Timeline for What is the intersection of Spin(7) and U(4)?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 18, 2015 at 21:22 | comment | added | seub | @Robert Bryant: thank you very much! | |
| Mar 18, 2015 at 20:26 | comment | added | Robert Bryant | The maximal abelian subgroups of $\mathrm{Spin}(7)$ all have dimension $3$, so it cannot contain even the maximal torus of $\mathrm{U}(4)$, let alone the whole thing. Yes, there are Ricci-flat Riemannian $8$-manifolds with holonomy not contained in $\mathrm{Spin}(7)$. I'm not aware of a compact one, though. | |
| Mar 18, 2015 at 19:22 | comment | added | seub | So Spin(7) does not contain U(4) right? And if I am allowed to ask more: I suppose there are Ricci-flat 8-manifolds that are not Spin(7)? | |
| Mar 18, 2015 at 19:17 | comment | added | seub | @Bilateral and Robert Bryant: thank you both. | |
| Mar 18, 2015 at 19:07 | comment | added | Robert Bryant | Of course, this depends on which conjugates of $\mathrm{Spin}(7)$ and $\mathrm{U}(4)$ in $\mathrm{SO}(8)$ you choose to intersect. For instance, is not true that all the conjugates of $\mathrm{SU}(4)$ lie inside some fixed conjugate of $\mathrm{Spin}(7)$ | |
| Mar 18, 2015 at 18:47 | comment | added | Bilateral | I would say it is $SU(4)$, since $SU(4)\subset Spin(7)$. In fact, $SU(4)$ is another of the few holonomy groups admissible for an irreducible Riemannian manifold of dimension eight. | |
| Mar 18, 2015 at 18:23 | history | asked | seub | CC BY-SA 3.0 |