Multaplicative Hom-Structures On Finite Dimensional Modular Lie Algebras Of Cartan Type

Let F be an algebraically closed field of characteristic p>3. This article considers the Hom-structures on the finite dimensional graded Lie algebras of Cartan type. Lie algebras of Cartan type play an important role in the theory of modular Lie algebras. They are consist of four infinite families, namely W, S, H and K. We first determine the actions on the-1components for the Horn-structures and then determine the actions on the0components for the Hom-structures. Finally, using the invariance of the natural filtrations of Lie algebras under consideration, we prove.that there exists only the trivial multiplicative Hom-structure on the finite dimensional Z-graded Lie algebras of Cartan type.