Cleft extensions of Koszul twisted Calabi–Yau algebras

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.

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Cleft Extensions of Hopf Algebras, II

Proceedings of the London Mathematical Society ◽

10.1112/plms/s3-67.2.277 ◽

1993 ◽

Vol s3-67 (2) ◽

pp. 277-304 ◽

Cited By ~ 7

Author(s):

N. P. Byott

Keyword(s):

Hopf Algebras ◽

Cleft Extensions

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Cleft extensions for quasi-entwining structures

Mathematica Slovaca ◽

10.1515/ms-2017-0105 ◽

2018 ◽

Vol 68 (2) ◽

pp. 339-352

Author(s):

J. N. Alonso Álvarez ◽

J. M. Fernández Vilaboa ◽

R. González Rodríguez

Keyword(s):

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Cleft Extension ◽

Cleft Extensions ◽

Necessary And Sufficient

Abstract In this paper we introduce the notions of quasi-entwining structure and cleft extension for a quasi-entwining structure. We prove that if (A, C, ψ) is a quasi-entwining structure and the associated extension to the submagma of coinvariants AC is cleft, there exists an isomorphism ωA between AC ⊗ C and A. Moreover, we define two unital but not necessarily associative products on AC ⊗ C. For these structures we obtain the necessary and sufficient conditions to assure that ωA is a magma isomorphism, giving some examples fulfilling these conditions.

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Comparison of two notions of weak crossed product

Journal of Algebra and Its Applications ◽

10.1142/s0219498821500596 ◽

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].

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Invertible Lax Entwining Structures and C-Cleft Extensions

Non-Associative and Non-Commutative Algebra and Operator Theory - Springer Proceedings in Mathematics & Statistics ◽

10.1007/978-3-319-32902-4_18 ◽

2016 ◽

pp. 239-254 ◽

Cited By ~ 1

Author(s):

Carlos Soneira Calvo

Keyword(s):

Cleft Extensions

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

Glasgow Mathematical Journal ◽

10.1017/s0017089500032468 ◽

1998 ◽

Vol 40 (2) ◽

pp. 147-160 ◽

Cited By ~ 1

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.

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