Classification of Hom--Lie structures on finite-dimensional simple Lie algebras

We classify the Hom–Lie structures on finite-dimensional simple Lie algebras over an algebraically closed field of characteristic different from $2$ and $3$. Except for the three-dimensional simple Lie algebra and the Zassenhaus algebras, every such structure is a scalar multiple of the identity. In the two exceptional cases, we determine the full Hom–Lie space: it has dimension six for the three-dimensional algebra, while for a Zassenhaus algebra it consists precisely of the multiplication operators arising from the corresponding divided-power algebra. This proves a conjecture of Makhlouf and Zusmanovich.