The Minimal Non-Koszul A(Gamma)

TL;DR

This paper presents the smallest known example of a directed layered graph whose associated algebra is non-Koszul, advancing understanding of the algebraic properties related to graph structures.

Contribution

It provides the first minimal example of a non-Koszul algebra A(Γ) associated with a directed layered graph, refining previous counterexamples.

Findings

Identifies a minimal graph with 13 edges and 9 vertices producing a non-Koszul algebra.

Demonstrates this is the smallest such graph with this property.

Contributes to the classification of algebras based on graph structures.

Abstract

The algebras $A(\Gamma)$, where $\Gamma$ is a directed layered graph, were first constructed by I. Gelfand, S. Serconek, V. Retakh and R. Wilson. These algebras are generalizations of the algebras $Q_n$, which are related to factorizations of non-commutative polynomials. It was conjectured that these algebras were Koszul. In 2008, T.Cassidy and B.Shelton found a counterexample to this claim, a non-Koszul $A(\Gamma)$ corresponding to a graph $\Gamma$ with 18 edges and 11 vertices. We produce an example of a directed layered graph $\Gamma$ with 13 edges and 9 vertices which produces a non-Koszul $A(\Gamma)$. We also show this is the minimal example with this property.