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 a L-infinity structure on A[-1]. It is known, and it follows from the results of this paper, that the resulting L-infinity 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-infinity structures, in this way we also extend Voronov's construction when the assumption A abelian is dropped: the resulting formulas involve Bernoulli numbers. In the last section we consider some example and some further application.

arXiv commentv3: several changes in the exposition

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• Homotopy Loday Algebras and Symplectic $2$-Manifolds
• Equivalences of coisotropic submanifolds
• On the abstract Bogomolov-Tian-Todorov Theorem
• Cumulants, Koszul brackets and homological perturbation theory for commutative $B{V}_{\infty }$ and $IB{L}_{\infty }$ algebras