“Coalgebra” Structures on 1-Homological Models for Commutative Differential Graded Algebras

In [3] “small” 1-homological model H of a commutative differential graded algebra is described. Homological Perturbation Theory (HPT) [7-9] provides an explicit description of an A∞-coalgebra structure (Δ₁,Δ₂,Δ₃,...) of H. In this paper, we are mainly interested in the determination of the map Δ₂ : H - HH ⊗ 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 Δ₂ 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.