L ∞ -algebras and their cohomology

Abstract

We give an overview of different characterisations of L ∞ -structures in terms of symmetric brackets and (co)differentials on the symmetric (co)algebra. We then do the same for their representations (up to homotopy) and approach L ∞ -algebra cohomology using the commutator bracket on the space of coderivations of the symmetric coalgebra. This leads to abelian extensions of L ∞ -algebras by 2-cocycles.