Algebras associated to acyclic directed graphs

TL;DR

This paper introduces a new class of algebras linked to generalized layered graphs, providing bases and Hilbert series calculations, motivated by their applications in polynomial factorizations over noncommutative rings.

Contribution

It constructs and analyzes algebras associated with generalized layered graphs, including basis and Hilbert series computations, expanding understanding of their algebraic structure.

Findings

Constructed linear bases for the algebras.

Computed Hilbert series for these algebras.

Linked algebraic structures to polynomial factorizations over noncommutative rings.

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. Each finite directed acyclic 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.