Timeline for $Spin(6,2) = SU(2, 2, \mathbb{H})$
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 31, 2023 at 12:58 | answer | added | Qmechanic | timeline score: 0 | |
| Oct 3, 2022 at 15:37 | comment | added | Hinanana | @runway44 Yes now I agree Spin$(6,2)$ is not SU$(2,2,\mathbb{H})$. In fact we don't have a faithful representation of Spin$(6,2)$: all I know is that Spin(2,6) embeds inside SK$(4, \mathbb{H}) \times$ SK$(4,\mathbb{H})$ (SK$(4,\mathbb{H})$ = SO*$(8)$). See Harvey's book "Spinors and Calibrations". | |
| Oct 1, 2022 at 1:59 | comment | added | anon | Is there an elementary description of what this ${\rm SU}(2,2,\Bbb H)$ thing is? My understanding is that the $4\times4$ quaternionic matrices whose transformations of $\Bbb H^4$ preserve the form $|x_1|^2+|x_2|^2−|x_3|^2−|x_4|^2$ make up a $36$-dimensional group which I would call $\rm Sp(2,2)$, but $\rm Spin(6,2)$ is $28$-dimensional so it can't be that. I am particularly interested in if we can see $\rm Spin(5,2)$ or $\rm Spin(6,1)$ in this classical group, or see how $AB$ embeds in it for $A$, $B$ classical groups corresponding to ${\rm Spin}(p,q)$, ${\rm Spin}(r,s)$ and $(p+q,r+s)=(6,2)$. | |
| Feb 15, 2022 at 10:43 | vote | accept | Hinanana | ||
| Feb 14, 2022 at 21:27 | answer | added | Torsten Schoeneberg | timeline score: 5 | |
| Feb 14, 2022 at 16:08 | comment | added | Dietrich Burde | Or chapter $6$ here. | |
| Feb 14, 2022 at 15:53 | comment | added | Dietrich Burde | I have seen it before (perhaps in Helgason's book). We also have $Spin(6)=SU(4)$ and $Spin(4,2)≃SU(2,2)$. Now take $\Bbb H=\Bbb R^4$. See here. | |
| Feb 14, 2022 at 15:40 | comment | added | Hinanana | Don't think so: the reference it gives: Paul Garrett, Sporadic isogenies to orthogonal groups, 2015, does not cover this example. Plus I am not even sure about the authenticity of this statement on wikipedia | |
| Feb 14, 2022 at 15:36 | comment | added | Hinanana | Yes. I think they are all theories. Plus I saw another reference request question on here which asked the reference for Spin(4,1) <math.stackexchange.com/questions/1105200/spin-group-spin4-1> | |
| Feb 14, 2022 at 15:33 | comment | added | Dietrich Burde | Did you check the references $1.,2.,3.,4.$ of the wikipedia article? | |
| Feb 14, 2022 at 15:24 | history | asked | Hinanana | CC BY-SA 4.0 |