Coalgebra structures in algebras

Transcription

1 Palestine Journal of atheatics Vol. 5(Special Issue: 1) (2016), 1 12 Palestine Polytechnic University-PPU 2016 Coalgebra structures in algebras Robert Wisbauer Counicated by Jawad Abuhlail SC 2010 Classifications: 18A40, 18C20, 16T15. Keywords and phrases: Coalgebras, Hopf algebras, Azuaya algebras, separable algebras. Abstract. Historically, the interest in coalgebras grew out fro the study of the notion of Hopf algebras introduced in topology. Now, in the definition of Hopf algebras, the coalgebraic part is forulated explicitly and is by itself the basis of a rich theory. Separable, Azuaya, and Frobenius algebras are usually introduced as algebraic structures without referring to the notion of a coalgebra. In this survey we reveal the internal coalgebra structure in these algebras which ay also be used to characterise the and to describe their properties. All these classes of algebras including the Hopf algebras have an associative ultiplication and a coassociative coultiplication; they are distinguished by requiring different copatibility conditions and properties for units and counits. 1 odules and algebras over coutative rings Throughout this paper R will denote a coutative ring with unit. Firstly we recall the basic notions of algebras and odules over R in a way which allows for an easy translation to coalgebras and coodules. For this the language of category theory is extreely helpful and for convenience we provide the basic notions needed (fro [9], [21]) Categories. A categoryaconsists of a class of objects Obj(A) and, for anya,a Obj(A), a (possibly epty) set or A (A,A ) of orphiss which allow for an associative coposition. Furtherore, for any or A (A,A) the existence of an identity orphis is required which we denote by I A (or just I). A covariant functor F : A B between categories A and B consists of assignents Obj(A) Obj(B),A F(A), and for all A,A Obj(A), or A (A,A ) or B (F(A),F(A )),f F(f), respecting the identity orphis and the coposition of orphiss. F is a contravariant functor if it reverses the coposition of orphiss. Given two functors F, G : A B, a natural transforation γ : F G is defined by a faily of orphiss γ A : F(A) G(A), A Obj(A), such that any orphis f : A A in A induces coutativity of the diagra F(A) F(f) F(A ) γ A G(A) γ A G(f) G(A ). For functorsl : A B and R : B A between any categoriesaandb, a pairing is defined by aps, natural in A A and B B, or B (L(A),B) αa,b or A (A,R(B)) βa,b or B (L(A),B). Such a pairing is deterined by the iages of the identity orphiss of L(A) and R(B), respectively, η A := α A,L(A) (I L(A) ) : A RL(A), ε B := β R(B),B (I R(B) ) : LR(B) B,

2 2 Robert Wisbauer corresponding to natural transforations η : I A RL, ε : LR I B, which are called quasi-unit and quasi-counit of (L, R, α, β), respectively. They allow to reconstruct α and β. (L,R) is said to be an adjoint pair provided α β and β α yield the identity aps and this corresponds to the equalities known as triangular identities. L Lη LRL εl L = I L, R ηr RLR Rε R = I R, Notice that so far we only have put up a fraework without using deeper results fro category theory. In the course of this talk we will encounter several ore concrete exaples of these abstract notions Category of R-odules. For the ring R, denote by R the category of R-odules, taking for objects the R-odules and for orphiss the R-linear aps. This is a category with products and coproducts, kernels and cokernels, and the R-odule R as a projective generator. For any R-odules, N, there is the tensor product R N yielding the functors R : R R, N R N, Ho R (, ) : R R, N Ho R (,N), which for an adjoint pair by the bijection (property of tensor product) Ho R ( R N,K) Ho R (N, Ho R (,K)), ϕ [n ϕ( n)], (1.1) and unit and counit of this adjunction coe out as η N : N Ho R (, N), n [ n], ε N : Ho R (,N) N, f f(). The functors R and Ho R (, ) are naturally isoorphic if and only if is a finitely generated and projective R-odule: the isoorphis iplies that Ho R (, ) preserves epiorphiss and direct sus (since R does so) and hence has the properties required. Furtherore, and N can be interchanged by the twist ap τ,n : R N N R, n n, which obviously satisfies τ N, τ,n = I N. For odules over a field R all these properties are well-known fro eleentary linear algebra and the corresponding proofs hold for any coutative base rings Algebras over R. An R-algebra (A,, 1 A ) is defined as R-odule A with an associative R-bilinear ultiplication : A A A (usually written as (a,b) ab) and unit eleent 1 A satisfying 1 A a = a1 A for all a A. By the properties of the tensor product, the bilinear ap can be replaced by an R-linear ap : A R A A, and 1 A defines an R-linear ap η : R A,r r1 A. With this ters, associativity and unitality conditions required for an algebra are expressed by coutativity of the diagras (writing for R ) A A A IA I A A A A A A, η I A A R IA η A A R A = = A. Defining R-algebras (A,,η) in this way we are only using objects and orphiss in the category R.

3 Coalgebra structures 3 As well known, the tensor product A B of two R-algebras A and B is again an R-algebra by coponentwise ultiplication. For this definition, the twist ap τ B,A : B A A B is needed. Analysing the setting shows that there ay be other R-linear aps λ : B A A B leading to an associative algebra structure on A B. These are exaples of distributive laws known fro general category theory (e.g. [2]) Tensor product of algebras. Consider two R-algebras (A,,η) and (B,,η ). ultiplication and unit on A B can be defined by AB : A B A B A τb,a B A A B B A B, a b c d a c b d ab cd, η AB : R η η A B, 1 R 1 A 1 B, aking (A B, AB,η AB ) an associative unital algebra. Replacing τ B,A by soe R-linear ap λ : B A A B, we observe: 1.5. Distributive laws. Let (A,,η) and (B,,η ) be R-algebras with soe R-linear ap λ : B A A B. Defining a product on A B by λ : A B A B A λ B A A B B A B, the triple (A B, λ,η η ) is an (associative and unital) R-algebra if and only if it induces coutativity of the diagras B A A λ A A B A A λ A A B B A B A λ A B B A B B A B λ B A B λ B A B B, λ B A B η B A B η B 2, B A η A λ A A B 2 A η. odules over an A-algebra are defined by R-bilinear aps : A. Using the tensor product this can again be expressed by referring only to objects and orphiss fro the category R A-odules. Let (A,,η) be an R-algebra. A (unital) left A-odule is a pair (, ), where is an R-odule and : A A (written as (a ) a) is an R-linear ap leading to coutativity of the diagras A A I A R η I A I A A A, =.

4 4 Robert Wisbauer An A-odule orphis between two A-odules (, ) and (N, N) is an R-linear ap f : N with coutative diagra A IA f A N f N. N The category of left A-odules is denoted by A. Siilar to R, it also has products, coproducts, kernels and cokernels and a projective generator (= A) but it need not allow for a tensor product. For any R-odule X, A X is a left A-odule by ultiplication of A and this induces the free and forgetful functors, φ A : R A, X (A X, I X ), U A : A R, (, ). (φ A,U A ) for an adjoint pair by the bijections, for X R,(, ) A, Ho A (A X,) Ho R (X,), A X f X η IX A X f, Ho R (X,) Ho A (A X,), X g A X IA g A, and unit η and counit ε for this adjunction coe out as η X : X η IX A X, ε : A. The algebra structure on the tensor product A B of two algebras ay also be seen as a lifting of functors, investigated in a general categorical setting by P. Johnstone [8], which here coes out as follows Lifting of functors. Let (A,,η) and (B,,η ) be R-algebras and consider the diagra B  B U B R A R. U B The following are equivalent: (a) there exists a functor  aking the diagra coutative; (b) there is a distributive law λ : B A A B (see 1.5); (c) A B has an algebra structure induced by soe R-linear ap λ : B A A B. Hereby, for a B-odule (,ρ),â() is the object A with the B-odule structure and one ay write B A λ I A B I ρ A, Â( ) = (A B) B : B B. 2 Coalgebras over coutative rings In the paper [7] (1941), H. Hopf pointed out the rich structure of the hoology of anifolds which adit a product operation: it allows for a coproduct and a product satisfying certain

5 Coalgebra structures 5 copatibility properties. In [18] (1965), J.W. ilnor and J.C. oore analysed the algebraic parts of this structure and provided an introduction to the theory of coalgebras and coodules. The fraework we built up for algebras and odules in the preceding section is suitable for a natural transition to coalgebras and coodules. This will be described in the subsequent section. Again R will denote a coutative ring Coalgebras. A coalgebra over R is a triple (C,, ε) where C is an R-odule with coassociative product and counit, that is, there are R-linear aps inducing coutativity of the diagras : C C C, ε : C R, C C C I C C I C C C, C = R C C C ε I C = C R I C ε Siilar to the situation for algebras, the product of twor-coalgebras(c,,ε) and(d,,ε ) can be defined using the twist ap τ C,D : C D D C; the latter can be replaced by a distributive law ϕ : C D D C with coutative diagras (e.g. [24, 4.11]) C C D C ϕ C D C ϕ C D C C D C D C ϕ D C D C C D D ϕ D D C D D ϕ D D C, ϕ C D D C ε D D ε 1 D, C D C ε ϕ D C ε C C C-coodules. Let(C,,ε) be an R-coalgebra. A leftc-coodule is a pair(,ρ ) where is an R-odule and ρ : C is an R-linear ap with coutative diagras ρ ρ C I C I ρ C C, = ρ C. ε I A C-coodule orphis between two C-coodules (,ρ ) and (N,ρ N ) is an R-linear ap f : N with coutative diagra f N ρ ρ N C I f C N.

6 6 Robert Wisbauer These data for the category of left C-coodules, denoted by C. There are the forgetful and the cofree functors, U C : C R, (,ρ ), φ C : R C, X (C R X, I X,) and (U C,φ C ) is an adjoint pair by the bijection, for(,ρ ) C, X R, Ho C (,C X) Ho R (,X), f C X f C X ε IX X, Ho R (,X) Ho C (,C X), g X ρ C IC g C X, and unit η and counit ε of this adjunction coe out as η : ρ C, ε X : C X ε IX X. Fro this adjunction a nuber of properties of coodules and their categories can be derived. For exaple, choosing X = R and = C, we obtain the isoorphiss Ho C (,C) Ho R (,R), End C (C) = Ho C (C,C) Ho R (C,R), showing that the R-dual odules play a significant role here. 3 Frobenius and separable algebras In [6] (1903), F. Frobenius investigated finite diensional K-algebras A over a field K with the property that A A := Ho K (A,K) as left A-odules. They can also be characterised by the existence of a non-degenerate bilinear for σ : A A K with σ(ab,c) = σ(a,bc) for all a,b,c A. Such algebras A were naed Frobenius algebras by Brauer and Nesbitt (1937); their duality properties were pointed out by Nakayaa (1939); Eilenberg and Nakayaa observed (1955) that the notion akes sense over coutative rings, provided A is finitely generated and projective as an R-odule. Frobenius algebras are of considerable interest in representation theory of finite groups, nuber theory, cobinatorics, coding theory, etc. Their relation with coalgebras were entioned by Lawvere (1967), Quinn (1991), Abras (1999) e.a. As pointed out by Dijkgraaf (1989), Abras (1996), and others, they show up in the fraework of topological quantu field theory. An outline of their categorical forulation, the Frobenius onads, is given by Street in [20] Coalgebra structure of A. Let (A,,η) be an R-algebra and assue A to be finitely generated and projective as an R-odule. Then there is an R-linear isoorphis λ : A A and (A R A) A R A as R-odules. Applying( ) := Ho R (,R) to : A K A A and η : R A yields coultiplication and counit on A, A (A R A) A R A, A η R. Applying λ, the coproduct and counit of A can be transferred to A: A δ A K A λ λ 1 λ 1 A A K A, A ε λ A η R, aking (A, δ, ε) a counital coalgebra.

7 Coalgebra structures 7 Now, if we assue λ : A A to be left A-linear, a little coputation shows that δ is also left A-linear and - by syetry - also right A-linear and this eans that product and coproduct on A are related by the Frobenius conditions, that is, coutativity of the diagras A A A A A A (3.1) I δ A A A I A A, δ δ I A A A I A A. δ It follows fro general category theory (also shown in [1]) that the category A of left A- odules is isoorphic to the category A of left A-coodules: A A. This isoorphis can be seen as a characterising property of Frobenius algebras (e.g. [15, Theore 3.13]). The coutativity of the diagras can be read in different ways Reforulation of the Frobenius conditions. Let(A,,δ) be given as above. (1) The following are equivalent: (a) δ = ( I A ) (I A δ) (left hand diagra); (b) δ is a left A-odule orphis; (c) is a right A-coodule orphis. (2) The following are equivalent: (a) δ = (I A ) (δ I A ) (right hand diagra); (b) δ is a right A-odule orphis; (c) is a left A-coodule orphis. By these observations one obtains: 3.3. Characterisation of Frobenius algebras. For an R-odule A, let (A,, η) be an R- algebra and (A,δ,ε) a coalgebra. Then the following are equivalent: (a) (A,,δ) satisfies the Frobenius conditions; (b) δ is a left A-odule orphis and is a left A-coodule orphis; (c) δ is a left A-odule orphis and a right A-odule orphis; (d) is a left A-coodule and a right A-coodule orphis; (e) A R (equivalently R A) is adjoint to itself by the unit and counit I A η A δ A A, A A A ε IA. Notice that in (b) the conditions only refer to one side, no twist ap is needed for this property. Fro (c) it follows that δ(a) = aδ(1 A ) = δ(1 A )a, for all a A Frobenius biodules. Let (A,, η, δ, ε) be a Frobenius algebra. Then an R-odule is called a Frobenius biodule provided it has an A-odule and also an A-coodule structure, : A and ν : A, inducing coutativity of the diagras A A I ν A A I A, ν δ I A A I A. ν Taking for objects the Frobenius biodules and for orphiss the R-linear aps which are A-odule and A-coodule orphiss, one obtains the category of Frobenius biodules which we denote by A A. Obviously, (A,,δ) itself is a Frobenius biodule and there is a pair of functors A R : R A A, X (A X, A X,δ A X), Ho A A(A, ) : A A R, Ho A A(A,),

8 8 Robert Wisbauer which for an adjoint pair by the bijection, for X R,(,,ν) A A, Ho A A(A X,) Ho R (X, Ho A A(A,)), ϕ [x ϕ( x)], and unit η and counit ε are given by η X : X Ho A A(A,A X), x [a a x], ε : A Ho A A(A,), a f f(a). Coinvariants of a Frobenius odule are defined as the iage of Ho A A(A,), f f(1 A ), and this ap is indeed surjective, in particular one has End A A(A) A. Thus the pair of functors (A R, Ho A A(A, )) induces an equivalence between A and A A. This can be expressed by showing that, for any lefta-odule(, ), there is ana-coodule structure on, ν : η I A δ I A A I A, aking (,, ν) a Frobenius biodule, and is an isoorphis of categories. Ψ : A A A, (, ) (,ν), Siilarly, any left A-coodule (,ν) allows for a right coodule structure leading to the isoorphis of categories : A A ν A A A A ε, Φ : A A A, (,ν) (,ν). Cobining these functors, we obtain the isoorphiss of the A-odule and the A-coodule categories entioned before, A Ψ A A UA A, A Φ A A UA A. Because of these isoorphiss, the category of Frobenius biodules ay see to be of little interest for Frobenius algebras(a,, η, δ, ε). However, the approach sketched above also allows to deal with ore general situations, for exaple, when no counit (or unit) is at hand (see [26]) Separable algebras. An R-algebra(A,, η) is called separable if there is soe A-biodule apδ : A A A with δ = I A. This iplies that(a,,δ) satisfies the Frobenius condition (3.1) and yields a (coparison) functor K A : R A A, X (A X, A X), which is right adjoint to the functor A Ho A (A, ) : A A R by the bijection (derived fro (1.1)) AHo A (A X,) Ho R (X, A Ho A (A,)). Here the coinvariants of any A A are defined as the iage of AHo A (A,), f f(1 A ), and Z(A) := A End A (A) is the center of A leading to the equivalence A Z(A) : Z(A) A A, N (A Z(A) N, I N ). The R-algebra A is called central if the ap R A, r r1 A, induces an isoorphis R Z(A) and a central separable algebra is called Azuaya algebra. ore about this kind of algebras can be found, for exaple, in [22] and [16]. In general categories, separable functors are considered in [19]; for Azuaya onads and coonads we refer to [17] for a recent account.

9 Coalgebra structures 9 4 Bialgebras and Hopf algebras In this section, we will again consider R-odules endowed with an algebra and a coalgebra structure but with different copatibility conditions Bialgebras. LetB be anr-odule with an algebra structureb = (B,,η) and a coalgebra structure B = (B,,ε). Then (B,,η,,ε) is called a bialgebra if and ε are algebra orphiss, or, equivalently, µ and η are coalgebra orphiss. To ake an algebra orphis one needs coutativity of the outer path in the diagra Defining an R-linear ap B B B B B B B B B B B B B ω B B B B B B B B τ B B B B B B B. ω : B B B B B B B τ B B B B B, the condition reduces to coutativity of the upper rectangle. With the ap ω : B B B B B B τ B B B B B B one obtains a siilar rectangle (sides interchanged). These orphiss ay be considered as entwinings between algebras and coalgebras (see Section 5), ω : B B B B, ω : B B B B. They can be applied to define biodules which fit into the setting Hopf odules and algebras. Given a bialgebra(b,b,ω), anr-odule is called a Hopf odule provided it is ab-oduleρ : B and ab-cooduleν : B inducing coutativity of the diagra B ρ ν B B ν B ρ B B ω B B. The category of Hopf odules, denoted by B B, has the Hopf odules as objects and as orphiss those R-linear aps, which are B-odule as well as B-coodule orphiss. As can be shown easily, for any R-odule X, B R X is a Hopf odule and this observation leads to the functor B R : R B B, X (B X, X, X), which is left adjoint to Ho B B(B, ) : B B R by the bijection (derived fro (1.1)) Ho B B(B X,) Ho R (X, Ho B B(B,)). For any B B, the coinvariants are the iage of HoB B(B,), f f(1 B ), and the coinvariants of B coe out as End B B(B) R.

10 10 Robert Wisbauer A bialgebra (B,B,ω) is called a Hopf algebra provided B R : R B B is an equivalence (known as Fundaental Theore). This can be characterised by the existence of an antipode, and is also equivalent to require that the (fusion) orphis ( I B ) (I B ) : B B B B is an isoorphis (e.g. [5]). The corresponding constructions for onads and coonads on categories can be found in [12]. 5 Entwining algebras and coalgebras Suitable distributive laws (e.g. the twist ap) allow for giving the tensor product of two algebras an algebra structure and the tensor product of two coalgebras a coalgebra structure. The question arises: which structure can be given to the tensor product of an algebra and a coalgebra? This leads to the notions of ixed distributive laws and corings over non-coutative rings (e.g. [4], [5]). Let(A,, η) be an R-algebra and (C, δ, ε) an R-coalgebra Entwining fro A to C. An R-linear ap ω : A C C A is called an entwining fro the algebra A to the coalgebra C provided it induces coutativity of the diagras A A C A ω A C A ω A C A A C A C ω C A C A δ δ A A C C ω C C A C C ω C C A, ω A C η C C C A C η 2, A C A ε ω C A ε A A. The Hopf odules for bialgebras can be generalised to biodules for entwined structures Biodules for entwinings fro A to C. For an entwining ω : A C C A, an R-odule with an A-odule structure : A and a C-coodule structure : C is called an entwined odule if one gets coutativity of the diagra I A A A C ω I C I C C A. Taking as orphiss the R-linear aps which are A-odule as well as C-coodule orphiss defines the category C A of entwined odules. There is an (induction) functor (e.g. [5, 32.7]) C R : A C A, C R, that is right adjoint to the forgetful functor C U : C A A. Now assue A belongs to C A, that is, A is a C-coodule : A C R A with grouplike eleent (1 A ), and puts := End C A(A) (a subalgebra ofa). Then there is a (coparison) functor S C A, X (A S X, S I X, S I X ),

11 Coalgebra structures 11 and this is an equivalence provided C R is flat and C A is a Galois coring (see e.g. [5], [13], [23]). In case A = C = B, we get S = R and this brings us back to the Hopf odules and the Fundaental Theore (see 4.2) Entwining fro C to A. An R-linear ap ω : C A A C is an entwining fro the coalgebra C to the algebra A if it induces coutativity of the diagras C A A ω A A C A A ω A A C C C A ω A C C δ A A δ C C A C ω C A C ω C A C C, ω C A C η C A C η C 2, C A ε A ω A C A ε A. It was observed in Section 1.7 that the distributive laws between two algebras ay be understood as liftings of functors to odule categories. The situation for entwinings between algebras and coalgebras is quite siilar Liftings and entwinings fro A to C. An entwining ω : A C C A fro A to C corresponds to a lifting Ĉ of C R to A and also to a lifting  of A R to C, that is, there are coutative diagras A Ĉ A C  C U A R C U A R, U C R A R, U C where the U s denote the forgetful functors. The reader can find a ore detailed description of liftings for tensor functors in [25]. For entwinings fro a coalgebra to an algebra the situation is slightly different: they do not correspond to liftings to the (Eilenberg-oore) categories A and C but to extensions to the Kleisli categories A and C (which ay be seen as subcategories deterined by the (co)free objects of the Eilenberg-oore categories, e.g. [3]) Liftings and entwining fro C to A. An entwining ω : C A A C fro C to A corresponds to an extension C of C R to A and also to an extension à of A R to C, that is, there are coutative diagras R C R R A R φ A A C φ A A, φ C C à φ C C, where the φ s denote the (co)free functors.

12 12 Robert Wisbauer The notions in the preceding section can be readily transferred fro the category R of R-odules to arbitrary categories A. Hereby A R : R R is to be replaced by any onad F : A A and C R is to be replaced by any coonad G : A A. The role of an entwining ω : A R C C R A is taken by a natural transforation ω : FG GF requiring coutativity of the corresponding diagras and the definition of entwined odules is obvious. This allows to apply the basic theory in fairly general situations (e.g. [12], [13]). References [1] Abras, L., odules, coodules, and cotensor products over Frobenius algebras, J. Algebra 219(1) (1999), [2] Beck, J., Distributive laws, Lecture Notes in ath., 80, (1969). [3] Böh, G., Brzeziński, T. and Wisbauer, R., onads and coonads in odule categories, J. Algebra 322, (2009). [4] Brzeziński, T. and ajid, Sh., Coalgebra bundles, Co. ath. Phys. 191, (1998). [5] Brzeziński, T. and Wisbauer, R., Corings and Coodules, London ath. Soc. LNS 309, Cabridge University Press (2003). [6] Frobenius, F., Theorie der hypercoplexen Größen, Sitz. Kön. Preuss. Akad. Wiss., (1903); Gesaelte Abhandlungen, art. 70, [7] Hopf, H., Über die Topologie der Gruppen-annigfaltigkeiten und ihre Verallgeeinerungen, Ann. of ath. (2) 42, (1941). [8] Johnstone, P., Adjoint lifting theores for categories of algebras, Bull. London ath. Soc. 7, (1975). [9] ac Lane, S., Categories for the Working atheatician, 2nd edn, Springer-Verlag, New York (1998). [10] esablishvili, B., Entwining Structures in onoidal Categories, J. Algebra 319(6), (2008). [11] esablishvili, B., onads of effective descent type and coonadicity, Theory Appl. Categ. 16, 1 45 (2006). [12] esablishvili, B. and Wisbauer, R., Bionads and Hopf onads on categories, J. K-Theory 7(2), (2011). [13] esablishvili, B. and Wisbauer, R., Galois functors and entwining structures, J. Algebra 324, (2010). [14] esablishvili, B. and Wisbauer, R., Notes on bionads and Hopf onads, Theory Appl. Categ. 26, (2012). [15] esablishvili, B. and Wisbauer, R., QF functors and (co)onads, J. Algebra 376, (2013). [16] esablishvili, B. and Wisbauer, R., Azuaya algebras as Galois coodules, Translated fro Sovre. at. Prilozh. 83 (2012); J. ath. Sci. (N. Y.) 195(4) (2013). [17] esablishvili, B. and Wisbauer, R., Azuaya onads and coonads, Axios 4(1), 32 70, electronic only (2015). [18] ilnor, J.W. and oore, J.C., On the structure of Hopf algebras, Ann. of ath. (2) 81, (1965). [19] Rafael,.D., Separable functors revisited, Co. Algebra, 18, (1990). [20] Street, R., Frobenius onads and pseudoonoids, J. ath. Phys. 45(10) (2004), [21] Wisbauer, R., Foundations of odule and Ring Theory, Gordon & Breach, Philadelphia (1991). [22] Wisbauer, R., odules and algebras: biodule structure and group actions on algebras, Pitan onographs 81, Longan, Harlow (1996). [23] Wisbauer, R., On Galois coodules, Coun. Algebra 34, (2006). [24] Wisbauer, R., Algebra Versus Coalgebras, Appl. Categor. Struct. 16, (2008). [25] Wisbauer, R., Lifting theores for tensor functors on odule categories, J. Algebra Appl. 10(1), (2011). [26] Wisbauer, R., Weak Frobenius bionads and Frobenius biodules, arxiv: (2015). Author inforation Robert Wisbauer, Departent of atheatics, HHU, Düsseldorf, Gerany. E-ail: Û ÙÖÑغÙÒ¹Ù ÐÓÖº ÊÚ ÔØ ÙÙ Ø ½¼ ¾¼½º ÇØÓÖ ½ ¾¼½