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 [Formula: see text]. 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 [Formula: see text], whereas there is exactly one compatible root graded anti-pre-Lie algebraic structure on [Formula: see text].