Nonabelian higher derived brackets

Abstract

Let M be a graded Lie algebra, together with graded Lie subalgebras L and A such that as a graded space M is the direct sum of L and A, and A is abelian. Let D be a degree one derivation of M squaring to zero and sending L into itself, then Voronov's construction of higher derived brackets associates to D an L∞ structure on A[−1]. It is known, and it follows from the results of this paper, that the resulting L∞ algebra is a weak model for the homotopy fiber of the inclusion of differential graded Lie algebras i:(L,D,[⋅,⋅])→(M,D,[⋅,⋅]). We prove this fact using homotopical transfer of L∞ structures, in this way we also extend Voronov's construction when the assumption that A is abelian is dropped: the resulting formulas involve Bernoulli numbers. In the last section we consider some example and some further application.