Shuang-jian Guo, Xiu-li Chen
In this paper, 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 $(H, \a)$-coaction to be separable. This leads to a generalized notion of integrals.
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We give necessary and sufficient conditions for the functor that forgets the $(C, γ)$-coaction to be separable. This leads to a generalized notion of integrals. Finally, the applications of our results are considered.
We introduce separable functors of the second kind (or $H$-separable functors) and $H$-Maschke functors. $H$-separable functors are generalizations of separable functors. Various necessary and sufficient conditions for a functor to be $H$-separable or $H$-Maschke, in terms of generalized (co)Casimir elements (integrals, in the case of Hopf algebras), are given. An $H$-separable functor is always $H$-Maschke, but the converse holds in particular situations. A special role will be played by Frobenius functors and their relations to $H$-separability. Our concepts are applied to modules, comodules…
Entwined modules arose from the coalgebra-Galois theory. They are a generalisation of unified Doi-Hopf modules. In this paper, Frobenius properties and Maschke-type theorems, known for Doi-Hopf modules are extended to the case of entwined modules.
The left and right homological integrals are introduced for a large class of infinite dimensional Hopf algebras. Using the homological integrals we prove a version of Maschke's theorem for infinite dimensional Hopf algebras. The generalization of Maschke's theorem and homological integrals are the keys to study noetherian regular Hopf algebras of Gelfand-Kirillov dimension one.
There are theorems in the literature which are referred as Maschke type theorems, because of the attempt to generalize Maschke's classical theorem. Our remark is that all these theorems are in fact providing sufficient conditions for the separability of a certain forgetful functor. But, unlike classical Maschke's theorem, where the sufficient condition is also necessary, none of these theorems gives any necessary condition for the separability of the forgeful functor in question. Our job is to find a necessary and sufficient condition for the separability of the forgetful functor from the cate…
Given a ring $A$ and an $A$-coring $\cC$ we study when the forgetful functor from the category of right $\cC$-comodules to the category of right $A$-modules and its right adjoint $-\otimes_A\cC$ are separable. We then proceed to study when the induction functor $-\otimes_A\cC$ is also the left adjoint of the forgetful functor. This question is closely related to the problem when $A\to {}_A{\rm Hom}(\cC,A)$ is a Frobenius extension. We introduce the notion of a Galois coring and analyse when the tensor functor over the subring of $A$ fixed under the coaction of $\cC$ is an equivalence. We also …
In braided tensor categories we show the Maschke's theorem and give the necessary and sufficient conditions for double cross biproducts and crossbiproducts and biproducts to be bialgebras. We obtain the factorization theorem for braided Hopf algebras; In symmetric tensor categories, we show the duality theorem and construct quantum double; In ordinary vector space category with ordinary twist, we obtain the relation between the global dimension, the weak dimension, the Jacbson radical, Baer radical of algebra $R$ and its crossed product $R #_σH$. We also give the relation between the decomposi…
Study of the quotient module of a finite-dimensional Hopf subalgebra pair in order to compute its depth yields a relative Maschke Theorem, in which semisimple extension is characterized as being separable, and is therefore an ordinary Frobenius extension. We study the core Hopf ideal of a Hopf subalgebra, noting that the length of the annihilator chain of tensor powers of the quotient module is linearly related to the depth, if the Hopf algebra is semisimple. A tensor categorical definition of depth is introduced, and a summary from this new point of view of previous results are included. It i…
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.
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…
In the recent definition of Hom-Hopf algebras the antipode S is the relative Hominverse of the identity map with respect to the convolution product. We observe that some fundamental properties of the antipode of Hopf algebras and Hom-Hopf algebras, with the original definition, do not hold generally in the new setting. We show that the antipode is a relative Hom-anti algebra and a relative anti-coalgebra morphism. It is also relative Hom-unital, and relative Hom-counital. Furthermore if the twisting maps of multiplications and comultiplications are invertible then S is an anti-algebra and an a…
Let $(A,\a)$ and $(B,\b)$ be two Hom-Hopf algebras. In this paper, we construct a class of new Hom-Hopf algebras: $R$-smash product $(A\natural_R B,\aø\b)$. Moreover, necessary and sufficient conditions for $(A\natural_R B,\aø\b)$ to be a cobraided Hom-Hopf algebra are given.
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.
Let $(A,α)$ be a finite-dimensional Hom-Hopf algebra. In this paper we mainly construct the Drinfel'd double $D(A)=(A^{op}\bowtie A^{\ast},α\otimes(α^{-1})^{\ast})$ in the setting of Hom-Hopf algebras by two ways, one of which generalizes Majid's bicrossproduct for Hopf algebras (see \cite{M2}) and another one is to introduce the notion of dual pairs of of Hom-Hopf algebras. Then we study the relation between the Drinfel'd double $D(A)$ and Heisenberg double $H(A)=A\# A^{*}$, generalizing the main result in \cite{Lu}. Especially, the examples given in the paper are not obtained from the usual …
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.
Let $(H,\a)$ be a Hom-Hopf algebra and $(A,\b)$ be a Hom-algebra. In this paper we will construct the Hom-crossed product $(A\#_σH,\bø\a)$, and prove that the extension $A\subseteq A\#_σH$ is actually a Hom-type cleft extension and vice versa. Then we will give the necessary and sufficient conditions to make $(A\#_σH,\bø\a)$ a Hom-Hopf algebra. Finally we will study the lazy 2-cocycle on $(H,\a)$.
Hopf algebraic homogeneous spaces interpreted rationally: the Abe-Kanno theorem
将Abe-Kanno定理推广到Hopf代数，建立子Hopf代数与有理函数域不变子域之间的对应，并应用于剩余有限维Hopf代数的性质研究。
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.
In continuation of our recent work about smash product Hom-Hopf algebras in \cite{MLY}, we introduce Hom-Yetter-Drinfeld category $_H^H{\mathbb{YD}}$ via Radford biproduct Hom-Hopf algebra, and prove that the Hom-Yetter-Drinfeld modules can provide solutions of the Hom-Yang-Baxter equation and $_H^H{\mathbb{YD}}$ is a pre-braided tensor category, where $(H, \b, S)$ is a Hom-Hopf algebra. Furthermore, we obtain that $(A^{\natural}_{\diamond} H,\aø\b)$ is a Radford biproduct Hom-Hopf algebra if and only if $(A,\a)$ is a Hopf algebra in the category $_H^H{\mathbb{YD}}$. At last, some examples and…
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 $ 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 $…
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 $ 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 studied both the double cross product and the bicrossproduct constructions for the Hom-Hopf algebras of general $(α,β)$-type. This allows us to consider the universal enveloping Hom-Hopf algebras of Hom-Lie algebras, which are of $(α,{\rm Id})$-type. We show that the universal enveloping Hom-Hopf algebras of a matched pair of Hom-Lie algebras form a matched pair of Hom-Hopf algebras. We observe also that, the semi-dualization of a double cross product Hom-Hopf algebra is a bicrossproduct Hom-Hopf algebra. In particular, we apply this result to the universal enveloping Hom-Hopf algebras of a…
Some homological resultsin the category of colour $(A,H)$-Hopf modules
本文给出了颜色 $(A,H)$-Hopf 模作为分次 $H$-余模是内射的充分条件，并推广了 $(A,H)$-Hopf 模基本定理，证明了分次 $A^{coH}$-模范畴与颜色 $(A,H)$-Hopf 模范畴等价，以及 $A$ 作为分次右 $A^{coH}$-模是忠实平坦的，且是 $A^{coH}$ 的分次 Hopf-Galois 扩张，最后在假设下证明了 $A$ 的分次整体维数等于分次 $A$-模 $A^{coH}$ 的分次投射维数。
The aim of this paper is to generalize the concept of Lie-admissible coalgebra introduced by Goze and Remm to Hom-coalgebras and to introduce Hom-Hopf algebras with some properties. These structures are based on the Hom-algebra structures introduced by the authors.
We define a version of Hochschild homology and cohomology suitable for a class of algebras admitting compatible actions of bialgebras, called module algebras. We show this (co)homology, called Hopf--Hochschild (co)homology, can also be defined as a derived functor on the category of representations of a crossed product algebra. We investigate the relationship of our theory with Hopf cyclic cohomology and also prove Morita invariance of the Hopf--Hochschild (co)homology.
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.
Homological Integrals for Weak Hopf Algebras
本文引入了无限维弱Hopf代数的同调积分概念，并借此证明了若干结构定理，包括Artin-Schelter性质与Van den Bergh条件的等价性、对极映射的自动可逆性，以及关于仿射中心上有限弱Hopf代数的分解定理。
We introduce the Hom-analogue of the L-R-smash product and use it to define the Hom-analogue of the diagonal crossed product. When H is a finite dimensional Hom-Hopf algebra with bijective antipode and bijective structure map, we define the Drinfeld double of H; its algebra structure is a Hom-diagonal crossed product and it has all expected properties, namely it is quasitriangular and modules over it coincide with left-right Yetter-Drinfeld modules over H.
Hopf-Galois extensions of rings generalize Galois extensions, with the coaction of a Hopf algebra replacing the action of a group. Galois extensions with respect to a group $G$ are the Hopf-Galois extensions with respect to the dual of the group algebra of $G$. Rognes recently defined an analogous notion of Hopf-Galois extensions in the category of structured ring spectra, motivated by the fundamental example of the unit map from the sphere spectrum to $MU$. This article introduces a theory of homotopic Hopf-Galois extensions in a monoidal category with compatible model category structure that…
Measurings of Hopf algebroids and morphisms in cyclic (co)homology theories
本文研究Hopf代数胚及SAYD模上的测量，证明其诱导循环同调与循环上同调态射，并给出SAYD模范畴及非Σ operad上循环单位comp模的充实结构。
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}…
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。
Given a good homology theory E and a topological space X, the E-homology of X is not just an E_{*}-module but also a comodule over the Hopf algebroid (E_{*}, E_{*}E). We establish a framework for studying the homological algebra of comodules over a well-behaved Hopf algebroid (A, Gamma). That is, we construct the derived category Stable(Gamma) of (A, Gamma) as the homotopy category of a Quillen model structure on the category of unbounded chain complexes of Gamma-comodules. This derived category is obtained by inverting the homotopy isomorphisms, NOT the homology isomorphisms. We establish the…
The celebrated Milnor-Moore theorem and the more general Cartier-Kostant-Milnor-Moore theorem establish close relationships of a connected and a pointed cocommutative Hopf algebra with its Lie algebra of primitive elements and its group of group-like elements. Crossed homomorphisms for Lie algebras, groups and Hopf algebras have been studied extensively, first from a cohomological perspective and then more broadly, with an important case given by difference operators. This paper shows that the relationship among the different algebraic structures captured in the Milnor-Moore theorem can be str…
Hom-bialgebras and Hom-Hopf algebras are generalizations of bialgebra and Hopf algebra structures, where associativity and coassociativity conditions are twisted by a homomorphism. The purpose of this paper is to define a Gerstenhaber-Schack cohomology complex for Hom-bialgebras and then study one parameter formal deformations.
Integrals for Bialgebras
本文提出一种基于范畴方法的双代数积分新概念，证明其在存在antipode时退化为经典定义，并通过对多种特殊双代数的测试，建立了对偶Maschke型定理及有限维情况下的积分存在唯一性定理。
The notions of Hom-coring, Hom-entwining structure and associated entwined Hom-module are introduced. A theorem regarding base ring extension of a Hom-coring is proven and then is used to acquire the Hom-version of Sweedler coring. Motivated by the work of T. Brzezinski, a Hom-coring associated to an entwining Hom-structure is constructed and an identification of entwined Hom-modules with Hom-comodules of this Hom-coring is shown. The dual algebra of this Hom-coring is proven to be a $ψ$-twisted convolution algebra. By a construction, it is shown that a Hom-Doi-Koppinen datum comes from a Hom-…
Following the idea of an invariant differential complex, we construct general-type cyclic modules that provide the common denominator of known cyclic theories. The cyclicity of these modules is governed by Hopf-algebraic structures. We prove that the existence of a cyclic operator forces a modification of the Yetter-Drinfeld compatibility condition leading to the concept of a stable anti-Yetter-Drinfeld module. This module plays the role of the space of coefficients in the thus obtained cyclic cohomology of module algebras and coalgebras, and the cyclic homology and cohomology of comodule alge…
In this paper, we construct a bialgebraic and further a Hopf algebraic structure on top of subgraphs of a given graph. Further, we give the dual structure of this Hopf algebraic structure. We study the algebra morphisms induced by graph homomorphisms, and obtain a covariant functor from a graph category to an algebra category.
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