Kaj Börjeson: A-infinity Algebras Derived from Associative Algebras with a Non-Derivation Differential

Consider a graded associative algebra A equipped with a degree +1 differential D. In many natural situations D acts as a derivation, however, if this is not the case one may ask how far away it is from being a derivation. In case A is commutative, there is a Lie-infinity structure that is non-zero if and only if D is a derivation. The structure maps are actually identities for being a differential operator of a certain order and in that sense it gives a measure how much it deviates from being a derivation. We present a similar construction for not necessarily commutative algebras yielding A-infinity structures measuring the failure of D being a derivation. The structure maps can be interpreted as identities for being an operator of "associatative order n".