arXiv:0707.3607v1 [math.CO] 24 Jul 2007 ALGEBRAS ASSOCIATED TO DIRECTED ACYCLIC GRAPHS VLADIMIR RETAKH AND ROBERT LEE WILSON Abstract. We construct and study a class of algebras associ- ated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices and edges. Each finite directed acyclic graph admits a structure 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 fac- torizations of polynomials over noncommutative rings. 0. Introduction In this paper we construct and study a class of algebras A(Γ) associ- ated to generalized layered graphs Γ, i.e. directed graphs with a ranking function |.| on their vertices. Therefore, each edge has a length l; if an edge e goes from a vertex v to a vertex w then l(e)= |v|−|w|. Each finite directed acyclic graph admits a structure of a generalized layered graph. Generators of our algebras are elements a 1 (e),a 2 (e),...,a |e| (e) associated to edges e of Γ. The relations are defined as follows. Let sequences of edges e 1 , e 2 , ... , e p and f 1 ,f 2 ,...,f q define paths with the same end and the same origin. Then they define a relation given by the identity U e 1 (τ )U e 2 (τ ) ...U ep (τ )= U f 1 (τ )U f 2 (τ ) ...U fq (τ ), where τ is a formal central variable and U e (τ )= τ |e| − a 1 (e)τ |e|−1 + a 2 (e)τ |e|−2 −···± a |e| (e). for any edge e of Γ. Our interest to generalized layered graphs and algebras associated to those graphs is motivated by their relations to factorizations of poly- nomials over noncommutative rings. Let R be a unital algebra, P (τ ) a monic polynomial over R, and P be a set of monic right divisors of P (τ ), i.e. P consists of monic polynomials Q(τ ) ∈ R[τ ] such that 1991 Mathematics Subject Classification. 05E05; 15A15; 16W30. Key words and phrases. generalized layered graphs, Hilbert series, factorizations of noncommutative polynomials. 1