The sh-Lie algebra perturbation Lemma

TL;DR

This paper extends the sh-Lie algebra perturbation lemma to chain complexes over rings containing the rationals, providing a method to transfer sh-Lie structures via contractions and solving the master equation.

Contribution

It introduces a general framework for transferring sh-Lie algebra structures along contractions, including explicit constructions and solutions to the master equation.

Findings

Provides a method to transfer sh-Lie structures via contractions.

Constructs explicit extensions of perturbed retractions to sh-Lie maps.

Includes a general solution to the master equation.

Abstract

Let R be a commutative ring which contains the rationals as a subring and let g be a chain complex. Suppose given an sh-Lie algebra structure on g, that is, a coalgebra perturbation of the coalgebra differential on the cofree coaugmented differential graded cocommutative coalgebra T' on the suspension of g and write the perturbed coalgebra as T". Suppose, furthermore, given a contraction of g onto a 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 loop Lie algebra L on the perturbed coalgebra T", and an extension of this Lie algebra twisting cochain to a contraction of chain complexes from the Cartan-Chevalley-Eilenberg coalgebra on L onto S" which is natural in the data. For the special case where M and g are connected we also construct an explicit extension of the perturbed retraction to an sh-Lie map. This approach includes a very general solution of the master equation.