Cleft comodules over Hopf quasigroups

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

doi:10.1142/s0219199715500078

Cited by 7 publications

(12 citation statements)

References 9 publications

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“…As a Corollary of Theorem 5.1, for Hopf quasigroups we have a result which shows the close connection between the notion of cleft right H-comodule algebra (H-cleft extension for Hopf quasigroups), introduced in [5], and the one of H-Galois extension with normal basis introduced in this paper. Also, when A coH = K we have the equivalence proved in [6] because, in this case, i A = η A .…”

Section: Therefore γ −1supporting

confidence: 62%

“…In this section we introduce the notion of weak H-cleft extension associated to a weak Hopf quasigroup H. As a particular instances we recover the theory of cleft extensions associated to a weak Hopf algebra [1,2] and to a Hopf quasigroup [5,6].…”

Section: Cleft Extensions Associated To a Weak Hopf Quasigroupmentioning

confidence: 99%

“…, we recover the notion of cleft comodule algebra (or H-cleft extension for Hopf quasigroups) introduced in [5].…”

Section: Cleft Extensions Associated To a Weak Hopf Quasigroupmentioning

confidence: 99%

“…The first result linking Hopf Galois extensions with normal basis and cleft extensions in the Hopf quasigroup setting can be found in [6]. More specifically, in [5] we introduce the notion of cleft extension (cleft right H-comodule algebra) for a Hopf quasigroup H in a strict monoidal category C with tensor product ⊗ and unit object K. The notion of Galois extension with normal basis for H was introduced in [6], and we proved that, when the object of coinvariants is the unit object of the category, cleft extensions and Galois extension with normal basis and with the inverse of the canonical morphism almost lineal, are the same. Therefore, in [6], we extend the result proved by Doi and Takeuchi in [14] to the Hopf quasigroup setting, characterizing Galois extensions with normal basis in terms of cleft extensions when the object of coinvariants is K. The aim of this new paper is to show that all these results, that is, the one obtained for Hopf algebras in [14], the one obtained for weak Hopf algebras in [2], and the one proved for Hopf quasigroups in [6], are particular instances of a more general result that we can prove for weak Hopf quasigroups.…”

Section: Introductionmentioning

confidence: 99%

“…Section 3 is devoted to the study of weak H-cleft extensions for weak Hopf quasigroups. In particular we show that these kind of extensions contain as examples the notion of weak H-cleft extension associated to a weak Hopf algebra [1], as well as the notion of cleft right H-comodule algebra introduced in [5] for Hopf quasigroups. In the last section, we can find the main result of this paper, which assures that for any right H-comodule magma (A, ρ A ) such that A ⊗ − preserves coequalizers, under suitable conditions (see Theorem 5.1), the following assertions are equivalent:…”

Section: Introductionmentioning

confidence: 99%

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Cleft and Galois extensions associated with a weak Hopf quasigroup

Álvarez

Vilaboa

Rodríguez

2016

Journal of Pure and Applied Algebra

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Section: Therefore γ −1supporting

confidence: 62%

Section: Cleft Extensions Associated To a Weak Hopf Quasigroupmentioning

confidence: 99%

“…, we recover the notion of cleft comodule algebra (or H-cleft extension for Hopf quasigroups) introduced in [5].…”

Section: Cleft Extensions Associated To a Weak Hopf Quasigroupmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Cleft and Galois extensions associated with a weak Hopf quasigroup

Álvarez

Vilaboa

Rodríguez

2016

Journal of Pure and Applied Algebra

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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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