# On the perturbation algebra

## TL;DR

This paper introduces a differential graded bialgebra framework to systematically handle perturbations of differentials in complexes with Hodge decompositions, providing a conceptual approach to the Homological Perturbation Lemma and its applications.

## Contribution

It presents a novel differential graded bialgebra that unifies perturbation theory for complexes, leading to explicit forms of decomposition theorems for $A_infty$ structures.

## Key findings

- Explicit form of the decomposition theorem for $A_infty$ algebras and modules
- A conceptual framework for the Homological Perturbation Lemma
- Application to twisted objects in differential graded categories

## Abstract

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.