scholarly journals The Factorization for a Class of Hom-Coalgebras

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Lihong Dong ◽  
Shuan Xue ◽  
Xiaohui Zhang
Keyword(s):  
Hopf Algebra ◽  
Module Structure ◽  
Hopf Module

Let (H,α) be a Hom-Hopf algebra and (C,β) a Hom-coalgebra. In this paper, we first introduce the notions of Hom-crossed coproduct C ⋊λ H and cleft coextension and then discuss the equivalence between them. Furthermore, we discuss the relation between cleft coextension and Hom-module coalgebra with the Hom-Hopf module structure and obtain a Hom-coalgebra factorization for Hom-module coalgebra with the Hom-Hopf module structure.

scholarly journals Cohomology of Modules Over -categories and Co--categories

2019 ◽  
Vol 72 (5) ◽  
pp. 1352-1385
Author(s):  
Mamta Balodi ◽  
Abhishek Banerjee ◽  
Samarpita Ray
Keyword(s):  
Hopf Algebra ◽  
Module Category ◽  
Spectral Sequences ◽  
Locally Finite ◽  
Hopf Module ◽  
Comodule Algebra ◽  
Hopf Modules ◽  
Derived Functors

AbstractLet $H$ be a Hopf algebra. We consider $H$-equivariant modules over a Hopf module category ${\mathcal{C}}$ as modules over the smash extension ${\mathcal{C}}\#H$. We construct Grothendieck spectral sequences for the cohomologies as well as the $H$-locally finite cohomologies of these objects. We also introduce relative $({\mathcal{D}},H)$-Hopf modules over a Hopf comodule category ${\mathcal{D}}$. These generalize relative $(A,H)$-Hopf modules over an $H$-comodule algebra $A$. We construct Grothendieck spectral sequences for their cohomologies by using their rational $\text{Hom}$ objects and higher derived functors of coinvariants.

scholarly journals Varieties of Null-Filiform Leibniz Algebras Under the Action of Hopf Algebras

2021 ◽  
Author(s):  
Lucio Centrone ◽  
Chia Zargeh
Keyword(s):  
Hopf Algebra ◽  
Free Algebra ◽  
Hopf Algebras ◽  
Leibniz Algebra ◽  
Module Structure ◽  
Leibniz Algebras ◽  
Finite Dimensional ◽  
Filiform Leibniz Algebra ◽  
Algebra Over A Field ◽  
Cocommutative Hopf Algebra

AbstractLet L be an n-dimensional null-filiform Leibniz algebra over a field K. We consider a finite dimensional cocommutative Hopf algebra or a Taft algebra H and we describe the H-actions on L. Moreover we provide the set of H-identities and the description of the Sn-module structure of the relatively free algebra of L.

Total integrals of Doi Hom-Hopf modules

2016 ◽  
Vol 15 (04) ◽  
pp. 1650069
Author(s):  
Shuangjian Guo ◽  
Xiaohui Zhang ◽  
Shengxiang Wang
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Integral Formula ◽  
Hopf Module ◽  
Comodule Algebra ◽  
Hopf Modules ◽  
Total Integral

Let [Formula: see text] be a monoidal Hom-Hopf algebra, [Formula: see text] a right [Formula: see text]-Hom-comodule algebra and [Formula: see text] a right [Formula: see text]-Hom-module coalgebra. We first investigate the criterion for the existence of a total integral of [Formula: see text] in the setting of monoidal Hom-Hopf algebras. Also, we prove that there exists a total integral [Formula: see text] if and only if any representation of the pair [Formula: see text] is injective in a functorial way, which generalizes Menini and Militaru’s result. Finally, we extend to the category of [Formula: see text]-Doi Hom-Hopf modules a result of Doi on projectivity of every relative [Formula: see text]-Hopf module as an [Formula: see text]-module.

RELATIVE PROJECTIVITY AND RELATIVE INJECTIVITY IN THE CATEGORY OF DOI–HOPF MODULES

2011 ◽  
Vol 10 (05) ◽  
pp. 931-946 ◽  
Author(s):  
T. GUÉDÉNON
Keyword(s):  
Hopf Algebra ◽  
Fundamental Theorem ◽  
Hopf Module ◽  
Comodule Algebra ◽  
Hopf Modules

Let k be a field, H be a Hopf algebra, A be a right H-comodule algebra and C be a right H-module coalgebra. We extend to the category of (H, A, C)-Doi–Hopf modules a result of Doi on projectivity of every relative (A, H)-Hopf module as an A-module. We also extend the Fundamental Theorem of [C, H]-Hopf modules due to Doi to the category of (H, A, C)-Doi–Hopf modules. Then we discuss relative projectivity and relative injectivity in this category.

scholarly journals The DG Poisson Adjoint Action and its Application

2019 ◽  
Vol 12 (1) ◽  
pp. 14-24
Author(s):  
Xiaojie Li ◽  
Xianguo Hu ◽  
Jiafeng Lv ◽  
Xingting Wang
Keyword(s):  
Hopf Algebra ◽  
Adjoint Action ◽  
Module Structure

In this paper, the differential graded (DG for short) Poisson adjoint action on M is introduced, where M is a DG Poisson module over a DG Poisson Hopf algebra A. As an application, we give a new DG poisson module structure over the DG Poisson Hopf algebra A, which depends heavily on the structure of A.

The module structure of certain hopf algebra extensions

1982 ◽  
Vol 10 (2) ◽  
pp. 115-120 ◽  
Author(s):  
William C. Waterhouse
Keyword(s):  
Hopf Algebra ◽  
Module Structure

ON GORENSTEINNESS OF HOPF MODULE ALGEBRAS

2016 ◽  
Vol 59 (2) ◽  
pp. 299-321
Author(s):  
SERGE SKRYABIN
Keyword(s):  
Hopf Algebra ◽  
Additional Assumption ◽  
Zariski Topology ◽  
Module Algebra ◽  
Injective Dimension ◽  
Ground Field ◽  
Hopf Module ◽  
Maximal Ideals ◽  
Maximal Spectrum ◽  
Module Algebras

AbstractLet H be a Hopf algebra with a bijective antipode, A an H-simple H-module algebra finitely generated as an algebra over the ground field and module-finite over its centre. The main result states that A has finite injective dimension and is, moreover, Artin–Schelter Gorenstein under the additional assumption that each H-orbit in the space of maximal ideals of A is dense with respect to the Zariski topology. Further conclusions are derived in the cases when the maximal spectrum of A is a single H-orbit or contains an open dense H-orbit.

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