Weak Hopf algebras with projection and weak smash bialgebra structures

Álvarez, J. N. Alonso; Rodríguez, Ramón López; Vilaboa, J. M. Fernández; López, María Purificación López; Novoa, E. Villanueva

doi:10.1016/j.jalgebra.2003.05.003

Cited by 25 publications

(46 citation statements)

References 10 publications

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“…See Proposition 2.1 in [7]. Note that, as a consequence of the previous proposition, there are an object B D , an epimorphism…”

Section: Proposition 211mentioning

confidence: 71%

“…Now we give a slightly generalizations of some results of [7]. Having into account that the idea of the proofs is analogous, just changing the contexts where it is applied, they will be omitted.…”

Section: Definition 21mentioning

confidence: 99%

“…In the previous papers [7] and [1], the authors and their collaborators extended Radford's theory to weak Hopf algebra projections in a strict symmetric monoidal category C where every idempotent morphism splits. If H is a weak Hopf algebra in C and (B, f, g) is a weak Hopf algebra projection, i.e., B is a weak Hopf algebra, f : H → B and g : B → H are morphisms of weak Hopf algebras such that g • f = id H , we constructed an algebra-coalgebra object B H in C such that the weak Hopf algebra B is isomorphic to a weak crossed product B H × H where, in this case, the underlying object of B H × H is the image of a convenient idempotent endomorphism of B H ⊗ H that becomes an identity in the Hopf algebra setting.…”

Section: Introductionmentioning

confidence: 99%

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Projections of weak braided Hopf algebras

et al. 2011

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In this paper we study the projections of weak braided Hopf algebras using the notion of YetterDrinfeld module associated with a weak braided Hopf algebra. As a consequence, we complete the study of the structure of weak Hopf algebras with a projection in a braiding setting obtaining a categorical equivalence between the category of weak Hopf algebra projections associated with a weak Hopf algebra H living in a braided monoidal category and the category of Hopf algebras in the non-strict braided monoidal category of left-left Yetter-Drinfeld modules over H.

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“…See Proposition 2.1 in [7]. Note that, as a consequence of the previous proposition, there are an object B D , an epimorphism…”

Section: Proposition 211mentioning

confidence: 71%

Section: Definition 21mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Projections of weak braided Hopf algebras

et al. 2011

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“…Concerning the converse, we would like to warmly thank Gabriella Böhm for not only suggesting that it may be true, and pointing out that the ᐂ = Vect case appears in the PhD thesis of Imre Bálint [2008a], but for also helping us with the proof.…”

Section: Weak Hopf Monoids and Quantum Groupoidsmentioning

confidence: 99%

“…The dual of a bialgebroid is called a "bicoalgebroid" by Brzeziński and Militaru [2002] and further studied by Bálint [2008b]. In the terminology of [Day and Street 2004], these structures are quantum categories in the monoidal category of vector spaces.…”

Section: Theorymentioning

confidence: 99%

Weak Hopf monoids in braided monoidal categories

Pastro

Street

2009

ANT

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We develop the theory of weak bimonoids in braided monoidal categories and show them to be quantum categories in a certain sense. Weak Hopf monoids are shown to be quantum groupoids. Each separable Frobenius monoid R leads to a weak Hopf monoid R ⊗ R. Contents 1. Introduction 1 2. Weak bimonoids 5 3. Weak Hopf monoids 10 4. The monoidal category of A-comodules 13 5. Frobenius monoid example 23 6. Quantum groupoids 26 7. Weak Hopf monoids are quantum groupoids 30 Appendix A. String diagrams and basic definitions 36 Appendix B. Proofs of the properties of s, t, and r 42 References 45 = 10

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A Characterization of Weak Hopf (co) Quasigroups

2016

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In this paper we show that weak Hopf (co)quasigroups can be characterized by a Galois-type condition. Taking into account that this notion generalizes the ones of Hopf (co)quasigroup and weak Hopf algebra, we obtain as a consequence the first fundamental theorem for Hopf (co)quasigroups and a characterization of weak Hopf algebras in terms of bijectivity of a Galois-type morphism (also called fusion morphism).Keywords. Hopf algebra, weak Hopf algebra, Hopf (co)quasigroup, weak Hopf (co)quasigroup, Galois extension.MSC 2010: 18D10, 16T05, 17A30, 20N05.algebraic structures remain valid under this unified approach (in particular, the Fundamental Theorem of Hopf Modules associated to a weak Hopf quasigroup [4]), and it is very natural to ask for other wellknown properties related with Hopf algebras. In particular, Nakajima [10] gave a characterization of ordinary Hopf algebras in terms of bijectivity of right or left Galois maps (also called fusion morphisms in [14]). This result was extended by Schauenburg [13] to weak Hopf algebras, and by Brzeziński [7] to Hopf (co)quasigroups. The main purpose of this work is to give a similar characterization in the weak Hopf (co)quasigroup setting. More precisely, we state that a weak Hopf (co)quasigroup satisfies a right and left Galois-type condition, and these Galois morphisms must have almost right and left (co)linear inverses, and conversely. As a consequence we get the characterization of Hopf (co)quasigroups given by Brzeziński [7] (called the First Fundamental Theorem for Hopf (co)quasigroups), and the one obtained by Schauenburg [13] for weak Hopf algebras. A characterization of weak Hopf quasigroupsThroughout this paper C denotes a strict monoidal category with tensor product ⊗ and unit object K. For each object M in C, we denote the identity morphism by id M : M → M and, for simplicity of notation, given objects M , N , P in C and a morphism f :From now on we also assume that C admits equalizers and coequalizers. Then every idempotent morphism splits, i.e., for every morphismAlso we assume that C is braided, that is: for all M and N objects in C, there is a natural isomorphism c M,N : M ⊗ N → N ⊗ M , called the braiding, satisfying the Hexagon Axiom (see [8] for generalities). If the braiding satisfies c N,M • c M,N = id M⊗N , the category C will be called symmetric.By a unital magma in C we understand a triple A =, the unital magma will be called a monoid in C. Given two unital magmas (monoids) A = (A, η A , µ A ) and B = (B, η B , µ By duality, a counital comagma in C is a triple D = (D, ε D , δ D ) where D is an object in C and ε D :the counital comagma will be called a comonoid. If D = (D, ε D , δ D ) and E = (E, ε E , δ E ) are counital comagmas (comonoids),Moreover, if D is a comagma and A a magma, given two morphisms f, g : D → A we will denote by f * g its convolution product in C, that is f *We shall denote by C A the category of right A-modules. In an analogous way we can define the category of left A-modules and we denote it by A C.Let D be a comonoid. The...

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Weak Hopf algebras with projection and weak smash bialgebra structures

Cited by 25 publications

References 10 publications

Projections of weak braided Hopf algebras

Projections of weak braided Hopf algebras

Weak Hopf monoids in braided monoidal categories

A Characterization of Weak Hopf (co) Quasigroups

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