Algebras associated to directed acyclic graphs
Vladimir Solomonovich Retakh, Robert Lee Wilson · arXiv (Cornell University) · 2007
Abstract. 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 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 factorizations of polynomials over noncommutative rings. In this paper we construct and study a class of algebras A(Γ) associated 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