Generalized derivations and quasi derivations of current $\text{H}$om- $\text{L}$ie algebras

In this paper, we investigate the structure of generalized derivations and quasiderivations of the tensor product algebra $\mathcal{H}\otimes \mathcal{A}$, where $\mathcal{H}$ is a Hom-Lie algebra and $\mathcal{A}$ is a commutative associative algebra with unity over the field $\mathcal{K}$. We examine whether every generalized derivation (respectively, quasiderivation) of $\mathcal{H}\otimes \mathcal{A}$ can be expressed as a sum of a derivation and a linear map in the centroid of $\mathcal{H}\otimes \mathcal{A}$, provided that this property holds for $\mathcal{H}$. We establish a partial affirmative result under suitable conditions and discuss the algebraic implications of our findings.