Koszul duality theory for operads over Hopf algebras

TL;DR

This paper develops a Koszul duality framework for operads over Hopf algebras, simplifying the understanding of homotopy algebras and morphisms, with applications to Batalin-Vilkovisky algebras.

Contribution

It introduces a new Koszul duality theory for operads in modules over cocommutative Hopf algebras, enabling a clearer description of homotopy structures.

Findings

Simplifies the category of homotopy algebras and morphisms.

Provides a new description of the homotopy category of such algebras.

Main example includes Batalin-Vilkovisky algebras.

Abstract

The transfer of the generating operations of an algebra to a homotopy equivalent chain complex produces higher operations. The first goal of this paper is to describe precisely the higher structure obtained when the unary operations commute with the contracting homotopy. To solve this problem, we develop the Koszul duality theory of operads in the category of modules over a cocommutative Hopf algebra. This gives rise to a simpler category of homotopy algebras and infinity-morphisms, which allows us to get a new description of the homotopy category of algebras over such operads. The main example of this theory is given by Batalin-Vilkovisky algebras.