Weak Hopf algebras and weak Yang–Baxter operators

Álvarez, J. N. Alonso; Vilaboa, J. M. Fernández; Rodríguez, Ramón López

doi:10.1016/j.jalgebra.2008.02.026

Cited by 23 publications

(55 citation statements)

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“…Also, just using the definition (see Definition 1.2 and Proposition 1.3 of [5]), we get that that t D,D is a weak Yang-Baxter operator with associated idempotent ∇ D⊗D if and only if t D,D is a weak Yang-Baxter operator with associated idempotent ∇ D⊗D , and…”

Section: Definition 21mentioning

confidence: 99%

“…Weak Yang-Baxter operators are generalizations of Yang-Baxter operators (see [14]) and were introduced by the authors in [5]. The definition is the following:…”

Section: Definition 21mentioning

confidence: 99%

“…In [3] it is shown that B H is a Hopf algebra in the braided monoidal category of left-left Yetter-Drinfeld modules defined for the weak Hopf algebra H by Böhm in [9]. The study of the situation described in the first paragraph of this introduction was completed in [5] obtaining the connection between Hopf algebras in H H YD and weak Hopf algebra projections for H in the following way: there exists a categorical equivalence between the category of projections associated to H and the category of Hopf algebras in H H YD. Obviously, this completes our previous works and puts the theory of Radford and Majid in a proper categorical prospective.…”

Section: Introductionmentioning

confidence: 99%

“…The notion of weak braided Hopf algebra introduced in [5] and studied in detail in [4] appears in a natural way associated with the problem of projections of weak Hopf algebras and contains as particular instances Hopf algebras in a symmetric or in a braiding setting, weak Hopf algebras and Takeuchi's braided Hopf algebras. The definition of weak braided Hopf algebra was given in any strict monoidal category using the notion of weak Yang-Baxter operator introduced in [5] and many results and identities for weak Hopf algebras can be verified in this context.…”

Section: Introductionmentioning

confidence: 99%

“…The definition of weak braided Hopf algebra was given in any strict monoidal category using the notion of weak Yang-Baxter operator introduced in [5] and many results and identities for weak Hopf algebras can be verified in this context. For example, in [4], the authors obtain a version of the fundamental theorem for Hopf modules and show that weak Hopf algebras provide examples of weak entwining structures.…”

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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Section: Definition 21mentioning

confidence: 99%

“…Weak Yang-Baxter operators are generalizations of Yang-Baxter operators (see [14]) and were introduced by the authors in [5]. The definition is the following:…”

Section: Definition 21mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Projections of weak braided Hopf algebras

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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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