Koszul duality and homotopy theory of curved Lie algebras

TL;DR

This paper develops a homotopy theory framework for curved Lie algebras, establishing a model structure and extending Koszul duality, with applications to algebraic deformation theory over pseudo-compact algebras.

Contribution

It introduces a model category structure for marked curved Lie algebras and extends Koszul duality to this setting, connecting it with pseudo-compact commutative differential graded algebras.

Findings

Established a Quillen equivalence between curved Lie algebras and pseudo-compact CDGAs

Extended algebraic deformation theory to non-local pseudo-compact CDGAs

Provided a homotopy-theoretic framework for curved Lie algebras

Abstract

This paper introduces the category of marked curved Lie algebras with curved morphisms, equipping it with a closed model category structure. This model structure is---when working over an algebraically closed field of characteristic zero---Quillen equivalent to a model category of pseudo-compact unital commutative differential graded algebras; this extends known results regarding the Koszul duality of unital commutative differential graded algebras and differential graded Lie algebras. As an application of the theory developed within this paper, algebraic deformation theory is extended to functors on pseudo-compact, not necessarily local, commutative differential graded algebras.