Koszul calculus for N-homogeneous algebras

TL;DR

This paper generalizes the Koszul calculus from quadratic to N-homogeneous algebras, establishing associative products on (co)homology and computing examples for truncated polynomial algebras.

Contribution

It introduces a new N-Koszul calculus framework for N-homogeneous algebras, extending previous quadratic algebra methods.

Findings

Defined Koszul cup and cap products for N>2

Proved compatibility with Koszul differentials

Computed N-Koszul calculus for truncated polynomial algebras

Abstract

We extend the Koszul calculus defined on quadratic algebras by Berger, Lambre and Solotar, to N-homogeneous algebras. When N>2, the Koszul cup and cap products are defined by specific expressions, and they are compatible with the Koszul differentials, providing associative products on (co)homology classes. The N-Koszul calculus is calculated for the truncated polynomial algebras.