Abstract:

Our main interest in this work is to study the properties and basics of the Lie group. Through what we offer from  double representations of the Lie group, and we linked the tensor product and the double representations to obtain new characteristics. In this paper we   use  the  quadrilateral Z action ( QZA ). We   researched the relationship between (QZA) of  Lie group and  tensor product, we will put in this relationship new structures   consisting of five vector spaces by double representations. At the end of the research, we showed the relationship between the Lie group and the Lie algebra through the tables that were clarified.

These proposals are developed and substantiated through observations and sources.

Keywords: Lie group, tensor product, tensor product of Lie group , representation of Lie group.