Ultraproducts and quotients of infinite direct products of lie algebras / by Hussein Abdallah Awala.

Abstract

We investigate some applications of ultraproducts in Algebra. In particular, we first present the classical applications: Robinson Theorem and Malcev Theorem. Then we focus on the most recent application of ultraproducts by G. Bergman and N. Nahlus. For example, we show that any finite-dimensional quotient of an infinite direct product (over any arbitrary index set) of finite-dimensional solvable Lie algebras is also solvable. The same is true for nilpotent and semi-simple Lie algebras. However, the proof in the case of semisimple Lie algebras, requires the deep theorem that L=[x,L] +[y, L] for some x, y in L , or it requires Brown Theorem which we both prove. The general technique in all three cases requires an investigation of ultraproducts with (resp. without) countably-complete ultrafilters. For simplicity, we shall assume that the base field is algebraically closed of characteristic 0.