\(L_{\infty}\) algebra representations (Q1431353)

An \(L_\infty\)-algebra on a differential graded vector space \(L\) may be described by a coderivation \(D\) of degree \(+1\) on the cofree commutative coalgebra \(\wedge^* sL\), with \(D^2= 0\). A notion of left module over \(L_\infty\) algebras appeared in [\textit{T. Lada} and \textit{M. Markl}, Strongly homotopy Lie algebras, Commun. Algebra, 23, No. 6, 2147--2161 (1995; Zbl 0999.17019)]. In the paper under review, the author proves that, if \(L\) is an \(L_\infty\)-algebra and \(M\) a left \(L\)-module, then \(L\oplus M\) has an \(L_\infty\)-algebra structure. An explicit description in terms of a collection of linear maps \(j_k: \bigotimes^k(L\oplus M)\to L\oplus M\) is given. This is the homotopy theoretic version of a classical similar result in the setting of modules over Lie algebras.