Timeline for Is an associative division algebra required for this phenomenon?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| May 27, 2016 at 15:23 | comment | added | user21230 | @AdamPrzeździecki Cześć Adam ! wikipedia article en.wikipedia.org/wiki/Parallelizable_manifold says that it was proved in 1958 by Kervaire, Bott and Milnor that the only paralelizable spheres are $S^1$, $S^3$ and $S^7$. | |
| S Aug 19, 2015 at 14:55 | history | suggested | Māris Ozols | CC BY-SA 3.0 |
explicit solution for d=8
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| Aug 19, 2015 at 14:46 | comment | added | Māris Ozols | I checked that this indeed works, so I added an explicit expression to your answer. Do you have any good reference for why there is no solution other than $d \in \{1,2,4,8\}$? | |
| Aug 19, 2015 at 14:41 | review | Suggested edits | |||
| S Aug 19, 2015 at 14:55 | |||||
| Aug 19, 2015 at 14:02 | comment | added | Adam Przeździecki | You may view the octonions $\bf{O}$ as a vector space $\mathbb{R}^8$ plus a bilinear map $\bf{O}\times\bf{O}\to\bf{O}$. You also have the standard basis $\{e_i\mid i=0,1,\ldots,7\}$ of $\bf{O}$. The bilinear map is not associative, but all you need is that each $e_i$ acts linearly from the left on $\bf{O}$ and that $\forall_{v\in\bf{O}}\{e_0v,e_1v,\ldots,e_7v\}$ is an orthonormal basis. Thus you may put $R_i=e_{i-1}$. | |
| Aug 19, 2015 at 13:34 | comment | added | Māris Ozols | Could you elaborate more on how this works (e.g., what exactly are the matrices $R_i$)? Since octonions are not associative, I don't think you can represent them by matrices as matrix multiplication is associative. | |
| Aug 19, 2015 at 12:32 | history | answered | Adam Przeździecki | CC BY-SA 3.0 |