Timeline for The special embedding $\mathfrak{so}(7)\subset\mathfrak{so}(8)$
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Jun 11, 2015 at 12:19 | comment | added | Jjm | @RobertBryant Thank you...! Your answers are simply brilliant, they always help me a lot. | |
| Jun 11, 2015 at 11:49 | comment | added | Robert Bryant | Oh, also: Any Cartan subalgebra of ${\frak{su}}(4)$ is also a Cartan subalgebra of ${\frak{spin}}(7)$, since both these groups have rank $3$. | |
| Jun 11, 2015 at 11:42 | comment | added | Robert Bryant | Look at the material in Chapter 14 in the section "Triality", where $\mathrm{Spin}(7)\subset\mathrm{SO}(8)$ is described as the group generated by $R_u$ for $u\in S^6\subset\mathrm{Im}(\mathbb{O})$. (He's identifying $\mathbb{R}^8$ with $\mathbb{O}$, the octonions.) While the Lie algebra isn't written out explicitly there, it follows immediately from Harvey's Lemma 14.66 that ${\frak{spin}}(7)\subset{\frak{so}}(8)$ is the space of skew-symmetric matrices orthogonal to all the skew-symmetric matrices $R_u$ for $u\in\mathrm{Im}(\mathbb{O})$. | |
| Jun 11, 2015 at 6:31 | comment | added | Jjm | @RobertBryant Thank you very much!! Which chapter / page approximately? | |
| Jun 10, 2015 at 11:39 | comment | added | Robert Bryant | You'll find all of this and more in F. R. Harvey's book "Spinors and Calibrations". | |
| Jun 10, 2015 at 10:33 | history | asked | Jjm | CC BY-SA 3.0 |