Poincar\'e duality for Koszul algebras

TL;DR

This paper explores the implications of Poincaré duality for Koszul algebras, revealing how it characterizes algebra structures and generalizes Lie algebras, with a review of Koszul duality concepts.

Contribution

It demonstrates the role of Poincaré duality in characterizing Koszul algebras and extends the concept to nonhomogeneous cases, including a discussion on generalized Lie algebras.

Findings

Poincaré duality implies the existence of twisted potentials for Koszul algebras.

For quadratic linear Koszul algebras, Poincaré duality aids in generalizing universal enveloping algebras.

The paper reviews Koszulity and Koszul duality for N-homogeneous and nonhomogeneous algebras.

Abstract

We discuss the consequences of the Poincar\'e duality, versus AS- Gorenstein property, for Koszul algebras (homogeneous and non homogeneous). For homogeneous Koszul algebras, the Poincar\'e duality property implies the existence of twisted potentials which characterize the corresponding algebras while in the case of quadratic linear Koszul algebras, the Poincar\'e duality is needed to get a good generalization of universal enveloping algebras of Lie algebras. In the latter case we describe and discuss the corresponding generalization of Lie algebras. We also give a short review of the notion of Koszulity and of the Koszul duality for N-homogeneous algebras and for the corresponding nonhomogeneous versions.