The lie algebra perturbation lemma

Abstract

Let R be a commutative ring which contains the rational numbers as a subring. We shall establish the following. Theorem. Let (Formula presented) be a contraction of chain complexes and suppose that g is endowed with a bracket [.,.] turning it into differential graded Lie algebra. Then the given contraction and the bracket [.,.] determine an sh-Lie algebra structure on M, that is, a coalgebra perturbation D of the coalgebra differential d0 on (the cofree coaugmented differential graded cocommutative coalgebra) Sc[sM] (on the suspension sM of M), the coalgebra differential d0 being induced by the differential on M, a Lie algebra twisting cochain (Formula presented) and, furthermore (Formula presented) of chain complexes which are natural in terms of the data. Here C[g] refers to the classifying coalgebra of g.