Graded anti-pre-Lie algebraic structures on Witt and Virasoro algebras

We give the graded anti-pre-Lie algebraic structures on the Witt algebra $\mathcal{W}$ by the classification of certain indecomposable weight representations of $\mathcal{W}$. Their classification in the sense of isomorphism is also given. Furthermore, there does not exist a graded anti-pre-Lie algebraic structure on the Virasoro algebra $\mathcal{V}$ satisfying some natural conditions.