Lan You, Hui-Xiang Chen
In this article, we study Dorroh extensions of algebras and Dorroh extensions of coalgebras. Their structures are described. Some properties of these extensions are presented. We also introduce the finite duals of algebras and modules which are not necessarily unital. Using these finite duals, we determine the dual relations between the two kinds of extensions.
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In this paper, we study Dorroh extensions of bialgebras and Hopf algebras. Let $(H,I)$ be both a Dorroh pair of algebras and a Dorroh pair of coalgebras. We give necessary and sufficient conditions for $H\ltimes_dI$ to be a bialgebra and a Hopf algebra, respectively. We also describe all ideals of Dorroh extensions of algebras and subcoalgebras of Dorroh extensions of coalgebras and compute these ideals and subcoalgebras for some concrete examples.
Cohomologies, non-abelian extensions and Wells sequences of lambda-weighted Rota-Baxter Lie coalgebras
本文研究λ-权Rota-Baxter李余代数的上同调与非阿贝尔扩张，完成分类、推导Wells基本序列并讨论阿贝尔扩张情形下的对应结果。
Coalgebra-Galois extensions generalise Hopf-Galois extensions, which can be viewed as non-commutative torsors. In this paper it is analysed when a coalgebra-Galois extension is a separable, split, or strongly separable extension.
For any k-coalgebra C it is shown that similar quasi-finite C-comodules have strongly equivalent coendomorphism coalgebras; (the converse is in general not true). As an application we give a general result about codepth two coalgebra homomorphisms. Also a notion of codepth two bicomodule is introduced. The last section applies similarity to an endomorphism ring theorem for quasi-Frobenius (QF) bimodules and then to finite depth ring extensions. For QF extensions, we establish that left and right depth two are equivalent notions as well as a converse endomorphism theorem, and characterize depth…
The notion of a coalgebra-Galois extension is defined as a natural generalisation of a Hopf-Galois extension. It is shown that any coalgebra-Galois extension induces a unique entwining map $ψ$ compatible with the right coaction. For the dual notion of an algebra-Galois coextension it is also proven that there always exists a unique entwining structure compatible with the right action.
We use prime coalgebras as a generalization of simple coalgebras, and observe that prime subcoalgebras represent the structure of the coalgebra in a more efficient way than simple coalgebras. In particular, in this work we focus our attention on the study and characterization of prime subcoalgebras of path coalgebras of quivers and, by extension, of prime pointed coalgebras.
For a given Hopf algebra $A$ we classify all Hopf algebras $E$ that are coalgebra split extensions of $A$ by $H_4$, where $H_4$ is the Sweedler's 4-dimensional Hopf algebra. Equivalently, we classify all crossed products of Hopf algebras $A # H_4$ by computing explicitly two classifying objects: the cohomological 'group' ${\mathcal H}^{2} (H_4, A)$ and $\textsc{C}\textsc{r}\textsc{p} (H_4, A) :=$ the set of types of isomorphisms of all crossed products $A # H_4$. All crossed products $A #H_4$ are described by generators and relations and classified: they are parameterized by the set ${\mathcal…
Localisation is an important technique in ring theory and yields the construction of various rings of quotients. Colocalisation in comodule categories has been investigated by some authors where the colocalised coalgebra turned out to be a suitable subcoalgebra. Rather then aiming at a subcoalgebra we look at possible coalgebra covers p:D->>C that could play the role of a dual quotient object. Codense covers will dualise dense (or rational) extensions; a maximal codense cover construction for coalgebras with projective covers is proposed. We also look at a dual non-singularity concept for modu…
We prove a number of results concerning monomorphisms, epimorphisms, dominions and codominions in categories of coalgebras. Examples include: (a) representation-theoretic characterizations of monomorphisms in all of these categories that when the Hopf algebras in question are commutative specialize back to the familiar necessary and sufficient conditions (due to Bien-Borel) that a linear algebraic subgroup be epimorphically embedded; (b) the fact that a morphism in the category of (cocommutative) coalgebras, (cocommutative) bialgebras, and a host of categories of Hopf algebras has the same cod…
We investigate when the categories of all rational $A$-modules and of finite dimensional rational modules are closed under extensions inside the category of $C^*$-modules, where $C^*$ is the cofinite topological completion of $A$. We give a complete characterization of these two properties, in terms of a topological and a homological condition. We also give connections to other important notions in coalgebra theory such as coreflexive coalgebras. In particular, we are able to generalize many previously known partial results and answer some questions in this direction, and obtain large classes …
Dual coalgebras of twisted tensor products
研究扭曲张量积的有限双煤格是否为各自有限双煤格的cotwisted张量积，通过将有限双格视为拓扑对偶，证明连续对偶是线性拓扑向量空间的强单oidal函子，并给出应用条件。
We study rational modules over complete path and monomial algebras, and the problem of when rational modules over the dual $C^*$ of a coalgebra $C$ are closed under extensions, equivalently, when is the functor $Rat$ a torsion functor. We show that coreflexivity, closure under extensions of finite dimensional rational modules and of arbitrary modules are Morita invariant, and that they are preserved by subcoalgebras. We obtain new large classes of examples of coalgebras with torsion functor, coming from monomial coalgebras, and answer some questions in the literature.
We study generalized comatrix coalgebras and upper triangular comatrix coalgebras, which are not only a dualization but also an extension of classical generalized matrix algebras. We use these to answer several questions on Noetherian and Artinian type notions in the theory of coalgebras, and to give complete connections between these. We also solve completely the so called finite splitting problem for coalgebras: we show that a coalgebra $C$ has the property that the rational part of every finitely generated left $C^*$-module splits off if and only if $C$ has the form $C=\left(\begin{array}{c…
Quantum group coproducts and universality under scalar extensions
本文刻画了域上双代数、Hopf代数及余代数族乘积有限维的条件，并纠正了标量扩张保持乘积的文献错误，明确了有限与代数扩张的普适性特征。
The unified product was defined in \cite{am3} related to the restricted extending structure problem for Hopf algebras: a Hopf algebra $E$ factorizes through a Hopf subalgebra $A$ and a subcoalgebra $H$ such that $1\in H$ if and only if $E$ is isomorphic to a unified product $A \ltimes H$. Using the concept of normality of a morphism of coalgebras in the sense of Andruskiewitsch and Devoto we prove an equivalent description for the unified product from the point of view of split morphisms of Hopf algebras. A Hopf algebra $E$ is isomorphic to a unified product $A \ltimes H$ if and only if there …
A Morita context is constructed for any comodule of a coring and, more generally, for an $L$-$\cC$ bicomodule $Σ$ for a pure coring extension $(\cD:L)$ of $(\cC:A)$. It is related to a 2-object subcategory of the category of $k$-linear functors $\Mm^\Cc\to\Mm^\Dd$. Strictness of the Morita context is shown to imply the Galois property of $Σ$ as a $\cC$-comodule and a Weak Structure Theorem. Sufficient conditions are found also for a Strong Structure Theorem to hold. Cleft property of an $L$-$\cC$ bicomodule $Σ$ -- implying strictness of the associated Morita context -- is introduced. It is sho…
Let K be a comonad on a model category M. We provide conditions under which the associated category of K-coalgebras admits a model category structure such that the forgetful functor to M creates both cofibrations and weak equivalences. We provide concrete examples that satisfy our conditions and are relevant in descent theory and in the theory of Hopf-Galois extensions. These examples are specific instances of the following categories of comodules over a coring. For any semihereditary commutative ring R, let A be a dg R-algebra that is homologically simply connected. Let V be an A-coring that …
We introduce and study a general concept of integral of a threetuple (H, A, C), where H is a Hopf algebra acting on a coalgebra C and coacting on an algebra A. In particular, quantum integrals associated to Yetter-Drinfel'd modules are defined. Let A be an H-bicomodule algebra, $^H {\cal YD}_A$ be the category of (generalized) Yetter-Drinfel'd modules and $B$ the subalgebra of coinvariants of the Verma structure of $A$. We introduce the concept of quantum Galois extensions and we prove the affineness criterion in a quantum version.
In previous work, to each Hopf algebra H and each invertible right two-cocycle on H, Eli Aljadeff and the first-named author attached a subalgebra B of the free commutative Hopf algebra S generated by the coalgebra underlying H; the algebra B is the subalgebra of coinvariants of a generic Hopf Galois extension. In this paper we give conditions under which S is faithfully flat, or even free, as a B-module. We also show that B is generated as an algebra by certain elements arising from the theory of polynomial identities for comodule algebras developped jointly with Aljadeff.
Amenability of Lie, Group and Hopf algebras
本文通过泛包络代数中几乎不变的有限维子余代数定义Lie代数可均性，证明其与Elek定义等价，并推广至Hopf代数，验证Gromov猜想及多种封闭性质。
This paper is a written form of a talk. It gives a review of various notions of Galois (and in particular cleft) extensions. Extensions by coalgebras,bialgebras and Hopf algebras (over a commutative base ring) and by corings,bialgebroids and Hopf algebroids (over a non-commutative base algebra) are systematically recalled and compared. In the first version of this paper, the journal version of [15, Theorem 2.6] was heavily used, in two respects. First, it was applied to establish an isomorphism between the comodule categories of two constituent bialgebroids in a Hopf algebroid. Second, it was …
Wreath products of cocommutative Hopf algebras
本文定义余交换Hopf代数的圈积，证明其具有类似Kaloujnine-Krasner定理的泛性质，并用于分类群扩张和李代数扩张。
We classify the twisted tensor products of a finite set algebra with a two elements set algebra using colored quivers obtained through considerations analogous to Ore extensions. This provides also a classification of entwining structures between a finite set algebra and the grouplike coalgebra on two elements. The resulting 2-nilpotent algebras have particular features with respect to Hochschild (co)homology and cyclic homology.
Hopf formulae for cocommutative Hopf algebras
本文利用余代数与Hopf代数间的伴随关系，证明余交换Hopf代数具有足够的投射对象，并借助范畴Galois理论分类正规扩张，给出基本群的广义Hopf公式及同调5项正合序列。
We show that the functor of $p$-typical co-Witt vectors on commutative algebras over a perfect field $k$ of characteristic $p$ is defined on, and in fact only depends on, a weaker structure than that of a $k$-algebra. We call this structure a $p$-polar $k$-algebra. By extension, the functors of points for any $p$-adic affine commutative group scheme and for any formal group are defined on, and only depend on, $p$-polar structures. In terms of abelian Hopf algebras, we show that a cofree cocommutative Hopf algebra can be defined on any $p$-polar $k$-algebra $P$, and it agrees with the cofree co…
The $n$-Lie bialgebras are studied. In Section 2, the $n$-Lie coalgebra with rank $r$ is defined, and the structure of it is discussed. In Section 3, the $n$-Lie bialgebra is introduced. A triple $(L, μ, Δ)$ is an $n$-Lie bialgebra if and only if $Δ$ is a conformal $1$-cocycle on the $n$-Lie algebra $L$ associated to $L$-modules $(L^{\otimes n}, ρ_s^μ)$, $1\leq s\leq n$, and the structure of $n$-Lie bialgebras is investigated by the structural constants. In Section 4, two-dimensional extension of finite dimensional $n$-Lie bialgebras are studied. For an $m$ dimensional $n$-Lie bialgebra $(L, μ…
Twisted unipotent groups
本文研究仿射代数幺幂群上带Hopf 2-上循环形变的余三角Hopf代数的结构与表示，证明其为迭代导子型Hopf Ore扩张，给出交叉积分解、单模分层及有限维单模的群结构。
Recantly, William Crawley-Boevey proposed the definition of a Poisson structure on a noncommutative algebra $A$ based on the Kontsevich principle. His idea was to find the {\it weakest} possible structure on $A$ that induces standard (commutative) Poisson structures on all representation spaces $ \Rep_V(A) $. It turns out that such a weak Poisson structure on $A$ is a Lie algebra bracket on the 0-th cyclic homology $ \HC_0(A) $ satisfying some extra conditions; it was thus called in an {\it $ H_0$-Poisson structure}. This paper studies a higher homological extension of this construction. In ou…
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