Algebras associated to acyclic directed graphs

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 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.1991 Mathematics Subject Classification. 05E05; 15A15; 16W30.

Functions of random walks on hyperplane arrangements

“…We hope that our analysis will contribute to the understanding of these algorithms. For an introduction to the literature relating acyclic orientations to factoring noncommutative polynomials, see [34].…”

Hilbert series of algebras associated to directed graphs and order homology

“…In [13,5,21,22,23,24,26] we introduced and studied certain associative noncommutative algebras A(Γ) defined by layered graphs Γ (or ranked posets) and their generalizations. The algebras A(Γ) are related to factorizations of polynomials with noncommutative coefficients and we called them splitting algebras.…”

Representations of Aut(A(Γ)) acting on homogeneous components of A

“…We will now define a subalgebra of gr A(Γ ) (see [9] for a more generalized setting). Let σ be an automorphism of the layered graph Γ ; i.e.…”