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Hopf-Galois extensions for monoidal Hom-Hopf algebras

100%
Chen Y., Zhang L.
Colloquium Mathematicum
|
2016
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tom 143
|
nr 1
127-147
EN
Hopf-Galois extensions for monoidal Hom-Hopf algebras are investigated. As the main result, Schneider's affineness theorem in the case of monoidal Hom-Hopf algebras is shown in terms of total integrals and Hopf-Galois extensions. In addition, we obtain an affineness criterion for relative Hom-Hopf modules which is associated with faithfully flat Hopf-Galois extensions of monoidal Hom-Hopf algebras.
2
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The duality theorem for twisted smash products of Hopf algebras and its applications

100%
Wang Z., Zhang L.
Colloquium Mathematicum
|
2015
|
tom 141
|
nr 1
25-44
EN
Let $A #_{T} H$ denote the twisted smash product of an arbitrary algebra A and a Hopf algebra H over a field. We present an analogue of the celebrated Blattner-Montgomery duality theorem for $A #_{T} H$, and as an application we establish the relationship between the homological dimensions of $A #_{T} H$ and A if H and its dual H* are both semisimple.
3
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The fundamental theorem and Maschke's theorem in the category of relative Hom-Hopf modules

81%
Chen Y., Wang Z., Zhang L.
Colloquium Mathematicum
|
2016
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tom 144
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nr 1
55-71
EN
We introduce the concept of relative Hom-Hopf modules and investigate their structure in a monoidal category $𝓗̃ (ℳ_{k})$. More particularly, the fundamental theorem for relative Hom-Hopf modules is proved under the assumption that the Hom-comodule algebra is cleft. Moreover, Maschke's theorem for relative Hom-Hopf modules is established when there is a multiplicative total Hom-integral.
4
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Quasitriangular Hom-Hopf algebras

81%
Chen Y., Wang Z., Zhang L.
Colloquium Mathematicum
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2014
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tom 137
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nr 1
67-88
EN
A twisted generalization of quasitriangular Hopf algebras called quasitriangular Hom-Hopf algebras is introduced. We characterize these algebras in terms of certain morphisms. We also give their equivalent description via a braided monoidal category $𝓗̃ (_Hℳ )$. Finally, we study the twisting structure of quasitriangular Hom-Hopf algebras by conjugation with Hom-2-cocycles.
5
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Deformed commutators on comodule algebras over coquasitriangular Hopf algebras

81%
Wang Z., Zhang G., Zhang L.
Colloquium Mathematicum
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2015
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tom 139
|
nr 2
165-183
EN
We construct quantum commutators on comodule algebras over coquasitriangular Hopf algebras, so that they are quantum group coinvariant and have the generalized antisymmetry and Leibniz properties. If the coquasitriangular Hopf algebra is additionally cotriangular, then the quantum commutators satisfy a generalized Jacobi identity, and turn the comodule algebra into a quantum Lie algebra. Moreover, we investigate the projective and injective dimensions of some Doi-Hopf modules over a quantum commutative comodule algebra.
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