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. 1. Introduction Homological Perturbation Theory [52, 14, 8, 9, 18, 36, 19, 20, 29] is a set of technique...

Analyzing the transference of the coalgebra structure on the homology of CDGAs

In [3] "mall" 1-homological model H of a commutative differential graded algebra is described. Homological Perturbation Theory (HPT) [7-9] provides an explicit description of an A1-coalgebra structure ( 1 ; 2 ; 3 ; : : :) of H. In this paper, we are mainly interested in the determination of the map 2 : H ! H H as a first step in the study of this structure. Developing the techniques given in [20] (inversion theory), we get an important improvement in the computation of 2 with regard to the first formula given by HPT. In the case of purely quadratic algebras, we sketch a procedure for giving the complete Hopf algebra structure of its 1-homology.