From Hopf algebra to braided $L_\infty$-algebra

TL;DR

This paper develops a framework connecting $L_$-algebras with graded Hopf algebras, introducing a twisting procedure via Drinfel'd twists to define braided $L_$-algebras, expanding the algebraic structures and morphisms involved.

Contribution

It extends $L_$-algebras to Hopf algebras, applies Drinfel'd twists, and introduces braided $L_$-algebras with new morphism concepts.

Findings

Extended $L_$-algebras to graded Hopf algebras.

Defined braided $L_$-algebras via twisting.

Identified morphisms with new algebraic actions.

Abstract

We show that an $L_\infty$-algebra can be extended to a graded Hopf algebra with a codifferential. Then we twist this extended $L_\infty$-algebra with a Drinfel'd twist, simultaneously twisting its modules. Taking the $L_\infty$-algebra as its own (Hopf) module, we obtain the recently proposed braided $L_\infty$-algebra. The Hopf algebra morphisms are identified with the strict $L_\infty$-morphisms, while the braided $L_\infty$-morphisms define a more general $L_\infty$-action of twisted $L_\infty$-algebras.