Do any significant changes happen in hypercomplex numbers beyond the eight dimensions of the octonions?

Question

They continue in the fashion of powers of 2: reals (1), complex (2), quaternion (4), octonions (8), and then there is sedonions(16), right? And, this keeps going, right? Do any significant changes happen in hypercomplex numbers beyond the eight dimensions of the octonions, the way octonions mark where associativity is lost?

Answer 1

The process by which we go $$\mathbb{R}\leadsto\mathbb{C}\leadsto\mathbb{H}\leadsto\mathbb{O}$$ is called the Cayley-Dickson construction. We can keep going more-or-less indefinitely, the next step being the sedenions, $\mathbb{S}$.

• It's also worth noting that there's a lot of flexibility here: we could have also gone from $\mathbb{R}$ to the split-complex numbers instead of to $\mathbb{C}$ if we used $1$ instead of $-1$ in the Cayley-Dickson construction.

However, when we do this things get truly nasty; the obvious horror in $\mathbb{S}$ is the presence of zero divisors, so division breaks down. There are other nastinesses - we have even less associativity in $\mathbb{S}$ than we did in $\mathbb{O}$ (only the latter satisfies alternativity, a weakening of full associativity) - but to my mind that's the most dramatic one.

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An interesting question here is how much algebraic nastiness we will ever have to deal with - or, phrased more positively, what are some algebraic tameness properties which the Cayley-Dickson construction will never kill off? I believe there's no good general answer known, but the discussion here will be of interest; for example, we never lose power associativity (basically, that "$x^n$" is well-defined for all $n\in\mathbb{N}$ - this isn't trivial when things aren't associative!).

Answer 2

Yes. The Cayley–Dickson construction doubles the dimension indefinitely, from $\Bbb R$ to $\Bbb C$ to $\Bbb H$ to $\Bbb O$ to the $16$-dimensional sedenions $\Bbb S$ etc. But Hurwitz's theorem tells us $\Bbb O$ is the largest normed division algebra, which somewhat restricts the interest in sedenions. (They include zero divisors, e.g. $(e_3+e_{10})(e_6-e_{15})=0$.) Just as octonions lost associativity but keep alternativity, sedenions lose even this but keep power-associativity, which survives throughout the construction.

Answer 3

Since the OP indicated he is interested not only in Cayley-Dickson construction, here is an answer about other algebras.

In tessarines or multicomplex numbers (which are isomorphic to each other), nothing special happens after dimension $4$.

At dimension 4, zero divisors will appear, but in all further dimensions equal to powers of $2$ the algebras remain commutative, associative and algebraically closed in the sense that all polynomials with non-zero-divisor coefficients have roots.

In split (hyperbolic) construction, that skips usual complex numbers, zero divisors appear at dimension $2$. The algebra remains commutative and associative at any power-of-2 dimension, but it is not algebraically closed in the above-mentioned sense.