# Koszul duality, minimal model and $L_\infty$-structure for differential algebras with weight

## TL;DR

This paper extends Koszul duality theory to differential algebras with nonzero weight, constructing their minimal models and homotopy structures, and exploring the resulting deformation complexes.

## Contribution

It establishes Koszul duality for weighted differential algebras, a case not previously treated due to non-quadratic operads, and develops associated homotopy and deformation theories.

## Key findings

- Koszul duality is established for nonzero weight differential algebras.
- Minimal models and homotopy cooperads are explicitly constructed.
- Deformation complexes with $L_$-structure are derived.

## Abstract

A differential algebra with weight is an abstraction of both the derivation (weight zero) and the forward and backward difference operators (weight $\pm 1$). In 2010 Loday established the Koszul duality for the operad of differential algebras of weight zero. He did not treat the case of nonzero weight, noting that new techniques are needed since the operad is no longer quadratic. This paper continues Loday's work and establishes the Koszul duality in the case of nonzero weight. In the process, the minimal model and the Koszul dual homotopy cooperad of the operad governing differential algebras with weight are determined. As a consequence, a notion of homotopy differential algebras with weight is obtained and the deformation complex as well as its $L_\infty$-algebra structure for differential algebras with weight are deduced.