Koszul duality and Calabi-Yau structures

TL;DR

This paper explores how Koszul duality between certain algebraic structures swaps smooth and proper Calabi-Yau structures, providing a unified conceptual framework for their applications in Lie algebras and topology.

Contribution

It generalizes and explains the duality between smooth and proper Calabi-Yau structures via Koszul duality, connecting algebraic and topological examples.

Findings

Calabi-Yau structures on universal enveloping algebras relate to Poincare duality.

Calabi-Yau structures on chain coalgebras correspond to Poincare duality in spaces.

Duality exchanges smooth and proper Calabi-Yau structures in dg categories.

Abstract

We show that Koszul duality between differential graded categories and pointed curved coalgebras interchanges smooth and proper Calabi-Yau structures. This result is a generalization and conceptual explanation of the following two applications. For a finite-dimensional Lie algebra a smooth Calabi-Yau structure on the universal enveloping algebra is equivalent to a proper Calabi-Yau structure on the Chevalley-Eilenberg chain coalgebra, which exists if and only if Poincare duality is satisfied. For a topological space X having the homotopy type of a finite complex we show an oriented Poincare duality structure (with local coefficients) on X is equivalent to a proper Calabi-Yau structure on the dg coalgebra of chains on X and to a smooth Calabi-Yau structure on the dg algebra of chains on the based loop space of X.