Crossed Products in Weak Contexts

CROSSED PRODUCTS IN WEAK CONTEXTS J.N. Alonso ´ Alvarez2, J.M. Fern´andez Vilaboa1, R. Gonz´alez Rodr´ıguez3, and A.B. Rodr´ıguez Raposo1 1Departamento de ´ Alxebra. Universidad de Santiago de Compostela. Santiago de Compostela. E-15771. SPAIN. E-mail: [email protected], abrap[email protected] 2Departamento de Matem´aticas. Universidad de Vigo. Lagoas-Marcosende. Vigo. E-36280. SPAIN. E-mail: [email protected] 3Departamento de Matem´atica Aplicada II. Universidad de Vigo. Lagoas-Marcosende. Vigo. E-36310. SPAIN. E-mail: [email protected] Abstract. We define the general notion of crossed products in a weak context, which generalizes the ones defined by Blattner, Cohen and Montgomery, Doi and Takeuchi in the context of Hopf algebras and the one given by Brzezi´nski. Also, the crossed products obtained by the authors, for weak Hopf algebras living in a symmetric monoidal category and weak C-cleft extensions associated to weak entwined structures, are particular instances of this theory. MSC: 18D05, 16W30. Corresponding author: R. Gonz´alez Rodr´ıguez. 1 Introduction Hopf crossed products, that is smash products where the multiplication is twisted by a cocycle σ, were introduced independently by Blattner, Cohen and Montgomery [7] and Doi and Takeuchi [16], as a generalization of group crossed products to the context of Hopf algebras living in a category of vector spaces over a field K. These objects, which play an important role in the theory of extensions of Hopf algebras, are constructed in the following way: Let Hbe a Hopf algebra with unit ηH, product µH, counit εHand coproduct δH. Suppose that ϕA:H⊗A→A is a weak action of Hon the K-algebra Aand let σ:H⊗H→Abe a K-linear map. In the vector space A⊗H, denoted by A]σH, define the product (possible non-associative) µA]σH= (µA⊗H)◦(µA⊗σA H)◦(A⊗ψA H⊗H) where σA H= (σ⊗µH)◦(H⊗cH,H ⊗H)◦(δH⊗δH), ψA H= (ϕA⊗H)◦(H⊗cH,A)◦(δH⊗A), µAis the product of Aand cis the flip. If A]σHis associative with ηA⊗ηHas unity morphism, we call A]σHa crossed product. A necessary and sufficient conditions that A]σHbe a crossed 1 product was found by Blattner, Cohen and Montgomery [Corollary 4.6,[7]], and by Doi and Takeuchi [Lemma 10, [16]]. The result is the following: A]σHis a crossed product if and only if σis normal (σ◦(ηH⊗H) = εH⊗ηA=σ◦(H⊗ηA)), σsatisfy the twisted module condition µA◦(ϕA⊗A)◦(H⊗ϕA⊗A)◦(H⊗H⊗cA,A)◦(H⊗H⊗σA⊗A)◦(δH⊗H⊗A) = µA◦(A⊗ϕA)◦(σA H⊗A), and the cocycle condition ∂4(σA)∧∂2(σA) = ∂1(σA)∧∂3(σA), where the morphisms ∂iare defined by ∂1=ϕA◦(H⊗σA), ∂2=σA◦(µH⊗H), ∂3=σA◦(H⊗µH), ∂4=σA⊗εH, and ∧denotes the usual convolution in HomC(H⊗H⊗H, A). A more general notion of crossed product was introduced by Brzezi´nski in [8], as follows: Let Abe a K-algebra and Va vector space equipped with a distinguished morphism ηV:K→V. Given maps ψA V:V⊗A→A⊗Vand σA V:V⊗V→A⊗V, the object A]V, whose underlaying vector space is A⊗V, endowed with the product µA]V = (µA⊗V)◦(µA⊗σA V)◦(A⊗ψA V⊗V) is called a crossed product if it is associative with ηA⊗ηHas identity. In this case, to ensure that the product of A]V is associative and unitary, the morphisms ψA V(the twisting morphism) and σA V(the cocycle) must satisfy the following suitable conditions: The twisting morphism is compatible with the the algebra structure of A,ψA V◦(ηV⊗A) = A⊗ηV,σA Vis normal ( σA V◦(ηV⊗V) = ηA⊗V=σA V◦(V⊗ηV)), and it is a cocycle which satisfies the twisted module condition, that is: (µA⊗V)◦(A⊗σA V)◦(σA V⊗V) = (µA⊗V)◦(A⊗σA V)◦(ψA V⊗V)◦(V⊗σA V), (µA⊗V)◦(A⊗ψA V)◦(σA V⊗A) = (µA⊗V)◦(A⊗σA V)◦(ψA V⊗V)◦(V⊗ψA V). As a particular instance of the crossed product constructed in the last paragraph, we obtain the crossed products defined by Blattner, Cohen, Montgomery, Doi and Takeuchi. Also, the twisted tensor products or matched pairs, studied by Cap, Schichl, Vanzura and Tambara [14], [26] are examples of the Brzezi´nski’s crossed products. On the other hand, this notion of crossed products is needed in the theory of braided Hopf crossed products developed by J.A. Gucccione and J.J. Guccione in [18], which includes the classical type (Blattner-Cohen-Montgomery, DoiTakeuchi) and the automorphism Ore extensions type. Finally, Brzezi´nski’s theory can be generalized, in an straightforward form, to the context of braided monoidal categories, and then, we obtain as examples, the crossed products by braided groups defined by Majid and Bespalov in [21],[6]. Unfortunately, all these constructions are not valid when we want to extend the theory of crossed products to more general Hopf structures like weak Hopf algebras or weak entwined structures. The aim of this paper, inspired by the work of Brzezi´nski’s and by our own papers [2], [3], [4], [5], is to introduce a general notion of crossed products that includes the crossed 2 products described in the previous paragraphs and also the new crossed products that arises in weak contexts such as weak Hopf algebras or weak entwining structures. This paper is organized as follows: in Section 2, for an algebra Aand an object V, living in a strict monoidal category with equalizers and coequalizers, we introduce the notion of crossed product system and we prove that the product induced by it is associative if satisfies the twisted and the cocycle conditions. In Section 3 we obtain that the notion of weak C-cleft extension, introduced by us in [4], provides an example of crossed product system satisfying the twisted and the cocycle conditions. As a consequence, the crossed product defined in [4] is a particular instance of the product induced by a crossed product system. This crossed products are deeply connected with Galois theory as we can see in the intrinsic characterization of weak cleftness in terms of weak C-Galois extensions obtained in [5](see also [1] for the Hopf algebra case in braided categories). In Section 4, we define the notion of crossed product system with unity and we prove the main result of this paper, that is Theorem 4.6. As a particular case of this Theorem, we obtain Brzezi´nski’s characterization of crossed products and, of course, the classical characterizations related in the first paragraph of this Introduction. Finally, in Section 5, we apply our theory to the context of weak Hopf algebras in a symmetric monoidal category with split idempotents, obtaining that our construction is valid to develop a theory of crossed products for weak Hopf algebras. The final example of this section is especially interesting because we prove that for all morphism of weak Hopf algebras with coalgebra splitting, it is possible to obtain a crossed product system that is also an example of the cleft theory developed in Section 3. 2 Crossed product systems Throughout the paper Cdenotes a strict monoidal category with tensor product ⊗and base object K. Given objects A,B,Dand a morphism f:B→D, we write A⊗ffor idA⊗f and f⊗Afor f⊗idA. Also we assume that Cadmits equalizers and coequalizers. It is an easy exercise to prove that, under these conditions, all idempotent splits, i.e., for every morphism ∇Y:Y→Y, such that ∇Y=∇Y◦ ∇Y, there exist an object Zand morphisms iY:Z→Y and pY:Y→Zsatisfying ∇Y=iY◦pYand pY◦iY=idZ. We assume that the reader is familiar with the notions of algebra, coalgebra, module and comodule. Unless otherwise explicitly established, we assume that algebras are associative with unity and the coalgebras coassociative with counity. Given an algebra Aand a coalgebra C, we let ηA:K→A,µA:A⊗A→A,εD:D→K, and δD:D→D⊗Ddenote the unity, the product, the counity, and the coproduct respectively. Given two algebras Aand B,f:A→Bis an algebra morphism if µB◦(f⊗f) = f◦µA,f◦ηA=ηB. Also, if Cis braided with braiding c, given A,Bare algebras in C, the object A⊗Bis also an algebra in Cwhere ηA⊗B=ηA⊗ηBand µA⊗B= (µA⊗µB)◦(A⊗cB,A⊗B).If Dand Eare coalgebras, f:D→Eis a coalgebra morphism if (f⊗f)◦δD=δE◦f,εE◦f=εD.If Cis braided with braiding c, given D,Ecoalgebras in C,D⊗Eis a coalgebra in Cwhere εD⊗E=εD⊗εEand δD⊗E= (D⊗cD,E ⊗E)◦(δD⊗δE). Definition 2.1 An algebra Aand and object Vtogether with two morphisms ψA V:V⊗A→A⊗V, σA V:V⊗V→A⊗V is called a crossed product system if the following equalities hold: 3 (a1) (µA⊗V)◦(A⊗ψA V)◦(ψA V⊗A) = ψA V◦(V⊗µA), (a2) (µA⊗V)◦(A⊗σA V)◦(ψA V⊗V)◦(V⊗ ∇A⊗V) = ∇A⊗V◦(µA⊗V)◦(A⊗σA V)◦(ψA V⊗V), (a3) ∇A⊗V◦(µA⊗V)◦(A⊗σA V)◦(∇A⊗V⊗V) = ∇A⊗V◦(µA⊗V)◦(A⊗σA V). where the morphism ∇A⊗V:A⊗V→A⊗V, is defined by ∇A⊗V= (µA⊗V)◦(A⊗ψA V)◦(A⊗V⊗ηA). In what follows we denote the crossed product systems by (A, V, ψA V, σA V). For example, if for ψA Vthe equality ηA⊗V=ψA V◦(V⊗ηA) holds, then ∇A⊗V=idA⊗Vand therefore (A, V, ψA V, σA V) is a crossed product system for all morphism σA V:V⊗V→A⊗V. Remark 2.2 Note that if (A, V, ψA V, σA V) is a crossed product system, the morphism ∇A⊗Vis idempotent. Let pA⊗V:A⊗V→A×V,iA⊗V:A×V→A⊗Vbe the morphisms such that iA⊗V◦pA⊗V=∇A⊗V,pA⊗V◦iA⊗V=idA×Vwhere A×Vrepresents the image of the idempotent morphism ∇A⊗V. Composing with pA⊗Vin (a2) and (a3) we obtain the following: (a2’) pA⊗V◦(µA⊗V)◦(A⊗σA V)◦(ψA V⊗V)◦(V⊗∇A⊗V) = pA⊗V◦(µA⊗V)◦(A⊗σA V)◦(ψA V⊗V), (a3’) pA⊗V◦(µA⊗V)◦(A⊗σA V)◦(∇A⊗V⊗V) = pA⊗V◦(µA⊗V)◦(A⊗σA V). Similarly, composing with iA⊗Vin (a2) and (a3) we obtain (a2”) (µA⊗V)◦(A⊗σA V)◦(ψA V⊗V)◦(V⊗iA⊗V) = ∇A⊗V◦(µA⊗V)◦(A⊗σA V)◦(ψA V⊗V)◦(V⊗iA⊗V), (a3”) ∇A⊗V◦(µA⊗V)◦(A⊗σA V)◦(iA⊗V⊗V) = ∇A⊗V◦(µA⊗V)◦(A⊗σA V)◦(iA⊗V⊗V). Proposition 2.3 Let (A, V, ψA V, σA V)be a crossed product system. The following identities hold (b1) ∇A⊗V◦(ηA⊗V) = ψA V◦(V⊗ηA). (b2) (µA⊗V)◦(A⊗ ∇A⊗V) = ∇A⊗V◦(µA⊗V). (b3) ∇A⊗V◦ψA V=ψA V. (b4) (µA⊗V)◦(A⊗ψA V)◦(∇A⊗V⊗A) = (µA⊗V)◦(A⊗ψA V). Proof. The proof is a straightforward consequence of the definition of ∇A⊗V.2 Definition 2.4 We will say that a crossed product system satisfies the twisted condition if the following equality holds pA⊗V◦(µA⊗V)◦(A⊗ψA V)◦(σA V⊗A) = pA⊗V◦(µA⊗V)◦(A⊗σA V)◦(ψA V⊗V)◦(V⊗ψA V) or, equivalently, ∇A⊗V◦(µA⊗V)◦(A⊗ψA V)◦(σA V⊗A) = ∇A⊗V◦(µA⊗V)◦(A⊗σA V)◦(ψA V⊗V)◦(V⊗ψA V) 4 Definition 2.5 We will say that a crossed product system satisfies the cocycle condition if the following equality holds pA⊗V◦(µA⊗V)◦(A⊗σA V)◦(σA V⊗V) = pA⊗V◦(µA⊗V)◦(A⊗σA V)◦(ψA V⊗V)◦(V⊗σA V) or, equivalently, ∇A⊗V◦(µA⊗V)◦(A⊗σA V)◦(σA V⊗V) = ∇A⊗V◦(µA⊗V)◦(A⊗σA V)◦(ψA V⊗V)◦(V⊗σA V). Definition 2.6 Let (A, V, ψA V, σA V) be a crossed product system. The product µA×V:A×V⊗A×V→A×V, defined by µA×V=pA⊗V◦(µA⊗V)◦(A⊗µA⊗V)◦(A⊗A⊗σA V)◦(A⊗ψA V⊗V)◦(iA⊗V⊗iA⊗V), is called the product induced by (A, V, ψA V, σA V). Proposition 2.7 Let (A, V, ψA V, σA V)be a crossed product system satisfying the twisted and the cocycle conditions. The product induced by (A, V, ψA V, σA V)is associative. Proof. Composing µA×V◦(µA×V⊗A×V) with pA⊗V⊗pA⊗V⊗pA⊗Vwe obtain the following: µA×V◦(µA×V⊗A×V)◦(pA⊗V⊗pA⊗V⊗pA⊗V) =pA⊗V◦(µA⊗V)◦(A⊗µA⊗V)◦(A⊗A⊗σA V)◦(A⊗ψA V⊗V)◦(∇A⊗V⊗A⊗V)◦ (µA⊗V⊗A⊗V)◦(A⊗µA⊗V⊗A⊗V)◦(A⊗A⊗σA V⊗A⊗V)◦(A⊗ψA V⊗V⊗A⊗V)◦ (∇A⊗V⊗ ∇A⊗V⊗ ∇A⊗V) =pA⊗V◦(µA⊗V)◦ (A⊗[(µA⊗V)◦(µA⊗σA V)◦(A⊗ψA V⊗V)◦(σA V⊗A⊗V)])◦ (µA⊗V⊗V⊗A⊗V)◦(A⊗ψA V⊗V⊗A⊗V) =pA⊗V◦(µA⊗V)◦ (A⊗[(µA⊗V)◦(A⊗σA V)◦(∇A⊗V⊗V)◦(µA⊗V⊗V)◦(A⊗ψA V⊗V)◦(σA V⊗A⊗V)])◦ (µA⊗V⊗V⊗A⊗V)◦(A⊗ψA V⊗V⊗A⊗V) =pA⊗V◦(µA⊗V)◦ (A⊗[(µA⊗V)◦(A⊗σA V)◦(∇A⊗V⊗V)◦(µA⊗V⊗V)◦(A⊗σA V⊗V)◦(ψA V⊗V⊗V)◦ (V⊗ψA V⊗V)])◦ (µA⊗V⊗V⊗A⊗V)◦(A⊗ψA V⊗V⊗A⊗V) =pA⊗V◦(µA⊗V)◦ 5 (A⊗[(µA⊗V)◦(A⊗µA⊗V)◦(A⊗A⊗σA V)◦(A⊗σA V⊗V)◦(ψA V⊗V⊗V)◦(V⊗ψA V⊗V)])◦ (µA⊗V⊗V⊗A⊗V)◦(A⊗ψA V⊗V⊗A⊗V) =pA⊗V◦(µA⊗V)◦ (A⊗[(µA⊗V)◦(A⊗∇A⊗V)◦(A⊗µA⊗V)◦(A⊗A⊗σA V)◦(A⊗σA V⊗V)◦(ψA V⊗V⊗V)◦ (V⊗ψA V⊗V)])◦ (µA⊗V⊗V⊗A⊗V)◦(A⊗ψA V⊗V⊗A⊗V) =pA⊗V◦(µA⊗V)◦ (µA⊗[∇A⊗V◦(µA⊗V)◦(A⊗σA V)◦(ψA V⊗V)◦(V⊗σA V)])◦ (A⊗µA⊗V⊗V⊗V)◦(A⊗A⊗ψA V⊗V⊗V)◦(A⊗ψA V⊗ψA V⊗V) =pA⊗V◦(µA⊗V)◦(µA⊗σA V)◦(A⊗ψA V⊗V)◦(A⊗V⊗µA⊗V)◦(A⊗V⊗µA⊗σA V)◦ (A⊗V⊗A⊗ψA V⊗V) =µA×V◦(A×V⊗µA×V)◦(pA⊗V⊗pA⊗V⊗pA⊗V). In the last computations, the first equality follows by definition, the second one by (a2) and (b4), the third one by (a3) and (b2), the fourth one by the twisted condition, the fifth one by (a3) and (b2), the sixth one by (b2), the seventh one by the cocycle condition, the eighth one by (a1) and finally, the ninth one by (a2) and (b4). Therefore, µA×V◦(µA×V⊗A×V) = µA×V◦(A×V⊗µA×V).2 3 Weak cleft extensions and crossed product systems Weak entwining structures have been introduced by Caenepeel and De Groot [13] as a generalization of entwining structures defined by Brzezinski and Majid [9], [10]. They introduce the so-called entwining structures, consisting of an algebra A, a coalgebra C, and an entwining ψ:C⊗A→A⊗Csatisfying four technical conditions which have been replaced by weaker axioms in the definition of Caenepeel and De Groot. The definition in the monoidal setting is the following: Definition 3.1 A weak entwining structure on Cconsists of a triple (A, C, ψ), where Ais an algebra, Ca coalgebra, and ψ:C⊗A→A⊗Ca morphism satisfying the relations (c1) ψ◦(C⊗µA) = (µA⊗C)◦(A⊗ψ)◦(ψ⊗A), (c2) (A⊗δC)◦ψ= (ψ⊗C)◦(C⊗ψ)◦(δC⊗A), (c3) ψ◦(C⊗ηA) = (eRR ⊗C)◦δC, (c4) (A⊗εC)◦ψ=µA◦(eRR ⊗A), where eRR :C→Ais the morphism defined by eRR = (A⊗εC)◦ψ◦(C⊗ηA). The morphism ψis called entwining. 6 In the definition of entwining structure the morphism eRR =ηA⊗εCand, obviously, any entwining structure is a weak entwining structure. Moreover, a weak entwining structure is an entwining structure if and only if eRR =ηA⊗εC. Definition 3.2 Let (A, C, ψ) be a weak entwining structure in C. We denote by MC A(ψ) the category whose objects are triples (M, φM, ρM), where (M, φM) is a right A-module (i.e. φM◦ (φM⊗A) = φM◦(M⊗µA), idM=φM◦(M⊗ηA)), (M, ρM) is a right C-comodule (i. e. (ρM⊗C)◦ρM= (M⊗δC)◦ρM, (M⊗εC)◦ρM=idM), and ρM◦φM= (φM⊗C)◦(M⊗ψ)◦(ρM⊗A). The objects of MC A(ψ) will be called weak entwined modules and a morphism in MC A(ψ) is a morphism of A-modules and C-comodules. If (A, C, ψ) is an entwining structure then we find the category of entwined modules introduced by Brzezi´nski in [9]. 3.3 We have the following (see [4]). Let (A, C, ψ) be a weak entwining structure such that there exists a coaction ρAsatisfying that (A, µA, ρA) belongs to MC A(ψ). If for all (M, φM, ρM)∈ MC A(ψ), we denote by MCthe equalizer of ρMand ζM= (φM⊗C)◦(M⊗(ρA◦ηA)) and by iM Cthe injection of MCin M, then: i) The triple (AC, ηAC, µAC) is an algebra in C, where ηAC:K→ACand µAC:AC⊗AC→ ACare the factorizations of ηAand µA◦(iA C⊗iA C) respectively, through the equalizer iA C. ii) The pair (MC, φMC) is a right AC-module, where φMC:MC⊗AC→MCis the factorization of φM◦(iM C⊗iA C) through the equalizer iM C. Definition 3.4 Let (A, C, ψ) be a weak entwining structure and suppose that (A, ρA) is a right C-comodule. By RegW R(C, A) we denote the set of morphisms h∈HomC(C, A) such that there exists a morphism h−1∈HomC(C, A) (the left weak inverse of h) satisfying h−1∧h=eRR. Let Abe an algebra and Cbe a coalgebra in C. By Reg(C, A) we denote the set of morphisms h:C→Asuch that there exists a morphism h−1:C→A(the inverse of h) satisfying h−1∧h=h∧h−1=εC⊗ηA=ηA◦εC. Of course, if (A, C, ψ) is an entwining structure in C eRR =εC⊗ηAand Reg(C, A)⊂RegW R(C, A). Remark 3.5 Suppose that (A, C, ψ) is a weak entwining structure such that there exists a coaction ρAsatisfying that (A, µA, ρA) belongs to MC A(ψ). Then if h∈HomC(C, A) is a morphism of right C-comodules h∧eRR =h. Definition 3.6 Let (A, C, ψ) be a weak entwining structure and suppose that (A, µA, ρA)∈ MC A(ψ). We will say that AC,→Ais a weak C-cleft extension if there exists a morphism h∈RegW R(C, A) of right C-comodules, called weak cleaving morphism, such that ψ◦(C⊗h−1)◦δC=ζA◦(eRR ∧h−1) where ζA= (µA⊗C)◦(A⊗(ρA◦ηA)) is the morphism defined in 3.3. Observe that, if AC,→Ais a weak C-cleft extension with weak cleaving morphism h, the morphism g=eRR ∧h−1verifies g∧h=eRR,eRR ∧g=gand ψ◦(C⊗g)◦δC=ζA◦(eRR ∧g). Then, as a consequence, we can suppose without loss of generality that eRR ∧h−1=h−1. 7 The definition of weak C-cleft extension was introduced in [4] and is a generalization of the one used by Brzezi´nski [9] (see [15], [16], [17], [22] for the classical definitions) in the context of entwined modules but changing Reg(C, A) by RegW R(C, A) and adding a new condition. The explanation and the conceptual meaning of the last definition appear if we link it with Galois theory. An old result in this theory says that if B⊂Ais a finite Galois extension of fields with Galois group H, then A/B has a normal basis, i.e. there exists a∈Asuch that the set {x.a ;x∈H}is a basis for Aover B. The notion of normal basis for extensions, associated to Hopf algebras in categories of modules over a commutative ring, was introduced by Kreimer and Takeuchi in [19] and in [16] Doi and Takeuchi characterized the H-Galois extensions with normal basis in terms of H-cleft extensions. Recently, in the work of Brzezi´nski [9] we can find a more general formulation of these last results in the context of entwining structures. In [5], we formulate the definition of weak C-Galois extension with normal basis for a weak entwining structure living in a strict monoidal category with equalizers and coequalizers and we characterize this extensions using the notion of cleftness introduced in Definition 3.6. Of course, as a particular instances, we recover the results described in this paragraph. Remarks 3.7 i) Let AC,→Abe a weak C-cleft extension with weak cleaving morphism h. Then, the entwining ψis completely determined in the following form: ψ= (µA⊗C)◦(A⊗(ρA◦µA)) ◦(((h−1⊗h)◦δC)⊗A). ii) Let (A, C, ψ) be an entwined structure and suppose that (A, µA, ρA)∈ MC A(ψ). If h∈ Reg(C, A) is a morphism of right C-comodules we have that ψ◦(C⊗h−1)◦δC=ζA◦h−1=ζA◦(eRR ∧h−1). Then, as a consequence, a C-cleft extension for an entwining structure is a weak C-cleft extension. 3.8 Let AC,→Abe a weak C-cleft extension. The morphism qA C=µA◦(A⊗h−1)◦ρA:A→A factors through the equalizer iA C(see [4]). Therefore, there exists a morphism pA C:A→ACsuch that iA C◦pA C=qA C. On the other hand, the morphism ϕA:C⊗A→Adefined by ϕA=µA◦(µA⊗h−1)◦(h⊗ψ)◦(δC⊗A) factors through the equalizer iA C. Moreover, if ϕ0 Ais the factorization of ϕA, we have ϕ0 A= pA C◦µA◦(h⊗A) and the morphism ϕAC=ϕ0 A◦(C⊗iC A) : C⊗AC→ACverifies µAC◦(ϕ0 A⊗ ϕAC)◦(C⊗ψ⊗AC)◦(δC⊗iC A⊗AC) = ϕAC◦(C⊗µAC) [Proposition 1.15 of [4]]. Finally, (see [Proposition 1.17 of [4]]) the morphism σA:C⊗C→Adefined by σA= µA◦(µA⊗h−1)◦(h⊗ψ)◦(δC⊗h) factors through the equalizer iA C. If σACis the factorization of σA, then σAC=pA C◦µA◦(h⊗h). Lemma 3.9 Let AC,→Abe a weak C-cleft extension. The following identities hold (d1) µA◦(A⊗eRR)◦ρA=idA. 8 (d2) µA◦(qA C⊗h)◦ρA=idA. (d3) ρA◦µA= (µA⊗C)◦(qA C⊗(ρA◦µA◦(h⊗A))) ◦(ρA⊗A). (d4) µA◦(iA C⊗h) = µA◦(qA C⊗A)◦(µA⊗h)◦(iA C⊗(ρA◦h)). Proof. (d1) We have µA◦(A⊗eRR)◦ρA=µA◦(A⊗A⊗εC)◦(A⊗ψ)◦(ρA⊗ηA) = (A⊗εC)◦ρA◦µA◦(A⊗ηA) = idA. (d2) This equality follows from (d1). Indeed: µA◦(qA C⊗h)◦ρA=µA◦((µA◦(A⊗h−1)◦ρA)⊗h)◦ρA= µA◦(A⊗(h−1∧h)) ◦ρA=µA◦(A⊗eRR)◦ρA=idA. (d3) Using the condition of weak entwined module for Aand (d2) we have (µA⊗C)◦(qA C⊗(ρA◦µA◦(h⊗A))) ◦(ρA⊗A) = (µA⊗C)◦((µA◦(qA C⊗h)◦ρA)⊗ψ)◦(ρA⊗A) = (µA⊗C)◦(A⊗ψ)◦(ρA⊗A) =ρA◦µA. (d4) This equality is a consequence of the following computations: µA◦(qA C⊗A)◦(µA⊗h)◦(iA C⊗(ρA◦h)) =µA◦(µA⊗A)◦(µA⊗h−1⊗A)◦(A⊗ψ⊗A)◦((ρA◦iA C)⊗((A⊗h)◦ρA◦h)) =µA◦(µA⊗A)◦(µA⊗h−1⊗h)◦(µA⊗ψ⊗A)◦(iA C⊗(ρA◦ηA)⊗(ρA◦h)) =µA◦(µA⊗(µA◦(h−1⊗h))) ◦(A⊗(ρA◦µA)⊗C)◦(iA C⊗ηA⊗(ρA◦h)) =µA◦(iA C⊗(µA◦(A⊗eRR)◦ρA◦h)) =µA◦(iA C⊗h). In the previous series of equalities, the first an the third ones follow from the weak entwined module condition for A. In the second one we used the definition of iA Cand in the fourth one we applied the right C-comodule condition for h. Finally, the fifth one follows by (d1). 2 3.10 Let AC,→Abe a weak C-cleft extension with weak cleaving morphism h. The left AC-module and right C-comodule ( ϕAC⊗C=µAC⊗C,ρAC⊗C=AC⊗δC) morphisms ωA:AC⊗C→A, ω0 A:A→AC⊗C, defined by ωA=µA◦(iA C⊗h) and ω0 A= (pA C⊗C)◦ρAsatisfy the equality ωA◦ω0 A=idAbecause ωA◦ω0 A=µA◦(A⊗eRR)◦ρA=idA. As a consequence, the morphism ΩA=ω0 A◦ωAis an idempotent morphism and we have a commutative diagram 9 The first equality follows from the definition of the induced product, the second one by (a2)and (b4), the third one by the unity condition ∇A⊗V◦(A⊗ηV) = ψA V◦(ηV⊗A), the fourth one by (a3). Finally, in the last equality, we used the normality of the crossed product system. Therefore, we obtain µA×V◦(A×V⊗ηA×V) = idA×V=µA×V◦(ηA×V⊗A×V),and then A×Vis an algebra. i) =⇒ii) Assume that there exists an idempotent morphism ∇A⊗V:A⊗V→A⊗V, with image A×Vand factorization ∇A⊗V=iA⊗V◦pA⊗V, satisfying (µA⊗V)◦(A⊗ ∇A⊗V) = ∇A⊗V◦(µA⊗V), the object A×Vis an algebra with unit ηA×V=pA⊗V◦(ηA⊗ηV) and such that (f1) holds. Define morphisms ψA V:V⊗A→A⊗V, ψA V=iA⊗V◦µA×V◦((pA⊗V◦(ηA⊗V)) ⊗(pA⊗V◦(A⊗ηV))), σA V:V⊗V→A⊗V, σA V=iA⊗V◦µA×V◦((pA⊗V◦(ηA⊗V)) ⊗(pA⊗V◦(ηA⊗V))). Using the equality pA⊗V◦iA⊗V=idA⊗V, it is immediate to show that these morphisms satisfy ∇A⊗V◦ψA V=ψA V, and ∇A⊗V◦σA V=σA V.Also, the equality pA⊗V◦iA⊗V=idA⊗Vand (f1) implies that (f2) iA⊗V◦µA×V◦(pA⊗V⊗(pA⊗V◦(A⊗ηV))) = (µA⊗V)◦(A⊗ψA V), (f3) iA⊗V◦µA×V◦(pA⊗V⊗(pA⊗V◦(ηA⊗V))) = (µA⊗V)◦(A⊗σA V). Moreover, by (µA⊗V)◦(A⊗ ∇A⊗V) = ∇A⊗V◦(µA⊗V) we have ∇A⊗V= (µA⊗V)◦(A⊗ψA V)◦(A⊗V⊗ηA) and trivially ψA V◦(ηV⊗A) = ∇A⊗V◦(A⊗ηV), or equivalently, (A, V, ψA V, σA V) has unity. It remains to check that (A, V, ψA V, σA V) is a normal crossed product system which satisfy the twisted and cocycle conditions. To prove (a1) compute (µA⊗V)◦(A⊗ψA V)◦(ψA V⊗A) =iA⊗V◦µA×V◦(µA×V⊗A⊗V)◦((pA⊗V◦(ηA⊗V))⊗(pA⊗V◦(A⊗ηV))⊗(pA⊗V◦(A⊗ηV))) =iA⊗V◦µA×V◦(A⊗V⊗µA×V)◦((pA⊗V◦(ηA⊗V))⊗(pA⊗V◦(A⊗ηV))⊗(pA⊗V◦(A⊗ηV))) =iA⊗V◦µA×V◦((pA⊗V◦(ηA⊗V)) ⊗(pA⊗V◦(µA⊗ηV))) =ψA V◦(V⊗µA). The first equality follows from pA⊗V◦iA⊗V=idA⊗V, the second one by the associativity of µA×V, the third one by (f1) and finally the fourth one by definition. Using the same arguments and the equalities (µA⊗V)◦(A⊗ ∇A⊗V) = ∇A⊗V◦(µA⊗V), ∇A⊗V◦σA V=σA V, we obtain the proof for (a2) and (a3). Indeed, we have (µA⊗V)◦(A⊗σA V)◦(ψA V⊗V)◦(V⊗ ∇A⊗V) =iA⊗V◦µA×V◦((pA⊗V◦(ηA⊗V))⊗[µA×V◦((pA⊗V◦(A⊗ηV))⊗(pA⊗V◦(ηA⊗V)))◦∇A⊗V]) =iA⊗V◦µA×V◦((pA⊗V◦(ηA⊗V)) ⊗pA⊗V) 16 =iA⊗V◦µA×V◦((pA⊗V◦(ηA⊗V)) ⊗[µA×V◦((pA⊗V◦(A⊗ηV)) ⊗(pA⊗V◦(ηA⊗V)))]) =∇A⊗V◦(µA⊗V)◦(A⊗σA V)◦(ψA V⊗V), and ∇A⊗V◦(µA⊗V)◦(A⊗σA V)◦(∇A⊗V⊗V) = (µA⊗V)◦(A⊗σA V)◦(∇A⊗V⊗V) =iA⊗V◦µA×V◦(pA⊗V⊗(pA⊗V◦(ηA⊗V))) = (µA⊗V)◦(A⊗σA V) =∇A⊗V◦(µA⊗V)◦(A⊗σA V). On the other hand, the equalities µA×V◦(A×V⊗ηA×V) = idA×V=µA×V◦(ηA×V⊗A×V), imply pA⊗V◦σA V◦(ηV⊗V) = pA⊗V◦(ηA⊗V) = pA⊗V◦σA V◦(V⊗ηV) and then the crossed product system with unity (A, V, ψA V, σA V) is normal. To prove the twisted condition compute ∇A⊗V◦(µA⊗V)◦(A⊗σA V)◦(ψA V⊗V)◦(V⊗ψA V) = (µA⊗V)◦(A⊗σA V)◦(ψA V⊗V)◦(V⊗ψA V) =iA⊗V◦µA×V◦((pA⊗V◦(ηA⊗V)) ⊗[µA×V◦((pA⊗V◦(ηA⊗V)) ⊗(pA⊗V◦(A⊗ηV)))]) =iA⊗V◦µA×V◦([µA×V◦((pA⊗V◦(ηA⊗V)) ⊗(pA⊗V◦(ηA⊗V)))] ⊗(pA⊗V◦(A⊗ηV))) = (µA⊗V)◦(A⊗ψA V)◦(σA V⊗A) =∇A⊗V◦(µA⊗V)◦(A⊗ψA V)◦(σA V⊗A). The first equality follows by (µA⊗V)◦(A⊗∇A⊗V) = ∇A⊗V◦(µA⊗V) and ∇A⊗V◦σA V=σA V, the second one by pA⊗V◦iA⊗V=idA⊗V, (f3) and (f1). The third one follows by the associativity of µA×Vand the fourth one by (f2) and (f1). The last one follows by (µA⊗V)◦(A⊗ ∇A⊗V) = ∇A⊗V◦(µA⊗V) and ∇A⊗V◦ψA V=ψA V. Finally, one verifies the cocycle condition by the same arguments used in the proof of the twisted condition. Indeed: ∇A⊗V◦(µA⊗V)◦(A⊗σA V)◦(ψA V⊗V)◦(V⊗σA V) = (µA⊗V)◦(A⊗σA V)◦(ψA V⊗V)◦(V⊗σA V) =iA⊗V◦µA×V◦([µA×V◦((pA⊗V◦(ηA⊗V)) ⊗(pA⊗V◦(ηA⊗V)))] ⊗(pA⊗V◦(ηA⊗V))) =iA⊗V◦µA×V◦((pA⊗V◦(ηA⊗V)) ⊗[µA×V◦((pA⊗V◦(ηA⊗V)) ⊗(pA⊗V◦(ηA⊗V)))]) = (µA⊗V)◦(A⊗σA V)◦(σA V⊗V) =∇A⊗V◦(µA⊗V)◦(A⊗σA V)◦(σA V⊗V).2 17 Remarks 4.7 i) Note that if there exists an idempotent morphism ∇A⊗V:A⊗V→A⊗V, with image A×Vand factorization ∇A⊗V=iA⊗V◦pA⊗V, satisfying (µA⊗V)◦(A⊗ ∇A⊗V) = ∇A⊗V◦(µA⊗V) and the object A×Vis an algebra with unit ηA×V=pA⊗V◦(ηA⊗ηV) and such that (f1) holds, the product µA×Vis the one induced by the crossed product system (A, V, ψA V, σA V), where ψA Vand σA Vare the morphism defined in the proof of the last theorem. Using the usual arguments, one can verify this assertion computing: ∇A⊗V◦(µA⊗V)◦(A⊗µA⊗V)◦(A⊗A⊗σA V)◦(A⊗ψA V⊗V)◦(∇A⊗V⊗ ∇A⊗V) = (µA⊗V)◦(µA⊗σA V)◦(A⊗ψA V⊗V) =iA⊗V◦µA×V◦(pA⊗V⊗[µA×V◦((pA⊗V◦(A⊗ηV)) ⊗(pA⊗V◦(ηA⊗V)))]) =iA⊗V◦µA×V◦(pA⊗V⊗pA⊗V) Therefore, µA×V=pA⊗V◦(µA⊗V)◦(A⊗µA⊗V)◦(A⊗A⊗σA V)◦(A⊗ψA V⊗V)◦(iA⊗V⊗iA⊗V). ii) If Cis the category of vector spaces over a field Kand ∇A⊗V=idA⊗V, Theorem 4.6 is the result proved by Brzezi´nski in [8]. In this situation A×V=A⊗Vand Brzezi´nski’s Proposition describe conditions which allow to built an algebra structure on a tensor product of an algebra Aand a vector space V. iii) The referee has pointed out the paper of Wisbauer [27], where the author proves a result such that with a weak modification in the conditions has a strong similarity with Theorem 4.6. 5 Weak Hopf algebras and crossed product systems Weak Hopf algebras (or quantum groupoids in the terminology of Nikshych and Vainerman [23]) are generalizations of Hopf algebras that were defined by B¨ohm, Nill and Szlach´anyi in [11], [12]. The axioms are the same as the ones for a Hopf algebra, except that the coproduct of the unit, the product of the counit and the antipode condition are replaced by weaker properties. The main motivation for studying weak Hopf algebras comes from quantum field theory and operator algebras. Let Cbe a strict symmetric monoidal category with split idempotents. Bellow we collect the definition an basic properties of weak Hopf algebras. Definition 5.1 A weak Hopf algebra Hin Cis by definition an algebra (H, ηH, µH) and coalgebra (H, εH, δH) such that the following axioms hold: (g1) δH◦µH= (µH⊗µH)◦δH⊗H. (g2) εH◦µH◦(µH⊗H) = (εH⊗εH)◦(µH⊗µH)◦(H⊗δH⊗H) = (εH⊗εH)◦(µH⊗µH)◦(H⊗(cH,H ◦δH)⊗H). (g3) (δH⊗H)◦δH◦ηH= (H⊗µH⊗H)◦(δH⊗δH)◦(ηH⊗ηH) = (H⊗(µH◦cH,H )⊗H)◦(δH⊗δH)◦(ηH⊗ηH). 18 (g4) There exists a morphism λH:H→Hin C(called antipode of H) satisfying: (g4-1) µH◦(H⊗λH)◦δH= ((εH◦µH)⊗H)◦(H⊗cH,H )◦((δH◦ηH)⊗H). (g4-2) µH◦(λH⊗H)◦δH= (H⊗(εH◦µH)) ◦(cH,H ⊗H)◦(H⊗(δH◦ηH)). (g4-3) µH◦(µH⊗H)◦(λH⊗H⊗λH)◦(δH⊗H)◦δH=λH. Axioms (g2) and (g3) above are the weaker version to the usual bialgebra axioms of δH being a unit preserving map and εHbeing an algebra homomorphism. Axioms (g4-1), (g4-2) and (g4-3) generalize the properties of the antipode in a Hopf algebra with respect to the counit εH. Observe that in the definition of Hopf algebra, (g2-g4) are replaced by the conditions (g2’) εH◦µH=εH⊗εH, (g3’) δH◦ηH=ηH⊗ηH, (g4’) There exists a morphism λH:H→Hin Csatisfying: µH◦(H⊗λH)◦δH=µH◦(λH⊗H)◦δH=εH⊗ηH. Therefore, a Hopf algebra is always a weak Hopf algebra. Then, a weak Hopf algebra is a Hopf algebra if an only if the morphism δH(comultiplication) is unit-preserving and if and only if the counit is a homomorphism of algebras. If His a weak Hopf algebra, the antipode λHis unique, antimultiplicative, anticomultiplicative and leaves the unit ηHand the counit εHinvariant: λH◦µH=µH◦(λH⊗λH)◦cH,H , δH◦λH=cH,H ◦(λH⊗λH)◦δH, λH◦ηH=ηH, εH◦λH=εH. If we define the morphisms ΠL H, ΠR H, ΠL Hand ΠR Hby ΠL H= ((εH◦µH)⊗H)◦(H⊗cH,H )◦((δH◦ηH)⊗H) : H→H, ΠR H= (H⊗(εH◦µH)) ◦(cH,H ⊗H)◦(H⊗(δH◦ηH)) : H→H, ΠL H= (H⊗(εH◦µH)) ◦((δH◦ηH)⊗H) : H→H, ΠR H= ((εH◦µH)⊗H)◦(H⊗(δH◦ηH)) : H→H. it is straightforward to show (see [11]) that they are idempotent and ΠL H, ΠR Hsatisfy the equalities: ΠL H=µH◦(H⊗λH)◦δH,ΠR H=µH◦(λH⊗H)◦δH. Moreover, we have that (see [13]) ΠR H◦ΠL H= ΠL H,ΠL H◦ΠR H= ΠR H,ΠL H◦ΠR H= ΠR H,ΠR H◦ΠL H= ΠL H, ΠL H◦ΠL H= ΠL H,ΠL H◦ΠL H= ΠL H,ΠR H◦ΠR H= ΠR H,ΠR H◦ΠR H= ΠR H. 19 Also it is easy to show the formulas: ΠL H= ΠR H◦λH=λH◦ΠL H,ΠR H= ΠL H◦λH=λH◦ΠR H, ΠL H◦λH= ΠL H◦ΠR H=λH◦ΠR H,ΠR H◦λH= ΠR H◦ΠL H=λH◦ΠL H. Finally, if λHis bijective (for example, when His finite), we can find the equalities: ΠL H=µH◦(H⊗λ−1 H)◦cH,H ◦δH,ΠR H=µH◦(λ−1 H⊗H)◦cH,H ◦δH. A morphism between weak Hopf algebras Hand Bis a morphism f:H→Bwhich is both algebra and coalgebra morphism. If f:H→Bis a weak Hopf algebra morphism, then λB◦f=f◦λH[Proposition 1.4 [2]]. Definition 5.2 Let Hbe a weak Hopf algebra and Aan algebra. By a weak action of Hon A we mean a morphism ϕA:H⊗A→Ain Csuch that the following equalities hold: (h1) ϕA◦(ηH⊗A) = idA, (h2) µA◦((ϕA◦(H⊗ηA)) ⊗A) = ϕA◦(ΠL H⊗A), (h3) µA◦cA,A ◦((ϕA◦(H⊗ηA)) ⊗A) = ϕA◦(ΠL H⊗A), (h4) µA◦(ϕA⊗ϕA)◦(H⊗cH,A ⊗A)◦(δH⊗A⊗A) = ϕA◦(H⊗µA), Note that the equality (h2) implies (h5) ϕA◦(H⊗ηA) = ϕA◦(ΠL H⊗ηA), or, equivalently, (h50)ϕA◦(H⊗ηA) = ϕA◦(ΠL H⊗ηA). Note that, if His a Hopf algebra, replacing (h3-4) by (h30)ϕA◦(H⊗ηA) = εH⊗ηA. we obtain the classical definition of weak action (see [7]). Lemma 5.3 Let Hbe a weak Hopf algebra and Aan algebra. Given a weak action ϕA:H⊗A→ A, the morphism ψA H= (ϕA⊗H)◦(H⊗cH,A)◦(δH⊗A) : H⊗A→A⊗Hsatisfies (a1) and then the morphism ∇A⊗H= (µA⊗H)◦(A⊗ψA H)◦(A⊗H⊗ηA)is idempotent. Moreover, the unity condition holds, i.e., ∇A⊗H◦(A⊗ηH) = ψA H◦(ηH⊗A). Proof. By (h4) we have (µA⊗H)◦(A⊗ψA H)◦(ψA H⊗A) = ((µA◦(ϕA⊗ϕA)◦(H⊗cH,A ⊗A)◦(δH⊗A⊗A)) ⊗H)◦(H⊗A⊗cH,A)◦ (H⊗cH,A ⊗A)◦(δH⊗A⊗A) 20 = ((ϕA◦(H⊗µA)) ⊗H)◦(H⊗A⊗cH,A)◦(H⊗cH,A ⊗A)◦(δH⊗A⊗A) =ψA H◦(H⊗µA), and, as a consequence, ∇A⊗His idempotent. On the other hand, using (h3) we obtain the unity condition. Indeed: ψA H◦(ηH⊗A) = (ϕA⊗H)◦(H⊗cH,A)◦((δH◦ηH)⊗A) = ((ϕA◦(ΠL H⊗A)) ⊗H)◦(H⊗cH,A)◦((δH◦ηH)⊗A) = ((µA◦cA,A ◦((ϕA◦(H⊗ηA)) ⊗A)) ⊗H)◦(H⊗cH,A)◦((δH◦ηH)⊗A) = (µA⊗H)◦(A⊗ϕA⊗H)◦(A⊗H⊗cH,A)◦(A⊗(δH◦ηH)⊗ηA) =∇A⊗H◦(A⊗ηH).2 5.4 Let Hbe a weak Hopf algebra and Abe an algebra. Assume that there are a weak action ϕA:H⊗A→Aand a morphism σA:H⊗H→Asuch that for ψA Hand σA H= (σA⊗µH)◦δH⊗H:H⊗H→A⊗H, (a2) and (a3) hold. Then, by 5.3, we obtain that (A, H, ψA H, σA H) is a crossed product system with unity. Remark 5.5 Let Hbe a weak Hopf algebra and Aan algebra. If ϕA:H⊗A→Ais a weak action of Hon A, considering (h3), we have (ϕA⊗µH)◦(A⊗cH,A ⊗H)◦((δH◦ηH)⊗A⊗H) = ((ϕA◦cA,H ◦cH,A)⊗µH)◦(A⊗cH,A ⊗H)◦((δH◦ηH)⊗A⊗H) = ((ϕA◦cA,H)⊗µH)◦(A⊗(δH◦ηH)⊗H) = ((ϕA◦cA,H)⊗H)◦(A⊗((ΠL H⊗H)◦δH)) = (µA◦(A⊗(ϕA◦(H⊗ηA)))) ⊗H)◦(A⊗δH) =∇A⊗H. Therefore, if (A, ϕA) is also a left H-module (i.e., (A, ϕA) satisfies ϕA◦(ηH⊗A) = idAand ϕA◦(H⊗ϕA) = ϕA◦(µH⊗A), the image of ∇A⊗H, denoted by A×H, is the tensor product of A and Hin the representation category of H(the category of left H-modules), denoted by Rep(H). In [23] and [24] it is possible to find a detailed construction of a non-strict monoidal structure in Rep(H) for a weak Hopf algebra living in a category of vector spaces. This construction can be extended without any difficulty to the general categorical case of this paper, i.e., for a weak Hopf algebra in a strict symmetric monoidal category. In the following lines we give a brief resume of the monoidal structure of Rep(H). 21 For two left H-modules (M, ϕM), (N, ϕN) the tensor product is defined as object as the image M×Nof ∇M⊗N=ϕM⊗N◦(ηH⊗M⊗N) : M⊗N→M⊗Nwhere ϕM⊗N:H⊗M⊗N→M⊗N is defined by ϕM⊗N= (ϕM⊗ϕN)◦(H⊗cH,M ⊗N)◦(δH⊗M⊗N).As a consequence, M×N is a left H-module with the following action: ϕM×N=pM⊗N◦ϕM⊗N◦(H⊗iM⊗N) where pM⊗Nand iM⊗Nare the morphisms such that pM⊗N◦iM⊗N=∇M⊗Nand iM⊗N◦pM⊗N= idM×N The base object is HL=Im(ΠL H) or, equivalently, the equalizer of δHand ζ1 H= (H⊗ΠL H)◦δH or the equalizer of δHand ζ2 H= (H⊗ΠR H)◦δH. The structure of left H-module for HLis the one derived of the following morphism ϕHL=pL◦µH◦(H⊗iL), where pL:H→HLand iL:HL→Hare the morphism such that ΠL H=iL◦pLand pL◦iL=idHL. The unit constrains are: lM=ϕM◦(iL⊗M)◦iHL⊗M:HL×M→M, rM=ϕM◦cM,H ◦(M⊗(ΠL H◦iL)) ◦iM⊗HL:M×HL→M. These morphisms are isomorphisms with inverses: l−1 M=pHL⊗M◦(pL⊗ϕM)◦((δH◦ηH)⊗M) : M→HL×M, r−1 M=pM⊗HL◦(ϕM⊗pL)◦(H⊗cH,M )◦((δH◦ηH)⊗M) : M→M×HL. If M,N,Pare objects in the category Rep(H), the associativity constrains are defined by aM,N,P =p(M×N)⊗P◦(pM⊗N⊗P)◦(M⊗iN⊗P)◦iM⊗(N×P):M×(N×P)→(M×N)×P where the inverse is the morphism a−1 M,N,P =pM⊗(N×P)◦(M⊗pN⊗P)◦(iM⊗N⊗P)◦i(M×N)⊗P: (M×N)×P→M×(N×P). If γ:M→M0and φ:N→N0are morphisms in the category, then γ×φ=pM0×N0◦(γ⊗φ)◦iM⊗N:M×N→M0×N0 is a morphism in Rep(H) and (γ0×φ0)◦(γ×φ) = (γ0◦γ)×(φ0◦φ), where γ0:M0→M00 and φ0:N0→N00 are morphisms in Rep(H). Theorem 5.6 Let Hbe a weak Hopf algebra and Aan algebra. Assume that there are a weak action ϕA:H⊗A→Aand a morphism σA:H⊗H→Asuch that (A, H, ψA H= (ϕA⊗H)◦ (H⊗cH,A)◦(δH⊗A), σA H= (σA⊗µH)◦δH⊗H)is a crossed product system with unity. Then, A×Hwith the induced product µA×Hand unit ηA×H=pA⊗H◦(ηA⊗ηH), is an algebra if and only if (A, H, ψA H, σA H)is normal and satisfies the twisted and cocycle conditions. 22 Proof. Suppose that A×His an algebra with unity ηA×H=pA⊗H◦(ηA⊗ηH) and product µA×H. Under these conditions we know that µA×H◦(A×H⊗ηA×H) = idA×Hand then pA⊗H =µA×H◦(pA×H⊗(pA×H◦(ηA⊗ηH))) =pA×H◦(µA⊗H)◦(A⊗((µA⊗H)◦(A⊗σA H)◦(ψA H⊗H)◦(H⊗∇A⊗H)))◦(∇A⊗H⊗ηA⊗ηH) =pA×H◦(µA⊗H)◦(µA⊗σA H)◦(A⊗ψA H⊗H)◦(∇A⊗H⊗ηA⊗ηH) =pA×H◦(µA⊗H)◦(A⊗σA H)◦(∇A⊗H⊗ηH) =pA×H◦(µA⊗H)◦(A⊗σA H)◦(A⊗H⊗ηH). In the last computations, the second equality is simply the definition of µA×H, while the third one follows by (a2) and (b2). In the fourth one we used the idempotent character of ∇A⊗H and the fifth one follows by (a3) and (b2). Also, using the equality µA×H◦(ηA×H⊗A×H) = idA×Hwe obtain pA⊗H =µA×H◦((pA×H◦(ηA⊗ηH)) ⊗pA×H) =pA×H◦(µA⊗H)◦(A⊗((µA⊗H)◦(A⊗σA H)◦(ψA H⊗H)))◦((∇A⊗H◦(ηA⊗ηH))⊗∇A⊗H) =pA×H◦(µA⊗H)◦(µA⊗σA H)◦(A⊗ψA H⊗H)◦((ψA H◦(ηH⊗ηA)) ⊗A⊗H) =pA×H◦(µA⊗H)◦(A⊗σA H)◦((ψA H◦(ηH⊗A)) ⊗H) =pA×H◦(µA⊗H)◦(A⊗σA H)◦((∇A⊗H◦(A⊗ηH)) ⊗H) =pA⊗H◦(µA⊗H)◦(A⊗σA H)◦(A⊗ηH⊗H). As in the first computations of this proof, the second equality is simply the definition of µA×H. The third equality follows from the idempotent character of ∇A⊗Hand by (a2), while the fourth one follows by (a1). In the fifth one we used the unity condition and finally, the sixth one follows by (a3). Therefore, we have pA⊗H◦(µA⊗H)◦(A⊗σA H)◦(A⊗ηH⊗H) = pA⊗H=pA⊗H◦(µA⊗H)◦(A⊗σA H)◦(A⊗H⊗ηH) and, by 4.3, we obtain that (A, H, ψA H, σA H) is normal. By 4.6, to derive the twisted and cocycle conditions we only need to show (f1). Indeed, using the normality and the properties of (A, H, ψA H, σA H) we have iA⊗H◦µA×H◦((pA×H◦(A⊗ηH)) ⊗pA×H) =∇A⊗H◦(µA⊗H)◦(µA⊗σA H)◦(A⊗ψA H⊗H)◦((ψA H◦(ηH⊗A)) ⊗ ∇A⊗H) =∇A⊗H◦(µA⊗H)◦(A⊗σA H)◦(ψA H⊗H)◦(ηH⊗(∇A⊗H◦(µA⊗H))) 23 =∇A⊗H◦(µA⊗H)◦(A⊗(σA H◦(ηH⊗H))) ◦ ∇A⊗H◦(µA⊗H) =∇A⊗H◦(µA⊗H). Conversely, by Theorem 4.6, if (A, H, ψA H, σA H) is normal and satisfies the twisted and cocycle conditions we have that A×H, with the induced product µA×Hand unit ηA×H=pA⊗H◦(ηA⊗ ηH), is an algebra. 2 Remark 5.7 Let Hbe a weak Hopf algebra and Aan algebra. Assume that there are a weak action ϕA:H⊗A→Aand a morphism σA:H⊗H→Asuch that (A, H, ψA H= (ϕA⊗H)◦(H⊗cH,A)◦(δH⊗A), σA H= (σA⊗µH)◦δH⊗H) is a crossed product system with unity. Note that pA⊗H◦σA H◦(H⊗ηH) =pA⊗H◦(σA⊗µH)◦δH⊗H◦(H⊗ηH) =pA⊗H◦(σA⊗H)◦(H⊗((ΠR H⊗H)◦δH)) ◦δH =pA⊗H◦((σA◦(H⊗ΠR H)◦δH)⊗H)◦δH, and pA⊗H◦σA H◦(ηH⊗H) =pA⊗H◦(σA⊗µH)◦δH⊗H◦(ηH⊗H) =pA⊗H◦((σA◦cH,H )⊗H)◦(H⊗((ΠL H⊗H)◦δH)) ◦δH =pA⊗H◦((σA◦cH,H ◦(H⊗ΠL H)◦δH)⊗H)◦δH. Then, (A, H, ψA H, σA H) is normal if and only if pA⊗H◦((σA◦(H⊗ΠR H)◦δH)⊗H)◦δH=pA⊗H◦(ηA⊗H) = pA⊗H◦((σA◦cH,H ◦(H⊗ΠL H)◦δH)⊗H)◦δH. On the other hand, the twisted condition is equivalent to (i1) pA⊗H◦ ([µA◦(ϕA⊗A)◦(A⊗ϕA⊗A)◦(H⊗H⊗cA,A)◦(H⊗H⊗σA⊗A)◦(δH⊗H⊗A)]⊗cH,A)◦ (H⊗H⊗µH⊗A)◦(δH⊗H⊗A) =pA⊗H◦([µA◦(A⊗ϕA)◦(σA H⊗A)] ⊗cH,A)◦(H⊗H⊗µH⊗A)◦(δH⊗H⊗A), and the cocycle condition can be viewed in the following form: (i2) pA⊗H◦(∂4(σA)∧∂2(σA)⊗µH)◦(H⊗H⊗H⊗H⊗µH)◦δH⊗H⊗H =pA⊗H◦(∂1(σA)∧∂3(σA)⊗µH)◦(H⊗H⊗H⊗H⊗µH)◦δH⊗H⊗H, 24 where ∂1=ϕA◦(H⊗σA), ∂2=σA◦(µH⊗H), ∂3=σA◦(H⊗µH), ∂4=σA⊗εH, and ∧denotes the usual convolution in HomC(H⊗H⊗H, A). When His a Hopf algebra, the normal condition for (A, H, ψA H, σA H) is equivalent to σA◦(ηH⊗H) = σA◦(H⊗ηH) = ηA⊗εH because ΠR H= ΠL H=ηH⊗εHand ∇A⊗H=idA⊗H. Then, we have that (A, H, ψA H, σA H) is normal if and only if σAis normal in the classical sense (see [7]). Also, composing with A⊗εHin (i1) and (i2) we obtain that (A, H, ψA H, σA H) satisfy the twisted condition if and only if µA◦(ϕA⊗A)◦(H⊗ϕA⊗A)◦(H⊗H⊗cA,A)◦(H⊗H⊗σA⊗A)◦(δH⊗H⊗A) = µA◦(A⊗ϕA)◦(σA H⊗A), and satisfies the cocycle condition if and only if ∂4(σA)∧∂2(σA) = ∂1(σA)∧∂3(σA).Therefore, in the Hopf algebra case, (A, H, ψA H, σA H) satisfy the twisted and the cocycle conditions if and only if σAis a twisted cocycle (see also [7] for the definition). As a consequence, Theorem 5.6 is a generalization of the results obtained by Blattner, Cohen and Montgomery [7] and Doi and Takeuchi [16] in the study of crossed products in a category of vector spaces. Also, if Cis a braided category whose underlying monoidal category is of vector spaces, using a similar computations, we obtain the conditions described by Majid [21] which allow to built an algebra structure in the tensor product of an algebra Aand a Hopf algebra H. In these cases the algebra A×Hwas denoted by A]σAH(the crossed product of Aand H). Example 5.8 Let H,Bbe weak Hopf algebras in a strict symmetric monoidal category Cwith split idempotents. Let g:B→Hbe a morphism of weak Hopf algebras and f:H→Bbe a morphism of coalgebras such that g◦f=idHand f◦ηH=ηB. If we define ρB:B→B⊗H and the entwining ψ:H⊗B→B⊗Hby ρB= (B⊗g)◦δB, ψ = (B⊗µH)◦(cH,B ⊗H)◦(H⊗ρB) we have that (B, H, ψ) is a weak entwining structure where eRR = ΠR B◦f. The morphism qB H=µB◦(B⊗(λB◦f◦g)) ◦δB:B→Bis an idempotent in C[Proposition 2.1, [3]]. As a consequence, there exist an epimorphism pB H, a monomorphism iB Hand an object BHsuch that the diagram - HH Hj ½½> BB BH qB H pB HiB H commutes and pB H◦iB H=idBH. Also, -- - BHBB⊗H iB H ρB (B⊗ΠR H)◦ρB 25