Transactions of the American Mathematical Society Graphs with Relations, Coverings and Group-graded Algebras
Transactions of the American Mathematical Society Graphs with Relations, Coverings and Group-graded Algebras
复制标题
DOI：
--
复制
发表时间：
--
期刊：
影响因子：
--
通讯作者：
E. Green
中科院分区：
文献类型：
--
作者：
E. Green
研究了有限有向图的覆盖与有向图相关的路径代数的分级之间的相互关系。为了包括所有基本有限维代数在代数闭域上的分级，引入了带关系图的覆盖理论。本文的目的是将代数上的群分级与与代数相关的图的覆盖联系起来。这些理论的联系使人们能够将纯代数问题与代数拓扑、群论或组合学中的问题联系起来。在代数的表示理论中，有限有向图与代数的关联，称为代数的颤振，是一个有用的工具。这样一个代数的颤振之所以令人感兴趣，是因为存在一个颤振表示的自然定义，使得满足某些关系的颤振表示的范畴等价于代数上有限生成模的范畴。§1将这些概念扩展到有限生成代数。本文的主要重点是证明由C. Riedtmann[9]引入并由P. Gabriel[2]展开的带关系图的覆盖理论与群分级代数理论在本质上是相同的。虽然覆盖和分级之间最初的联系是受到z -分级Artin代数[3,4]和P. Gabriel公布的结果[2]的相似性的启发，但本文的背景更一般，涉及域上所有有限生成的代数。我们给每一个这样的代数关联一个有限有向图，我们仍然称它为代数的一个颤振。我们证明了对于代数a的颤振T0的每一个正则覆盖T，在一定的限定条件下，我们得到代数a的一个G-分级，其中G是覆盖Y / 0的自同构群。反之，给定a的某类G级，其中G是一个群，我们构造了代数的颤音T0的正则覆盖Y，使得G同构于T / 0的自同构群。进一步，如果Y是T0的正则覆盖，具有自同构群G，我们证明了Y满足一组关系的表示的范畴等价于有限维梯度G模的范畴。我们
The paper studies the interrelationship between coverings of finite directed graphs and gradings of the path algebras associated to the directed graphs. To include gradings of all basic finite-dimensional algebras over an algebraically closed field, a theory of coverings of graphs with relations is introduced. The object of this paper is to relate group gradings on algebras to coverings of a graph which is associated to the algebra. The linking of the theories allows one to relate purely algebraic questions to questions in algebraic topology, group theory or combinatorics. In the representation theory of Artin algebras the association to each algebra of a finite directed graph, called the quiver of the algebra, has been a useful tool. The reason that the quiver of such an algebra is of interest is that there is a natural definition of representations of the quiver so that the category of representations of the quiver satisfying certain relations is equivalent to the category of finitely generated modules over the algebra. §1 gives a slight extension of these concepts to finitely generated algebras. The main emphasis of the paper is to show that the theory of coverings of graphs with relations, introduced by C. Riedtmann [9] and expanded by P. Gabriel [2], and the theory of group-graded algebras are essentially the same. Although the original connection between coverings and gradings was inspired by the similarity of results for Z-graded Artin algebras [3,4] and P. Gabriel's announced results [2], the context of this paper is more general and deals with all finitely generated algebras over a field. We associate to each such algebra a finite directed graph which we still call a quiver of the algebra. We show that for each regular covering T of a quiver T0 of an algebra A, with certain prescribed restrictions, we get a G-grading of the algebra A, where G is the automorphism group of the covering Y over ro. Conversely, given a certain type of G-grading of A, where G is a group, we construct a regular covering Y of the quiver T0 of the algebra such that G is isomorphic to the automorphism group of T over ro. Furthermore, if Y is a regular covering of T0 with automorphism group G, we show that the category of representations of Y satisfying a certain set of relations is equivalent to the category of finite-dimensional graded G-modules. We