Dual representation of finite dimensional Lie groups

Abstract

Representation theory considered as one of the most important concepts in mathematics that makes the study of algebraic structure easier, and it has important practical applications, especially in quantum mechanics, and When we represent a group on the space , we replace each element with invertible operator or matrix.

The main purpose of this paper is to study the dual representation of finite dimensional Lie group on finite dimensional space, where we studied the sub representations of dual representation, thus we proved that if  is subrepresentation of , then  is not subrepresentation of , but we can extend it to subrepresentation.                                

We also proved that if  is reducible, completely reducible and unitary then  is reducible completely reducible, and unitary, and we found cases in which dual representations of Lie group are isomorphic with respect of the character,

and the weight of the representation.