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## The quaternions and Lie algebras I

Someone who has just read the previous post on how exponentiating quaternions gives a nice parameterization of $\text{SO}(3)$ might object as follows: “that’s nice and all, but there has to be a general version of this construction for more general Lie groups, right? You can’t always depend on the nice properties of division algebras.” And that someone would be right. Today we’ll begin to describe the appropriate generalization, the exponential map from a Lie algebra to its Lie group. To simplify the exposition, we’ll restrict to the case of matrix groups; that is, nice subgroups of $\text{GL}_n(\mathbb{F})$ for $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$, which will allow us to mostly avoid differential geometry.

The theory of Lie groups and Lie algebras is regarded to be one of the most beautiful in mathematics, and it is also fundamental to many areas, so today’s post is an extended discussion motivating the definition of a Lie algebra. In the next post we will actually do something with them.

For studying the hydrogen atom, our interest in Lie algebras comes from the following. If a Lie group $G$ acts smoothly on a smooth manifold $M$, its Lie algebra acts by differential operators on the space $C^{\infty}(M)$ of smooth functions, and these differential operators are the “infinitesimal generators” which give us conserved quantities for the evolution of a quantum system on $M$ (in the case that $G$ consists of symmetries of the Hamiltonian). Despite the fact that Lie algebras are commonly sold as a tool for understanding Lie groups, arguably in quantum mechanics the Lie algebra of symmetries of a Hamiltonian is more fundamental. This is important in sitations where the Lie algebra can sometimes exist without an associated Lie group.

## SU(2) and the quaternions

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 $\text{SO}(2)$.

In order to study the hydrogen atom, we’ll need to know something about the representation theory of the special orthogonal group $\text{SO}(3)$. This post consists of a few preliminaries along the way to doing this. I’ll be somewhat vague about a few things that 1) I don’t have much experience with, and 2) that would detract from the main narrative anyway.