A ∞ -Algebras Derived from Associative Algebras with a Non-Derivation

Given an associative graded algebra equipped with a degree +1 differential ∆ we define an A ∞ -structure that measures the failure of ∆ to be a derivation. This can be seen as a non-commutative analog of generalized BValgebras. In that spirit we introduce a notion of associative order for the operator ∆ and prove that it satisfies properties similar to the commutative case. In particular when it has associative order 2 the new product is a strictly associative product of degree +1 and there is compatibility between the products, similar to ordinary BV-algebras. We consider several examples of structures obtained in this way. In particular we obtain an A ∞ -structure on the bar complex of an A ∞ -algebra that is strictly associative if the original algebra is strictly associative. We also introduce strictly associative degree +1 products for any degree +1 action on a graded algebra. Moreover, an A∞-structure is constructed on the Hochschild cocomplex of an associative algebra with a non-degenerate inner product by using Connes’ B-operator.