Compatible root graded anti-pre-Lie algebraic structures on finite-dimensional complex simple Lie algebras

We investigate the compatible root graded anti-pre-Lie algebraic structures on any finite-dimensional complex simple Lie algebra by the representation theory of $\mathrm{s}{\mathrm{l}}_2(\text{C})$. We show that there does not exist a compatible root graded anti-pre-Lie algebraic structure on a finite-dimensional complex simple Lie algebra except $\mathrm{s}{\mathrm{l}}_2(\text{C})$, whereas there is exactly one compatible root graded anti-pre-Lie algebraic structure on $\mathrm{s}{\mathrm{l}}_2(\text{C})$.