Graphs with Relations , Coverings and Group - Graded Algebras

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 Received by the editors September 20, 1982. 1980 Mathematics Subject Classification. Primary 16A03, 57M10; Secondary 16A90, 05C20. 1 The author gratefully acknowledges the partial support of the National Science Foundation. ©1983 American Mathematical Society 0002-9947/83/0000-0962/S04.25