Noncommutative Koszul Algebras from Combinatorial Topology

TL;DR

This paper investigates when noncommutative graded quadratic algebras derived from layered graphs are Koszul, using topological methods and introducing new cohomology groups to analyze their properties.

Contribution

It provides a topological approach to determine the Koszul property of these algebras for specific classes of graphs, correcting previous misconceptions.

Findings

Identifies conditions under which these algebras are Koszul.

Introduces new cohomology groups H_X(n,k) for analysis.

Provides topological proofs of existing algebraic results.

Abstract

Associated to any uniform finite layered graph Gamma there is a noncommutative graded quadratic algebra A(Gamma) given by a construction due to Gelfand, Retakh, Serconek and Wilson. It is natural to ask when these algebras are Koszul. Unfortunately, a mistake in the literature states that all such algebras are Koszul. That is not the case and the theorem was recently retracted. We analyze the Koszul property of these algebras for two large classes of graphs associated to finite regular CW complexes, X. Our methods are primarily topological. We solve the Koszul problem by introducing new cohomology groups H_X(n,k), generalizing the usual cohomology groups H^n(X). Along with several other results, our methods give a new and primarily topological proof of a result of Serconek and Wilson and of Piontkovski.