Homological perturbation theory and associativity

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, SFAPL) 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.

Keywords

filtered algebra, graded algebra, resolutions, homological perturbation, differential homological algebra, augmented algebra, chain complex, contraction

2010 Mathematics Subject Classification

18G10, 18Gxx, 55U15, 55Uxx

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Published 1 January 2000