Operad and cohomology of associative algebras with generalized derivations

Abstract: An associative algebra with a generalized derivation is called an AsGDer triple. We introduce the operad that encodes AsGDer triples, and prove it is a Koszul operad. Using its Koszul dual cooperad, we introduce the homotopy version of AsGDer triples. As an application, we construct the AsGDer cohomology theory for AsGDer triples, and show that the formal deformation of an AsGDer triple is controlled by the AsGDer cohomology.