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In this paper, we first introduce the notion of D-homoderivations in Lie algebras, which includes establishing explicit structural results for D-homoderivations in Lie algebras over arbitrary fields. It shows that the set of D-homoderivations forms a Lie algebra that decomposes into the sum of derivations and centroids, with their intersection precisely equal to the space of central derivations.
Lie algebras and their derivations are central to many areas of mathematics and theoretical physics. The study of derivations, defined as linear maps that satisfy the Leibniz rule with respect to the Lie bracket, has expanded to include generalized derivations, quasi-derivations, centroids, and central derivations. The study of derivation was initiated during the 1950s and 1960s. Strictly, derivations of rings got a tremendous development in 1957. Based on the fundamental definition of derivation $ζ \colon G \to G$ satisfies $ζ(xy) = ζ(x)y +xζ(y)$ where $x, y\in G$.In the following, we introduce the main definition:
Definition: A linear mapping $\zeta\colon L\to L$ for any Lie algebra $L$ is a $D$-homoderivations Lie algebra for which there exists a homoderivation $D$ such that $$\zeta([x,y])=[\zeta(x),D(y)]+[\zeta(x),y]+[x,D(y)]$$ where $x, y \in L.$ We denote by $Der_{D}(L)$ for a $D$-homoderivations Lie algebra.
Theorem: Let $L$ be a Lie algebra and $D \in \mathrm{End}(L)$ be a central homoderivation. Then $\mathrm{Der}_{D}(L) = \mathrm{Der}(L) + C(L),$ where $\mathrm{Der}(L)$ is the set of derivations of $L$ and $C(L)$ is the set of central maps.