pr o of. If follo ws d irectly from Prop osition 3, Lemma 2 applied to S : H cop → H and L emma 4.  4. The fundament al t heorem o f Hopf modules In th is section w e presen t a generalization to the case of coquasi–Hopf algebras of the fu n damen tal theo rem on Hopf mo d ules in trod uced by S weedler in [25]. F or a m o dern p r esen tation of this general case see also [21, 22]. 4.1. The funda mental theorem. In this section we establish the basic set up in order to state and pro v e the fun damen tal theorem on Hopf mo du les in our con text. In this section, in order to use (formal) du alit y argumen ts, we w ork with a braided monoidal categ ory that we call V and w ith unit ob j ect k (see [12] as a general reference). Moreo ve r we assume that V h as equalizers and that the tensor pro du ct preserv es equalizers in eac h v ariable. The br aiding γ is a natural isomorphism γ X,Y : X ⊗ Y → Y ⊗ X and its existence ensures that the n atural transformations: ( X ⊗ Y ) ⊗ ( Z ⊗ W ) ∼ = − → ( X ⊗ ( Y ⊗ Z )) ⊗ W (id ⊗ γ Y ,Z ) ⊗ id − − − − − − − − → ( X ⊗ ( Z ⊗ Y )) ⊗ W ∼ = − → ( X ⊗ Z ) ⊗ ( Y ⊗ W ) deﬁne a monoidal s tructure on the functor ⊗ : V × V → V . Here we are endowing the catego ry V × V with its usual monoidal s tructure: ( X , Y ) ⊗ ( X ′ , Y ′ ) = ( X ⊗ X ′ , Y ⊗ Y ′ ). Using the coherence results in [12], we ma y assume without lo ss of generalit y that V is a strict monoidal cate gory . In the ab o v e set u p, give n a braided monoidal categ ory V , one can deﬁne coalge bra, como dule, co quasi bialgebra and co quasi Hopf algebra ob jects in V . The braiding ensures that the tensor p ro duct of t wo coalg ebras is a co algebra, and lik ewise with como du les. T he fact that the tensor p ro duct in V preserve s MONOIDAL CA TEGORIES OF COM ODULES 15 equalizers in eac h v ariable, allo w s us to d eﬁ ne the cotensor pro d uct of bicomo du les in exactly the same manner than in the case th at V is the category of v ector s paces. If C is a co quasi bialgebra w e d enote as V C , C V , C V C , C C V C and C V C C the catego ries of righ t, left and bicomod ules, and the categ ory of left and right Hopf mo dules in V resp ectiv ely . The ﬁrst thr ee cate gories hav e m on oidal structures induced by the tensor pro duct of V . The catego ry of bicomo dules has also a monoidal structur e giv en by the cotensor pro du ct  C . In the same manner than in Deﬁn ition 4, we can deﬁne the bicomo dules f + and f + for a morphism f : C → D of coalgebra ob jects in V . F or a co quasi bialgebra C in V , we call u : k → C the unit morphism. I n this situation w e can consider a pair of adjoin t functors ( u +  k − ) ⊣ ( u +  C − ) : C V C → V C . Explicitly , if M is a bicomo dule, u +  C M is the righ t co mo du le of left coin v ari- an ts co C M and if N is a right como d ule, u +  k N is th e basic ob ject N ∈ V with the same r igh t coaction than N and with left coaction giv en b y u ⊗ id N : N → C ⊗ N . Deﬁnition 8. If C is a coquasi bialgebra in V , w e deﬁne the free modu le fu nctor F : C V C → C C V C as F ( M ) = C ⊗ M for M ∈ C V C . T his functor, together with the f orgetful functor U : C C V C → C V C constitute a pair of adjoin t functors: F ⊣ U : C C V C → C V C . Deﬁne the functor L : V C → C C V C as the comp osition L = F ◦ ( u +  k − ). Clearly L will ha ve a righ t adjoin t given as ( u +  C − ) ◦ U . The monoidal structure  C , lifts from C V C to the category of Ho pf mo dules in suc h a w a y that if U : C C V C → C V C is the forgetful fu nctor, then the adjunction F ⊣ U : C C V C → C V C is monoidal ( i.e. , U is lax monoidal, F is strong monoidal and the unit and counit of the adju nction are monoidal natural transformations). W e wan t to sho w that under certain additional hypothesis, the fun ctor L is a monoidal equ iv alence. Without imp osing furth er r estrictions on the category V , it is not hard to pro ve the follo wing r esu lt, that is analogous to [21, Prop. 3.6] where th e diﬀerence b eing th at in the ab ov e men tioned pap er the result is form ulated for the case that V = V ect (see also [2 1, Lemma 2.1]). A general theorem that yields in p articular the lemma we presen t b elo w, app ears in [18, Pr op. 3.4]. In this lemma w e will need the follo wing piece of nota tion. If C, D , C ′ , D ′ are coalge bras in V and U ∈ C V D and V ∈ C ′ V D ′ , w e will d en ote by U • V ∈ C ⊗ C ′ V D ⊗ D ′ the ob j ect U ⊗ V equipp ed with the ob vious b icomod ule stru cture. Lemma 6. The functor L : ( V C , k , ⊗ ) → ( C C V C , C,  C ) is ful ly faithful and monoidal . Pr o of. It is w ell kno w n that the f unctor L is fu lly f aithful if and only if the unit of the adjunction L ⊣ ( u +  C − ) U is an isomorphism. It follo ws from the dual of [11, Lemma A1 .1.1], that it is enough to exhibit a natural isomorph ism b etw een ( u +  C − ) U L and the identit y fun ctor of V C . The comp osition U L : V C → C V C can b e written as U L ( M ) = ( C • M )  C ⊗ 2 p + . W e ha v e natural isomorp hisms u +  C ( C • M )  C ⊗ 2 p + ∼ = ( u + • M )  C ⊗ 2 p + ∼ = M  C ( u + • C )  C ⊗ 2 p + ∼ = M where the last isomorph ism is in duced b y ( u + • C )  C ⊗ 2 p + ∼ = (( u ⊗ id C ) p ) + ∼ = (id C ) + = C . This sho ws that there is a n atural isomorphism u +  C U L ( M ) ∼ = M . 16 W AL TER FER R ER SANTOS AND IGNACIO LOPEZ FRA NCO W e will no w exh ibit a canonical m onoidal s tr ucture on L . The basic observ ation is that, for M ∈ M C , L ( M ) = C ⊗ ( u +  k M ) is isomorphic to ( C • M )  C ⊗ 2 p + , where C ∈ C M C is the regular bicomo dule. Th en , we can form the comp osition L ( M )  C L ( N ) ∼ = (( C • M )  C ⊗ 2 p + )  C (( C • M )  C ⊗ 2 p + ) ∼ = ((( C • M )  C ⊗ 2 p + ) • N )  C ⊗ 2 p + ∼ = ( C • M • N )  C ⊗ 3 ( p + • C )  C ⊗ 2 p + (27) ∼ = ( C • M • N )  C ⊗ 3 ( C • p + )  C ⊗ 2 p + (28) ∼ = ( C • ( M ⊗ N ))  C ⊗ 2 p + ∼ = L ( M ⊗ N ) . All the isomorphisms ab ov e follo w easily f orm the d eﬁnition of the con tensor pro du ct except for the isomorphism b et ween (27) and (28 ), whic h is induced by the isomorphism ( p + • C )  C ⊗ 2 p + ∼ = (( p ⊗ id C ) p ) + ∼ = ((id C ⊗ p ) p ) + ∼ = ( C • p + )  C ⊗ 2 p + that is induced b y th e asso ciator φ . The isomorp hism describ ed ab o v e together with th e obvious isomorph ism k ∼ = L ( k ) pro vide a m onoidal structure for L . The axioms of a monoidal fu nctor follo w easily from the axioms satisﬁed by the asso ciator φ .  In the presence of an anti p od e, one obtains the follo wing strengthening of the ab o v e resu lts. This form of the fu n damen tal theorem on Hopf mo d ules for co quasi Hopf algebras is a consequence of [18 , Theorem 7.2]. It f ollo ws easily by a simple adaptation of the argumen ts of the Section 11 of the same w ork. Theorem 5. F or an arbitr ary c o quasi Hopf algebr a in V , the asso ciate d functor L is a mono idal e quivalenc e. Observe that we do not ask V to b e ab elian or additiv e. Neither w e assu m e an ything ab out the existence of duals in V . The only requiremen ts on V are that it is braided monoidal, with equalizers that are preserv ed b y the tensor pro du ct. F or a version of Theorem 5 o v er V ect , see for example [22]. In order to app ly this theorem to our cont ext, we need its righ t v ersion that will b e dedu ced b elo w. Observ ation 12. Consider the braided monoidal category V rev , whic h has the same underlyin g category as V , the s ame unit ob ject but the rev erse tensor pro du ct X ⊗ rev Y = Y ⊗ X , see Section 1. If γ X,Y : X ⊗ Y → Y ⊗ X is th e braiding in V , th en the braiding in V rev is γ rev X,Y = γ Y ,X . T he symmetry in the deﬁnition of co quasi b ialgebra implies that if ( C, p, u, φ ) is a coquasi bialgebra in V , th en ( C, p, u, φ ) is a co quasi bialgebra in V rev . Moreo v er, if ( S, α, β ) is an anti p o d e for the coquasi bialgebra H in V , th en ( S, β , α ) is an an tip o d e for H in V rev . Corollary 2. Supp ose H is a c o quasi Hopf algebr a with invertible antip o de in V . Then the f unctor R : H V → H V H H deﬁne d as the c omp osition of −  k u + : H V → H V H with the fr e e right Hopf mo dule functor H V H → H V H H is a monoida l e quivalenc e fr om H V to H V H H . Pr o of. Let us d enote V rev b y W . If ( H , u, p, φ ) is a coqu asi b ialgebra in V , as we sa w in Observ ation 12, ( H, u, p, φ ) is a coquasi bialgebra in W , and if ( S, α, β ) is an antipo d e f or H in V then ( S, β , α ) is an antipo d e for H in W . Clearly , as monoidal categorie s w e ha ve that H V = ( W H ) rev , H V H = ( H W H ) rev and H V H H = ( H H W H ) rev . The functor −  k u + corresp onds to u +  − : W H → MONOIDAL CA TEGORIES OF COM ODULES 17 H W H and the free H -mo d ule functor H V H → H V H H to the free H -mo d ule f unctor H W H → H H W H . Then R is just L rev for the co quasi Hopf algebra H in W , and hence it is a monoidal equiv alence as we wan ted to guaran tee.  4.2. The case of inv ertible an tip o de. In this section w e prov e that the com- p osition of th e fu nctors u +  k − : M H → H M H and the fr ee right H –mo dule functor F : H M H → H M H H is a monoidal equiv alence when H has an inv ertible an tip o de. First we deal with the general case of a co quasi bialgebra. Lemma 7. F or a c o quasi bi algebr a C , the functor F ( u +  k − ) : M C → C M C C is lef t adjoint to ( u +  H − ) U : C M C C → M C wher e U : C M C C → C M C is the for getful functor. Mor e over, the c ounit tr ansforma tion c orr esp onding to the ab ove adjunction is the fol lowing: if M ∈ C M C C , then ε M : 0 ( co C M ) ⊗ C → M is the map ε M ( m ⊗ c ) = m · c . Pr o of. The assertion ab out the adjunction is clear. If ε ′ and ε ′′ are the counits of F ⊣ U and ( u +  k − ) ⊣ ( u +  C − ) resp ectiv ely , then the counit of the comp osition of these functors is ε : F ( u +  k − )( u +  C − ) U F ε ′′ U − − − → F U ε ′ − → id . F or M ∈ C M C , the tr an s formation ε ′′ M is the inclusion of 0 ( co C M ) in M , while for N ∈ C M C C , ε ′ N : N ⊗ C → N is giv en b y th e righ t ac tion of C on N . Therefore ε is indeed giv en by the ab o v e form ula.  Deﬁnition 9. Let H b e a co quasi Hopf algebra. Deﬁne a functor I : M H → H M as the comp osition M H ( − ) ◦ − − − → H cop M S +  H cop − − − − − − − − → H M . In other w ords, on ob j ects I ( M , χ ) = ( M , ( S ⊗ id )sw χ ), and on arro ws I is the identit y . Corollary 3. In the situation ab ove the functor I : M H rev → H M is mono idal Pr o of. It follo ws imm ediately from: a) the equ alit y I = ( S +  H cop − ) ◦ ( − ) ◦ : M H ( − ) ◦ − − − → H cop M S +  H cop − − − − − − − − → H M ; b) the fact that ( − ) ◦ is monoidal –see the comments after Deﬁn ition 7–; c) The- orem 2.  The follo wing natural transf orm ation will b e crucial in the pro of of Radford’s form ula. Theorem 6. L et H b e a c o quasi Hopf algebr a with inve rtible antip o de. F or M ∈ M H the arr ows τ M : M ⊗ H → M ⊗ H deﬁne d as τ M ( m ⊗ h ) = P m 0 φ − 1 ( m 1 ⊗ S ( m 3 ) ⊗ h 2 ) β ( m 2 ) ⊗ S ( m 4 ) h 1 ar e the c omp onents of a natur al tr ansforma tion b etwe e n the functors F ◦ ( − ) 0 ◦ I a nd F ◦ 0 ( − ) : M H → H M H H . Mor e over, the natur al tr ansformation τ is inve rtible and its inve rse is given for al l M ∈ M H by the formula: τ − 1 M ( m ⊗ h ) = P φ ( S ( m 1 ) ⊗ m 3 ⊗ h 1 ) α ( m 2 ) m 0 ⊗ m 4 h 2 . Pr o of. It is con v enien t for the p ro of to split the map τ M as follo ws. First deﬁ ne the map π M : ( I M ) 0 → 0 M ⊗ H as π M ( m ) = P m 0 ⊗ β ( m 1 ) S ( m 2 ). An elemen tary computation sh o ws that τ M : ( I M ) 0 ⊗ H π M ⊗ id − − − − → ( 0 M ⊗ H ) ⊗ H Φ 0 M ,H,H − − − − − − → 0 M ⊗ ( H ⊗ H ) id ⊗ p − − − → 0 M ⊗ H 18 W AL TER FER R ER SANTOS AND IGNACIO LOPEZ FRA NCO The structure of H –bicomod ule on ( F ◦ ( − ) 0 ◦ I )( M ) = M ⊗ H is m ⊗ h 7→ P S ( m 1 ) h 1 ⊗ m 0 ⊗ h 2 ⊗ h 3 , while the stru ctur e on ( F ◦ 0 ( − ))( M ) = M ⊗ H is m ⊗ h 7→ P h 1 ⊗ m 0 ⊗ h 2 ⊗ m 1 h 3 . A direct v eriﬁcation shows that π M is a morph ism of bicomo dules. Hence τ M is a comp osition of morphisms of bicomo dules. The compatibilit y of τ M with the r igh t ac tion of H is deduced directly fr om the fact that π M ⊗ id and (id ⊗ p )Φ 0 M ,H ,H are m orphisms of right H –mo dules. The H – equiv ariance of the ﬁr st morphism is ob vious, while the equiv ariance of the second is a co nsequence of the follo wing general fact –that we a pp ly in the situation that C = H M H and A = H –. If ( A, p , u ) is an algebra –also called a monoid– in an arbitrary monoidal category C (that in accordance with [12] can b e assumed to b e strict) then, for any ob ject X the arro w id ⊗ p : X ⊗ A ⊗ A → X ⊗ A is a morphism of right A –mo dules. Finally , the v eriﬁcation of that the maps τ M and τ − 1 M are indeed in v erses to eac h other is a direct computation.  A v ersion of the ab ov e lemma for qu asi Hopf algebras app ears in [22]. Our pro of is similar. The f ollo win g result is an immediate consequen ce of Theorem 6 and Corollary 2. Corollary 4. In the situation ab ove, the functor F ( u +  k − ) : ( M H ) rev → H M H H has a unique monoidal str uctur e such that τ is a mo noidal natur al tr ansformation. In p articular, with this structur e F ( u +  k − ) is a mono idal e quivalenc e . W e end the section with the follo w ing observ ation, that will b e used in Section 6.2. Observ ation 13. If M , N ∈ M H and P , Q ∈ H M , there exist canonical isomor- phisms of bicomo du les (( u +  k M ) ⊗ H )  H (( u +  k N ) ⊗ H ) ∼ = ( u +  k N ) ⊗ (( u +  k M ) ⊗ H ) (( P  k u + ) ⊗ H )  H (( Q  k u + ) ⊗ H ) ∼ = ( P  k u + ) ⊗ (( Q  k u + ) ⊗ H ) If f : ( P  k u + ) → ( u +  k M ) and g : ( Q  k u + ) → ( u +  k N ) are m orphisms of bicomo dules, then the follo w ing diagram comm u tes (( P  k u + ) ⊗ H )  H (( Q  k u + ) ⊗ H ) ∼ = / / f  H g   ( P  k u + ) ⊗ (( Q  k u + ) ⊗ H ) id ⊗ g   ( P  k u + ) ⊗ (( u +  k N ) ⊗ H ) Φ − 1   (( P  k u + ) ⊗ ( u +  k N ) ) ⊗ H sw ⊗ id   (( u +  k N ) ⊗ ( P  k u + )) ⊗ H id ⊗ f   (( u +  k M ) ⊗ H )  H (( u +  k N ) ⊗ H ) ∼ = / / ( u +  k N ) ⊗ (( u +  k M ) ⊗ H ) MONOIDAL CA TEGORIES OF COM ODULES 19 5. The Frobenius iso morphism and t he o bject of c ointegrals Supp ose th at H is a coquasi Hopf alge bra with inv ertible antipo d e. If M is a left H –mo dule in the catego ry H M H f its left dual ∗ M , is an ob ject in H M H H in a functorial w a y . If a M : H ⊗ M → M is a H –mo du le structure of M in H M H , the corresp ondin g s tr ucture a ∗ M : ∗ M ⊗ H → ∗ M is giv en by ∗ M ⊗ H id ⊗ coev ℓ − − − − − → ( ∗ M ⊗ H ) ⊗ ( M ⊗ ∗ M ) ∼ = − → ( ∗ M ⊗ ( H ⊗ M )) ⊗ ∗ M (id ⊗ a M ) ⊗ id − − − − − − − → ( ∗ M ⊗ M ) ⊗ ∗ M ev ℓ ⊗ 1 − − − − → ∗ M . (29) F rom no w on w e assume that H is a ﬁ nite dimensional coqu asi Hopf algebra. If we tak e H ∈ H M H f as a left H –mod ule with resp ect to the regular act ion, its righ t dual ∗ H is canonically an ob ject in H M H H . An explicit description of the righ t H –stru cture deﬁned ab ov e for ∗ H is th e follo wing: if f ∈ ∗ H and x, y ∈ H , a ∗ H ( f ⊗ x )( y ) = ( f · x )( y ) is equal to ( f · x )( y ) = X φ − 1 ( S ( x 5 y 7 ) x 1 ⊗ y 3 ⊗ S ( y 1 )) αS ( y 2 ) φ ( S ( x 4 y 6 ) ⊗ x 2 ⊗ y 4 ) β S ( x 3 y 5 ) f ( x 6 y 8 ) φ ( S ( x 7 y 9 ) x 11 ⊗ y 13 ⊗ S ( y 15 )) φ − 1 ( S ( x 8 y 10 ) ⊗ x 10 ⊗ y 12 ) α ( x 9 y 11 ) β ( y 14 ) . (30) It is imp ortant to notice that in the ab o v e formula –a nd in the form ula for the F rob enius isomorphism– we obtain the expression for f · x ∈ ∗ H in te rms of the standar d ev aluation of v ector spaces H ∨ ⊗ H → k . Theorem 7. If H is a ﬁnite dimensional c o quasi H opf algebr a, then ther e exists a unique up to isomorph ism one dimensional obje ct W ∈ M H such that ther e is an isomorph ism 0 W ⊗ H ∼ = ∗ H ∈ H M H H . Mor e over, W c an b e ta ken as the sp ac e of left c ointe gr als of the Hopf algebr a H and the isomorp hism –c al le d the F r ob e ni u s isomorph ism– is the map F given by F ( ϕ ⊗ x ) = ϕ · x (31) wher e the action use d is the one deﬁne d in (29) and app lie d to M = ∗ H in ac c or danc e to the formula (30) . Pr o of. The existence and uniqueness of W follo ws immediately fr om the fu n- damen tal th eorem on Hopf mo dules–see more sp eciﬁcally Corollary 2 –. Th e c haracterizati on of W as a space of left coint egrals is ded u ced dir ectly from the explicit description of the inv erse fun ctor of F ◦ 0 ( − ) as the comp osition of the forgetful functor U : H M H H → M H H with the left ﬁxed part fun ctor –see the considerations pr evious to Lemma 6 –. Thus, W is the space of left coin v ari- an ts of ∗ H with r esp ect to the coaction describ ed in (24). In explicit terms W = { ϕ ∈ ∗ H : ∗ χ ( ϕ ) = 1 ⊗ ϕ } . Using the d escription of ∗ χ , app earing in Observ ation 11, w e conclude th at ϕ ∈ W if a nd only if for all x ∈ H , ϕ ( x )1 = P S ( x 1 ) ϕ ( x 2 ). In other words ϕ ( x )1 = P x 1 ϕ ( x 2 ) and th en ϕ ∈ ∗ H is a left coin tegral. The descrip tion of the counit of the adjun ction as the map giv en b y the action –see Lemma 7– will yield the c haracterization (31).  Observ ation 14. In th e same manner than in the classica l case, from the ex- istence of th e isomorp h ism F we conclude that W , the sp ace of left coin tegrals, 20 W AL TER FER R ER SANTOS AND IGNACIO LOPEZ FRA NCO is one d imensional. Hence, one can pro v e the existe nce of a group lik e elemen t a ∈ H s uc h that P ϕ ( x 1 ) x 2 = ϕ ( x ) a for all ϕ ∈ W and x ∈ H . T h e element a ∈ H is called the mo dular elemen t. Observ ation 15. Using the deﬁnition of the mod ular element a ju st presen ted as w ell as f orm ula (30) applied to the situation that f = ϕ is a coin tegral, w e obtain the follo wing explicit form ula for the F rob enius isomorphism. F ( ϕ ⊗ x )( y ) = α ( a ) β (1) X φ − 1 ( x 1 ⊗ y 2 ↼ α ⊗ S y 1 ) ϕ ( x 2 y 3 ) φ ( x 3 ⊗ β ⇀ y 4 ⊗ S y 5 ) φ ( a − 1 ⊗ a ⊗ S y 6 ) (32) Lemma 8. The c o action χ W : W → W ⊗ H is of the form χ W ( ϕ ) = ϕ ⊗ a − 1 for a ∈ H as ab ove. Pr o of. As W is one dim en sional, the coaction χ W is of the form χ W ( ϕ ) = ϕ ⊗ b for b ∈ H . T he deﬁnition of the right comodu le structur e on W (see Ob serv ation 11) yields –for x ∈ H – the formula: P ϕ ( x 1 ) S ( x 2 ) = ϕ ( x ) b . It follo ws then that S ( a ) = a − 1 = b –see Observ ation 4–.  Observ ation 16. a) T he como du le W is isomorphic to a − 1 + ∈ M H , where a − 1 : k → H is the coalge bra morphism indu ced by the multiplicati on by a − 1 . b) Similarly , the coaction in ∗ W is giv en as χ ∗ W ( t ) = t ⊗ a for t ∈ ∗ W . Hence, ∗ W is isomorphic to a + ∈ M H . Recall that we ab b reviated the fu n ctors u +  k − and −  k u + b y 0 ( − ) and ( − ) 0 resp ectiv ely . Observ ation 17. W ∈ M H as w ell as 0 W ∈ H M H are in v ertible ob jects in the corresp ondin g monoidal categories. In other words, the functors − ⊗ W : M H → M H and − ⊗ 0 W : H M H → H M H are equiv alences and it is clear that the in v erse equiv alences are obtained by tensoring w ith the corresp ond ing duals. F or use in the next section w e wr ite down the follo w ing deﬁnitions. Deﬁnition 10. Deﬁne the follo wing fun ctors c l W , c r W : H M H → H M H as f ollo ws: c l W = ( 0 W ⊗ − ) ⊗ 0 ∗ W and c r W = 0 W ⊗ ( − ⊗ 0 ∗ W ) Observ ation 18. It is clear that c l W and c r W are monoidal f u nctors that are nat- urally isomorphic via the natural transformation giv en by the ob vious asso ciator. In the notations of Theorem 7 and using the fact th at c l W , c r W are monoidal func- tors, we conclude that ( 0 W ⊗ H ) ⊗ ∗ 0 W and ( 0 W ⊗ H ) ⊗ ∗ 0 W are algebras in the catego ry H M H . 6. R a dford’s formula In this section we use catego rical metho ds to prov e Radford’s formula expressing S 4 in terms of co nju gation with a fu n ctional and a group like element . In the second part of this section w e pro v e the m on oidalit y of the fu nctional. 6.1. Radford’s formula. W e use the notations of the last section and assu me that H is a ﬁn ite d imensional coquasi Hopf algebra. W e will tak e basis elemen ts ϕ ∈ W , t ∈ ∗ W n ormalized in suc h a wa y that t ( ϕ ) = 1. MONOIDAL CA TEGORIES OF COM ODULES 21 Lemma 9. In the notations of The or em 7 the isomorphism in H M H γ : ( 0 W ⊗ H ) ⊗ ∗ 0 W F ⊗ i d − − − → ∗ H ⊗ ∗ 0 W ∼ = ∗ ( 0 W ⊗ H ) ( ∗ F ) − 1 − − − − → ∗∗ H is a morphism of algebr as. Mor e over, i f we deﬁne the N akayama i somorphism N : H → ∗∗ H by the formula: N ( x ) = γ (( ϕ ⊗ x ) ⊗ t ) , then the c ommutativity of the diagr am b elow char acterizes N : H ⊗ ( W ⊗ H ) N ⊗F   ∼ = / / ( H ⊗ W ) ⊗ H F sw ⊗ id   ∗∗ H ⊗ ∗ H ev ℓ ∗ H / / k ∗ H ⊗ H ev ℓ H o o Pr o of. The multiplicat ivit y of γ follo ws immediately fr om the f act th at F is a morphism of H –mo dules and fr om the comm utativit y of the f ollo win g diagram that is a direct consequence of the deﬁnition of the action a ∗ H : ∗ H ⊗ H → ∗ H –see Deﬁnition (29)–. ( ∗ H ⊗ H ) ⊗ H a ∗ H ⊗ id   ∼ = / / ∗ H ⊗ ( H ⊗ H ) id ⊗ p   ∗ H ⊗ H ev ℓ H / / k ∗ H ⊗ H ev ℓ H o o The assertion concerning N follo w s directly from the deﬁnitions.  Observ ation 19. a) The fact that γ is a morph ism of algebras is v alid in the follo w ing general con text. Let C b e a rigid monoidal ca tegory and let a, w ∈ C b e resp ectiv ely an algebra and an arbitrary ob ject. Let F : w ⊗ a → ∗ a b e an in v ertible morphism of a –mo dules in C , then the ob ject ( w ⊗ a ) ⊗ ∗ w is an algebra in C and the map γ : ( w ⊗ a ) ⊗ ∗ w F ⊗ i d − − − → ∗ a ⊗ ∗ w ∼ = ∗ ( w ⊗ a ) ( ∗ F ) − 1 − − − − → ∗∗ a is a morphism of algebras. b) In the case of ordinary Hopf algebras, the commutat ivit y of the d iagram that c haracterizes N after id en tifying H with its doub le dual reads as ϕ ( y N ( x )) = ϕ ( xy ) that is the u sual d eﬁ nition of the Nak ay ama automo rph ism. Lemma 10. In the notations of The or em 6 and The or em 7 if M is an obje c t i n M H , then the morp hism ξ M : ∗∗ I ( M ) 0 ⊗ (( 0 W ⊗ H ) ⊗ ∗ 0 W ) → ∗∗ 0 M ⊗ (( 0 W ⊗ H ) ⊗ ∗ 0 W ) deﬁne d by the c ommutativity of the diagr am b elow is a morphism of right ( 0 W ⊗ H ) ⊗ ∗ 0 W –mo dules. ∗∗ ( I ( M ) 0 ⊗ H ) ∼ =   ∗∗ τ M / / ∗∗ ( 0 M ⊗ H ) ∼ =   ∗∗ I ( M ) 0 ⊗ ∗∗ H id ⊗ γ − 1   ω M / / ∗∗ 0 M ⊗ ∗∗ H id ⊗ γ − 1   ∗∗ I ( M ) 0 ⊗ (( 0 W ⊗ H ) ⊗ ∗ 0 W ) ξ M / / ∗∗ 0 M ⊗ (( 0 W ⊗ H ) ⊗ ∗ 0 W ) (33) 22 W AL TER FER R ER SANTOS AND IGNACIO LOPEZ FRA NCO Pr o of. Being τ M a m orphism of H –mo du les it is clear that ω M is a morphism of ∗∗ H –mod ules. Then, from the fact that γ is an alge bra morphism and that al l th e mo dules inv olved are free o v er the corresp ondin g algebra ob jects, it follo w s that ξ M is a morphism of ( W 0 ⊗ H ) ⊗ ∗ W 0 –mo dules.  Deﬁnition 11. Deﬁne ν M : ∗∗ I ( M ) 0 ⊗ ( 0 W ⊗ H ) → ∗∗ 0 M ⊗ ( 0 W ⊗ H ) as the unique morphism su c h that the d iagram b elo w commutes. ∗∗ I ( M ) 0 ⊗ (( 0 W ⊗ H ) ⊗ ∗ 0 W ) ξ M / / ∼ =   ∗∗ 0 M ⊗ (( 0 W ⊗ H ) ⊗ ∗ 0 W ) ∼ =   ( ∗∗ I ( M ) 0 ⊗ ( 0 W ⊗ H )) ⊗ ∗ 0 W ν M ⊗ id ∗ 0 W / / ( ∗∗ 0 M ⊗ ( 0 W ⊗ H )) ⊗ ∗ 0 W The existence of ν M and the fact that it is a morphism in H M H follo w s imme- diately from the considerations of Observ ation 17. Moreo v er from th e fact that ξ M is a morp h ism in H M H ( 0 W ⊗ H ) ⊗ ∗ 0 W , it follo ws that ν M is a morp h ism in H M H H . The m on oidalit y of I giv es canonical isomorp h isms I ( M ∗∗ ) ∼ = ∗∗ I ( M ). Com- p osing with ν M w e get an isomorphism b ν M : I ( M ∗∗ ) 0 ⊗ ( 0 W ⊗ H ) → ∗∗ 0 M ⊗ ( 0 W ⊗ H ). F or M ∈ M H , consider the f ollo win g comp osition or arro ws in H M H H , from 0 W ∗ ⊗ (( ∗∗ 0 M ⊗ 0 W ) ⊗ H ) to 0 M ∗∗ ⊗ H . ζ M : 0 W ∗ ⊗ ∗∗ 0 M ⊗ 0 W ⊗ H id ⊗ b ν − 1 M − − − − → 0 W ∗ ⊗ I ( M ∗∗ ) 0 ⊗ 0 W ⊗ H → id ⊗ sw ⊗ id − − − − − − → 0 W ∗ ⊗ 0 W ⊗ I ( M ∗∗ ) 0 ⊗ H ev ⊗ τ M ∗∗ − − − − − − → 0 M ∗∗ ⊗ H (34) Here w e omitted the asso ciativit y constraint s for simplicit y . Ho w ev er th is do es not introd uce an y am biguit y as long as we kno w ho w to asso ciate the domain and co domain, b y the coherence theorem for monoidal categories. The comp osition ζ M is a morphism in H M H H . Ind eed, b ν M and τ M ∗∗ are mor- phisms of Hopf mo du les; the morphism id ⊗ sw ⊗ id is the im age under the free Hopf mo dule functor H M H → H M H H of the morphism of bicomod ules id ⊗ s w : 0 W ∗ ⊗ I ( M ∗∗ ) 0 ⊗ 0 W → 0 W ∗ ⊗ 0 W ⊗ I ( M ∗∗ ) 0 . Observe that sw is a morphism of bicomo du les b ecause the trivial comodule structures in eac h tensor factor are added on opp osite sides. Deﬁnition 12. Denote by µ M : W ∗ ⊗ ( ∗∗ M ⊗ W ) → M ∗∗ the unique morp hism in M H suc h that µ M ⊗ id H = ζ M . It is clear that µ is a natur al isomorphism b et w een the fu nctors W ∗ ⊗ ( ∗∗ ( − ) ⊗ W ) and ( − ) ∗∗ : M H → M H . Corollary 5. The c anonic al line ar isomorphisms M ∼ = M ∨∨ to gether with µ give a natur al isomorph ism in M H M  H p ( a − 1 ⊗ p ( S 2 ⊗ a )) + → M  H S 2 + . (35) Pr o of. First w e use Theorem 4, and sub stitute ∗∗ M by M  H S 2 + and M ∗∗ b y M  H S 2 + . In this manner we obtain from µ M an isomorph ism W ∗ ⊗ (( M  H S 2 + ) ⊗ W ) → M  H S 2 + . MONOIDAL CA TEGORIES OF COM ODULES 23 W ∗ ⊗ ∗∗ M ⊗ W ⊗ W ∗ ⊗ ∗∗ N ⊗ W µ M ⊗ µ N / / id ⊗ id ⊗ ev ⊗ id ⊗ i d   ∗∗ M ⊗ ∗∗ N W ∗ ⊗ M ⊗ N ⊗ W µ M ⊗ N 3 3 g g g g g g g g g g g g g g g g g g g k id   y y r r r r r r r r r r r W ∗ ⊗ k ⊗ W µ k / / k (37) Figure 2. No w using the fact that W ∼ = a − 1 + –Observ ation 16– and the conclusions of O b- serv ation 10 part b) w e d educe our result.  Theorem 8 (Radford’s formula) . Ther e exists an invertible functional σ : H → k such that for al l x ∈ H a − 1 ( S 2 ( x ) a ) = S 2 ( σ ⇀ x ↼ σ − 1 ) . Pr o of. It easily follo ws form Corollary 5 and Theorem 1 . In deed in the situation of Corollary 5 the theorem guaran tees the existe nce of a f unctional σ s uc h that –see Observ ation 8, (13)– p ( a − 1 ⊗ p ( S 2 ⊗ a ))( x ) = S 2 ( σ ⇀ x ↼ σ − 1 ).  The fu nctional σ deﬁned in the theorem ab o v e is the analog ue for ﬁnite d imen- sional coq u asi Hopf alg ebras of the mo du lar function of a ﬁn ite dimensional Hopf algebra. S ee S ection 7. Observ ation 20. The ab o v e f ormula can b e transformed in to another similar to the classica l formula: S 4 ( x ) = ( a − 1 ( b σ ⇀ x ↼ b σ − 1 )) a (36) where b σ is another inv ertible functional th at can b e computed explicitly in terms of the ab o ve information. 6.2. Monoidalit y. In this section w e prov e that the natural isomorphism µ of Deﬁnition 12 is monoidal. W e shall w ork as if the monoidal ca tegory ( H M H , k , ⊗ ) w ere strict, and hence ignore the asso ciativit y and un it constraints. This can b e formalized b y passing to an monoidally equiv alent strict m on oidal category . Indeed, our pro of do es n ot dep end on the f act that we are working with the catego ry of como dules, bu t only on certain prop erties satisﬁed by the seve ral arro ws w e consider. The fun ctor M H → M H giv en b y M 7→ W ∗ ⊗ M ⊗ W has a canonical monoidal structure giv en by the constrain ts W ∗ ⊗ M ⊗ W ⊗ W ∗ ⊗ N ⊗ W id ⊗ id ⊗ ev ⊗ id ⊗ id − − − − − − − − − − → W ∗ ⊗ M ⊗ N ⊗ W k coev − − − → W ∗ ⊗ W ∼ = − → W ∗ ⊗ k ⊗ W These morphisms are isomorph isms b ecause W is an in v ertible ob j ect –it has dimension one–. Theorem 9. The natur al tr ansformat ion µ in Deﬁnition 12 is monoidal. The assertion that µ is a monoidal natural transformation is expressed in the comm utativit y of the diagrams in Figure 2. 24 W AL TER FER R ER SANTOS AND IGNACIO LOPEZ FRA NCO ( 0 W ∗ )( ∗∗ 0 M )( 0 W )( 0 W ∗ )( ∗∗ 0 N )( 0 W ) H (id)ev(id) / / (id) ˆ ν − 1 N   ( 0 W ∗ )( ∗∗ 0 M )( ∗∗ 0 N )( 0 W ) H ∼ = / / (id) ˆ ν − 1 N   (C) ( 0 W ∗ )( ∗∗ 0 ( M N ))( 0 W ) H (id) ˆ ν − 1 M N   ( 0 W ∗ )( ∗∗ 0 M )( 0 W )( 0 W ∗ )( I ( N ∗∗ ))( 0 W ) H / / (id)sw(id)   ( 0 W ∗ )( ∗∗ 0 M )( I ( N ∗∗ ))( 0 W ) H (id)sw(id)   ( 0 W ∗ )( ∗∗ 0 M )( 0 W )( 0 W ∗ )( 0 W )( I ( N ∗∗ )) H (id)(id)(id)ev(i d)(id)   / / ( 0 W ∗ )( ∗∗ 0 M )( 0 W )( I ( N ∗∗ ) 0 ) H sw(id)   ( 0 W ∗ )( I ( M N ) 0 )( 0 W ) H (sw)(id)(id)   ( 0 W ∗ )( ∗∗ 0 M )( 0 W )( I ( N ∗∗ ) 0 ) H g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g id ˆ τ N ∗∗   ( I ( N ∗∗ ) 0 )( 0 W ∗ )( ∗∗ 0 M )( 0 W ) H id ˆ ν − 1 M   ( I ( N ∗∗ ) 0 )( 0 W ∗ )( I ( M ∗∗ ) 0 )( 0 W ) H   / / ( I (( M N ) ∗∗ ) 0 )( 0 W ∗ )( 0 W ) H (id)ev(id)   ( 0 W )( ∗∗ 0 M )( 0 W )( 0 N ∗∗ ) H ( 0 µ M )(id)(id)   ( I ( N ∗∗ ) 0 )( I ( M ∗∗ ) 0 )( 0 W ∗ )( 0 W ) H (id)(id)ev(id)   ( I ( N ∗∗ ) 0 )( I ( M ∗∗ ) 0 ) H (id) τ M ∗∗   ∼ = / / ( I (( M N ) ∗∗ ) 0 ) H τ ( M N ) ∗∗   ( I ( N ∗∗ ) 0 )( 0 M ∗∗ ) H sw(id)   ( 0 M ∗∗ )( I ( N ∗∗ ) 0 ) H id τ N ∗∗ s s g g g g g g g g g g g g g g g g g g g g ( 0 M ∗∗ )( 0 N ∗∗ ) H ∼ = / / (A) ( 0 ( M N ) ∗∗ ) H (B) Figure 3. ( I ( M ) 0 ⊗ H )  H ( I ( N ) 0 ⊗ H ) ∼ = / / τ M  H τ N   (D) I ( N ) 0 ⊗ I ( M ) 0 ⊗ H / / id ⊗ τ M   (E) I ( M ⊗ N ) 0 ⊗ H τ M ⊗ N   I ( N ) 0 ⊗ 0 M ⊗ H sw ⊗ id   0 M ⊗ I ( N ) 0 ⊗ H id ⊗ τ N   ( 0 M ⊗ H )  H ( 0 N ⊗ H ) ∼ = / / 0 M ⊗ 0 N ⊗ H / / 0 ( M ⊗ N ) ⊗ H Figure 4. MONOIDAL CA TEGORIES OF COM ODULES 25 0 W ∗ ⊗ ∗∗ 0 M ⊗ 0 W ⊗ I ( N ∗∗ ) 0 ⊗ H sw ⊗ id / / id ⊗ id ⊗ id ⊗ τ N ∗∗   0 µ M ⊗ id ⊗ id S S S S S S S S S S S S S S S S S ) ) S S S S S S S S S S S S S S S S S I ( N ∗∗ ) 0 ⊗ 0 W ∗ ⊗ ∗∗ 0 M ⊗ 0 W ⊗ H id ⊗ 0 µ M ⊗ id   0 W ∗ ⊗ ∗∗ 0 M ⊗ 0 W ⊗ 0 N ∗∗ ⊗ H 0 µ M ⊗ id ⊗ id   I ( N ∗∗ ) 0 ⊗ 0 M ∗∗ ⊗ H sw ⊗ id   0 M ∗∗ ⊗ 0 N ∗∗ ⊗ H 0 M ∗∗ ⊗ I ( N ∗∗ ) 0 ⊗ H id ⊗ τ N ∗∗ o o Figure 5. ∗∗ 0 M ⊗ ∗∗ 0 N ⊗ 0 W ⊗ H ∼ = / / id ⊗ ν − 1 N   ∗∗ 0 ( M ⊗ N ) ⊗ 0 ( M ⊗ N ) ⊗ 0 W ⊗ H ν − 1 M ⊗ N   ∗∗ 0 M ⊗ ∗∗ I ( N ) 0 ⊗ 0 W ⊗ H sw ⊗ id   ∗∗ I (( M ⊗ N ) ∗∗ ) 0 ⊗ 0 W ⊗ H ∗∗ I ( N ) 0 ⊗ ∗∗ 0 M ⊗ 0 W ⊗ H id ⊗ ν − 1 M / / ∗∗ I ( N ) 0 ⊗ ∗∗ I ( M ) 0 ⊗ 0 W ⊗ H ∼ = O O Figure 6. ∗∗ 0 M ⊗ ∗∗ 0 N ∗∗ H ∼ = / / id ⊗ ∗∗ τ − 1 N   ∗ ∗ 0 ( M ⊗ N ) ⊗ ∗∗ H ∗∗ τ − 1 M ⊗ N   ∗∗ 0 M ⊗ ∗∗ I ( N ) 0 ⊗ ∗∗ H sw ⊗ id   ∗∗ I ( M ⊗ N ) 0 ⊗ ∗∗ H ∗∗ I ( N ) 0 ⊗ ∗∗ 0 M ⊗ ∗ ∗ H id ⊗ ∗∗ τ − 1 M / / ∗∗ I ( N ) 0 ⊗ ∗∗ I ( M ) 0 ⊗ ∗∗ H ∼ = O O Figure 7. Pr o of. W e d ivide the pro of in tw o parts. In some diagrams, we omit the symbol ⊗ as a saving sp ace measure, adding p aren thesis when necessary . First axiom. The image of the diagram on th e left hand side of Figure 2 is the exterior rectangle in Fig ure 3. So it is enough to sho w the latter comm utes, as the functor M 7→ 0 M ⊗ H is an equiv alence b y Corollary 4. T he sub diagrams left b lank commute trivially . The diagram marked by (A) is just the commutativ e r ectangle in Figure 5. This is ea sy to sho w using the natur alit y of sw and the deﬁnition of µ . The diagram (B) in Figure 3 comm utes if the d iagram marked by (E) in Figure 4 comm u tes f or all M , N . T o sho w this, observ e that the exterio r r ectangle in Figure 4 comm u tes b y monoidalit y of τ an d that the s u b diagram (D) commutes by Observ ation 13. Finally , the diagram marke d b y (C ) in Figure 4 comm utes if and only if the diagram in Figure 6 do es. If we tensor this diagram with 0 W ∗ on the righ t, after 26 W AL TER FER R ER SANTOS AND IGNACIO LOPEZ FRA NCO ( 0 W ∗ )( ∗∗ 0 k )( 0 W ) H id ν − 1 k + + V V V V V V V V V V V V V V V V V V V (a) ( 0 W ∗ )( 0 k )( 0 W ) H (id) j (id ) / / (id) j (id)(id) 4 4 i i i i i i i i i i i i i i i i i ( 0 W ∗ )( ∗∗ 0 k )( 0 W ) (id)sw(id)   ( 0 W ∗ )( ∗∗I ( k ) 0 )( 0 W ) H (id)sw(id)   (id) ∗∗ θ (id)(id ) o o ( 0 W ∗ )( 0 W )( ∗∗ 0 k ) H ev(id)   ( 0 W ∗ )( 0 W )( ∗∗ I ( k ) 0 ) H (ev)(id)   (id)(id) ∗∗ θ (id) o o ( 0 k ) H ∼ = O O j id / / τ k = θ (id) U U U U U U U U * * U U U U U U U U j id % % L L L L L L L L L L L L L L L L L L L L L L L L L L ( ∗∗ 0 k ) H ∼ = + + V V V V V V V V V V V V V V V V V V V V V V (b) ( ∗∗ I ( k ) 0 ) H ∗∗ θ (id) o o δ (id )   ( I ( k ) 0 ) H ( I ( j ) 0 )(id) / / (c) ( I ( k ∗∗ ) 0 ) H τ k ∗∗ s s h h h h h h h h h h h h h h h h h h h h h ( 0 k ∗∗ ) H Figure 8. comp osing with the isomorph ism γ : 0 W ⊗ H ⊗ 0 W ∗ ∼ = ∗∗ H of Lemma 9, we get the diagram in Figure 7, whic h comm utes as the diagram (E) referred to abov e do es. Se c ond axiom. No w w e p r o v e the comm u tativit y of the d iagram in vo lving k in Figure 2. F or this w e w ill need some notation. The symb ol k will denote the trivial left H -como dule. Let us denote the canonical isomorphism s b et w een k and b oth ∗∗ k and k ∗∗ b y j . As I is a monoidal functor –se e Corollary 3–, w e hav e canonical isomorphisms δ : ∗∗ I ( k ) → I ( k ∗∗ ) in M H and θ : I ( k ) 0 → 0 k in H M H . One usefu l observ ation is that, as the monoidal stru ctur e on I is the unique one making τ a monoidal n atural transform ation, we ha ve θ ⊗ id = τ k : I ( k ) 0 ⊗ H → 0 k ⊗ H . The diagram we w ant to show that comm utes lies in the category M H ; hence, w e ma y equiv alen tly sho w that its image under th e fu nctor M 7→ 0 M ⊗ H of Corollary 4 comm utes. This new diagram is the one outer diagram in Figure 8. Indeed, the image of µ k is the arro w ζ in (34), that, by naturalit y of sw, is equal to the composition ( δ ⊗ id)(ev ⊗ id)(id ⊗ sw ⊗ id)(id ⊗ ν − 1 k ) on the r ight h and side of the diagram in Figure 8. The only sub d iagrams wh ose comm utativit y is not ob vious are the ones mark ed with (a), (b) and (c). The comm utativit y of the d iagram (a) f ollo ws form the follo wing observ ation. By deﬁn ition, ν − 1 k ⊗ id ∗ 0 W corresp onds, up to composing with certain canonica l isomorphisms, to ∗∗ τ k –see Lemma 10–. On the ot her hand , ∗∗ θ ⊗ id ⊗ id ⊗ id : ∗∗ I ( k ) 0 ⊗ 0 W ⊗ H ⊗ 0 W ∗ → ∗∗ 0 k ⊗ 0 W ⊗ H ⊗ 0 W ∗ also corresp ond s, up to comp osing with th e same canonical isomorphism s, to ∗∗ τ k (this b ecause θ ⊗ id H = τ k ). It follo w s that the diagram (a) comm utes. The diagram made by (b) comm utes b ecause θ is in duced by the monoidal structure of I , and monoidal functors pr eserv e du als. Finally , th e diagram (c) comm u tes b y natur ality of τ .  MONOIDAL CA TEGORIES OF COM ODULES 27 The monoidalit y of the natur al transformation µ just pro ve d translates in to prop erties of the functional σ in Theorem 8. W e compu te the b elo w in an explicit w a y the m onoidal stru cture of σ . Observ ation 21. The functional σ indu ces the n atural isomorphism of Corollary 5, w h ic h is monoidal since µ is. Therefore, if we kno w the monoidal structures of the morphisms p ( a − 1 ⊗ p ( ¯ S 2 ⊗ a )) and S 2 , we can deduce the equations satisﬁed b y σ . More explicitly , if these morp hisms hav e monoidal structures ( χ 1 , ρ 1 ) an d ( χ 2 , ρ 2 ) resp ectiv ely , then σ satisﬁes χ 1 ⋆ σ p = ( σ ⊗ σ ) ⋆ χ 2 ρ 1 σ (1) = ρ 2 . (38) The an tip o de S : H ◦ → H has a monoidal structure ( χ S , 1), where χ S is giv en explicitly in Prop osition 3. By O b serv ation 2 w e ha ve that S 2 , whic h is the comp osition of S cop : H → H ◦ and S : H ◦ → H , has ( χ S ( S ⊗ S ) ⋆ ( χ S ) − 1 sw , 1) as monoidal structure. This is b ecause S cop has a monoidal structure (( χ S ) − 1 sw , 1). The inv erse of th e an tip o de ¯ S : H ◦ → H has a ca nonical monoidal structure giv en in terms of χ S b y (( χ S ) − 1 ( ¯ S ⊗ ¯ S ) , 1). Thus, ¯ S 2 , this is, the comp osition of ¯ S with ¯ S cop : H ◦ → H , h as a monoidal structure ( χ S ( ¯ S 2 ⊗ ¯ S 2 )sw ⋆ ( χ S ) − 1 ( ¯ S ⊗ ¯ S ) , 1). The morph ism p ( a − 1 ⊗ p (id ⊗ a )) : H → H is monoidal with a monoidal structure giv en by ( χ 0 , 1) where χ 0 is the follo wing pro du ct in ( H ⊗ H ) ∨ . φ − 1 ( a − 1 ⊗ ( − ) a ⊗ a − 1 ((?) a )) ⋆ φ (( − ) a ⊗ a − 1 ⊗ (?) a ) ⋆ φ − 1 ( − ⊗ a ⊗ a − 1 ) ⋆ φ ( −⊗ ? ⊗ a ) Then, the monoidal structure ( χ 1 , ρ 1 ) of the comp osition of ¯ S 2 with p ( a − 1 ⊗ p (id ⊗ a )) is giv en by χ 1 = χ 0 ( ¯ S 2 ⊗ ¯ S 2 ) ⋆ χ S ( ¯ S 2 ⊗ ¯ S 2 )sw ⋆ ( χ S ) − 1 ( ¯ S ⊗ ¯ S ) and ρ 1 = 1. W e deduce th at σ satisﬁes σ (1) = 1 an d χ 0 ( ¯ S 2 ⊗ ¯ S 2 ) ⋆ χ S ( ¯ S 2 ⊗ ¯ S 2 )sw ⋆ ( χ S ) − 1 ( ¯ S ⊗ ¯ S ) ⋆ σ p = ( σ ⊗ σ ) ⋆ χ S ( S ⊗ S ) ⋆ ( χ S ) − 1 sw . (39) 7. The case of a Hop f algebr a W e brieﬂy men tion the needed adjustments to the pro of ab ov e in order to get the classica l Radf ord’s f ormula for S 4 . W e assu me that H is a ﬁnite dimensional Hopf algebra and deﬁne the follo wing functions. Denote by ω ∈ H ∨ , the mo du lar fun ction of H that w e k n o w it is an algebra homomorphism. It can b e deﬁned as the mod ular elemen t in the Hopf alg ebra H ∨ (the linear dual of H ). In particular if i ∈ H is a right in tegral, the fu nctional ω is c haracterized b y the p rop ert y that for all x ∈ H we ha v e that xi = ω ( x ) i . W e will also consider the automorph ism of Nak a ya ma N , that is charact erized b y the equation ϕ ( xy ) = ϕ ( y N ( x )) for all x, y ∈ H wher e ϕ is as b efore a righ t coin tegral for H . Next w e sho w ho w to obtain an expression of the in v erse of Nak a y ama’s auto- morphism in terms of S and ω . F or all x, y ∈ H , we ha v e that: X ϕ ( y 1 x ) y 2 = X ϕ ( y 1 x 1 ) y 2 ε ( x 2 ) = X ϕ ( y 1 x 1 ) y 2 x 2 S ( x 3 ) = X ϕ ( y x 1 ) S ( x 2 ) (40) If we tak e y = i in the abov e equalit y and assu m e that ϕ ( i ) = 1 we obtain that S ( x ) = P ϕ ( i 1 x ) i 2 or equiv alen tly that x = P ϕ ( i 1 x ) S i 2 . Hence, it follo ws that N ( x ) = P ϕ ( i 1 N x ) S i 2 = S 2 ( P ϕ ( xi 1 ) S i 2 ). Now, using us ing again the equ ation 28 W AL TER FER R ER SANTOS AND IGNACIO LOPEZ FRA NCO (40) we conclude that N ( x ) = S 2 ( X ϕ ( x 1 i ) x 2 ) = S 2 ( X ω ( x 1 ) x 2 ) = S 2 ( x ↼ ω ) . T aking the in ve rse m ap s in the ab o v e equation we conclude that S 2 x ↼ ω − 1 = N − 1 x . T hen, it follo ws that ε N − 1 = ω − 1 . In the case of a Hopf alge bra, for M ∈ M H then τ M : M ⊗ H → M ⊗ H is giv en b y the formula τ M ( m ⊗ h ) = P m 0 ⊗ S ( m 1 ) h . Th is is easily obtained from Theorem 6 substituting the asso ciators as well as α and β by ε . T h e in v erse of τ M is giv en as τ − 1 M ( m ⊗ h ) = P m 0 ⊗ m 1 h . With resp ect to the p rop erties of du ality , the same formulæ (19) and (24) yields the comodu le structure on the dual sp aces. Th e ev aluation and co ev aluation in this case are the same than the us u al ones in the catego ry of vec tor spaces (see form ulæ (20), (21), (25), (26 )). If M is a left H –mo du le the n atural righ t H –mo dule structure on the left dual –see (19)– a ∗ M : ∗ M ⊗ H → ∗ M is giv en b y ∗ M ⊗ H id ⊗ id ⊗ c − − − − − → ∗ M ⊗ H ⊗ M ⊗ ∗ M id ⊗ a M ⊗ id − − − − − − → ∗ M ⊗ M ⊗ ∗ M e ⊗ id − − − → ∗ M . F or the right dual the formula is similar –see (19). In particular, in the case we consider H ∈ H M H as a left mo du le with resp ect to the regular action, the righ t H –structure considered abov e in this situation is simply th e follo w in g: f ↼ h ∈ ∗ H , ( f ↼ h )( x ) = f ( hx ). The F rob enius map F , that w as given in Theorem 7, is F ( ϕ ⊗ h ) = ϕ ↼ h for ϕ ∈ W and h ∈ H . In particular as we ment ioned in Ob serv ation 19 part b), the m orp hism of algebras γ deﬁned in Lemma 9: γ : 0 W ⊗ H ⊗ 0 ∗ W F ⊗ i d − − − → ∗ H ⊗ 0 ∗ W ∼ = ∗ ( 0 W ⊗ H ) ( ∗ F ) − 1 − − − − → ∗∗ H , is giv en by the form ula: γ ( ϕ ⊗ x ⊗ t ) = ev( − ⊗ N x ) where N : H → H is the Nak a y ama morph ism, that in this case is an algebra automorphism . The map ξ M considered in Lemma 10, can b e describ ed explicitly by ξ M (ev m ⊗ ϕ ⊗ h ⊗ t ) = P ev m 0 ⊗ ϕ ⊗ N − 1 ( S ( m 1 )) h ⊗ t . In deed, fr om the co mmutati ve diagram (33) we deduce that ω M (ev m ⊗ ev h ) = P ev m 0 ⊗ ev S ( m 1 ) h . T o pro v e the form ula for ξ M w e pro v e that (id ⊗ γ )( X ev m 0 ⊗ ϕ ⊗ N − 1 ( S ( m 1 )) h ⊗ t ) = ω M (id ⊗ γ )(ev m ⊗ ϕ ⊗ h ⊗ t ) . The left hand side of the ab ov e equation is: (id ⊗ γ )( X ev m 0 ⊗ ϕ ⊗ N − 1 ( S ( m 1 )) h ⊗ t ) = X ev m 0 ⊗ ev S ( m 1 ) N ( h ) , while the right hand sid e can b e computed as: ω M (id ⊗ γ )(ev m ⊗ ϕ ⊗ h ⊗ t ) = ω M (ev m ⊗ ev N ( h ) ) = X ev m 0 ⊗ ev S ( m 1 ) N ( h ) . Hence the m ap ν M : ∗∗ I ( M ) 0 ⊗ 0 W ⊗ H → ∗∗ 0 M ⊗ 0 W ⊗ H introd uced in Deﬁnition 11 is giv en by: ν M (ev m ⊗ ϕ ⊗ h ) = P ev m 0 ⊗ ϕ ⊗ N − 1 ( S ( m 1 )) h . Moreo ver, the map b ν M has exactly the same expression than ν M . F or later u se w e record the follo wing form ula for b ν − 1 M that can b e prov ed by a direct computation: b ν − 1 M (ev m ⊗ ϕ ⊗ h ) = X ev m 0 ⊗ ϕ ⊗ N − 1 ( m 1 ) h. (41) MONOIDAL CA TEGORIES OF COM ODULES 29 Th us, th e m orphism ζ M deﬁned in (34) is giv en as: ζ M ( t ⊗ ev m ⊗ ϕ ⊗ h ) = X ev m 0 ⊗ S ( m 1 ) N − 1 ( m 2 ) h. Indeed it f ollo ws from equation (19) that the righ t coaction in M ∗∗ is χ M ∗∗ (ev m ) = P ev m 0 ⊗ S 2 ( m 1 ). Thus, ζ M ( t ⊗ ev m ⊗ ϕ ⊗ h ) = (ev ⊗ τ M ∗∗ )(id ⊗ sw ⊗ id)(id ⊗ b ν − 1 M )( t ⊗ ev m ⊗ ϕ ⊗ h ) = X (ev ⊗ τ M ∗∗ )(id ⊗ sw ⊗ id)( t ⊗ ev m 0 ⊗ ϕ ⊗ N − 1 ( m 1 ) h ) = τ M ∗∗ ( X ev m 0 ⊗ N − 1 ( m 1 ) h ) = X ev m 0 ⊗ S ( m 1 ) N − 1 ( m 2 ) h. Then, as ζ M = µ M ⊗ id H , w ith µ M : W ∗ ⊗ ∗∗ M ⊗ W → M ∗∗ , it is clear that µ M satisﬁes the follo wing equali t y: µ M ( t ⊗ ev m ⊗ ϕ ) ⊗ h = P ev m 0 ⊗ S ( m 1 ) N − 1 ( m 2 ) h . If we apply id ⊗ id ⊗ ε to the equalit y abov e w e obtain: µ M ( t ⊗ ev m ⊗ ϕ ) = X ev m 0 ( ε N − 1 )( m 1 ) = X ev m 0 ω − 1 ( m 1 ) W e ha v e used ab o v e the equalit y ε N − 1 = ω − 1 pro v ed b efore. Hence w e deduce that µ M ( t ⊗ ev m ⊗ ϕ ) = ev ω − 1 ⇀m . Next, w e observe that the natural isomorphism constructed in Corollary 5 is simply the map m 7→ ( ω − 1 ⇀ m ). Applying the bijections pro ve d in Theorem 1, w e ﬁ nd that th e map σ app earing in Radford’s form ula –Theorem 8– is sim p ly σ ( h ) = ε ( ω − 1 ⇀ h ) = ω − 1 ( h ). Hence, w e deduce the classical Radford’s formula a − 1 S 2 ( x ) a = ω − 1 ⇀ S 2 ( x ) ↼ ω or S 4 ( x ) = ω ⇀ a − 1 xa ↼ ω − 1 . This shows that the functional σ is indeed the co quasi Hopf algebra analogue of the mod ular fun ction. Next we explain ho w the m on oidalit y of σ prov ed at the end of the previous section generalize s the m ultiplicativit y of the mo d ular fun ction ω ∈ H ∨ . Recall th at in the case of a Hopf al gebra the asso ciativit y of the pro du ct and the fact that S is a m orphism of algebras are expressed as φ = ε ⊗ ε ⊗ ε and χ S = ε ⊗ ε . Th er efore, the equ alit y (39 ) simp liﬁes to σ p = σ ⊗ σ , this is, σ is m ultiplicativ e. Th e equalit y σ (1) = 1 wa s sh o wn for an arb itrary co quasi Hopf algebra. Hence ω = σ − 1 is a morphism of algebras. An imp ortant p oin t is that in the pro of of the monoidalit y of σ we did not use in tegrals (only cointe grals, presented as the como dule W ). Then, in the case of a ﬁnite d im en sional Hopf algebra, if we use σ ins tead of the mod u lar function, we can a void men tioning in tegrals altog ether in the pro of of the classical Radford’s form ula. 8. Ap pendix: ca t egorical backgr ound This app endix is an accoun t of some basic r esu lts on functors b et w een cate- gories of como du les. Th ese results are not unkno wn, but the pr o ofs foun d in the literature are usually ad ho c . The uniﬁed presen tation b elo w is based on densit y of functors and completio ns of catego ries under certain classes of colimits. W e will w ork with categories enric hed in the category of vect or sp aces o ve r a ﬁeld k , sometimes called k –linear categories. Although one has to b e careful w hen dealing with enriched categories, in our case the sub tleties of the theory disapp ear. This is a consequence of the fact that the u nderlying set fu nctor V ect ( k , − ) : V ect → Set is conserv ativ e ( i.e. , reﬂects isomorphisms). W e denote by [ A , B ] 30 W AL TER FER R ER SANTOS AND IGNACIO LOPEZ FRA NCO the k –linear category of k –linear functors A → B and n atural transformations b et we en them. Recall the notion of dens e fun ctor (see [16] for a co mplete exp osition on the sub ject). Let A b e a small category . A fu nctor K : A → C is dense if the functor ˜ K : C → [ A op , V ect ], giv en by C 7→ C ( K − , C ), is fully faithful. In other w ords, K is d ense if ev ery natural transformation C ( K − , C ) ⇒ C ( K − , D ) is of the form C ( K − , f ) for a unique f : C → D in C . When K is the in clusion of a full su b category , sa y that A is den se in C . A colimit in C is K - absolute if it is preserved b y ˜ K . In elemen tary terms, a colimit σ j : P ( j ) → colim P of a fu nctor P : J → C is K -absolute if for all ob jects A ∈ A the trans formation C ( K ( A ) , P ( j )) → C ( K ( A ) , colim P ) is a colimit in V ect . No w sup p ose K is the inclusion of a full su b category A into C . Consider a family of f unctors Φ = { P γ : J γ → C } γ ∈ Γ . W e say that C is the cl osure of A under the family Φ if there is no p r op er replete full su b category D of C conta ining (the image of ) A suc h that colim P γ ∈ D whenev er P γ tak es facto rs through D . If Φ is the family of all th e functors with small (ﬁn ite) domine in to C , we sa y that C is the closure of A under sm all (ﬁn ite) colimits. W e s a y th at Φ is a density pr esentation for K if the colimit of eac h P γ exists and is K -absolute, and C is the closure of A und er the family Φ. Th e functor K is dense when it has a densit y presen tation (although densit y can b e deﬁn ed in other w a ys, our c hoice here is justiﬁ ed b y [16, Theorem 5.35 ]). W rite Co cts K [ C , B ] for the full sub- k -catego ry of [ C , B ] of those functors that preserve K -absolute colimits. The follo wing is a particular instance of [16, Th m. 5.31]. Theorem 10. L et Φ = { P γ : J γ → C } γ ∈ Γ b e a density prsentation of the ful ly faithful f u nctor K : A → C . Supp ose e ach J γ is smal l and that B admits al l smal l c olimits. Then pr e c omp osing with K yie lds an e quivalenc e Co cts K [ C , B ] ≃ [ A , B ] with pseudoinverse g iven by taking lef t Kan e xtensions along K . A basic exa mple of dense su b category is p ro vided by the category of mo d ules o v er a ring R . If w e tak e C = R M and A the fu ll sub category determined b y the R -mo dule R , then A is dense in C . A d ensit y presen tation is giv en by the family of fu n ctors with small domain in to R M . This is so b ecause all colimits are K -absolute: after id entifying [ A op , V ect ] with R M , K is isomorp hic to the iden tit y f u nctor. The category A is also dense in the categ ory R M fp of ﬁnitely present ed R -mo dules. Indeed, R M fp is the closure of A u nder ﬁn ite colimits and these are K -absolute, where K is the inclus ion of A in to R M fp . Observe that ˜ K in this case is isomorphic to the in clusion of R M fp in to R M , and hence it preserv es colimits. As a consequen ce of Theorem 10 we ha v e equiv alences Co cts[ R M , B ] ≃ [ A , B ] = R op - B Re x[ R M fp , D ] ≃ [ A , D ] = R op - D for an y categ ories B and D with small colimits and ﬁnite colimits resp ectiv ely . Here R op - B denotes the category with ob j ects B of B equipp ed with an action or R op , that is, a ring morphism R op → B ( B , B ), and eviden t morphisms. Sligh tly more general, th e Y oneda em b edd ing K : A → [ A op , V ect ] is dense for an y small category A . MONOIDAL CA TEGORIES OF COM ODULES 31 A second exa mple of interest for u s is sub category of ﬁnite-dimensional como d- ules M C f . Let K : M C f → M C b e the inclusion functor. Given a C -como du le M , consider the c omma c ate gory K / M . That is, the category w hose ob jects are pairs ( N , f ) where N ∈ M C f and f : N → M , and whose arr ows ( N , f ) → ( N ′ , f ′ ) are the arro w s g : N → N ′ suc h that f ′ g = f . The functor P M : K/ M → M C sending ( N , f ) to its N is the base of a cone of v ertex M , w ith comp onen ts σ ( N ,f ) = f : N → M . Clearly K/ M is small and ﬁltered (since M C f has ﬁnite co l- imits and K p reserv es them) and σ is a colimiting cone. The family of fu nctors P M with M a C -como dule is a density presentati on for K : clearly M C is the closure of M C f under ﬁltered colimits and ﬁltered co limits are preserv ed b y ˜ K since ﬁn ite dimensional como dules are ﬁnitely presentable ( i.e. , M C ( N , − ) preserv es ﬁ ltered colimits wh enev er N is ﬁnite-dimensional). Since K pr eserv es ﬁnite colimits, it is clear that the image of ˜ K : M C → [( M C f ) op , V ect ] lie s in the full sub category Lex[( M C f ) op , V ect ] of left exact fun ctors; moreo ve r, the replete image of ˜ K can b e sho wn to b e exactly this sub category . This yields an equiv alence Fin[ M C , B ] ≃ [ M C f , B ] for an y category B with ﬁltered colimits, w here the category on the left hand sid e is the category of ﬁn itary ( i.e. , ﬁltered colimit -preservin g) fu nctors. Our next examp le is the full sub catego ry A of M C determined b y the regular comod ule C . Recall that eac h comodu le M is the equ alizer of the canonical pair of como dule morphisms b et w een free como du les M ⊗ C → M ⊗ C ⊗ C . Clearly , M C is th e closure of A under s m all limits. F or, every fr ee como dule has to b e in the closure under small limits, and hence eac h comodule d o es to o. No w consider th e inclusion functor K : A op ֒ → ( M C ) op . W e shall sho w that the functor ˜ K : ( M C ) op → [ A , V ect ] p reserv es small colimits. T o do this, it is enough to sho w that the composition of ˜ K with the “forgetful” functor [ A , V ect ] → V ect (recall that A has just o ne ob ject) preserv es small colimits. The resulting f unctor ( M C ) op → V ect is simply M C ( − , C ), whic h is isomorphic to V ect ( U ( − ) , k ), where U denotes the forgetful functor M C → V ect . Clearly U p r eserv es limits and V ect ( − , k ) con v erts them into coli mits. W e ha v e sh o wn, then, that small colimits in ( M C ) op are K -absolute. This, together with the fact that ( M C ) op is the completio n of A op under small colimit s, shows that K is dense. W e get, then , an equiv alence Cts[ M C , B ] ≃ [ A , B ] for any categ ory B with sm all limits; here the category on the left hand side is the catego ry of con tinuous ( i.e. small limit-preserving) functors and transformations b et we en them. Finally , for a ﬁnite d imensional coal gebra C , co nsider the full sub category A of M C giv en by the single ob ject C . As ( M C f ) op is equiv alen t to ( M C ∨ ) f , the functors J → A op form a ﬁnite catego ry J constitute a den sit y present ation for A op ֒ → ( M C f ) op . F urth ermore, w e h av e equiv alences Lex[ M C f , B ] ∼ = Rex[( M C f ) op , B op ] ≃ [ A op , B op ] ∼ = [ A , B ] for an y category B w ith ﬁnite limits. Recall th at there is an equiv alence [ A , V ect ] ∼ = M C ∨ . F or, to give a fu nctor A → V ect is to giv e a vect or space with a left actio n of th e algebra M C ( C, C ), which is isomorph ic to ( C ∨ ) op via the map send in g 32 W AL TER FER R ER SANTOS AND IGNACIO LOPEZ FRA NCO γ : C → k to ( γ ⊗ id)∆. No w it is easy to d ed uce that for a ﬁnite-dimensional coalge bra C there are equiv alences L ex[ M C f , M D ] ≃ C M D , send in g a righ t exact functor F to the bicomo dule F ( C ). A pseu d oin v erse for th is equiv alence is the functor sending a bicomo dule M to −  C M . In Section 2 we used the follo wing easy observ ation. Observ ation 22. Let C b e a ﬁn ite d im en sional coa lgebra, K : M C f → M C b e the inclusion f unctor and M , N ∈ C M D . Th en w e hav e a string of canonical isomorphisms C M D ( M , N ) ∼ = Lex[ M C f , M D ]( K ( − )  C M , K ( − )  C N ) = [ M C f , M D ]( K ( − )  C M , K ( − )  C N ) ∼ = Fin[ M C , M D ]( −  C M , −  C N ) = [ M C , M D ]( −  C M , −  C N ) The last piece of categorical bac kgroun d we will need is the tensor pr o duct of catego ries w ith ﬁ nite limits. This is closely relate d to Del igne’s tensor pro duct of abelian categories of [5]. W e only need the case of categories of ﬁnite dimen - sional comod ules o ver ﬁnite d imensional coal gebras, th ough, and in this case the existence of this pro du ct reduces to few sim p le ob s erv ations. Recall that the category C ⊗ D has as ob jects p airs of ob jects ( c, d ) w ith c ∈ C and d ∈ D , and homs C ⊗ D (( c, d ) , ( c ′ , d ′ )) = C ( c, c ′ ) ⊗ k D ( d, d ′ ). If C , D , E are catego ries with ﬁnite limits, a functor F : C ⊗ D → E is left exact in e ach variable if for eac h c ∈ C and d ∈ D the functors F ( c, − ) : D → E and F ( − , d ) : D → E are left exact. These functors, tog ether with the natural transformations b et w een them, form a cate gory Lex[ C , D ; E ]. A tensor pro duct of C with D as categ ories with ﬁnite limits is a ca tegory with ﬁnite limits C ⊠ D to gether with a functor C ⊗ D → C ⊠ E left exact in eac h v ariable that in duces equ iv alences Lex [ C ⊠ D , E ] ≃ L ex[ C , D ; E ] f or eac h E . The case of in terest for u s in this work is the one of categorie s of ﬁnite di- mensional como dules o v er ﬁnite dimens ional coalg ebras, dual to th e case consid- ered in [5]. If C , D are ﬁ nite dimensional coalge bras, we claim that the functor ⊗ k : M C f ⊗ M D f → M C ⊗ D f is a tensor p r o duct of M C f with M D f as catego ries with ﬁnite limits. T o see this, let u s call C ⊂ M C f , D ⊂ M D f and B ⊂ M C ⊗ D f the full sub categories determined b y the r esp ectiv e regular como du le, and observe that there is a comm utativ e diagram as depicted b elo w . Lex[ M C ⊗ D f , E ] / / ≃   Lex[ M C f , M D f ; E ] ≃   [ B , E ] / / [ C ⊗ D , E ] The horizon tal arrows are indu ced b y ⊗ k and the ob vious functor C ⊗ D → B whic h at the lev el of the u nique hom-space is just M C f ( C, C ) ⊗ M D f ( D , D ) → M C ⊗ D f ( C ⊗ D , C ⊗ D ). T his last linear transformation is an isomorphism, as a consequence of the ﬁn iteness of C and D , and th en the b ottom ro w of the d iagram is an isomorphism. It f ollo ws that the top row is an equiv alence. MONOIDAL CA TEGORIES OF COM ODULES 33 Referen ces [1] B ´ enab ou, J. I ntr o duction to bic ate gories. 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Monoidal categories of comodules for coquasi Hopf algebras and Radfords formula

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