Hilbert C*-模上框架(强)可补的充要条件

Hilbert C*-模上框架的框架变换的实质是将该模进行膨胀,使得该框架变换的值域存在标准正交基,以便于Hilbert C*-模上不同框架之间关系的研究.受此启发,本文引入了Hilbert C*-模上框架(强)可补的概念,给出并证明了Hilbert C*-模上有限个框架(强)可补的充要条件.     

半连续格上的半基和局部半基

在完备格上引入了半基和局部半基的概念,给出了半基和局部半基的性质及若干等价刻画,证明了一个完备格是半连续格当且仅当它具有半基也当且仅当它每点有局部半基。在此基础上定义了半连续格的权和特征,探讨了半连续格的权和特征与其上赋予半Scott拓扑和半Lawson拓扑时的拓扑空间的权和特征的关系。解决了文献[8](赵彬,刘妮.连续Domain的特征和浓度,陕西师范大学学报,2002,30(2):1~6)中提出的一个问题。     

K[X]~m中模的生成基及其应用

本文研究了多项式环上的素模中的生成基理论和方法．通过建立新的约化准则,得到了模中生成基的结构和机械化计算方法．对于低维情形给出了素模中生成基的充分必要条件．文中的方法本质地简化了传统的模中Ｇｒｏｂｎｅｒ基方法．文中同时介绍了该方法在样条理论研究中的应用,并给出了一些计算例子．     

Young表与Uq（osp（1｜2n））的晶体基

设Uq（osp（1｜2n））是对应Lie超代数osp（1｜2n）的量子包络超代数．利用满足一定条件的半标准Young表,给出有限维既约Uq（osp（1｜2n））模晶体图的实现．建立晶体图张量积分解的广义Littlewood—Richardson法则．     

K[X]^m中模的生成基及其应用

本文研究了多项式环上的素模中的生成基理论和方法。通过建立新的约化准则,得到了模中生成基的结构和机械计算方法。对于低维情形给出了素右生成基的充分必要条件。文中的方法本质地简化了传统的模中Grobner基方法。文中同时介绍了该方法在样条理论研究中的应用,并给出了一些计算例子。     

循环差分-微分模上双变元维数多项式的Gr?bner基算法北大核心CSCD

Gr?bner基算法是在计算机辅助设计和机器人学、信息安全等领域广泛应用的重要工具.文章在周梦和Winkler(2008)给出的差分-微分模上Gr?bner基算法和差分-微分维数多项式算法基础上,进一步研究了分别差分部分和微分部分的双变元维数多项式算法.在循环差分-微分模情形,构造和证明了利用差分-微分模上Gr?bner基计算双变元维数多项式的算法.     

差分-微分模上的Grbner基及差分-微分维数多项式

通过引入广义单项式序把Grbner基理论拓展到差分-微分模上,构造和证明了差分-微分模上Grbner基算法.然后利用差分-微分模上的Grbner基构造了线性差分-微分方程系的维数多项式算法.     

差分-微分模上的Gröbner基及差分-微分维数多项式

通过引入广义单项式序把Gröbner基理论拓展到差分\!-\!微分模上, 构造和证明了差分\!-\!微分模上Gröbner基算法. 然后利用差分-微分模上的Gröbner基构造了线性差分-微分方程系的维数多项式算法.     

Young表与$\textrm{U}_{q}(\mathrm{osp}(1|2n))$的晶体基

设 $U_q(\mathrm{osp}(1|2n))$是对应Lie超代数$\mathrm{osp}(1|2n)$的量子包络超代数. 利用满足一定条件的半标准Young表, 给出有限维既约$U_q(\mathrm{osp}(1|2n))$模晶体图的实现 . 建立晶体图张量积分解的广义Littlewood-Richardson法则.     

加法范畴的Jacobson结构定理

本文从环论的角度来研究加法范畴的结构,对加法范畴完整地给出相应于环的Jacobson理论的结构定理。定义加法范畴上的模与Jacdbson根,证明本原加法范畴都是稠密线性变换加法范畴等。     

Standard Bases of a Vector Space Over a Linearly Ordered Incline

Inclines are additively idempotent semirings, in which the partial order ≤ : x ≤ y if and only if x + y = y is defined and products are less than or equal to either factor. Boolean algebra, max-min fuzzy algebra, and distributive lattices are examples of inclines. In this article, standard bases of a finitely generated vector space over a linearly ordered commutative incline are studied. We obtain that if a standard basis exists, then it is unique. In particular, if the incline is solvable or multiplicatively-declined or multiplicatively-idempotent (i.e., a chain semiring), further results are obtained, respectively. For a chain semiring a checkable condition for distinguishing if a basis is standard is given. Based on the condition an algorithm for computing the standard basis is described.     

半环上矩阵的秩和坡上矩阵可逆的条件

研究了交换半环上矩阵的秩和坡上矩阵的可逆条件.利用Beasley的引理以及不变式,获得了交换半环上正则矩阵的行秩、列秩与Schein秩三者相等,以及坡上矩阵可逆的充要条件.推广模糊代数和分配格上矩阵的结果.     

Algorithmic Complexity of a Problem of Idempotent Convex Geometry

Properties of the idempotently convex hull of a two-point set in a free semimodule over the idempotent semiring R _{max min} and in a free semimodule over a linearly ordered idempotent semifield are studied. Construction algorithms for this hull are proposed.     

Direct sums of injective semimodules and direct products of projective semimodules over semirings

We prove that if the direct sum of a family of semimodules over a semiring S is an injective semimodule or if the direct product of a family of semimodules over S is a projective semimodule, then the cardinality of the subfamily consisting of all semimodules which are not modules is strictly less than the cardinality of S. As a consequence, we obtain semiring analogs of well-known characterizations of classical semisimple, quasi-Frobenius, and one-sided Noetherian rings.     

Embedding Sums of Cancellative Modes into Semimodules

A mode (idempotent and entropic algebra) is a Lallement sum of its cancellative submodes over a normal band if it has a congruence with a normal band quotient and cancellative congruence classes. We show that such a sum embeds as a subreduct into a semimodule over a certain ring, and discuss some consequences of this fact. The result generalizes a similar earlier result of the authors proved in the case when the normal band is a semilattice. The paper was written within the framework of COST Action 274. Research supported by Warsaw University of Technology under grant number 504G/1120/0008/000.     

Graded central polynomials for the matrix algebra of order two

Let K be an infinite integral domain, and let A = M ₂(K) be the matrix algebra of order two over K. The algebra A can be given a natural -grading by assuming that the diagonal matrices are the 0-component while the off-diagonal ones form the 1-component. In this paper we study the graded identities and the graded central polynomials of A. We exhibit finite bases for these graded identities and central polynomials. It turns out that the behavior of the graded identities and central polynomials in the case under consideration is much like that in the case when K is an infinite field of characteristic 0 or p > 2. Our proofs are characteristic-free so they work when K is an infinite field, char K = 2. Thus we describe finite bases of the graded identities and graded central polynomials for M ₂(K) in this case as well. A. Krasilnikov has been partially supported by CNPq and FINATEC.     

Composition-diamond lemma for modules

We investigate the relationship between the Gröbner-Shirshov bases in free associative algebras, free left modules and “double-free” left modules (that is, free modules over a free algebra). We first give Chibrikov’s Composition-Diamond lemma for modules and then we show that Kang-Lee’s Composition-Diamond lemma follows from it. We give the Gröbner-Shirshov bases for the following modules: the highest weight module over a Lie algebra sl ₂, the Verma module over a Kac-Moody algebra, the Verma module over the Lie algebra of coefficients of a free conformal algebra, and a universal enveloping module for a Sabinin algebra. As applications, we also obtain linear bases for the above modules.     

Based algebras and standard bases for quasi-hereditary algebras

Quasi-hereditary algebras can be viewed as a Lie theory approach to the theory of finite dimensional algebras. Motivated by the existence of certain nice bases for representations of semisimple Lie algebras and algebraic groups, we will construct in this paper nice bases for (split) quasi-hereditary algebras and characterize them using these bases. We first introduce the notion of a standardly based algebra, which is a generalized version of a cellular algebra introduced by Graham and Lehrer, and discuss their representation theory. The main result is that an algebra over a commutative local noetherian ring with finite rank is split quasi-hereditary if and only if it is standardly full-based. As an application, we will give an elementary proof of the fact that split symmetric algebras are not quasi-hereditary unless they are semisimple. Finally, some relations between standardly based algebras and cellular algebras are also discussed.     

On the First Hochschild Cohomology linebreak of Admissible Algebras

The aim of this paper is to investigate the first Hochschild cohomology of admissible algebras which can be regarded as a generalization of basic algebras. For this purpose, the authors study differential operators on an admissible algebra. Firstly, differential operators from a path algebra to its quotient algebra as an admissible algebra are discussed. Based on this discussion, the first cohomology with admissible algebras as coefficient modules is characterized, including their dimension formula. Besides, for planar quivers, the $k$-linear bases of the first cohomology of acyclic complete monomial algebras and acyclic truncated quiver algebras are constructed over the field $k$ of characteristic $0$.