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?
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1Not sure why this got a downvote ... – Noah Schweber Feb 23 '20 at 21:07
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Sorry, I accidentally posted that and tried to start editing right after with bad wifi connection. Should I change it back, and then write the edited version in a different post, since there are already two answers? – user3146 Feb 23 '20 at 21:09
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1Well, both the answers mention such a significant change, namely the appearance of zero divisors, so I think the situation is more-or-less fine. – Noah Schweber Feb 23 '20 at 21:11
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1Related (but unanswered): https://mathoverflow.net/questions/347002/equational-theory-in-the-signature-0-1-of-sedenions-and-beyond – Eric Wofsey Feb 23 '20 at 21:14
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@EricWofsey Thanks! – user3146 Feb 23 '20 at 21:22
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Are you interested only in Cayley-Dickson hypercomplex numbers? – Anixx Mar 19 '21 at 13:15
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@Anixx No, please share. – user3146 Mar 27 '21 at 17:34
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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.
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!).
Noah Schweber
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Thanks very much for thinking ahead. So, do properties keep breaking down? I imagine at some point no, right? How many properties can there be, right? Or, do some new ones start emerging, only to be lost later? – user3146 Feb 23 '20 at 21:20
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@user3146 New properties can't emerge, because each new number system includes the old one, do e.g. what's true of all sedenions is true of all real numbers as well. – J.G. Feb 23 '20 at 21:58
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2@J.G. Strictly speaking that's not quite true - that only holds for universal properties, but properties of the form $\forall x\exists y...$ (or similar) could hold in $\mathbb{S}$ but not $\mathbb{O}$. That said, purely universal properties are of course of special interest, and I don't know any natural property $\mathbb{S}$ has that $\mathbb{O}$ doesn't. – Noah Schweber Feb 23 '20 at 22:44
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1This is perhaps a vague question, but I wonder if you know of any natural way in which the properties of commutativity, associativity, alternativity, etc. fit into a sequence, like the corresponding algebras do? I mean, is there any uniform way to describe them? (I know of one, but it's a bit ad-hoc) – pregunton Feb 24 '20 at 08:15
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2@pregunton I don't, but that's a good question - it might be worth asking as a separate MSE question. – Noah Schweber Feb 24 '20 at 14:37
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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.
J.G.
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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.
Anixx
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