Confluence algebras and acyclicity of the Koszul complex

TL;DR

This paper introduces a method to explicitly construct a contracting homotopy for the Koszul complex of N-Koszul algebras using confluence algebra representations, enhancing understanding of their homological properties.

Contribution

It provides a new explicit construction of the contracting homotopy for N-Koszul algebras based on confluence algebra representations, extending previous computational approaches.

Findings

Constructed explicit contracting homotopy for N-Koszul algebras.

Linked side-confluence property to homotopy construction.

Validated approach with multiple examples.

Abstract

The $N$-Koszul algebras are $N$-homogeneous algebras which satisfy an homological property. These algebras are characterised by their Koszul complex: an $N$-homogeneous algebra is $N$-Koszul if and only if its Koszul complex is acyclic. Methods based on computational approaches were used to prove $N$-Koszulness: an algebra admitting a side-confluent presentation is $N$-Koszul if and only if the extra-condition holds. However, in general, these methods do not provide an explicit contracting homotopy for the Koszul complex. In this article we present a way to construct such a contracting homotopy. The property of side-confluence enables us to define specific representations of confluence algebras. These representations provide a candidate for the contracting homotopy. When the extra-condition holds, it turns out that this candidate works. We explicit our construction on several examples.