Dual representation of finite dimensional Semisimple Lie algebra

Abstract

 The main purpose of this paper is to study the dual representation of finite dimensional Semisimple Lie algebra on finite dimensional space, and to know if the dual representation  has the same properties of main representation , thus we proved that if  is reducible then  is reducible. We also proved that any dual representation of finite dimensional Semisimple Lie algebra is faithful and completely reducible, then we found the weight and the highest weigh of  with depending on the weight and the highest weight of , and we found cases in which dual representations of Semisimple Lie algebra are isomorphic.