On the perturbation algebra (Q1628499)

The Homological Perturbation Lemma (HPL) is a kind of homotopy transfer theorem: it allows one to transfer a perturbed differential of a (co)chain complex onto another one, that is a strong deformation retract of it. The category of modules over a dg algebra \(A\) is, essentially, the category of strong homotopy retractions with a perturbed differential. This category is non-symmetric monoidal, which corresponds to a certain comultiplication on \(A\) turning it into a noncommutative and noncocommutative bialgebra. The authors introduce a certain localization \(\hat{A}\) of \(A\) which turns out also to be a bialgebra which is a central object of study in this paper. The algebraic structure of \(\hat{A}\) is responsible for all of the homological perturbation theory and as such, this algebra is implicitly present in many of the earlier papers dealing with this subject, e.g., [\textit{V. K. A. M. Gugenheim}, Ill. J. Math. 16, 398--414 (1972; Zbl 0238.55015)]. A certain remarkable endomorphism of \(\hat{A}\) is described that leads to a strengthened version of the HPL, as well as its multiplicative version [\textit{J. Huebschmann} and \textit{T. Kadeishvili}, Math. Z. 207, No. 2, 245--280 (1991; Zbl 0723.57030)]. As a corollary, a decomposition theorem for \(A_\infty\) algebras is obtained: every such algebra is isomorphic to the direct sum of a minimal \(A_\infty\) algebra and a linear contractible one. A similar result is also obtained for \(A_\infty\) modules over an \(A_\infty\) algebra.