The sh-Lie algebra perturbation lemma

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.