Classification, ${\alpha }^k$-Derivations and Centroids of $4$-dimensional Complex Hom-associative Dialgebras

The basic objective of the current research paper is to investigate the structure and the algebraic varieties of Hom-associative dialgebras. We elaborate a classification of $n$-dimensional Hom-associative dialgebras for $n\le 4$. Additionally, using the classification result of Hom-associative dialgebras, we characterize the $\alpha$-derivations and centroids of low-dimensional Hom-associative dialgebras. Furthermore, we equally tackle certain features of derivations and centroids in the light of associative dialgebras and compute the centroids of low-dimensional associative dialgebras.