Abstract
The Baker–Campbell–Hausdorff (BCH) formula expresses the product of two exponentials in a Lie group as the exponential of a Lie algebra element involving iterated commutators. Beyond its computational role, the BCH formula provides a canonical way to endow a Lie algebra with a local group structure depending only on its bracket. In this talk, we’ll explain how the BCH formula encodes the infinitesimal multiplication of a Lie group and serves as the key ingredient in the integration of Lie algebras. We then outline how this local group law can be globalized to obtain a connected, simply connected Lie group with a prescribed Lie algebra, culminating in Lie’s Third Theorem.