From Koszul duality to Poincar\'e duality

TL;DR

This paper explores the relationship between Poincaré duality and Koszul duality in graded algebras, highlighting their significance in algebraic structures and potential generalizations.

Contribution

It establishes connections between Poincaré duality and Koszul duality, and discusses their implications for algebraic potentials and generalizations of Lie algebras.

Findings

Poincaré duality is relevant for twisted potentials in Koszul algebras

Connections between dualities inform algebraic structure generalizations

Provides insights into quadratic-linear algebra extensions

Abstract

We discuss the notion of Poincar\'e duality for graded algebras and its connections with the Koszul duality for quadratic Koszul algebras. The relevance of the Poincar\'e duality is pointed out for the existence of twisted potentials associated to Koszul algebras as well as for the extraction of a good generalization of Lie algebras among the quadratic-linear algebras.