Inanna and the Algebras
Once again this month I found myself thinking about quaternions and related matters–partly for their own sake, and perhaps partly to take my mind off the news. You probably know something about this too. This editorial is not intended as a lesson, just a meditation on how elegant some bits of mathematics can be.
You know that the complex numbers are generated from the real numbers by throwing in a square root for -1, or, equivalently, modding out the ideal \langle x^2+1 \rangle from the ring of polynomials. Amazingly, by assuming a solution for one previously-insoluble polynomial, we’ve solved them all — and laid the foundations for much of the important applied math of the twentieth century. The complex numbers behave very much like the real numbers, except that they can’t be ordered.
If instead we mod out <x^2> we get the \textit{dual numbers}, with “infinitesimal” elements: these can be considered as an alternative basis for at least som calculus, and are useful in geometry. If we mod out <x^2-1> we get the \textit{double numbers} with applications to special relativity. Neither of these rings is quite as useful as \mathbb{C}, but that’s a high standard!
This is such a good trick that people wanted to try it again. Hamilton, after fruitless efforts to devise a three-dimensional division algebra, tried four dimensions and developed the quaternions. It’s a surprise today to realize that quaternions were in use before vectors: vectors in fact supplanted them for much of the twentieth century, but quaternions staged a comeback recently as a blisteringly fast way to handle rotations of three-dimensional objects in graphics processing units.
There are two obvious ways to think of the quaternions. We can start with the real numbers, and create an associative algebra with {two} square roots of -1, with ij = – ji. We then have a symmetry between i, j, and k := ij. This is the Clifford algebra approach. Alternatively, we can use the Cayley-Dickson construction, and add another imaginary element to \mathbb{C}, constructing the quaternions as \mathbb{H} := \mathbb{C}+j\mathbb{C}. I’m omitting important details in both cases; but, either way, we get the quaternions. And they behave {somewhat} like the complex numbers, but we have to give up commutativity.
If we repeat the Cayley-Dickson construction, we get the octonions \mathbb{O} := \mathbb{H}+k\mathbb{H}. This time the new algebra is nonassociative. (Like Inanna descending into the underworld, our algebra must surrender a property at each gateway.) As the Clifford algebras are constructed to be associative, the Clifford algebra with three generators cannot be the octonions, though they are related. However, the octonions do preserve a vestige of associativity: they are an alternative algebra, meaning that triples of the forms (xx)y=x(xy) and (xy)y=x(yy) associate.
Repeat the construction one more time, and Inanna loses her last attribute: the sedenions, \mathbb{S} := \mathbb{O}+\ell\mathbb{O}, are merely power-associative, with (xx)x = x(xx). (You’ve been familiar with another non-alternating but power-associative operation — the mean, \frac{x+y}{2}— since elementary school!) Worse, the sedenion algebra has zero-divisors, and thus has no multiplicative norm. There’s little left to be lost in further repetitions of the construction.
Meanwhile, the Clifford algebras are just getting warmed up for a pattern of Bott periodicity. And there are hybrids between the two families of algebras. Fascinating stuff, isn’t it?
