A cup-cap duality in Koszul calculus

TL;DR

This paper introduces a duality in Koszul calculus for N-homogeneous algebras, linking cup and cap products, and explores their graded symmetry and commutativity properties using derived categories.

Contribution

It presents a novel cup-cap duality in Koszul calculus and proposes a conceptual approach to prove graded commutativity via derived categories.

Findings

Established a cup-cap duality in Koszul calculus

Linked graded symmetry of cap product to cup product commutativity

Developed enriched structures for N>2 cases

Abstract

We introduce a cup-cap duality in the Koszul calculus of N-homogeneous algebras. As an application, we prove that the graded symmetry of the Koszul cap product is a consequence of the graded commutativity of the Koszul cup product. We propose a conceptual approach that may lead to a proof of the graded commutativity, based on derived categories in the framework of DG algebras and DG bimodules. Various enriched structures are developed in a weaker situation corresponding to N>2.