斜群代数与平凡扩张的表示维数

设A是域k上的一个有限维自入射代数,G是一个有限群,且G的阶在A中可逆,A*G是斜群代数,T(A)是平凡扩张代数,∧V是外代数.本文证明了A的稳定范畴与A*G的稳定范畴的三角维数相等,得到了∧V*G及T(∧V*G)的表示维数.     

完备广义代数群的构造

除了一个例外, 有限维可换连通代数群的全形都被证明是完备广义代数群. 这个结果与Lie代数的相应结果基本一致.     

素特征域上广义Witt李超代数的自同构群

设W是素特征域上无限维或有限维广义Witt李超代数.本文利用W的自然滤过不变性和W的底代数的不变维数性质,证明了W的自同构群AutW同构于W的底代数的容许自同构群,还证明了在此群同构之下,AutW的标准正规列恰好对应W的底代数的容许自同构群的标准正规列,并给出AutW若干较为细致的性质.     

交叉积群代数有限维数问题的一点注记

设G是一个有限群,k是一个代数闭域且k的特征不整除G的阶.Λ是一个扭kG-模代数,Λ*G是一个交叉积代数.该文证明Λ*G和Λ具有相同的有限维数,且同时满足有限维数猜想定理.     

广义弱群像代数

本文首先介绍了广义弱群像代数的概念, 并构造了一系列具体的例子, 通过例子可以看出有限群胚代数和群像代数都可以看作是广义弱群像代数; 接着本文证明了广义弱群像代数也可以看作是一类广义弱双Frobenius代数, 并探讨了广义弱双Frobenius代数成为有限群胚代数的条件; 最后本文给出了低维广义弱群像代数的具体结构.     

有限维李代数的结构常数与立方阵

用有限维李代数的结构常数及相对应的立方阵来刻画李代数的若干性质,给出了求李代数的自同构群和导子群的新方法以及李代数的结构常数法扩张.     

非单W和S型李超代数的滤过和分类

设F是特征p3的域,g是F上外代数与有限维广义Witt李代数或特殊李代数的张量积所构成的有限维非单李超代数.本文通过确定ad幂零元的方法证明了g的标准滤过是g的自同构群的不变量,进而证明了定义这类李超代数的参数组是内蕴的,从而给出了这两类李超代数在同构意义下的分类.     

有限自动机的积与路代数的张量积

讨论了两个有限自动机积的路代数,得出了这些积的路代数与这两个有限自动机路代数的张量积的一些关系.     

有限维代数扩张的同构与维数群

对于有限维C＊-代数A，证明了其本质扩张的同构与酉等价是一致的，由此证明了扩张群Ext（A）中的等价类是区分该类扩张代数的完全不变量，并利用Bratteli图计算出它们的维数群．     

(AF)实C~*-代数的等价定义

本文给出了有限维实Ｃ~*-代数复化中标准矩阵单位基的描述,继而给出了（ＡＦ）实Ｃ~*-代数的等价定义．     

Κ1 Group of Finite Dimensional Path Algebra

In this paper, by calculating the commutator subgroup of the unit group of finite path algebra κΔ and the unit group abelianized, we explicitly characterize the Κ₁ group of finite dimensional path algebra over an arbitrary field. Received November 2, 1998, Accepted May 7, 1999     

The Natural Quiver of an Artinian Algebra

The motivation of this paper is to study the natural quiver of an artinian algebra, a new kind of quivers, as a tool independing upon the associated basic algebra. In Li (J Aust Math Soc 83:385–416, 2007), the notion of the natural quiver of an artinian algebra was introduced and then was used to generalize the Gabriel theorem for non-basic artinian algebras splitting over radicals and non-basic finite dimensional algebras with 2-nilpotent radicals via pseudo path algebras and generalized path algebras respectively. In this paper, firstly we consider the relationship between the natural quiver and the ordinary quiver of a finite dimensional algebra. Secondly, the generalized Gabriel theorem is obtained for radical-graded artinian algebras. Moreover, Gabriel-type algebras are introduced to outline those artinian algebras satisfying the generalized Gabriel theorem here and in Li (J Aust Math Soc 83:385–416, 2007). For such algebras, the uniqueness of the related generalized path algebra and quiver holds up to isomorphism in the case when the ideal is admissible. For an artinian algebra, there are two basic algebras, the first is that associated to the algebra itself; the second is that associated to the correspondent generalized path algebra. In the final part, it is shown that for a Gabriel-type artinian algebra, the first basic algebra is a quotient of the second basic algebra. In the end, we give an example of a skew group algebra in which the relation between the natural quiver and the ordinary quiver is discussed.     

Idempotent-Conjugate Monoids with Nilpotent Unit Groups

Let M be a finite monoid with unit group G such that J-related idempotents in M are conjugate. If G is nilpotent, we prove that the complex monoid algebra CM of M is semisimple if and only if M is an inverse monoid. Conversely let G be a finite group such that for any finite idempotent-conjugate monoid M with unit group G, CM semisimple implies that M is an inverse monoid. We then show that G is a nilpotent group.     

The structure of Leavitt path algebras and the Invariant Basis Number property

In this paper, we provide the structure of the Leavitt path algebra of a finite graph via some step-by-step process of source eliminations, and restate Kanuni and Özaydin's nice criterion for Leavitt path algebras of finite graphs having Invariant Basis Number via matrix-theoretic language. Consequently, we give a matrix-theoretic criterion for the Leavitt path algebra of a finite graph having Invariant Basis Number in terms of a sequence of source eliminations. Using these results, we show certain classes of finite graphs for which Leavitt path algebras have Invariant Basis Number, as well as investigate the Invariant Basis Number property of Leavitt path algebras of certain Cayley graphs of finite groups.     

Unit groups of group algebras of some small groups

Let FG be a group algebra of a group G over a field F and U (FG) the unit group of FG. It is a classical question to determine the structure of the unit group of the group algebra of a finite group over a finite field. In this article, the structure of the unit group of the group algebra of the non-abelian group G with order 21 over any finite field of characteristic 3 is established. We also characterize the structure of the unit group of FA ₄ over any finite field of characteristic 3 and the structure of the unit group of FQ ₁₂ over any finite field of characteristic 2, where Q ₁₂ = 〈x, y; x ⁶ = 1, y ² = x ³, x ^y = x ^?1〉.     

The structure of the unit group of the group algebra $\mathbb{F}_{2^k } A_4 $

The structure of the unit group of the group algebra of the group A4 over any finite field of characteristic 2 is established in terms of split extensions of cyclic groups.     

The graded structure of Leavitt path algebras

A Leavitt path algebra associates to a directed graph a ?-graded algebra and in its simplest form it recovers the Leavitt algebra L(1, k). In this note, we first study this ?-grading and characterize the (?-graded) structure of Leavitt path algebras, associated to finite acyclic graphs, C _n -comet, multi-headed graphs and a mixture of these graphs (i.e., polycephaly graphs). The last two types are examples of graphs whose Leavitt path algebras are strongly graded. We give a criterion when a Leavitt path algebra is strongly graded and in particular characterize unital Leavitt path algebras which are strongly graded completely, along the way obtaining classes of algebras which are group rings or crossed-products. In an attempt to generalize the grading, we introduce weighted Leavitt path algebras associated to directed weighted graphs which have natural ⊕?-grading and in their simplest form recover the Leavitt algebras L(n, k). We then show that the basic properties of Leavitt path algebras can be naturally carried over to weighted Leavitt path algebras.     

标度广义效应代数和标度效应代数的结构

研究了标度广义效应代数与标度效应代数的代数结构,给出了比较完整的结果.通过引入全标度广义代数的概念,本文证明了区间[0,1)上的标度广义效应代数和单位区间[0,1]上的标度效应代数完全由单位区间上的阿基米德余模确定,标度广义效应代数恰同构于全标度广义代数的下集.若标度广义代数满足局部有限条件,则它同构于实数加法群的子群代数.满足(S)条件的标度效应代数同构于实数加法群的子群代数和全标度广义代数的字典序乘积的子代数.     

Hereditary crossed products

We characterize when a crossed product order over a maximal order in a central simple algebra by a finite group is hereditary. We need only concentrate on the cases when the group acts as inner automorphisms and when the group acts as outer automorphisms. When the group acts as inner automorphisms, the classical group algebra result holds for crossed products as well; that is, the crossed product is hereditary if and only if the order of the group is a unit in the ring. When the group is acting as outer automorphisms, every crossed product order is hereditary, regardless of whether the order of the group is a unit in the ring.

     

Torsion Units in the Integral Group Ring of the Alternating Group of Degree 6

It was conjectured by H. Zassenhaus that a torsion unit of an integral group ring ?G of a finite group G conjugates to a group element within the rational group algebra ?G. We investigate the Zassenhaus Conjecture (ZC) and a conjecture by W. Kimmerle about prime graph in the normalized unit group of ?A₆.