From Hopf algebra to braided $L_\infty$-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, while the braided L ∞ -morphisms define a more general L ∞ -action of twisted L ∞ -algebras.

Braided symmetries in noncommutative field theory

“…as the degree 0 cohomology of the corresponding differential graded braided cocommutative coalgebra (Sym R (V [1]), D ). The coalgebra formulation also provides a natural interpretation of a braided L ∞ -algebra as a deformation of an L ∞ -algebra in the category Uv M [52]: the classical differential graded coalgebra (Sym(V [1]), D) naturally inherits the structure of a cocommutative Hopf algebra compatible with the differential D, 31 which can be twisted to a new noncocommutative Hopf algebra by the techniques of section 4.2 with a compatible differential D F . The braided L ∞ -algebra (V[[h]], D ) may then be interpreted as an L ∞ -module for the resulting twisted L ∞ -algebra.…”