Compatible $L_{\infty }$-algebras

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.

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