A construction of the Hom-Yetter-Drinfeld category

Haiying Li; Tianshui Ma

Displaying similar documents to “A construction of the Hom-Yetter-Drinfeld category”

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.

Yetter-Drinfeld-Long bimodules are modules

Daowei Lu, Shuan Hong Wang (2017)

Czechoslovak Mathematical Journal

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Let $H$ be a finite-dimensional bialgebra. In this paper, we prove that the category $ℒℛ (H)$ of Yetter-Drinfeld-Long bimodules, introduced by F. Panaite, F. Van Oystaeyen (2008), is isomorphic to the Yetter-Drinfeld category $_{H \otimes H^{*}}^{H \otimes H^{*}} 𝒴𝒟$ over the tensor product bialgebra $H \otimes H^{*}$ as monoidal categories. Moreover if $H$ is a finite-dimensional Hopf algebra with bijective antipode, the isomorphism is braided. Finally, as an application of this category isomorphism, we give two results.

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.

Monomorphisms of coalgebras

A. L. Agore (2010)

Colloquium Mathematicae

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We prove new necessary and sufficient conditions for a morphism of coalgebras to be a monomorphism, different from the ones already available in the literature. More precisely, φ: C → D is a monomorphism of coalgebras if and only if the first cohomology groups of the coalgebras C and D coincide if and only if $\sum_{i \in I} ε (a^{i}) b^{i} = \sum_{i \in I} a^{i} ε (b^{i})$ for all $\sum_{i \in I} a^{i} \otimes b^{i} \in C ◻_{D} C$ . In particular, necessary and sufficient conditions for a Hopf algebra map to be a monomorphism are given.

The duality theorem for twisted smash products of Hopf algebras and its applications

Zhongwei Wang, Liangyun Zhang (2015)

Colloquium Mathematicae

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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.

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.

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Czechoslovak Mathematical Journal

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Colloquium Mathematicae

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Let Λ be an artin algebra. We prove that for each sequence ${(h_{i})}_{i \in ℤ}$ of non-negative integers there are only a finite number of isomorphism classes of indecomposables $X \in^{b} (Λ)$ , the bounded derived category of Λ, with $l e n g t h_{E (X)} H^{i} (X) = h_{i}$ for all i ∈ ℤ and E(X) the endomorphism ring of X in $^{b} (Λ)$ if and only if $^{b} (M o d Λ)$ , the bounded derived category of the category $M o d Λ$ of all left Λ-modules, has no generic objects in the sense of [4].

$n$ -angulated quotient categories induced by mutation pairs

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Czechoslovak Mathematical Journal

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Czechoslovak Mathematical Journal

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Two results of $n$ -exangulated categories

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Czechoslovak Mathematical Journal

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M. Herschend, Y. Liu, H. Nakaoka introduced $n$ -exangulated categories, which are a simultaneous generalization of $n$ -exact categories and $(n + 2)$ -angulated categories. This paper consists of two results on $n$ -exangulated categories: (1) we give an equivalent characterization of axiom (EA2); (2) we provide a new way to construct a closed subfunctor of an $n$ -exangulated category.

The categories of presheaves containing any category of algebras

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ContentsIntroduction.................................................................................................................................................. 5I. Preliminaries........................................................................................................................................... 6II. Main theorem.......................................................................................................................................... 8III. The...

On $n$ -exact categories

Said Manjra (2019)

Czechoslovak Mathematical Journal

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An $n$ -exact category is a pair consisting of an additive category and a class of sequences with $n + 2$ terms satisfying certain axioms. We introduce $n$ -weakly idempotent complete categories. Then we prove that an additive $n$ -weakly idempotent complete category together with the class $𝒞_{n}$ of all contractible sequences with $n + 2$ terms is an $n$ -exact category. Some properties of the class $𝒞_{n}$ are also discussed.

Gorenstein dimension of abelian categories arising from cluster tilting subcategories

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Czechoslovak Mathematical Journal

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Let $𝒞$ be a triangulated category and $𝒳$ be a cluster tilting subcategory of $𝒞$ . Koenig and Zhu showed that the quotient category $𝒞 / 𝒳$ is Gorenstein of Gorenstein dimension at most one. But this is not always true when $𝒞$ becomes an exact category. The notion of an extriangulated category was introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. Now let $𝒞$ be an extriangulated category with enough projectives and enough injectives, and...

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Justyna Kosakowska, Daniel Simson (2008)

Colloquium Mathematicae

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Let C be a coalgebra over an arbitrary field K. We show that the study of the category C-Comod of left C-comodules reduces to the study of the category of (co)representations of a certain bicomodule, in case C is a bipartite coalgebra or a coradical square complete coalgebra, that is, C = C₁, the second term of the coradical filtration of C. If C = C₁, we associate with C a K-linear functor $ℍ_{C} : C - C o m o d \to H_{C} - C o m o d$ that restricts to a representation equivalence $ℍ_{C} : C - c o m o d \to H_{C} - c o m o d_{s p}^{•}$ , where $H_{C}$ is a coradical square complete hereditary...

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Czechoslovak Mathematical Journal

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For any positive integer $n$ , let $A_{n}$ be a linearly oriented quiver of type $A$ with $n$ vertices. It is well-known that the quotient of an exact category by projective-injectives is an extriangulated category. We show that there exists an extriangulated equivalence between the extriangulated categories $ℳ_{n + 1}$ and $ℱ_{n}$ , where $ℳ_{n + 1}$ and $ℱ_{n}$ are the two extriangulated categories corresponding to the representation category of $A_{n + 1}$ and the morphism category of projective representations of $A_{n}$ , respectively. As a...

Higher-dimensional Auslander-Reiten sequences

Jiangsha Li, Jing He (2024)

Czechoslovak Mathematical Journal

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Zhou and Zhu have shown that if $𝒞$ is an $(n + 2)$ -angulated category and $𝒳$ is a cluster tilting subcategory of $𝒞$ , then the quotient category $𝒞 / 𝒳$ is an $n$ -abelian category. We show that if $𝒞$ has Auslander-Reiten $(n + 2)$ -angles, then $𝒞 / 𝒳$ has Auslander-Reiten $n$ -exact sequences.

On the structure of triangulated categories with finitely many indecomposables

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Bulletin de la Société Mathématique de France

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We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field $k$ . We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category $𝒯$ is of the form $ℤ Δ / G$ where $Δ$ is a disjoint union of simply-laced Dynkin diagrams and $G$ a weakly admissible group of automorphisms of $ℤ Δ$ . Then we prove that for ‘most’ groups $G$ , the category $𝒯$ is standard, ...

Stratified modules over an extension algebra

Erzsébet Lukács, András Magyar (2018)

Czechoslovak Mathematical Journal

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Let $A$ be a standard Koszul standardly stratified algebra and $X$ an $A$ -module. The paper investigates conditions which imply that the module ${Ext}_{A}^{*} (X)$ over the Yoneda extension algebra $A^{*}$ is filtered by standard modules. In particular, we prove that the Yoneda extension algebra of $A$ is also standardly stratified. This is a generalization of similar results on quasi-hereditary and on graded standardly stratified algebras.