Simple Associative Algebras and their Corresponding Finitary Special Linear Lie Algebras

Let F be a eld of any characteristic. Given any associative algebra A over F, one can render it into a Lie algebra by de ning a new product, 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 natural to except that the Lie algebra so obtained has a structure which is closely connected with the associative structure of A. In this paper, we study the relation between simple associative algebras and their related nitary Special Linear Lie algebras.