Quaternions and Lie groups (part 4)
Automorphisms of the real normed division algebras
(Disclaimer: This series assumes some familiarity with linear algebra and Lie groups.)
In the previous part, we introduced the compact symplectic group $\mathrm{Sp}(n)$, the symmetry group of the symplectic inner product on $\mathbb{H}^n$. We also discussed the classification of simple Lie groups, and how the compact symplectic groups are as fundamental as the unitary and orthogonal groups. In this final part, we will look at the automorphisms of quaternions, along with those of the other normed division algebras.
An automorphism of an algebra is an isomorphism from the algebra to itself. Thus, it’s a transformation that leaves the algebraic structure invariant. These transformations are of great interest, as they capture the symmetries of the algebra.
The set of automorphisms of an algebra $A$ is a group under composition,(a) the composition of automorphisms is an automorphism, (b) the identity map is an automorphism, (c) the inverse of an automorphism is an automorphism, and (d) the composition of automorphisms is associative. named $\mathrm{Aut}(A)$. Because automorphisms preserve the underlying vector space structure, an automorphism is determined by its action on a basis of the algebra. Such action is subject to some constraints for the algebraic structure to be preserved: the element $1$ must be invariant, and, because automorphisms commute with conjugation, the norms must also be preserved.
Finally, the automorphism groups of the four real normed division algebras are:
- $\mathrm{Aut}(\mathbb{R})=1$, the trivial group containing only the identity, because the automorphisms must leave the unit invariant.
- $\mathrm{Aut}(\mathbb{C})=\mathbb{Z}_2$, acting by complex conjugation. The only non-trivial transformation allowed is $i \mapsto -i$.
- $\mathrm{Aut}(\mathbb{H})=\mathrm{SO}(3)$, acting by its canonical representation on the pure imaginary quaternions. These are three-dimensional rotations acting on $i,j,k$ as if they were a right-handed orthonormal triple of three-dimensional vectors.
- $\mathrm{Aut}(\mathbb{O})=G_2$, one of the five exceptional simple Lie groups.For those familiar with Riemannian geometry: this misteryous Lie group has another realization as a subgroup of $\mathrm{GL}(7,\mathbb{R})$ leaving a certain $3$-form on $\mathbb{R}^7$ invariant (via the canonical representation of $\mathrm{GL}(7,\mathbb{R})$ on $\Lambda^3(\mathbb{R}^7)$). Curiously, $G_2$ is useful somewhere in physics. Specifically, in M-theory, a conjectured theory of fundamental physics. M-theory hypothesizes that spacetime is eleven-dimensional, where seven dimensions are compactified and unobservable at large scale. In this theory, $G_2$ is the holonomy group of a Riemannian metric on the compactified seven-dimensional component of spacetime.
The fact that $\mathrm{Aut}(\mathbb{H}) = \mathrm{SO}(3)$ sheds some more light on the relation between quaternions and three-dimensional rotations. The imaginary units $i, j, k$ are in some way akin to the canonical versors from vector calculus, $\hat{\imath}, \hat{\jmath}, \hat{k} \in \mathbb{R}^3$. Indeed, the transformation group of the set of right-handed orthonormal bases of $\mathbb{R}^3$ is also $\mathrm{SO}(3)$. The usual names of the versors actually have their origins in quaternionic notation.
We have come to the end of this series. I hope you’ve enjoyed it, and I hope I’ve been able to convey some of the beauty in the natural way quaternions extend the real numbers, the deep connections they have to Lie groups, and how useful they are in solving many real-world problems.