李Rinehart代数的子代数若干性质

本文主要把李代数的c-可补、E-代数的性质以及Frattini理论推广到更为广泛的李Rinehart代数,得到它们的若干性质,给出了可解李Rinehart代数的一个必要条件.同时,分别获得判断c-可补李Rinehart代数和E-李Rinehart代数的一个充分必要条件.     

n+1维n-Lei代数的导子代数

本文给出了n 1维n-Lie代数的导子代数的结构，以及导子的具体表示形式．     

Kleene代数的诱导子代数及理想

由于逻辑学研究的需要 ,Kleene代数理论的重要性已越来越明显 .本文主要研究了Kleene代数的子代数及理想的一些性质 ,得到了一些比较好的结果 .     

三元代数的乘子代数

将L.Harris在[1]中关于三元代数的概念一般化，并且提出了相对近似单位元，相对Arens乘积，相对m-对称等概念，最后得到一个同构定理。     

5维Bernstein-Jordan代数的导子代数

Ⅰ．Ｃｏｒｒｅａ和Ｌ．Ａ．Ｐｅｒｅｓｉ对实数域上的５维Ｂｅｒｎｓｔｅｉｎ－Ｊｏｒｄａｎ代数进行了分类．本文在此基础上，确定了这些代数的导子代数     

三维Leibniz代数的分类

Leibniz代数是比Lie代数更广泛的一类代数,它通常不满足反交换性.在这篇文章里我们确定了维数等于3的Leibniz代数的同构类.     

辛三代数的Frattini子代数

本文研究了辛三代数的Frattini子代数和基本辛三代数的问题.利用Frattini子代数和基本辛三代数的性质,得到了辛三代数的非嵌入定理,从而推广了李三系中关于Frattini子系的结果.     

辫子Hopf代数的因子分解

我们获得了辫子双代数和辫子Hopf代数的双重分解, 给出了辫子Hopf代数的积分和半单性与它的因子的积分和半单性之间的关系.     

A Class of Multiplier Hopf Algebras

We compute the Drinfel’d double for the bicrossproduct multiplier Hopf algebra A = k[G] ⋊ K(H) associated with the factorization of an infinite group M into two subgroups G and H. We also show that there is a basis-preserving self-duality structure for the multiplier Hopf algebra A = k[G] ⋊ K(H) if there is a factor-reversing group isomorphism. Presented by A. Verschoren.     

Pairing and Quantum Double of Multiplier Hopf Algebras

We define and investigate pairings of multiplier Hopf (*-)algebras which are nonunital generalizations of Hopf algebras. Dual pairs of multiplier Hopf algebras arise naturally from any multiplier Hopf algebra A with integral and its dual Â. Pairings of multiplier Hopf algebras play a basic rôle, e.g., in the study of actions and coactions, and, in particular, in the relation between them. This aspect of the theory is treated elsewhere. In this paper we consider the quantum double construction out of a dual pair of multiplier Hopf algebras. We show that two dually paired regular multiplier Hopf (*-)algebras A and B yield a quantum double which is again a regular multiplier Hopf (*-)algebra. If A and B have integrals, then the quantum double also has an integral. If A and B are Hopf algebras, then the quantum double multiplier Hopf algebra is the usual quantum double. The quantum double construction for dually paired multiplier Hopf (*-)algebras yields new nontrivial examples of multiplier Hopf (*-)algebras.     

Constructing Quasitriangular Multiplier Hopf Algebras By Twisted Tensor Coproducts

Let A and B be multiplier Hopf algebras, and let R ∈ M(B ? A) be an anti-copairing multiplier, i.e, the inverse of R is a skew-copairing multiplier in the sense of Delvaux [5 Delvaux , L. ( 2004 ). Twisted tensor coproduct of multiplier Hopf (*)-algebras . J. Algebra 274 : 751 – 771 . [Google Scholar]]. Then one can construct a twisted tensor coproduct multiplier Hopf algebra A ? _R  B. Using this, we establish the correspondence between the existence of quasitriangular structures in A ? _R  B and the existence of such structures in the factors A and B. We illustrate our theory with a profusion of examples which cannot be obtained by using classical Hopf algebras. Also, we study the class of minimal quasitriangular multiplier Hopf algebras and show that every minimal quasitriangular Hopf algebra is a quotient of a Drinfel’d double for some algebraic quantum group.     

Traces on Multiplier Hopf Algebras

In this article we lay the algebraic foundations to establish the existence of trace functions on infinite-dimensional (multiplier) Hopf algebras. We solve the problem within the framework of multiplier Hopf algebra with integrals. By applying this theory to group-cograded multiplier Hopf algebras, we prove the existence of group-traces on group-cograded multiplier Hopf algebras with possibly infinite-dimensional components. We generalize the results as obtained by Virelizier in the case of finite-type Hopf group-coalgebras.     

On the Twisting and Drinfel'd Double for Multiplier Hopf Algebras

We define the concept of “semiprime” for preradicals and for submodules, and we prove some properties that relate both of them. Related concepts are defined in article by Bican et al. [2 Bican , L. , Jambor , P. , Kepka , T. , Nemec , P. ( 1980 ). Prime and coprime modules . Fundamenta Mathematicae CVII , 33 – 45 . [Google Scholar]] and by Van den Berg and Wisbauer [9 Van den Berg , J. , Wisbauer , R. ( 2001 ). Duprime and dusemiprime modules . Journal of Pure and Applied Algebra 165 : 337 – 356 .[Crossref], [Web of Science ®] , [Google Scholar]]. For any ring, we compare the least semiprime preradical, the Jacobson radical and the join of all nilpotent preradicals, and we characterize V-rings in terms of these three preradicals. We study the least semiprime preradical above any preradical and we prove some of its properties. Using “Amitsur constructions” we define another related operators and prove some of their properties.     

Ore Extensions of Multiplier Hopf Algebras

The goal of this article is to generalize the theory of Hopf–Ore extensions on Hopf algebras to multiplier Hopf algebras. First the concept of a Hopf–Ore extension of a multiplier Hopf algebra is introduced. We give a necessary and sufficient condition for Ore extensions to become a multiplier Hopf algebra. Finally, *-structures are constructed on Hopf–Ore extensions, and certain isomorphisms between Hopf–Ore extensions are discussed.     

A Lot of Quasitriangular Group-cograded Multiplier Hopf Algebras

A new method of constructing quasitriangular group-cograded multiplier Hopf algebras is provided. For a multiplier Hopf dual pairing σ between regular multiplier Hopf algebras A and B, we introduce the concept of a σ-compatible pairing (Φ, Ψ, σ ), where Φ and Ψ are actions of the twisted semi-direct group of a group G on A and B, respectively. We construct a twisted double group-cograded multiplier Hopf algbera D(A, B; σ, Φ, Ψ). Furthermore, if there is a canonical multiplier in M(B ⊗ A) we show existence of quasitriangular structure on D(A, B; σ, Φ, Ψ). As an application, special cases and examples are given.     

乘子Hopf代数Ore扩张的余积分

余积分是Hopf代数和乘子Hopf代数中的一类特殊元素,它的良好性质在研究Hopf代数的半单和余半单中有着很重要的作用.研究了乘子Hopf代数Ore扩张上的余积分,给出余积分的存在形式及其存在性.     

Galois Theory for Multiplier Hopf Algebras with Integrals

In this paper, we introduce a generalized Hopf Galois theory for regular multiplier Hopf algebras with integrals, which might be viewed as a generalization of the Hopf Galois theory of finite-dimensional Hopf algebras. We introduce the notion of a coaction of a multiplier Hopf algebra on an algebra. We show that there is a duality for actions and coactions of multiplier Hopf algebras with integrals. In order to study the Galois (co)action of a multiplier Hopf algebra with an integral, we construct a Morita context connecting the smash product and the coinvariants. A Galois (co)action can be characterized by certain surjectivity of a canonical map in the Morita context. Finally, we apply the Morita theory to obtain the duality theorems for actions and coactions of a co-Frobenius Hopf algebra.