Extensions, deformation and categorification of $\text{AssDer}$ pairs

Abstract: In this paper, we consider associative algebras equipped with derivations. Such a pair of an associative algebra with a derivation is called an AssDer pair. Using the Hochschild cohomology for associative algebras, we define cohomology for an AssDer pair with coefficients in a representation. We study central extensions and abelian extensions of AssDer pairs. Moreover, we consider extensions of a pair of derivations in central extensions of associative algebras. Next, we study formal one-parameter deformations of AssDer pair by deforming both the associative product and the derivation. They are governed by the cohomology of the AssDer pair with representation in itself. In the next part, we study $2$-term $A_\infty$-algebras with homotopy derivations considered by Loday and Doubek-Lada. Finally, we introduce $2$-derivations on associative $2$-algebras and show that the category of associative $2$-algebras with $2$-derivations are equivalent to the category of $2$-term $A_\infty$-algebras with homotopy derivations.