Generalized derivations and Hom-Lie algebra structures on $\mathfrak{sl}_2$

The purpose of this paper is to show that there are Hom-Lie algebra structures on $\mathfrak{sl}_2(\mathbb{F}) \oplus \mathbb{F}D$, where $D$ is a special type of generalized derivation of $\mathfrak{sl}_2(\mathbb{F})$, and $\mathbb{F}$ is an algebraically closed field of characteristic zero. It is shown that the generalized derivations $D$ of $\mathfrak{sl}_2(\mathbb{F})$ that we study in this work, satisfy the Hom-Lie Jacobi identity for the Lie bracket of $\mathfrak{sl}_2(\mathbb{F})$. We study the representation theory of Hom-Lie algebras within the appropriate category and prove that any finite dimensional representation of a Hom-Lie algebra of the form $\mathfrak{sl}_2(\mathbb{F}) \oplus \mathbb{F}D$, is completely reducible, in analogy to the well known Theorem of Weyl from the classical Lie theory. We apply this result to characterize the non-solvable Lie algebras having an invertible generalized derivation of the type of $D$. Finally, using root space decomposition techniques we provide an intrinsic proof of the fact that $\mathfrak{sl}_2(\mathbb{F})$ is the only simple Lie algebra admitting non-trivial Hom-Lie structures.