  We introduce a certain differential graded bialgebra, neither commutative nor
cocommutative, that governs perturbations of a differential on complexes
supplied with an abstract Hodge decomposition. This leads to a conceptual
treatment of the Homological Perturbation Lemma and its multiplicative version.
As an application we give an explicit form of the decomposition theorem for
$A_\infty$ algebras and $A_\infty$ modules and, more generally, for twisted
objects in differential graded categories.
