We now know what a Lie algebra is and we know they are abstractions of infinitesimal symmetries, which are given by derivations. Today we will see what we can say about associating infinitesimal symmetries to continuous symmetries: that is, given a matrix Lie group , we will describe its associated Lie algebra of infinitesimal elements and the exponential map which promotes infinitesimal symmetries to real ones.

As in the other post, I will be ignoring some technical details for the sake of exposition. For example, I am generally not specifying how I’m topologizing various objects, and this is because of the general fact that a finite-dimensional real vector space has a unique Hausdorff topology compatible with addition and scalar multiplication. Whenever I talk about limits in such a vector space, I therefore don’t need to specify how I’m imposing a topology, although it will generally be convenient to induce it via a norm (which I am also not specifying).