Extensions and deformations of algebras with higher derivations

Higher derivations on an associative algebra generalizes higher order derivatives. We call a tuple consisting of an algebra and a higher derivation on it by an AssHDer pair. We define a cohomology for AssHDer pairs with coefficients in a representation. Next, we study central extensions, abelian extensions of AssHDer pairs and relate them to the second cohomology group with appropriate coefficients. Deformations of AssHDer pairs are also considered which are governed by the cohomology with self coefficient.