The fundamental theorem and Maschke's theorem in the category of relative Hom-Hopf modules

Yuanyuan Chen; Zhongwei Wang; Liangyun Zhang

Displaying similar documents to “The fundamental theorem and Maschke's theorem in the category of relative Hom-Hopf modules”

A Maschke type theorem for relative Hom-Hopf modules

Shuangjian Guo, Xiu-Li Chen (2014)

Czechoslovak Mathematical Journal

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Let $(H, α)$ be a monoidal Hom-Hopf algebra and $(A, β)$ a right $(H, α)$ -Hom-comodule algebra. We first introduce the notion of a relative Hom-Hopf module and prove that the functor $F$ from the category of relative Hom-Hopf modules to the category of right $(A, β)$ -Hom-modules has a right adjoint. Furthermore, we prove a Maschke type theorem for the category of relative Hom-Hopf modules. In fact, we give necessary and sufficient conditions for the functor that forgets the $(H, α)$ -coaction to be separable. This leads...

Braided monoidal categories and Doi-Hopf modules for monoidal Hom-Hopf algebras

Shuangjian Guo, Xiaohui Zhang, Shengxiang Wang (2016)

Colloquium Mathematicae

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We continue our study of the category of Doi Hom-Hopf modules introduced in [Colloq. Math., to appear]. We find a sufficient condition for the category of Doi Hom-Hopf modules to be monoidal. We also obtain a condition for a monoidal Hom-algebra and monoidal Hom-coalgebra to be monoidal Hom-bialgebras. Moreover, we introduce morphisms between the underlying monoidal Hom-Hopf algebras, Hom-comodule algebras and Hom-module coalgebras, which give rise to functors between the category of...

Parametric representations of BiHom-Hopf algebras

Xiaohui Zhang, Wei Wang, Juzhen Chen (2024)

Czechoslovak Mathematical Journal

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The main purpose of the present paper is to study representations of BiHom-Hopf algebras. We first introduce the notion of BiHom-Hopf algebras, and then discuss BiHom-type modules, Yetter-Dinfeld modules and Drinfeld doubles with parameters. We get some new $n$ -monoidal categories via the category of BiHom-(co)modules and the category of BiHom-Yetter-Drinfeld modules. Finally, we obtain a center construction type theorem on BiHom-Hopf algebras.

Separable functors for the category of Doi Hom-Hopf modules

Shuangjian Guo, Xiaohui Zhang (2016)

Colloquium Mathematicae

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Let $̃ (_{k}) {(H)}_{A}^{C}$ be the category of Doi Hom-Hopf modules, $̃ {(_{k})}_{A}$ be the category of A-Hom-modules, and F be the forgetful functor from $̃ (_{k}) {(H)}_{A}^{C}$ to $̃ {(_{k})}_{A}$ . The aim of this paper is to give a necessary and suffcient condition for F to be separable. This leads to a generalized notion of integral. Finally, applications of our results are given. In particular, we prove a Maschke type theorem for Doi Hom-Hopf modules.

The category of partial Doi-Hopf modules and functors

Q.-G. Chen, D.-G. Wang (2013)

Rendiconti del Seminario Matematico della Università di Padova

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A construction of the Hom-Yetter-Drinfeld category

Haiying Li, Tianshui Ma (2014)

Colloquium Mathematicae

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In continuation of our recent work about smash product Hom-Hopf algebras [Colloq. Math. 134 (2014)], we introduce the Hom-Yetter-Drinfeld category $_{H}^{H}$ via the Radford biproduct Hom-Hopf algebra, and prove that Hom-Yetter-Drinfeld modules can provide solutions of the Hom-Yang-Baxter equation and $_{H}^{H}$ is a pre-braided tensor category, where (H,β,S) is a Hom-Hopf algebra. Furthermore, we show that $(A ♮_{⋄} H, α \otimes β)$ is a Radford biproduct Hom-Hopf algebra if and only if (A,α) is a Hom-Hopf algebra in the category...

Restriction to Levi subalgebras and generalization of the category $𝒪$

Guillaume Tomasini (2013)

Annales de l’institut Fourier

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The category of all modules over a reductive complex Lie algebra is wild, and therefore it is useful to study full subcategories. For instance, Bernstein, Gelfand and Gelfand introduced a category of modules which provides a natural setting for highest weight modules. In this paper, we define a family of categories which generalizes the BGG category, and we classify the simple modules for a subfamily. As a consequence, we show that some of the obtained categories are semisimple. ...

T-Rickart modules

S. Ebrahimi Atani, M. Khoramdel, S. Dolati Pish Hesari (2012)

Colloquium Mathematicae

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We introduce the notions of T-Rickart and strongly T-Rickart modules. We provide several characterizations and investigate properties of each of these concepts. It is shown that R is right Σ-t-extending if and only if every R-module is T-Rickart. Also, every free R-module is T-Rickart if and only if $R = Z ₂ (R_{R}) \oplus R^{'}$ , where R’ is a hereditary right R-module. Examples illustrating the results are presented.

The affineness criterion for quantum Hom-Yetter-Drinfel'd modules

Shuangjian Guo, Shengxiang Wang (2016)

Colloquium Mathematicae

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Quantum integrals associated to quantum Hom-Yetter-Drinfel’d modules are defined, and the affineness criterion for quantum Hom-Yetter-Drinfel’d modules is proved in the following form. Let (H,α) be a monoidal Hom-Hopf algebra, (A,β) an (H,α)-Hom-bicomodule algebra and $B = A^{c o H}$ . Under the assumption that there exists a total quantum integral γ: H → Hom(H,A) and the canonical map $β : A \otimes_{B} A \to A \otimes H$ , $a \otimes_{B} b \mapsto S^{- 1} (b_{[1]}) α (b_{[0] [- 1]}) \otimes β^{- 1} (a) β (b_{[0] [0]})$ , is surjective, we prove that the induction functor $A \otimes_{B} - : ̃ {(_{k})}_{B} \to_{A}^{H}$ is an equivalence of categories.

On the relation between maximal rigid objects and τ-tilting modules

Pin Liu, Yunli Xie (2016)

Colloquium Mathematicae

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This note compares τ-tilting modules and maximal rigid objects in the context of 2-Calabi-Yau triangulated categories. Let be a 2-Calabi-Yau triangulated category with suspension functor S. Let R be a maximal rigid object in and let Γ be the endomorphism algebra of R. Let F be the functor $H o m_{} (R, -) : \to m o d Γ$ . We prove that any τ-tilting module over Γ lifts uniquely to a maximal rigid object in via F, and in turn, that projection from to mod Γ sends the maximal rigid objects which have no direct summands...

Quasitriangular Hom-Hopf algebras

Yuanyuan Chen, Zhongwei Wang, Liangyun Zhang (2014)

Colloquium Mathematicae

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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 $̃ (_{H} ℳ)$ . Finally, we study the twisting structure of quasitriangular Hom-Hopf algebras by conjugation with Hom-2-cocycles.

A decomposition theorem for complete comodule algebras over complete Hopf algebras

Andrzej Tyc (1998)

Colloquium Mathematicae

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A family of noetherian rings with their finite length modules under control

Markus Schmidmeier (2002)

Czechoslovak Mathematical Journal

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We investigate the category $mod Λ$ of finite length modules over the ring $Λ = A \otimes_{k} Σ$ , where $Σ$ is a V-ring, i.e. a ring for which every simple module is injective, $k$ a subfield of its centre and $A$ an elementary $k$ -algebra. Each simple module $E_{j}$ gives rise to a quasiprogenerator $P_{j} = A \otimes E_{j}$ . By a result of K. Fuller, $P_{j}$ induces a category equivalence from which we deduce that $mod Λ ≃ ∐_{j} b a d h b o x P_{j}$ . As a consequence we can (1) construct for each elementary $k$ -algebra $A$ over a finite field $k$ a nonartinian noetherian ring $Λ$ such that $mod A ≃ mod Λ$ ,...

Lannes' T functor on Hopf algebras over the Steenrod algebra with applications to Carlsson modules and twisted algebras.

R. James Shank (1992)

Mathematische Zeitschrift

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Cobraided smash product Hom-Hopf algebras

Tianshui Ma, Haiying Li, Tao Yang (2014)

Colloquium Mathematicae

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Let (A,α) and (B,β) be two Hom-Hopf algebras. We construct a new class of Hom-Hopf algebras: R-smash products $(A ♮_{R} B, α \otimes β)$ . Moreover, necessary and sufficient conditions for $(A ♮_{R} B, α \otimes β)$ to be a cobraided Hom-Hopf algebra are given.

Grothendieck ring of quantum double of finite groups

Jingcheng Dong (2010)

Czechoslovak Mathematical Journal

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Let $k G$ be a group algebra, and $D (k G)$ its quantum double. We first prove that the structure of the Grothendieck ring of $D (k G)$ can be induced from the Grothendieck ring of centralizers of representatives of conjugate classes of $G$ . As a special case, we then give an application to the group algebra $k D_{n}$ , where $k$ is a field of characteristic $2$ and $D_{n}$ is a dihedral group of order $2 n$ .