The Courant sigma model is a 3-dimensional topological sigma model of AKSZ type which has been used for the systematic description of closed strings in non-geometric flux backgrounds. In particular, the expression for the fluxes and their Bianchi identities coincide with the local form of the axioms of a Courant algebroid. On the other hand, the axioms of a Courant algebroid also coincide with the conditions for gauge invariance of the Courant sigma model. In this paper we embed this interplay between background fluxes of closed strings, gauge (or more precisely BRST) symmetries of the Courant sigma model and axioms of a Courant algebroid into an L ∞ -algebra structure. We show how the complete BV-BRST formulation of the Courant sigma model is described in terms of L ∞ -algebras. Moreover, the morphism between the L ∞ -algebra for a Courant algebroid and the one for the corresponding sigma model is constructed.

A double field theory algebroid (DFT algebroid) is a special case of the metric (or Vaisman) algebroid, shown to be relevant in understanding the symmetries of double field theory. In particular, a DFT algebroid is a structure defined on a vector bundle over doubled spacetime equipped with the C-bracket of double field theory. In this paper, we give the definition of a DFT algebroid as a curved L∞-algebra and show how implementation of the strong constraint of double field theory can be formulated as an L∞-algebra morphism. Our results provide a useful step toward coordinate invariant descriptions of double field theory and the construction of the corresponding sigma-model.

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.

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.

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