An Introduction of Basic Lie theory

Lie theory, initially developed by Sophus Lie, is a fundamental part of mathematics. The areas it touches contain classical, differential, and algebraic geometry, topology, ordinary and partial differential equations, complex analysis and ect. And it is also an essential chapter of contemporary mathematics. A development of it is the Uniformization Theorem for Riemann surface. The final proof of such theorem is the invention from Einstein to the special theory of relativity and the Lorentz transformation. The application of Lie thoery is astonishing. In this paper, we are going to follow the work by Roger Howe to show the essential phenomenon of the theory that Lie groups G may be associated naturally by it Lie algebra g. And we will go through the proof of the fact that Lie group G is determined by g and its Lie bracket. And g is a vector space which is endowed with a bilinear nonassociative product called the Lie bracket. G is a complicated nonlinear object. Thus for many purposes we can replace G with g.