Timeline for Largest subgroup of $SU(n)$ for which the adjoint action preserves specific inner product on $\mathfrak{su}(N)$
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 19, 2015 at 17:26 | vote | accept | Benjamin | ||
| Apr 24, 2015 at 18:18 | comment | added | Benjamin | So, it seems that for $SU(2)$ with $K(A,B)=0$ the product $PQ$ (which is a torus as you pointed out) is the whole of $S_{A,B}$. However $PQ$ in general will have dimension $2n+2$ (if $A,B$ both regular) so it will not be codim $1$. I'm failing to understand what else could be in $S_{A,B}$, can even a single example be found? | |
| Apr 23, 2015 at 20:46 | comment | added | Benjamin | Ok, thanks for all you help! Hopefully this is related to quantum control somehow! We'll see.... | |
| Apr 23, 2015 at 20:38 | comment | added | Robert Bryant | @Benjamin: Oh, sorry. That's was a typo; it should have been 'codimension-$1$'. | |
| Apr 23, 2015 at 18:15 | comment | added | Benjamin | It's very likely that I missed something, however you said "it follows that $S_{A,B}$, which is easily seen to be a codimension-2 subset.." in a comment on the original post. I may be confused. It is clear that in the $SU(2)$ case $P$ and $Q$ are both co dim 2. | |
| Apr 23, 2015 at 18:03 | comment | added | Robert Bryant | Yes, co-dim 1 is correct for $S_{A,B}$. I don't know why you were thinking co-dim 2; nothing I wrote would have suggested that. | |
| Apr 23, 2015 at 17:33 | comment | added | Benjamin | Actually, isn't is always co dim 1? Any torus in $SU(2)$ would be? | |
| Apr 23, 2015 at 17:11 | comment | added | Benjamin | So now my only point of confusion which remains is why $S_{A,B}$ is co-dim 2 in the case of $SU(2)$ (with $A,B$ orthogonal) as you say above but is co-dim 1 in general. The only case I'm actually interested in is the case where they are orthogonal so if this is the cause then that would be interesting. | |
| Apr 23, 2015 at 16:37 | comment | added | Robert Bryant | @Benjamin: Correct. For example, when $n=2$, $S_{A,B}$ is a torus embedded in $\mathrm{SU}(2)\simeq S^3$, while the exponentials of linear subspaces of ${\frak{su}}(2)\simeq\mathbb{R}^3$ are either great circles or spheres in $\mathrm{SU}(2)\simeq S^3$. As for characterizing the preimage of $S_{A,B}$ under the exponential map in general, this is probably going to be somewhat messy. | |
| Apr 23, 2015 at 14:43 | comment | added | Benjamin | I guess that $\log$ of $S_{A,B}$ will not be a vector subspace of $\mathfrak{su}(n)$ but this seems tricky. It's very unclear to me which subsets of $SU(n)$ have the property that their $\log$ is a vector subspace of $\mathfrak{su}(n)$. | |
| Apr 23, 2015 at 13:25 | comment | added | Benjamin | Ok, the -1 is because $A$ is also traceless, I see. Thanks for your input, that issue is now much clearer. | |
| Apr 23, 2015 at 9:08 | comment | added | Robert Bryant | @Benjamin: Actually, for regular elements, the dimension of $P$ and $Q$ will be $n{-}1$. As long as $[A,B]\not=0$, the set $S_{A,B}$ will be smooth at the identity and its tangent space there will be the hyperplane in ${\frak{su}}(n)$ perpendicular to $[A,B]$. | |
| Apr 23, 2015 at 4:45 | comment | added | Benjamin | I meant commute with $A$. Assuming $A,B$ are regular elements, dim $P,Q$ will be $n$. As $S_{A,B}$ is co-dimension $1$ it's dim will (only in places where this makes sense, I'm not exactly sure how to make the precise) be $n^2 - 2$ right? So $P \cup Q$ is far from being the whole set $S_{A,B}$ it seems. | |
| Apr 23, 2015 at 1:26 | comment | added | Robert Bryant | @Benjamin: The Lie algebra of $P$ will be all the things that commute with $A$ in ${\frak{su}}(n)$. It's hard to say what else will be in $S_{A,B}$ since $S_{A,B}$ is of codimension $1$ in $\mathrm{SU}(n)$. It will surely be larger than $P\cup Q$. | |
| Apr 22, 2015 at 22:19 | comment | added | Benjamin | Thanks, a great answer as usual. Am I correct in saying that the Lie algebra of $P$ will simply by all the things that commute with $P$ in $\mathfrak{su}(n)$?. I'm trying to understand what else will be in $S_{A,B}$ in the case $K(A,B)=0$ other than $P \cup Q$. | |
| Apr 22, 2015 at 17:17 | history | edited | Robert Bryant | CC BY-SA 3.0 |
added 226 characters in body
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| Apr 22, 2015 at 17:12 | history | answered | Robert Bryant | CC BY-SA 3.0 |