The minimal non-Koszul A(Γ)

ABSTRACT

The algebras A(Γ), where Γ is a directed layered graph, were first constructed by Gelfand et al. [5]. These algebras are generalizations of the algebras Q_n, which are related to factorizations of non-commutative polynomials. It was originally conjectured that these algebras were Koszul. In 2008, Cassidy and Shelton found a counterexample to this claim, a non-Koszul A(Γ) corresponding to a graph Γ with 18 edges and 11 vertices. We produce an example of a directed layered graph Γ with 13 edges and 9 vertices, which produces a non-Koszul A(Γ). We also show this is the minimal example with this property.

KEYWORDS:

• Koszulity
• layered graphs
• quantum algebra

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16W50
• 05E05

Acknowledgments

We would like to thank Robert Wilson for his guidance and advice. We would also like to thank John Yeung for his introduction to, and help with Python, the language HiLGA was written in, which led us to the discovery of the poset H.

Notes

¹The name HiLGA comes from “Hilbert Series of Layered Graph Algebras”.

²There is still room, however, to generalize here using one of the different definitions of Koszulity for non-quadratic algebras.