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## The quaternions and Lie algebras II

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 $G$, we will describe its associated Lie algebra $\mathfrak{g}$ of infinitesimal elements and the exponential map $\mathfrak{g} \to G$ 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).

The simplest compact Lie group is the circle $S^1 \cong \text{SO}(2)$. Part of the reason it is so simple to understand is that Euler’s formula gives an extremely nice parameterization $e^{ix} = \cos x + i \sin x$ of its elements, showing that it can be understood either in terms of the group of elements of norm $1$ in $\mathbb{C}$ (that is, the unitary group $\text{U}(1)$) or the imaginary subspace of $\mathbb{C}$.
The compact Lie group we are currently interested in is the $3$-sphere $S^3 \cong \text{SU}(2)$. It turns out that there is a picture completely analogous to the picture above, but with $\mathbb{C}$ replaced by the quaternions $\mathbb{H}$: that is, $\text{SU}(2)$ is isomorphic to the group of elements of norm $1$ in $\mathbb{H}$ (that is, the symplectic group $\text{Sp}(1)$), and there is an exponential map from the imaginary subspace of $\mathbb{H}$ to this group. Composing with the double cover $\text{SU}(2) \to \text{SO}(3)$ lets us handle elements of $\text{SO}(3)$ almost as easily as we handle elements of $\text{SO}(2)$.