L$\mskip1mu _{\infty}$ structures on mapping cones

Abstract

We show that the mapping cone of a morphism of differential graded Lie algebras, $\chi :L\to M$, can be canonically endowed with an $L_{\infty }$-algebra structure which at the same time lifts the Lie algebra structure on $L$ and the usual differential on the mapping cone. Moreover, this structure is unique up to isomorphisms of $L_{\infty }$-algebras.