Biderivations of Simple Modular Lie Algebras of Cartan-Type

Assume that L is a simple Lie algebra of Cartan-type over an algebraically closed field with a characteristic p>3. We demonstrate that all symmetric biderivations vanish by using weight space decompositions relative to a suitable torus and the standard Z-grading structures of L. We then conclude that every biderivation of L is inner, based on a general result concerning skew-symmetric biderivations. As the direct applications, we determine the linear commuting maps and commutative post-Lie algebra structures on L completely.