Moduli problems for operadic algebras

TL;DR

This paper generalizes the correspondence between formal moduli problems and Lie algebras to operadic settings, linking algebraic structures over Koszul operads and their duals, with applications to pre-Lie and permutative cases.

Contribution

It extends the Pridham-Lurie theorem to operadic algebra, establishing a new equivalence between formal moduli problems and operad-based algebras.

Findings

Established a correspondence between operadic formal moduli problems and algebras over Koszul dual operads.

Connected pre-Lie structures to permutative formal moduli problems.

Provided a framework linking operadic formal moduli problems to augmented operads.

Abstract

A theorem of Pridham and Lurie provides an equivalence between formal moduli problems and Lie algebras in characteristic zero. We prove a generalization of this correspondence, relating formal moduli problems parametrized by algebras over a Koszul operad to algebras over its Koszul dual operad. In particular, when the Lie algebra associated to a deformation problem is induced from a pre-Lie structure it corresponds to a permutative formal moduli problem. As another example we obtain a correspondence between operadic formal moduli problems and augmented operads.