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

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.

to appear in the Special Issue of Universe on "Dualities and Geometry"; v2: typos in eqs.(2.8) and (3.11) fixed