Welcome octonions friends !

Long time ago when I travelled through octonion land, I conjectured that every $SO_8$ element can be expressed as product $L_a L_b R_c R_d$ for unit octonions $a$, $b$, $c$, $d$. Is this true ?

It can be seen as generalization of the fact that $SO_4$=$S^3 \otimes S^3$ i.e. every element in $SO_4$ can be seen as product $L_h R_\bar{k}$ for unit quaternions $h$, $k$.

Next step is to see what elements in $SO_{16}$ we obtain by multiplying $L_u$ and $R_v$ for unit sedenions $u, v \in \mathbb S$. We need at least 8 such elements sincce 15*8=120. Sedenions are not so nice as octonions since we have zero divisors there. Therefore we need to check the rank of matrix $L_u$ for unit sedenion $u$. There is work of Moreno showing that set $\{(a,b): ab=0; a,b \in S^{15} \subset \mathbb S\}$ is homeomorphic to $G_2$. Indeed, the subset of zero divisors on sphere $S^{15}$ is 11-dimensional set $\{a+b \iota: a,b \in S^6 \subset \mathbb O; a \perp b; a \perp 1; b \perp 1 \}$ (a,b are perpendicular imaginary unit octonions; $\iota$ is extra element in Cayley-Dickson formula defining sedenion multiplication). (The incorrect statement in wikipedia article on sedenions was corrected on June 28th, 2016 by John Baez). The space of zero-divisors on unit sphere $S^{15}$ is 11-dimensional.

EDIT 0 2018-04-27 I am trying to write article about octonions to magazine for high school students in Poland. While doing this I discovered that Moreno was right. There is no such thing as "zero divisor" in sedenions. Zero divisor should have norm equal to zero. In sedenions there is norm defined as square root of $x\bar x$ where conjugation is defined as in octonions from Cayley-Dickson formula $\bar x=\bar a-b\iota$. In this case sedenion norm is just length of the vector. So Moreno correctly considered pairs of elements whose product is zero.

EDIT 1: Since some people have justified doubts whether element $L_u$ can be in $SO_{16}$ for unit sedenion $u$ I present following.

EDIT 2: I extended 7-dimensional to 9-dimensional set of unitary sedenions u such that $L_u \in SO_{16}$.

$\underline{Fact}$: $L_x$ is in $SO_{16}$ for $x=(u+v\iota)/\sqrt 2$ such that $u,v \in \mathbb C \subset \mathbb O$. $\underline {Proof}:$ From Cayley-Dickson formula: $$L_u=\pmatrix{ L_u & \\ & R_u}$$ $$L_{v\iota}=\pmatrix{ 0 & -R_v S \\L_v S & 0}$$. Calculate the transposed matrix: $$L_x^T=\pmatrix{ L_{\bar u} & R_v S \\ -L_v S & R_\bar u }$$ Now $$L_x L_x^T=\pmatrix{2 & 0 \\ 0 & 2 }$$ because position 1,1 of the matrix is $L_u L_\bar u+R_v R_\bar v$ and position 1,2 of the matrix is $(L_u R_v-R_v L_u)S$. Elements $L_u$ and $R_v$ commutes only when $u,v$ are in the same complex subalgebra of octonions. I am using following facts in calculations: $L_u S=S R_\bar u$ $L_u^T=L_\bar u$ and similar. $\square$

The set defined in the Fact is roughly bundle $S^6 \times \times S^3$, because for each imaginary octonion $a$ there is sphere $S^3$ spanned by $<1,a,\iota,a\iota>$ belonging to this set. I said "roughly", becuase points {1,-1} belong to all fibers $S^3$.

Next, we can continue to trentaduonions $\mathbb T$. I claim that set of zero divisors is 27-dimensional subset of $S^{31}$. We need 16 elements of shape $L_u$ or $R_u$, since 16*31=496.

EDIT 3: Regarding trentaduonions I don't have ready theory yet. I tested in GAP, so I present following experimental results. For all 32 base elements $e_k$ the matrices $L_{e_k}$ and $R_{e_k}$ are in $SO_{32}$. Among 496 pairs of 32 base elements there 202 such pairs $k$, $l$ that $L_{e_k+e_l}$/length($e_k+e_l$) is in SO(32). 550 out 4960 triples satisfy it; 1002 out of 35960 fours; 1274 out of 201376 fives; 1218 out of 906192 sixs; 970 out of 3 365 856 sevens; 740 out of 10 518 300 eights. If I try to prove it then I would see whether similar formulas for conjugation $S$ multiplied by $L_u$ are valid in sedenions.

... and so on ...:)

I have also some generalization proposal for any Lie group, which I post in separate question.

  • "There is incorrect statement in wikipedia article on sedenions". Have you considered editing the article? – Andrei Smolensky Apr 14 '16 at 13:02
  • @AndreiSmolensky, when I was younger :) I edited article about octonions adding explicit definition of integral octonions. After some time I see that my additions are deleted. It is probably, because I prefer 123, 145, 167, 246, 275, 374, 365 octonions coming from Cayley-Dicksen formula and quaternions. While most mathematicions prefer triples starting from 124 and next shifting t->t+1 mod 7. Anyway, now I am too lazy (probably) to edit wikipedia articles, so I leave it to ones having more enthusiasm for it. Of course, if you insist more, then maybe I can do it. Feel free to edit the sentence. – Marek Mitros Apr 14 '16 at 13:10
  • The sedenions are not a composition algebra, so what do you even mean by $L_u$? Even if it has full rank, it has no reason to be a rotation matrix, does it? Is it even ever a rotation matrix? – Gro-Tsen Apr 14 '16 at 13:26
  • Good point ! I edited my question by proving that $L_x$ is in $SO_{16}$ for $x=(u+v\iota)/\sqrt 2$ and unit octonions $u$, $v$ lying in complex subalgebra of $\mathbb O$. For trentaduonions I only present some experimental results from GAP. – Marek Mitros Apr 15 '16 at 8:27
  • 1
    I proudly announce that my comment was considered on wikipedia page on sedenions. John Baez has corrected the article on June 28th this year. Now it is clearly stated that "space of pairs of norm-one sedenions that multiply to zero " is homeomorphic to Lie group $G_2$. – Marek Mitros Aug 19 '16 at 10:09
up vote 10 down vote accepted

Your conjecture on octonions is false. In Conway and Smith's book On Quaternions and Octonions (A.K. Peters 2003), §8.5 theorem 9 on page 94, it is shown that the set of $L_a L_b R_c$ for $a,b,c$ unit octonions is $20$-dimensional. So the set of $L_a L_b R_c R_d$ for $a,b,c,d$ unit octonions is at most $27$-dimensional and cannot be $SO_8$. Actually, based on the dimensions given in the above-cited theorem, I would conjecture that it is $26$-dimensional, but it seems, based on the comments the authors make near the end of the section (just before the conjecture they state), that they do not know the dimension.

On the other hand, they make a conjecture about products of five multiplications, namely that if $V,W,X,Y,Z$ are any five letters from the set $\{L,R,B\}$, not all equal, then $V_a W_b X_c Y_d Z_e$ gives all of $SO_8$.

  • Thank you very much ! I wish I have this book in my library. I see that I can find used one for 21 USD on Amazon, 159 pages ? There is also Tevian Dray book on geometry of octonions. Anyone here read this ? – Marek Mitros Apr 14 '16 at 15:16
  • @MarekMitros If you just want a quick look, you can try to search for 98cef04354cb47e2b1c868621e701bba in Google. – Gro-Tsen Apr 14 '16 at 17:48
  • I wonder whether this is legal - I mean downloading books without paying for it ? – Marek Mitros Apr 14 '16 at 19:03
  • @MarekMitros In many places, uploading is not legal, but downloading is, but I do not mean to provide legal counsel, let alone moral guidance. (And I certainly don't encourage you not to buy the book.) – Gro-Tsen Apr 14 '16 at 22:37

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