Real algebras and Hurwit's theorem
(Disclaimer: This series assumes some familiarity with linear algebra and Lie groups.)
Welcome to this expedition. In this series, we explore quaternions, a natural extension of real and complex numbers. Quaternions have broad applications in physics and engineering, solving many real-world problems. They are a very interesting mathematical structure, and they have deep connections to Lie groups and symmetries. The goal of this series is to convey some of their beauty and their practical relevance.
Here I adopt an unconventional approach to introduce them. We begin with an abstract discussion on real algebras and their properties, and then present Hurwit’s theorem, which states that there are essentially only four real normed division algebras (this will become meaningful in a moment). Then, quaternions appear as one of these rare examples. With this perspective, I aim to motivate quaternions as a natural and somewhat inevitable extension of the real numbers. An explicit introduction of quaternions will follow this motivation in the next part.
Let’s define some concepts. A real algebra is a finite-dimensional real vector space $A$ equipped with a bilinear product, and a unit element $1$. The product must be distributive over the sum, just as real and complex multiplication. Indeed, it’s easy to see that $\mathbb{R}$ and $\mathbb{C}$ are real algebras under their usual multiplication. Typically, the product of $a \in A$ and $b \in A$ is simply written $ab$, as is standard notation in $\mathbb{R}$ and $\mathbb{C}$ as well. Notice that a real algebra is not necessarily commutative, and it’s not necessarily associative either. For instance, the algebra of $M_2(\mathbb{R})$ under matrix multiplication is not commutative, and the algebra of $\mathbb{R}^3$ under the vector cross product is not associative.
An algebra is said to be a division algebra if it contains no zero divisors, meaning $ab = 0$ implies either $a=0$ or $b=0$ for $a,b \in A$. For instance, $\mathbb{R}$ and $\mathbb{C}$ are division algebras, while $M_2(\mathbb{R})$ is not. A division algebra is said to be normed when it’s equipped with a norm that is compatible with the product (i.e. \(\lVert a b \rVert = \lVert a \rVert \lVert b \rVert\) for all $a,b \in A$). The dimension of a real algebra is its dimension as a real vector space.
The last crucial ingredient we need is the concept of isomorphism, indicating when two algebras are fundamentally identical. Specifically, an isomorphism is a bijective map $ f \colon A \to B$ between algebras that preserves products, meaning $f(ab) = f(a)f(b)$ for all $a,b \in A$. Two algebras are said to be isomorphic when there exists an isomorphism between them. Isomorphic algebras are, essentially, different manifestations of the same underlying abstract algebra.
Within this context, elements of real normed division algebras are occasionally referred to as “numbers”. Remarkably, examples of this mathematical structure are very rare, as shown by the following theorem:
Theorem (Hurwit’s theorem) There are only four real normed division algebras (up to isomorphism):
That’s it. Ask a real algebra to be normed and have no zero divisors—a modest request for a suitable extension of the real numbers—and only four examples are left (up to isomorphism). In particular, this result shows that quaternions are not a capricious construction, but rather a very natural and unique generalization of the real numbers.
The proof of Hurwit’s theorem is quite involved, and it can be found in any standard book on the subject, such as Conway & Smith (2003)
It’s interesting to look at the hierarchical structure that these algebras form based on their properties. Notably, only $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{H}$ are associative, while the octonions, $\mathbb{O}$ are non-associative. Furthermore, commutativity holds only in $\mathbb{R}$ and $\mathbb{C}$.
Beyond the hypotheses of Hurwit’s theorem, relaxing the normed condition introduces an additional algebra: $\mathbb{S}$, the sedenions, of dimension 16. They are non-associative, non-commutative, and they contain zero divisors so they can’t be normed.
We have characterized the quaternions as an associative, non-commutative, real normed division algebra, being a natural and exceptional extension of the real numbers. In the next part, we will introduce them explicitly, and relate them to Lie groups, and in particular to three-dimensional rotations.