Nonexistence of Surjective Lie Homomorphism Between Complex Special and General Linear Groups

Abstract

Lie groups are very important objects in algebra, topology and geometry, as it is naturally endowed with two special structures: the algebraic structure of group and the geometric structure of differential manifold. Hence, it is quite meaningful to better know the structure of Lie groups and the actions, and the transformations between them. Two specific theorems about Lie groups  and their homomorphism and Lie homomorphism are proved in this paper. First, the fact that there does not exist a surjective Lie group homomorphism between  and  is introduced. This is done by first having a trial of constructing Lie group surjective homomorphism but fails. Next, this result is further proved by applying topological method that to construct a covering map between  and , followed by computing the fundamental group of . Finally, by applying that  is a simple group, the fact that there does not exist a surjective group homomorphism between  and  is demonstrated, together with the statement that  is a simple group.