Nonabelian higher derived brackets

TL;DR

This paper extends Voronov's higher derived brackets construction to non-abelian cases, providing a homotopical proof and explicit formulas involving Bernoulli numbers, with applications to homotopy theory.

Contribution

It generalizes the higher derived brackets construction beyond abelian subalgebras using homotopical transfer methods.

Findings

The L-infinity algebra models the homotopy fiber of a Lie algebra inclusion.

Formulas involve Bernoulli numbers when dropping abelian assumptions.

Provides examples and applications in homotopy theory.

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.