Compatible $L_\infty$-algebras

TL;DR

This paper introduces the concept of compatible $L_ abla$-algebras, constructs a graded Lie algebra framework for them, and explores their examples, cohomology, deformations, classifications, and relation to compatible Lie 2-algebras.

Contribution

It develops the theory of compatible $L_ abla$-algebras, including their construction, cohomology, deformation theory, and classification, and links them to compatible Lie 2-algebras.

Findings

Constructed a graded Lie algebra for compatible $L_ abla$-algebras.

Provided examples from Nijenhuis operators, $V$-datas, and Courant algebroids.

Classified strict and skeletal compatible $L_ abla$-algebras via cohomology and crossed modules.

Abstract

A compatible $L_\infty$-algebra is a graded vector space together with two compatible $L_\infty$-algebra structures on it. Given a graded vector space, we construct a graded Lie algebra whose Maurer-Cartan elements are precisely compatible $L_\infty$-algebra structures on it. We provide examples of compatible $L_\infty$-algebras arising from Nijenhuis operators, compatible $V$-datas and compatible Courant algebroids. We define the cohomology of a compatible $L_\infty$-algebra and as an application, we study formal deformations. Next, we classify `strict' and `skeletal' compatible $L_\infty$-algebras in terms of crossed modules and cohomology of compatible Lie algebras. Finally, we introduce compatible Lie $2$-algebras and find their relationship with compatible $L_\infty$-algebras.