simple associative algebras and their corresponding finitary special linear lie algebras

Let f be a eld of any characteristic. given any associativealgebra a over f, one can render it into a lie algebra by dening a newproduct, the lie product, for any two elements a and b in a by means of[a; b] = ab .. ba, where ab is the associative product in a. it is naturalto except that the lie algebra so obtained has a structure which is closelyconnected with the associative structure of a. in this paper, we study therelation between simple associative algebras and their related nitary speciallinear lie algebras.