Compatible $L_\infty$-algebras

Abstract

A compatible L∞-algebra is a graded vector space together with two compatible L∞-algebra structures on it. Given a graded vector space, we construct a graded Lie algebra whose Maurer-Cartan elements are precisely compatible L∞-algebra structures on it. We provide examples of compatible L∞-algebras arising from Nijenhuis operators, compatible V -datas and compatible Courant algebroids. We define the cohomology of a compatible L∞-algebra and as an application, we study formal deformations. Next, we classify 'strict' and 'skeletal' compatible L∞-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∞-algebras.