Algebras Associated to Acyclic Directed Graphs

We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices and edges. Each finite acyclic directed graph admits countably many structures of a generalized layered graph. We construct linear bases in such algebras and compute their Hilbert series. Our interest to generalized layered graphs and algebras associated to those graphs is motivated by their relations to factorizations of polynomials over noncommutative rings. 0. Introduction By a generalized layered graph we mean a pair Γ = (G, |.|) where G = (V,E) is a directed graph and |.| : V → Z≥0 satisfies |v| > |w| whenever v, w ∈ V and there is an edge e ∈ E from v to w. We call |.| the rank function of Γ. We write l(e) = |v| − |w| and call this the length of the edge e. We will see that if G is any acyclic directed graph then there are countably many rank functions |.| such that (G, |.|) is a generalized layered graph. In this paper we construct and study a class of algebras A(Γ) associated to generalized layered graphs Γ. Generators of our algebras are elements a1(e), a2(e), . . . , al(e)(e) associated to edges e of Γ. The relations are defined as follows. Let sequences of edges e1, e2, . . . , ep and f1, f2, . . . , fq define paths with the same end and the same origin. Then they define a relation given by the identity Ue1(τ)Ue2(τ) . . . Uep(τ) = Uf1(τ)Uf2(τ) . . . Ufq(τ), where τ is a formal central variable and Ue(τ) = τ l(e) − a1(e)τ l(e)−1 + a2(e)τ l(e)−2 − · · · ± al(e)(e) for any edge e of Γ. We will show that if Γ = (G, |.|) the structure of A(Γ) depends on the rank function |.| as well as directed graph G. 1991 Mathematics Subject Classification. 05E05; 15A15; 16W30.