The Lie algebra perturbation lemma

TL;DR

This paper extends the Lie algebra perturbation lemma by constructing an sh-Lie algebra structure on a chain complex M from a contraction of a differential graded Lie algebra g onto M, generalizing previous results.

Contribution

It introduces a method to derive an sh-Lie algebra structure on M from a contraction of g, broadening the applicability of the perturbation lemma.

Findings

Constructs an sh-Lie algebra structure on M

Defines a coalgebra perturbation of the differential

Extends a Lie algebra twisting cochain to a contraction

Abstract

Let g be a differential graded Lie algebra and suppose given a contraction of chain complexes of g onto a general chain complex M. We show that the data determine an sh-Lie algebra structure on M, that is, a coalgebra perturbation of the coalgebra differential on the cofree coaugmented differential graded cocommutative coalgebra S' on the suspension of M, a Lie algebra twisting cochain from the perturbed coalgebra S" to the given Lie algebra g, and an extension of this Lie algebra twisting cochain to a contraction of chain complexes from the Cartan-Chevalley-Eilenberg coalgebra on g onto S" which is natural in the data. This extends a result established in a joint paper of the author with J. Stashef [Forum math. 14 (2002), 847-868, math.AG/9906036] where only the particular where M is the homology of g has been explored.