The Baker–Campbell–Hausdorff Formula and Its Consequences

Abstract

Consider three elementary results from the preceding chapters of this book: (1) Every matrix Lie group G has a Lie algebra \(\mathfrak{g}\). (2) A continuous homomorphism \(\Phi\) between matrix Lie groups G and H gives rise to a Lie algebra homomorphism \(\phi : \mathfrak{g} \rightarrow \mathfrak{h}\). (3) If G and H are matrix Lie groups and H is a subgroup of G, then the Lie algebra \(\mathfrak{h}\) of H is a subalgebra of the Lie algebra \(\mathfrak{g}\) of G. Each of these results goes in the “easy” direction, from a group notion to an associated Lie algebra notion. In this chapter, we attempt to go in the “hard” direction, from the Lie algebra to the Lie group. We will investigate three questions relating to the preceding three theorems.