弱群余代数的交叉积

该文引入弱群交叉积的概念,并给出弱群交叉积代数和通常的张量积余代数构成弱半Hopf群余代数的充要条件,接着证明了弱群交叉积上的对偶定理,推广了沈和王~([7-8])的主要结果.     

Equivalent Crossed Products and Cross Product Bialgebras

In a previous article we proved a result of the type “invariance under twisting” for Brzeziński's crossed products. In this article we prove a converse of this result, obtaining thus a characterization of what we call equivalent crossed products. As an application, we characterize cross product bialgebras (in the sense of Bespalov and Drabant) that are equivalent (in a certain sense) to a given cross product bialgebra in which one of the factors is a bialgebra and whose coalgebra structure is a tensor product coalgebra.     

A class of new braided Hopf algebras

We give the necessary and sufficient conditions for a general crossed product algebra equipped with the usual tensor product coalgebra structure to be a Hopf algebra. Furthermore, we obtain the necessary and sufficient conditions for the general crossed product Hopf algebra to be a braided Hopf algebra which generalizes some known results.     

Deformation of Tensor Product (Co)Algebras via Non-(Co)Normal Twists

We study new coalgebra structures on the tensor product of two coalgebras C and D by twisting the tensor product coalgebra via a twist map Ψ: C ? D → D ? C. We deal with the general case in which the counit of the tensor product coalgebra is deformed as well. Some classes of such deformations are analysed, and a notion of equivalence of twists is discussed. We also present the dual deformation of tensor product algebras and provide examples.     

Hopf Algebra Structure Over Crossed Coproducts

The smash coproduct coalgebra has been generalized to crossed coproduct coalgebra in [3]. It is natural to replace the smash coproduct by the crossed coproduct and consider the conditions under which the smash product algebra structure and the crossed coproduct coalgebra structure will inherit a bialgebra structure or a Hopf algebra structure. We derive necessary and sufficient conditions for this problem. This generalizes the corresponding results in [7]. Finally, we characterize this new structure by introducing a concept of (H, )-comodule and prove that Heisenberg double [4] and smash coproduct do not make a bialgebra.AMS Subject Classification (1991): 16S40 16W30The first and second authors were partially supported by NNSF of China. The first author was also supported by the NSF of Henan Province and the second author by the YSF of Shandong Province (No. Q98A05113).     

The crossed coproduct theorem and Galois cohomology

We give a cohomological interpretation of the Brauer group of a coalgebra in terms of Galois coextensions and Galois cohomology. There is a crossed coproduct structure theorem, and the co-version of the classical splitting theorem holds for the Brauer group of an irreducible coreflexive coalgebra but it does not hold in general.     

交叉余积上的Hopf代数结构

本文给出了Ｓｍａｓｈ积代数结构和交叉余积余代数结构构成双代数的一个充分必要条件．另外,还给出了这一新的双代数成为Ｈｏｐｆ代数的一个充分条件．     

Morita Contexts and Partial Group Galois Extensions for Hopf Group Coalgebras

We first introduce the notion of partial group twisted smash product and construct a Morita context for partial coaction of co-Frobenius Hopf group coalgebra relating generalized partial group smash product and partial coinvariants. Furthermore, we study group-Galois extensions and reobtain some classical results in the partial case. Finally, we prove that any partial group Galois extension induces a unique partial group entwining map compatible with the partial right coaction.     

Preunits and weak crossed products

Using the notion of a preunit and the properties of idempotent morphisms, we give a general notion of a crossed product of an algebra A and an object V both living in a monoidal category C. We endow A⊗V with a multiplication and an idempotent morphism, whose image inherits the multiplication. Sufficient conditions for these multiplications to be associative are given. If the product on A⊗V has a preunit, the related idempotent is given in terms of the preunit, and its image has an algebra structure. A characterization of crossed products with preunit is given, and it is used to recover classical examples of crossed products and to study crossed products in weak contexts. Finally crossed products of an algebra by a weak bialgebra are recovered using this theory.     

The Notion of Vertex Operator Coalgebra and a Geometric Interpretation

The notion of vertex operator coalgebra is presented and motivated via the geometry of conformal field theory. Specifically, we describe the category of geometric vertex operator coalgebras, whose objects have comultiplicative structures meromorphically induced by conformal equivalence classes of worldsheets. We then show this category is isomorphic to the category of vertex operator coalgebras, which is defined in the language of formal algebra. The latter has several characteristics which give it the flavor of a coalgebra with respect to the structure of a vertex operator algebra and several characteristics that distinguish it from a standard dual—both of them will be highlighted.     

Cofree Compositions of Coalgebras

We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and in the theory of species. We prove that the composition of two cofree coalgebras is again cofree, and we give sufficient conditions that ensure the composition is a one-sided Hopf algebra. We show that these conditions are satisfied when one coalgebra is a graded Hopf operad ${\mathcal{D}}$ and the other is a connected graded coalgebra with coalgebra map to ${\mathcal{D}}$ . We conclude by computing the primitive elements for compositions of coalgebras built on the vertices of multiplihedra, composihedra, and hypercubes.     

(*)-Serial coalgebras

In this paper, we introduce the notion of (*)-serial coalgebras which is a generalization of serial coalgebras. We investigate the properties of (*)-serial coalgebras and their comodules, and obtain sufficient and necessary conditions for a basic coalgebra to be (*)-serial.     

Valued Gabriel quiver of a wedge product and semiprime coalgebras

We describe the valued Gabriel quiver of a wedge product of coalgebras and study the category of comodules of a semiprime coalgebra. In particular, we prove that any monomial semiprime k-tame fc-tame coalgebra is string. We also prove a version of Eisenbud-Griffith theorem for coalgebras, namely, any hereditary semiprime strictly quasi-finite coalgebra is serial.     

Antipodes and incidence coalgebras

We introduce the notion of a reduced incidence coalgebra of a family of locally finite partially ordered sets and describe how bialgebras and Hopf algebras arise as such coalgebras. In the case when a reduced incidence coalgebra is a Hopf algebra, we give explicit recursive and closed formulas for the antipode, each of which generalizes a corresponding formula from the theory of Möbius functions. We give applications to the inversion of ordinary formal power series and inversion of Dirichlet series. We also formulate the classical Lagrange inversion formula as a combinatorial formula for the antipode of a Hopf algebra arising from the family of finite partition lattices.     

Associated Graded Algebras and Coalgebras

We investigate the notion of associated graded coalgebra (algebra) of a bialgebra with respect to a subbialgebra (quotient bialgebra) and characterize those which are bialgebras of type one in the framework of abelian braided monoidal categories.     

On the wedge product and coprime coalgebras

The wedge product of subcoalgebras of a coalgebra can be used to define coprime coalgebras. On the other hand, coprime elements in the big lattice of preradicals in module categories also lead to the definition of coprime modules. Considering a coalgebra C as a module over its dual algebra C*, this yields another notion of coprimeness for coalgebras. Under special conditions, the two definitions coincide. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 45–49, 2005.     

Cotensor Coalgebras in Monoidal Categories

We introduce the concept of cotensor coalgebra for a given bicomodule over a coalgebra in an Abelian monoidal category ?. If ? is also cocomplete, complete, and AB5, we show that such a cotensor coalgebra exists and satisfies a meaningful universal property which resembles the classical one. Here the lack of the coradical filtration is filled by considering a direct limit of a filtration consisting of wedge products. We prove that this coalgebra is formally smooth whenever the comodule is relative injective and the coalgebra itself is formally smooth.     

Evolution coalgebras

We introduce the notion of evolution coalgebra as an algebraic structure to describe the backwards evolution of non-Mendelian genetic systems and establish their connection with evolution algebras. We study their main properties and consider evolution coalgebras endowed with genetic realization.     

余代数上的Green等价

本文在余代数上定义了五类等价关系,它们是Ｇｒｅｅｎ等价Ｇ,Ｄ,Ｌ,Ｒ,Ｈ．然 后给出了这些等价关系一些基本性质和结构特点．在每个Ｇｒｅｅｎ等价的等价类集上构 造了一种偏序并给出了偏序上半格和格结构的刻划．用Ｇ－,Ｌ－,Ｒ－类分别给出了子余 代数、左余理想、右余理想的结构刻划．进一步地,本文研究了张量积上的Ｇｒｅｅｎ等 价以及余代数同态的Ｇｒｅｅｎ保持性和提升性．作为应用,本文得到了两个不可约余代 数的张量积仍为不可约余代数的一个条件;证明了不可约余代数在Ｇ－保持余代数同 态下的同态像是不可约的．