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.

v3: several changes in the exposition

References in corpus (10)

• Non-Commutative Batalin-Vilkovisky Algebras, Homotopy Lie Algebras and the Courant Bracket
• Deformations and obstructions of pairs (X,D)
• Higher derived brackets
• Courant Algebroids. A Short History
• L-infinity algebra actions
• Deformations of coisotropic submanifolds for fibrewise entire Poisson structures
• Combinatorics and formal geometry of the master equation
• Infinitesimal Symmetries of Dixmier-Douady Gerbes
• L-infinity algebras, Cartan homotopies and period maps
• Continuous and Twisted L_infinity Morphisms

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• Universal Lie Formulas for Higher Antibrackets
• On coisotropic deformations of holomorphic submanifolds
• Homotopy relative Rota-Baxter Lie algebras, triangular $L_{\infty }$-bialgebras and higher derived brackets
• 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