Leibniz Central Extensions Over Some Lie Algebras

In this paper different Lie algebra will be studied:the Witt algebra,Block type algebra and sl?.The goal of this paper is to determine their(Leibniz)central extensions by determination of the possible(Leibniz)2-cocycles.We will first focus on the Witt algebra and provide a new proof of the existence of its universal central extension the Virasoro algebra,which is unique up to isomorphism.Then,we will study a Block type algebra and give an other proof of the already known result that states its central extension.Finally,thanks to the determination of Leibniz 2-cocycles,it will be proved that sl? doesn't have any non trivial 1-dimensional Leibniz central extension.