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We classify the compatible left-symmetric algebraic structures on the Witt algebra satisfying certain non-graded conditions. It is unexpected that they are Novikov algebras. Furthermore, as applications, we study the induced non-graded modules of the Witt algebra and the induced Lie algebras by Novikov-Poisson algebras' approach and Balinskii-Novikov's construction.

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Abstract. In this paper, we prove that a biderivation of a finite dimensional complex simple Lie algebra without the restriction of skewsymmetric is inner. As an application, the biderivation of a general linear Lie algebra is presented. In particular, we find a class of a non-inner and non-skewsymmetric biderivations. Furthermore, we also get the forms of linear commuting maps on the finite dimensional complex simple Lie algebra or general linear Lie algebra.