A Lie group is a topological group that at the same time is a smooth manifold in which multiplication and inversion are smooth.  The study of Lie groups originally arose from the work of Sophus Lie on separation of variables in partial differential equations.  This work was deep and brought to bear mathematics from quite different areas.  It showed that the multiplication law in a Lie group was completely determined in a neighborhood of the identity by the multiplication in a certain nonassociative finite-dimensional algebra that came to be known as the “Lie algebra” of the Lie group.  In 1897 J. E. Campbell began work on obtaining an explicit formula for the multiplication near the identity in the Lie group in terms of the operations in the Lie algebra.  The resulting conjectural formula came to be known as the Campbell-Baker-Hausdorff formula.  On some level it was proved within 10 years, but the formula was not fully useful until about 1947, when E. Dynkin exhibited a grouping of it that made it more manageable.

With Dynkin’s formula in hand, one could prove what are here called the four Fundamental Theorems of Lie Theory.  In turn three of these Fundamental Theorems of Lie Theory are sufficient to derive classification results and structure theorems for Lie algebras and connected Lie groups. These classification results and structure theorems appear in the sequel to this volume, Lie Groups Beyond an Introduction.