scholarly journals Cleft extensions for a Hopf algebrakq[X,X−1, Y]

1998 ◽  
Vol 40 (2) ◽  
pp. 147-160 ◽  
Author(s):  
Hui-Xiang Chen
Keyword(s):  
Hopf Algebra ◽  
Primitive Element ◽  
Normal Basis ◽  
Crossed Products ◽  
Galois Extensions ◽  
Infinite Dimensional ◽  
Cleft Extensions ◽  
Cocommutative Hopf Algebra

The concept of cleft extensions, or equivalently of crossed products, for a Hopf algebra is a generalization of Galois extensions with normal basis and of crossed products for a group. The study of these subjects was founded independently by Blattner-Cohen-Montgomery [1] and by Doi-Takeuchi [4]. In this paper, we determine the isomorphic classes of cleft extensions for a infinite dimensional non-commutative, non-cocommutative Hopf algebra kq[X, X–l, Y], which is generated by a group-like element X and a (1,X)-primitive element Y. We also consider the quotient algebras of the cleft extensions.

Cleft extensions and Galois extensions for Hom-associative algebras

2016 ◽  
Vol 27 (03) ◽  
pp. 1650025 ◽  
Author(s):  
J. N. Alonso Álvarez ◽  
J. M. Fernández Vilaboa ◽  
R. González Rodríguez
Keyword(s):  
Associative Algebra ◽  
Galois Extension ◽  
Classical Case ◽  
Normal Basis ◽  
Associative Algebras ◽  
Galois Extensions ◽  
Cleft Extension ◽  
Cleft Extensions

In this paper, we consider Hom-(co)modules associated to a Hom-(co)associative algebra and define the notion of Hom-triple. We introduce the definitions of cleft extension and Galois extension with normal basis in this setting and we show that, as in the classical case, these notions are equivalent in the Hom setting.

scholarly journals Comparison of two notions of weak crossed product

2020 ◽  
pp. 2150059
Author(s):  
Jorge A. Guccione ◽  
Juan J. Guccione
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Crossed Product ◽  
Weak Hopf Algebra ◽  
Crossed Products ◽  
Hopf Algebroid ◽  
Hopf Algebroids ◽  
Weak Crossed Product ◽  
Cleft Extensions ◽  
Weak Hopf Algebras

We compare the restriction to the context of weak Hopf algebras of the notion of crossed product with a Hopf algebroid introduced in [Cleft extensions of Hopf algebroids, Appl. Categor. Struct. 14(5–6) (2006) 431–469] with the notion of crossed product with a weak Hopf algebra introduced in [Crossed products for weak Hopf algebras with coalgebra splitting, J. Algebra 281(2) (2004) 731–752].

scholarly journals On Hopf-Galois extensions of linear categories

2012 ◽  
Vol 20 (3) ◽  
pp. 111-130
Author(s):  
Anca Stănescu
Keyword(s):  
Hopf Algebra ◽  
Duality Theorem ◽  
Crossed Products ◽  
Galois Extensions ◽  
Finite Dimensional

AbstractWe continue the investigation of H-Galois extensions of linear categories, where H is a Hopf algebra. In our main result, the Theorem 2.2, we characterize this class of extensions in the case when H is finite dimensional. As an application, we prove a version of the Duality Theorem for crossed products with invertible cocycle.

scholarly journals Finitistic dimension of weak Hopf-Galois extensions

Filomat ◽  
2016 ◽  
Vol 30 (10) ◽  
pp. 2825-2828
Author(s):  
Xiao-Yan Zhou ◽  
Qiang Li
Keyword(s):  
Hopf Algebra ◽  
Galois Extension ◽  
Duality Theorem ◽  
Weak Hopf Algebra ◽  
Crossed Products ◽  
Finitistic Dimension ◽  
Galois Extensions ◽  
Finite Dimensional ◽  
Dimension Conjecture

Let H be a finite dimensional weak Hopf algebra and A/B be a right faithfully flat weak H-Galois extension. Then in this note, we first show that if H is semisimple, then the finitistic dimension of A is less than or equal to that of B. Furthermore, using duality theorem, we obtain that if H and its dual H* are both semisimple, then the finitistic dimension of A is equal to that of B, which means the finitistic dimension conjecture holds for A if and only if it holds for B. Finally, as applications, we obtain these relations for the weak crossed products and weak smash products.

scholarly journals Weak quasi-entwining structures

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6229-6252
Author(s):  
Álvarez Alonso ◽  
Vilaboa Fernádez ◽  
González Rodríguez
Keyword(s):  
Galois Extension ◽  
Hopf Algebras ◽  
Normal Basis ◽  
Galois Extensions ◽  
Cleft Extension ◽  
Cleft Extensions ◽  
Weak Hopf Algebras

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.

Certain properties for crossed products by automorphisms with a certain non-simple tracial Rokhlin property

2012 ◽  
Vol 33 (5) ◽  
pp. 1391-1400 ◽  
Author(s):  
XIAOCHUN FANG ◽  
QINGZHAI FAN
Keyword(s):  
Finite Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Infinite Dimensional ◽  
Crossed Product Algebra ◽  
A Finite Group ◽  
Product Algebra ◽  
Tracial Rokhlin Property

AbstractLet $\Omega $ be a class of unital $C^*$-algebras. Then any simple unital $C^*$-algebra $A\in \mathrm {TA}(\mathrm {TA}\Omega )$ is a $\mathrm {TA}\Omega $ algebra. Let $A\in \mathrm {TA}\Omega $ be an infinite-dimensional $\alpha $-simple unital $C^*$-algebra with the property SP. Suppose that $\alpha :G\to \mathrm {Aut}(A)$ is an action of a finite group $G$ on $A$ which has a certain non-simple tracial Rokhlin property. Then the crossed product algebra $C^*(G,A,\alpha )$ belongs to $\mathrm {TA}\Omega $.

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