Talk:Cayley-Dickson construction

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Geometry guy 14:40, 21 May 2007 (UTC) article has some tone and style issues--Cronholm144 02:38, 22 May 2007 (UTC)

Cayley is surely Arthur Cayley, 1821-1895, Dickson must be Leonard Dickson, 1874-1954.

But who invented it, and why is it called Cayley-Dickson?

[edit] Multiplication formula

In the (german, transl. from russian) book of Kantor and Solodownikow, I found the following formula for the multiplication of pairs: $(a, b)(c, d) = (a c - d * b, d a + b c *)$ which is different from the formula given here. How to decide which one describes the Cayley-Dickson construction correctly? --J"org Knappen

Found it out: The two formulæ are æquivalent -- JKn.

The argument for equivalence was wrong, thus the problem still remains -- JKn

They generate isomorphic structures (identical up to signs of bases). Taking i, k, l as (0, 1), (0, 1), (0, 1) resp. and the others by analogy, the K-S formula generates the standard basis (ij=k, etc.) on $\mathbb{H}$ (the Wikipedia formula gives ij=-k) but nonstandard on $\mathbb{O}$ (il=(0,i) where the Wikipedia formula gives il=(0,-i)=-li). I rattled off a quick Python script to generate the sedenion multiplication tables for the two formulas and it turns out the table in Sedenion is that generated by the K-S formula. The two tables are identical except that the one has opposite sign to the other off the leading diagonal, row and column; equivalently, replacing all the non-1 units with their negatives.

The inductive proof is fairly easy (RTP: $x*_Wy=-x*_{KS}y, \pm 1 \neq (x = (a,b)) \neq (y = (c,d)) \neq \pm 1$ . Assumption: this holds for the previous level (i.e. for x=a, y=c, etc. Method: each of a, b, c, d can be 0, 1 or another complex unit from the previous level, etc.,)

Of course, this raises the question: which formulas give the correct algebraic structure? The main conditions are that norms work, that 1 works and that addition distributes over multiplication. These are satisfied by keeping the shape of the formula, but applying three (independent, Abelian) transformations: moving the conjugate operator on the db term (d*b/db*), swapping the order of the terms in the second part (da+bc*/ad+c*b), and moving the conjugate operator between the terms in the second part (da+bc*/da*+bc). This gives 8 isomorphic constructions; the same inductive proof demonstrates that the resulting structures are again equivalent up to replacing some bases with their negatives. Yet more isomorphic structures are available, however, by using different formulæ for the construction at different levels. Not that many, though; only 8^n as opposed to 2^(2^n-1) possibilities flipping bases. Which 8^n, and are all different? Uh... wish I could answer that. Not my field. Fun, though. -- EdC 02:26, 25 April 2006 (UTC)

The octonions have two natural bases, ${1, i, j, k, l, i l, j l, k l}$ , and ${1, i, j, k, l, l i, l j, l k}$ , each of which has a different multiplication table, and therefore needs a different formula for the C-D construction. It becomes more clear if you explicitly interpret the pairs as addition: the formula given in the article means