From Hopf Algebra to Braided L∞-Algebra

Abstract

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

A Matrix Model of Four-Dimensional Noncommutative Gravity

“…In this work a braided version of the Einstein-Cartan-Palatini gravitational theory is formulated. Moreover, in [83], braided L ∞ algebras are produced making use of the Drinfel'd twist of the graded Hopf algebra, which, in presence of a compatible codifferential, consists an alternative form of the underlying L ∞ algebra.…”