Timeline for Automorphism group of the special unitary group $SU(N)$
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 11, 2019 at 22:38 | comment | added | anon | Do all real forms of a complex simple lie algebra have the same outer automorphism group as the complex simple lie algebra itself? | |
| May 7, 2019 at 18:05 | comment | added | annie marie cœur | answer accepted, thanks, I must not know enough about Out so I ask a new one here: math.stackexchange.com/questions/3217455/… | |
| May 7, 2019 at 17:36 | vote | accept | annie marie cœur | ||
| May 7, 2019 at 17:05 | comment | added | Sean Lawton | @annieheart I have answered your question and indulged your further queries in the comments. If you have further questions you are welcome to make additional posts on MathOverFlow or on MathStackExchange. The comments section is not the appropriate place for new questions. | |
| May 7, 2019 at 15:25 | comment | added | annie marie cœur | also $$\mathrm{Out}(\mathrm{PSU}(n))\cong\left\{\begin{array}{ll}1 ,&\text{ if } n\geq 3\\ 1,&\text{ if }n=2. \end{array}\right. ?$$ | |
| May 7, 2019 at 15:10 | comment | added | annie marie cœur | Thanks very much, what can we say about Inn, Out, and Aut(PSU(N)) then? Is that true: $$\mathrm{Aut}(\mathrm{PSU}(n))\cong\left\{\begin{array}{ll}\mathrm{PSU}(n) ,&\text{ if } n\geq 3\\ \mathrm{PSU}(2),&\text{ if }n=2. \end{array}\right.$$ | |
| May 6, 2019 at 21:17 | comment | added | Sean Lawton | I am not sure. I think it is true for simple, simply-connected compact Lie groups. "Proof:" Since $G$ is simple, the Lie algebra of $\mathrm{Inn}(\mathfrak{g})$ is $\mathfrak{g}$ itself. Thus, since $G$ is simply-connected and compact, the abstract Lie group structure on $\mathrm{Inn}(\mathfrak{g})$ is a finite central quotient of $G$. Since the center is in the kernel, it must be $PG$. And $\mathrm{Inn}(G)\cong PG$ always. $\Box$ | |
| May 6, 2019 at 21:09 | comment | added | annie marie cœur | Thanks, Inn(𝐺)≅Inn(𝔤) also for simply-connected Lie groups? | |
| May 6, 2019 at 21:08 | comment | added | Sean Lawton | $PU(n)\cong PSU(n)$ in general. For simply-connected Lie groups, $\mathrm{Aut}(G)\cong \mathrm{Aut}(\mathfrak{g})$ and $\mathrm{Out}(G)\cong \mathrm{Out}(\mathfrak{g})$.The same argument does not work for $\mathrm{Aut}(PG)$ since $PG$ is not generally simply-connected. The inner automorphisms of the Lie algebra are the image of $G$ via the adjoint. I think you need to generally figure out the kernel. | |
| May 6, 2019 at 21:07 | comment | added | annie marie cœur | If these are true and I understand well, I will accept your answer. Thanks! | |
| May 6, 2019 at 19:43 | comment | added | annie marie cœur | Do we have $$\mathrm{Out}(G)\cong\mathrm{Out}(\mathfrak{g})?$$ $$\mathrm{Inn}(G)\cong\mathrm{Inn}(\mathfrak{g})?$$ $$\mathrm{Aut}(G)\cong\mathrm{Aut}(\mathfrak{g})?$$ in general? | |
| May 6, 2019 at 19:35 | comment | added | annie marie cœur | Does it mean that the $\mathrm{Aut}(\mathrm{PU}(n))=\mathrm{Aut}(\mathrm{PSU}(n))=\mathrm{Aut}(\mathrm{SU}(n))$, since they have the same Lie algebra: $$\mathrm{Aut}(\mathrm{PSU}(n))\cong\left\{\begin{array}{ll}\mathrm{PSU}(n)\rtimes \mathbb{Z}/2\mathbb{Z},&\text{ if } n\geq 3\\ \mathrm{PU}(2),&\text{ if }n=2. \end{array}\right. ?$$ | |
| May 6, 2019 at 19:27 | comment | added | annie marie cœur | PSU(2) = PU(2), yes? | |
| May 4, 2019 at 1:24 | history | answered | Sean Lawton | CC BY-SA 4.0 |