Homological perturbation theory and associativity

Abstract

In this paper, we prove various results concerning DGA-algebras in the context of the Homological Perturbation Theory. We distinguish two class of contractions for algebras: full algebra contractions and semi-full algebra contractions. A full algebra contraction is, in particular, a semi-full algebra contraction. Taking a full algebra contraction and an "algebra perturbation" as data of the Basic Perturbation Lemma, the Algebra Perturbation Lemma (or simply, F-APL) of [20] and [29 appears in a natural way. We establish here a perturbation machinery, the Semi-Full Algebra Perturbation Lemma (or, simply, SF-APL) that is a generalization of the previous one in the sense that the application range of SF-APL is wider than that of F-APL. We show four important applications in which this result is essential for the construction of algebra or coalgebra structures in various chain complexes.