$\mathcal{O}$-operators on Lie $\infty$-algebras with respect to Lie $\infty$-actions

TL;DR

This paper introduces $\\mathcal{O}$-operators on Lie $\infty$-algebras relative to actions, characterizes them via Maurer-Cartan elements, and studies their deformation theory within a higher algebraic framework.

Contribution

It defines and characterizes $\mathcal{O}$-operators on Lie $\infty$-algebras using Maurer-Cartan elements and constructs a controlling Lie $\infty$-algebra for their deformations.

Findings

$\mathcal{O}$-operators characterized as Maurer-Cartan elements

Deformation of $\mathcal{O}$-operators controlled by a specific Lie $\infty$-algebra

Higher derived brackets used to construct the controlling algebra

Abstract

We define $\mathcal{O}$-operators on a Lie $\infty$-algebra $E$ with respect to an action of $E$ on another Lie $\infty$-algebra and we characterize them as Maurer-Cartan elements of a certain Lie $\infty$-algebra obtained by Voronov's higher derived brackets construction. The Lie $\infty$-algebra that controls the deformation of $\mathcal{O}$-operators with respect to a fixed action is determined.