Cleft and Galois extensions associated with a weak Hopf quasigroup

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

doi:10.1016/j.jpaa.2015.08.005

Cited by 11 publications

(14 citation statements)

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“…Also, we introduce the definition of H-Galois extension with normal basis, and we proved that, under the suitable conditions, H-cleft extensions are the same that H-Galois extensions with normal basis and such that the inverse of the canonical morphism is almost lineal. Therefore, in [6], we extend the result proved by Doi and Takeuchi in [18] to the weak Hopf quasigroup setting and, as a consequence, for Hopf quasigroups. Of course, if H is a weak Hopf algebra we recover the result proved in [1] for weak Hopf algebras because, in an associative context, the conditions assumed in the main theorem of [6] hold trivially.…”

Section: Introductionsupporting

confidence: 59%

Section: Introductionmentioning

confidence: 99%

“…The first result linking Hopf Galois extensions with normal basis and cleft extensions in a non-associative setting can be found in [6]. More specifically, in [6] we introduce the notion of weak H-cleft extension, for a weak Hopf quasigroup H in a strict monoidal category C with tensor product ⊗, which generalizes the one introduced for Hopf quasigroups in [4] with the name of cleft H-comodule algebra. Also, we introduce the definition of H-Galois extension with normal basis, and we proved that, under the suitable conditions, H-cleft extensions are the same that H-Galois extensions with normal basis and such that the inverse of the canonical morphism is almost lineal.…”

Section: Introductionmentioning

confidence: 99%

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Weak quasi-entwining structures

Alonso¹,

Fernádez²,

Rodríguez³

2018

Filomat

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In this paper we introduce the notion of weak quasi-entwining structure as a generalization of quasi-entwining structures and weak entwining structures. Also, we formulate the notions of weak cleft extension, weak Galois extension, and weak Galois extension with normal basis associated to a weak quasientwining structure. Moreover, we prove that, under some suitable conditions, there exists an equivalence between weak Galois extensions with normal basis and weak cleft extensions. As particular instances, we recover some results previously proved for Hopf quasigroups, weak Hopf quasigroups and weak Hopf algebras.

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Section: Introductionsupporting

confidence: 59%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Weak quasi-entwining structures

Alonso¹,

Fernádez²,

Rodríguez³

2018

Filomat

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“…In this case, if we define M coH in the same way as in the weak Hopf algebra setting, we obtain a version of the Fundamental Theorem of Hopf Modules in the following way: all Hopf module M is isomorphic to M coH × H as Hopf modules, where M coH × H is the image of the same idempotent ∇ M used for Hopf modules associated to a weak Hopf algebra. Moreover, in [3] we proved that H L , the image of the target morphism, is a monoid and then it is possible to take into consideration the category C HL , to construct the tensor product M coH ⊗ HL H, and, if the functor − ⊗ H preserves coequalizers, to endow this object with a Hopf module structure. Unfortunately, it is not possible to assure that M coH ⊗ HL H is isomorphic to M coH × H as in the weak Hopf algebra case.…”

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confidence: 98%

Strong Hopf modules for weak Hopf quasigroups

2017

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In this paper we introduce the category of strong Hopf modules for a weak Hopf quasigroup H in a braided monoidal category. We also prove that this category is equivalent to the category of right modules over the image of the target morphism of H.Let F be a field and C = F − V ect. Let M be a right H-module and a right H-comodule. If, for all m ∈ M and h ∈ H, we write m.h for the action and we use the Sweedler notation ρ M (m) = m [0] ⊗ m [1] for the coaction, we will say that M is a Hopf module if the equalitythe coproduct of H and m [1] h (2) the product in H of m [1] and h (2) . A morphism between two Hopf modules is a F-linear map that is H-linear and H-colinear. Hopf modules and morphisms of Hopf modules constitute the category of Hopf modules denoted by M H H . In 1969 Larsson and Sweedler proved a result, called Fundamental Theorem of Hopf Modules, that asserts the following: If M ∈ M H H and M coH = {m ∈ M | ρ M (m) = m ⊗ 1 H } are the coinvariants of H in M , M is isomorphic to M coH ⊗ H as Hopf modules (see [11] and [16]). On the other hand, if N is a F-vector space, the tensor product N ⊗ H, with the action and coaction induced by the product and the coproduct of H, is a Hopf module. This construction is functorial, so we have a functor F = − ⊗ H : C → M H H . Also, for all M ∈ M H H , the construction of M coH is functorial and we have a functor G = ( ) coH : M H H → C and F ⊣ G. Moreover, F and G is a pair of inverse equivalences, and therefore, M H H is equivalent to the category of F-vector spaces. The Fundamental Theorem of Hopf Modules also holds for weak Hopf algebras as was proved by Böhm, Nill and Szlachányi in [5]. In this case, if H is a weak Hopf algebra, the category of Hopf modules is defined in the same way as in the Hopf algebra setting. For M ∈ M H H , the coinvariants of H in M are defined by M coH = {m ∈ M | ρ M (m) = m [0] ⊗ Π L H (m [1] )}, where Π L H is the target morphism associated to H. Then, Böhm, Nill and Szlachányi proved that M is isomorphic to M coH ⊗ HL H as Hopf modules, where H L is the image of Π L H . Moreover, if C HL is the category of right H L -modules, there exist two functors F = − ⊗ HL H : C HL → M H H and G = ( ) coH : M H H → C HL such that F is left adjoint of G and they induce a pair of inverse equivalences (see [7]). Therefore, in the weak setting, M H H is equivalent to C HL . In this case, is a relevant fact the following property: the tensor product M coH ⊗ HL H is isomorphic as Hopf modules to M coH × H where M coH × H is the image of a suitable idempotent ∇ M : M coH ⊗ H → M coH ⊗ H. Note that, as a consequence, in the weak framework the Fundamental Theorem of Hopf Modules can be written using M coH × H instead of M coH ⊗ HL H.

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“…Note that the morphism Ω 1 R is the same that the one defined in[5] by the name of ∇ H . The following Lemma gives an explanation of the meaning of the objects H × 1 L H, H × 1 R H, H × 2 L H and H × 2 R H by using equalizer and coequalizer diagrams.…”

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confidence: 99%

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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Cleft and Galois extensions associated with a weak Hopf quasigroup

Cited by 11 publications

References 20 publications

Weak quasi-entwining structures

Weak quasi-entwining structures

Strong Hopf modules for weak Hopf quasigroups

A Characterization of Weak Hopf (co) Quasigroups

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