S. Caenepeel, I. Goyvaerts
Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras) have been investigated in the literature recently. We study Hom-structures from the point of view of monoidal categories; in particular, we introduce a symmetric monoidal category such that Hom-algebras coincide with algebras in this monoidal category, and similar properties for coalgebras, Hopf algebras and Lie algebras.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We introduce the notion of an oplax Hopf monoid in any braided monoidal bicategory, generalizing that of a Hopf monoid in a braided monoidal category in an appropriate way. We show that Hopf V-categories introduced in [BCV16] are a particular type of oplax Hopf monoids in the monoidal bicategory Span|V described in [Böh17]. Finally, we introduce Frobenius V-categories as the Frobenius objects in the same monoidal bicategory.
We first introduce the notion of Doi Hom-Hopf modules and find the sufficient condition for the category of Doi Hom-Hopf modules to be monoidal. Also we obtain the condition for the monoidal Hom-algebra and monoidal Hom-coalgebra to be monoidal Hom-bialgebras. Second, we give the maps between the underlying monoidal Hom-Hopf algebras, Hom-comodule algebras and Hom-module coalgebras give rise to functors between the category of Doi Hom-Hopf modules and study tensor identities for monodial categories of Doi Hom-Hopf modules. Furthermore, we construct a braiding on the category of Doi Hom-Hopf mo…
Hopf algebras are determined by their monoidal derived categories
该研究证明有限维Hopf代数的规范等价与有界导出范畴的幺半三角等价等价，并推广到局部有限张量Grothendieck范畴的幺半导出等价可诱导幺半阿贝尔等价。
We investigate the theory of Hopf-Galois extensions for monoidal Hom-Hopf algebras. As the main result of this paper, we prove the Schneider's affineness theorems in the case of monoidal Hom-Hopf algebras in terms of the theory of the total integral and Hom-Hopf Galois extensions. In addition, we obtain the affineness criterion for relative Hom-Hopf module associated with faithfully flat Hom-Hopf Galois extensions.
Hopf algebras are closely related to monoidal categories. More precise, $k$-Hopf algebras can be characterized as those algebras whose category of finite dimensional representations is an autonomous monoidal category such that the forgetful functor to $k$-vectorspaces is a strict monoidal functor. This result is known as the Tannaka reconstruction theorem (for Hopf algebras). Because of the importance of both Hopf algebras in various fields, over the last last few decades, many generalizations have been defined. We will survey these different generalizations from the point of view of the Tanna…
In this paper we mainly construct bicrossproduct for finite-dimensional monoidal Hom-Hopf algebra $(H,α)$, generalizing the Majid's bicrossproduct. Naturally the Hom-type bicrossproduct leads to Drinfel'd double $(H^{op}\bowtie H^{\ast}, α\otimes (α^{-1})^\ast)$ with a quasitriangular structure $R$ satisfying the quantum Hom-Yang-Baxter equations.
We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele in [A. Van Daele, Multiplier Hopf algebras, {\em Trans. Amer. Math. Soc.}, {\bf 342}(2), (1994) 917--932.] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly non-unital, idempotent, non-degenerate, $k$-projective) algebra over a commutative ring $k$ is a multip…
Let $(H,\a_H)$ be a Hom-Hopf algebra, $(A,\a_A)$ a right $H$-comodule algebra and $(C,\a_C)$ a left $H$-module coalgebra. Then we have the category $_A\mathcal{M}(H)^C$ of Hom-type Doi-Hopf modules. The aim of this paper is to make the category $_A\mathcal{M}(H)^C$ into a braided monoidal category. Our construction unifies quasitriangular and coquasitriangular Hom-Hopf algebras and Hom-Yetter-Drinfeld modules. We study tensor identities for monoidal categories of Hom-type Doi-Hopf modules. Finally we show that the category $_A\mathcal{M}(H)^C$ is isomorphic to $A\#C^*$-module category.
Quantum Hopf rigidity in symmetric monoidal categories
该研究在对称monoidal范畴中证明，保持结合代数结构与余乘法的Hopf代数余作用自动保持其余单位等结构，推广统一了量子群仅具经典对称性的相关结果。
Let $(H,α)$ be a monoidal Hom-Hopf algebra, and $(A,β)$ a Hom-algebra. In this paper we will introduce the crossed product $(A\#_σH,β\otimesα)$, which is a Hom-algebra. Then we will introduce the notions of cleft extensions and Galois extensions respectively, and prove that a crossed product is equivalent to a cleft extension and a cleft extension is equivalent to a Galois extension with normal bases property.
Let $ Aut_{mHH}(H)$ denote the set of all automorphisms of a monoidal Hopf algebra $H$ with bijective antipode in the sense of Caenepeel and Goyvaerts \cite{CG2011}. The main aim of this paper is to provide new examples of braided $T$-category in the sense of Turaev \cite{T2008}. For this, first we construct a monoidal Hom-Hopf $T$-coalgebra $\mathcal{MHD}(H)$ and prove that the $T$-category $Rep(\mathcal{MHD}(H))$ of representation of $\mathcal{MHD}(H)$ is isomorphic to $\mathcal {MHYD}(H)$ as braided $T$-categories, if $H$ is finite-dimensional. Then we construct a new braided $T$-category $…
This paper introduces group-cograded monoidal Hom-Hopf algebras, and shows that this kind of group-cograded monoidal Hom-Hopf algebras are monoidal Hom-Hopf algebras in the Turaev category $\mathcal{J}_{k}$ introduced by Canepeel and De Lombaerde. Then we define the $p$-Yetter-Drinfeld category over a group-cograded monoidal Hom-Hopf algebra, and construct a new kind of braided $T$-categories.
Let $H$ and $L$ be two Hopf algebras such that their comodule categories are monoidal equivalent. We prove that if $H$ is a twisted Calabi-Yau (CY) Hopf algebra, then $L$ is a twisted CY algebra when it is homologically smooth. Especially, if $H$ is a Noetherian twisted CY Hopf algebra and $L$ has finite global dimension, then $L$ is a twisted CY algebra.
Let $ Aut_{mHH}(H)$ denote a set of all automorphisms of a monoidal Hopf algebra $H$ with bijective antipode in the sense of Caenepeel S. and Goyvaerts I. (Commun. Algebra 39, 2216-2240, 2011) and let $G$ be a crossed product group $ Aut_{mHH}(H)\times Aut_{mHH}(H)$. The main aim of this paper is to provide further examples of braided $T$-category in the sense of Turaev (1994, 2008). For this purpose, we first introduce a class of new categories $_{H}\mathcal {MHYD}^{H}(A, B)$ of monoidal Hom $(A, B)$-Yetter-Drinfeld modules with $A, B \in Aut_{mHH}(H)$. Then we show that the category ${\cal M…
We discuss the question of whether the global dimension is a monoidal invariant for Hopf algebras, in the sense that if two Hopf algebras have equivalent monoidal categories of comodules, then their global dimensions should be equal. We provide several positive new answers to this question, under various assumptions of smoothness, cosemisimplicity or finite dimension. We also discuss the comparison between the global dimension and the Gerstenhaber-Schack cohomological dimension in the cosemisimple case, obtaining equality in the case the latter is finite. One of our main tools is the new conce…
We develop the Tannaka-Krein duality for monoidal functors with target in the categories of bimodules over a ring. The $\coend$ of such a functor turns out to be a Hopf algebroid over this ring. Using the result of a previous paper we characterize a small abelian, locally finite rigid monoidal category as the category of rigid comodules over a transitive Hopf algebroid.
Four-angle Hopf modules for Hom-Hopf algebras
本文引入了同态霍普夫代数四角霍普夫模块的概念，并展示了其范畴是一个具有同态张量积或同态余张量积的单子范畴。还证明了四角霍普夫模块的单子范畴与Yetter-Drinfel'd模块的单子范畴等价，并给出了其单子结构。
Free and co-free constructions for Hopf categories
本文证明在温和条件下Hopf范畴的范畴是局部可表示的，推导出自由与余自由Hopf范畴的存在性，给出半Hopf范畴上的显式构造，并引入“非常平坦幺半积”概念在模范畴中进行研究。
Hopf bimodules for bialgebroids
本文在双代数胚框架下定义四类Hopf模及双边双余边Hopf模（Hopf双模），通过Hopf-Galois余模证明基本定理，并构造两种辫子幺半范畴等价于Yetter-Drinfeld模范畴。
Projections of Hopf braces
本文在幺半范畴设定下研究 Hopf brace 的投影，定义了左 Yetter-Drinfeld 模范畴，引入可玻色化 Hopf brace 概念，并证明其与 v$_{4}$-强投影范畴之间存在范畴等价。
Monoidal invariance of the cohomological dimension of Hopf algebras: the finite case
本研究通过直接证明展示了有限霍普夫代数在特定条件下全局维数的不变性，并去除了可逆反自同态的假设。
The Drinfel'd double D(A) of a finite-dimensional Hopf algebra A is a Hopf algebraic counterpart of the monoidal center construction. Majid introduced an important representation of the Drinfel'd double, which he called the Schrödinger representation. We study this representation from the viewpoint of the theory of monoidal categories. One of our main results is as follows: If two finite-dimensional Hopf algebras A and B over a field k are monoidally Morita equivalent, i.e., there exists an equivalence F from the module category over A to the module category over B of k-linear monoidal categor…
Gorenstein homological invariants and monoidal model categories of Hopf algebras
本文研究了霍普夫代数的戈林斯泰因同调不变量，并证明了在特定条件下霍普夫代数的模块范畴是单向模型范畴。
Distributive laws and Hopf quasigroups
本文提出 Hopf 拟群间的 $a$-monoidal 分配律，证明其可诱导使 $A\otimes H$ 成为 Hopf 拟群的圈积，并统一解释多种叉积与碎积构造。
Categorical isomorphisms for Hopf braces
本文在辫幺半范畴中引入括号三元组范畴，证明其与 s-Hopf 括号范畴同构，并将有限余交换 Hopf 括号与余交换 post-Hopf 代数子范畴的等价推广到辫幺半范畴环境。
Artin-Schelter Gorenstein property of Hopf Galois extensions
本文探讨了Hopf Galois扩展的AS Gorenstein性质，证明了在特定条件下该性质可被继承，并揭示了AS Gorenstein Hopf代数的投射维数单体不变量特性。
Let $A=\bigoplus_{p\in G}A_{p}$ be a multiplier Hopf $T$-coalgebra over a group $G$, in this paper we give the definition of the crossed left $A$-$G$-modules and show that the category of crossed left $A$-$G$-modules is a monoidal category. Finally we show that a family of multipliers $R = \{R_{p, q} \in M(A_{p}\otimes A_{q})\}_{p, q\in G}$ is a quasitriangular structure of a multiplier $T$-coalgebra $A$ if and only if the crossed left $A$-$G$-module category over $A$ is a braided monoidal category with the braiding $c$ defined by $R$, generalizing the main results in \cite{ZCL11} to the more …
The aim of this paper is to extend the classical Larson-Sweedler theorem, namely that a k-bialgebra has a non-singular integral (and in particular is Frobenius) if and only if it is a finite dimensional Hopf algebra, to the `many-object' setting of Hopf categories. To this end, we provide new characterizations of Frobenius V-categories and we develop the integral theory for Hopf V-categories. Our results apply to Hopf algebras in any braided monoidal category as a special case, and also relate to Turaev's Hopf group algebras and particular cases of weak and multiplier Hopf algebras.
In general, representation rings are well-known as Green rings from module categories of Hopf algebras. In this paper, we study Green rings in the context of monoidal categories such that representations of Hopf algebras can be investigated through Green rings of various levels from module categories to derived categories in the unified view-point. Firstly, as analogue of representation rings of Hopf algebras, we set up the so-called Green rings of monoidal categories, and then list some such categories including module categories, complex categories, homotopy categories, derived categories an…
Let $k$ be a field, $k^*=k\setminus\{0\}$ and $C_2$ the cyclic group of order 2. In this note we compute all the braided monoidal structures on the category of $k$-vector spaces graded by the Klein group $C_2\times C_2$. Actually, for the monoidal structures we will compute the explicit form of the 3-cocycles on $C_2\times C_2$ with coefficients in $k^*$, while for the braided monoidal structures we will compute the explicit form of the abelian 3-cocycles on $C_2\times C_2$ with coefficients in $k^*$. In particular, this will allow us to produce examples of quasi-Hopf algebras and weak braided…
We introduce the category of set-theoretic representations of a matched pair of groupoids. This is a monoidal category endowed with a monoidal functor to the category of quivers over the common base of the groupoids in the matched pair (the forgetful functor). We study monoidal functors between two such categories of representations which preserve the forgetful functor. We show that the centralizer of such a monoidal functor is the category of representations of a new matched pair, which we construct explicitly. We introduce the notions of {\em double} of a matched pair of groupoids and {\em g…
The aim of this paper is to define and study Yetter-Drinfeld modules over weak Hom-Hopf algebras. We show that the category ${}_H{\cal WYD}^H$ of Yetter-Drinfeld modules with bijective structure maps over weak Hom-Hopf algebras is a rigid category and a braided monoidal category, and obtain a new solution of quantum Hom-Yang-Baxter equation. It turns out that, If $H$ is quasitriangular (respectively, coquasitriangular)weak Hom-Hopf algebras, the category of modules (respectively, comodules) with bijective structure maps over $H$ is a braided monoidal subcategory of the category ${}_H{\cal WYD}…
We introduce the concept of Hopf-Galois system, a reformulation of the notion of Galois extension of the base field for a Hopf algebra. The main feature of our definition is a generalization of the antipode of an ordinary Hopf algebra. The main application of Hopf-Galois systems is the construction of monoidal equivalences between comodule categories over Hopf algebras.
The aim of this paper is to define and study Drinfeld twists for monoidal Hom-bialgebras. We show that a new Hom-bialgebra could be constructed by changing the coproduct of a monoidal Hom-bialgebra via a Drinfeld twist, and this construction preserves $R$-matrixes if there exist one. Moreover, their representation categories are monoidal isomorphic.
Opposite brace triples, Hopf braces and matched pairs of Hopf algebras
本文引入了在一般辫状单子设置下的反 brace 三元组范畴，在余交换条件下证明其与 Hopf braces 等价。进一步，当固定其中一个底层 Hopf 代数时，这些范畴也与该 Hopf 代数上的匹配对范畴等价。
By work of Farinati, Solberg, and Taillefer, it is known that the Hopf algebra cohomology of a quasi-triangular Hopf algebra, as a graded Lie algebra under the Gerstenhaber bracket, is abelian. Motivated by the question of whether this holds for nonquasi-triangular Hopf algebras, we show that Gerstenhaber brackets on Hopf algebra cohomology can be expressed via an arbitrary projective resolution using Volkov's homotopy liftings as generalized to some exact monoidal categories. This is a special case of our more general result that a bracket operation on cohomology is preserved under exact mono…
We introduce "geometric" partial comodules over coalgebras in monoidal categories, as an alternative notion to the notion of partial action and coaction of a Hopf algebra introduced by Caenepeel and Janssen. The name is motivated by the fact that our new notion suits better if one wants to describe phenomena of partial actions in algebraic geometry. Under mild conditions, the category of geometric partial comodules is shown to be complete and cocomplete and the category of partial comodules over a Hopf algebra is lax monoidal. We develop a Hopf-Galois theory for geometric partial coactions to …
The present article is devoted to introduce the notion of Hopf bracoid in the braided monoidal framework as the quantum version of skew bracoids, which have been presented by Martin-Lyons and Paul J. Truman. Taking into account that Hopf braces are particular examples of Hopf bracoids, in this paper we generalize many properties of Hopf braces to the Hopf bracoid setting and we obtain a categorical isomorphism between certain full subcategories of the Hopf bracoids category and the 1-cocycles category.
D-convolution categories and Hopf algebras
本文通过结合D-模理论与霍普夫代数，证明了D-卷积范畴与加权霍普夫代数模的等价性，并给出了导出D-模范畴的显式辫状结构。
Let $(H, \a)$ be a monoidal Hom-Hopf algebra and $(A, \b)$ a right $(H, \a)$-Hom-comodule algebra. We first investigate the criterion for the existence of a total integral of $(A, \b)$ in the setting of monoidal Hom-Hopf algebras. Also we prove that there exists a total integral $ϕ: (H, \a)\rightarrow (A, \b)$ if and only if any representation of the pair $(H,A)$ is injective in a functorial way, as a corepresentation of $(H, \a)$, which generalizes Doi's result. Finally, we define a total quantum integral $\g: H\rightarrow Hom(H, A)$ and prove the following affineness criterion: if there exis…
Let $(H,α)$ be a monoidal Hom-Hopf algebra and $^{H}_{H}\mathcal{HYD}$ the Hom-Yetter-Drinfeld category over $(H,α)$. Then in this paper, we first introduce the definition of braided Hom-Lie algebras and show that each monoidal Hom-algebra in $^{H}_{H}\mathcal{HYD}$ gives rise to a braided Hom-Lie algebra. Second, we prove that if $(A,β)$ is a sum of two $H$-commutative monoidal Hom-subalgebras, then the commutator Hom-ideal $[A,A]$ of $A$ is nilpotent. Also, we study the central invariant of braided Hom-Lie algebras as a generalization of generalized Lie algebras. Finally, we obtain a constru…
We show that Turaev's group-coalgebras and Hopf group-coalgebras are coalgebras and Hopf algebras in a symmetric monoidal category, which we call the Turaev category. A similar result holds for group-algebras and Hopf group-algebras. As an application, we give an alternative approach to Virelizier's version of the Fundamental Theorem for Hopf algebras. We introduce Yetter-Drinfeld modules over Hopf group-coalgebras using the center construction.
Let $H$ be a dual quasi-Hopf algebra. In this paper we will firstly introduce all possible categories of Yetter-Drinfeld modules over $H$, and give explicitly the monoidal and braided structure of them. Then we prove that the category $^H_H\mathcal{YD}^{fd}$ of finite-dimensional left-left Yetter-Drinfeld modules is rigid. Finally we will study the braided cocommunitivity of $H_0$ in $^H_H\mathcal{YD}$.
A heap is a structure with a ternary operation which is intuitively a group with forgotten unit element. Quantum heaps are associative algebras with a ternary cooperation which are to the Hopf algebras what heaps are to groups, and, in particular, the category of copointed quantum heaps is isomorphic to the category of Hopf algebras. There is an intermediate structure of a cop in monoidal category which is in the case of vector spaces to a quantum heap about what is a coalgebra to a Hopf algebra. The representations of Hopf algebras make a rigid monoidal category. Similarly the representations…
Parabosonic algebra in finite or infinite degrees of freedom is considered as a $\mathbb{Z}_{2}$-graded associative algebra, and is shown to be a $\mathbb{Z}_{2}$-graded (or: super) Hopf algebra. The super-Hopf algebraic structure of the parabosonic algebra is established directly without appealing to its relation to the $osp(1/2n)$ Lie superalgebraic structure. The notion of super-Hopf algebra is equivalently described as a Hopf algebra in the braided monoidal category ${}_{\mathbb{CZ}_{2}}\mathcal{M}$. The bosonisation technique for switching a Hopf algebra in the braided monoidal category $…
On the semi-abelianness of cocommutative Hopf monoids
本文通过推广Newman的双射对应关系，证明在满足忠实（余）平坦条件时，任何交换单则范畴中的余交换单体范畴是半阿贝尔的，统一并推广了余交换Hopf代数和余交换彩色Hopf代数的半阿贝尔性。
Hopf and Lie algebras in semi-additive Varieties
本文研究熵半加性簇中的Hopf幺半群，通过阿贝尔核及其幺半结构，探讨包络与本原元素函子间的伴随关系。
We consider Frobenius algebras in the monoidal category of right comodules over a Hopf algebra $H$. If $H$ is a group Hopf algebra, we study a more general Frobenius type property and uncover the structure of graded Frobenius algebras. Graded symmetric algebras are also investigated.
Let $H$ be a finite dimensional quasi-Hopf algebra over a field $k$ and ${\mathfrak A}$ a right $H$-comodule algebra in the sense of Hausser and Nill. We first show that on the $k$-vector space ${\mathfrak A}\ot H^*$ we can define an algebra structure, denoted by ${\mathfrak A}\ovsm H^*$, in the monoidal category of left $H$-modules (i.e. ${\mathfrak A}\ovsm H^*$ is an $H$-module algebra. Then we will prove that the category of two-sided $({\mathfrak A}, H)$-bimodules $\hba $ is isomorphic to the category of relative $({\mathfrak A}\ovsm H^*, H^*)$-Hopf modules, as introduced in by Hausser and…
We introduce Yetter-Drinfeld modules over a weak Hopf algebra $H$, and show that the category of Yetter-Drinfeld modules is isomorphic to the center of the category of $H$-modules. The categories of left-left, left-right, right-left and right-right Yetter-Drinfeld modules are isomorphic as braided monoidal categories. Yetter-Drinfeld modules can be viewed as weak Doi-Hopf modules, and, a fortiori, as weak entwined modules. If $H$ is finitely generated and projective, then we introduce the Drinfeld double using duality results between entwining structures and smash product structures, and show …
Ştefan spectral sequence for faithfully flat Hopf Galois extensions is multiplicative
本文证明忠实平坦Hopf Galois扩张的Ştefan谱序列具有乘法性，引入相关形式体系构造对应谱序列，结合比较定理完成等同，在多类代数结构上得到新乘法性结果。
For any Koszul Artin-Schelter regular algebra A, we consider a version of the universal Hopf algebra aut(A) coacting on A, introduced by Manin. To study the representations (i.e. finite dimensional comodules) of this Hopf algebra, we use the Tannaka-Krein formalism. Specifically, we construct an explicit combinatorial rigid monoidal category U, equipped with a functor M to finite dimensional vector spaces such that aut(A)= coend_U(M). Using this pair (U,M) we show that aut(A) is quasi-hereditary as a coalgebra and in addition is derived equivalent to the representation category of U.
Let $A$ be an algebra in a monoidal category $\Cc$, and let $X$ be an object in $\Cc$. We study $A$-(co)ring structures on the left $A$-module $A\ot X$. These correspond to (co)algebra structures in $EM(\Cc)(A)$, the Eilenberg-Moore category associated to $\Cc$ and $A$. The ring structures are in bijective correspondence to wreaths in $\Cc$, and their category of representations is the category of representations over the induced wreath product. The coring structures are in bijective correspondence to cowreaths in $\Cc$, and their category of corepresentations is the category of generalized en…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。