All Derivations of Semisimple Lie Algebras are Inner
It turns out that all the derivations on a semisimple Lie algebra 
are inner derivations. That is, they’re all of the form 
for some 
. We know that the homomorphism 
is injective when is semisimple. Indeed, its kernel is exactly the center 
, which we know is trivial. We are asserting that it is also surjective, and thus an isomorphism of Lie algebras.
If we set 
and 
, we can see that ![[D,M=I]\subseteq I](https://s0.wp.com/latex.php?latex=%5BD%2CM%3DI%5D%5Csubseteq+I&bg=e6e6e6&fg=333333&s=0&c=20201002)
. Indeed, if 
is any derivation and , then we can check that
&=\delta([\mathrm{ad}(x)](y))-[\mathrm{ad}(x)](\delta(y))\\&=\delta([x,y])-[x,\delta(y)]\\&=[\delta(x),y]+[x,\delta(y)]-[x,\delta(y)]\\&=[\mathrm{ad}(\delta(x))](y)\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Cleft%5B%5Cdelta%2C%5Cmathrm%7Bad%7D%28x%29%5Cright%5D%28y%29%26%3D%5Cdelta%28%5B%5Cmathrm%7Bad%7D%28x%29%5D%28y%29%29-%5B%5Cmathrm%7Bad%7D%28x%29%5D%28%5Cdelta%28y%29%29%5C%5C%26%3D%5Cdelta%28%5Bx%2Cy%5D%29-%5Bx%2C%5Cdelta%28y%29%5D%5C%5C%26%3D%5B%5Cdelta%28x%29%2Cy%5D%2B%5Bx%2C%5Cdelta%28y%29%5D-%5Bx%2C%5Cdelta%28y%29%5D%5C%5C%26%3D%5B%5Cmathrm%7Bad%7D%28%5Cdelta%28x%29%29%5D%28y%29%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0&c=20201002)
This makes 
an ideal, so the Killing form 
of 
is the restriction of 
of the Killing form of 
. Then we can define 
to be the subspace orthogonal (with respect to ) to , and the fact that the Killing form is nondegenerate tells us that 
, and thus ![[I,I^\perp]=0](https://s0.wp.com/latex.php?latex=%5BI%2CI%5E%5Cperp%5D%3D0&bg=e6e6e6&fg=333333&s=0&c=20201002)
.
Now, if is an outer derivation — one not in — we can assume that it is orthogonal to , since otherwise we just have to use to project onto and subtract off that much to get another outer derivation that is orthogonal. But then we find that
![\displaystyle\mathrm{ad}(\delta(x))=[\delta,\mathrm{ad}(x)]=0](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cmathrm%7Bad%7D%28%5Cdelta%28x%29%29%3D%5B%5Cdelta%2C%5Cmathrm%7Bad%7D%28x%29%5D%3D0&bg=e6e6e6&fg=333333&s=0&c=20201002)
since this bracket is contained in ![[I^\perp,I]=0](https://s0.wp.com/latex.php?latex=%5BI%5E%5Cperp%2CI%5D%3D0&bg=e6e6e6&fg=333333&s=0&c=20201002)
. But the fact that 
is injective means that 
for all , and thus 
. We conclude that 
and that 
, and thus that is onto, as asserted.
September 11, 2012 -
Posted by John Armstrong |
Algebra, Lie Algebras
The Unapologetic Mathematician 
I have a basic understanding of the nature of (finite) groups. I have played around with S3 for a few hours and days, and have respect for the depth of its properties. I experience both fear and awe when trying to think about larger and larger S groups.
Algebra seems like an infinite maze. The fact there are only countably many possible algebraic expressions is some comfort, but not that much, because my brain feels decidedly finite.
I am trying to get a grip on implications and applications. Does this theorem lead one (eventually) to a better understanding of polynomial equations? Is there something geometrical one can infer? Can I use it to write interesting computer programs?
It helps simplify the project of classifying Lie algebras and their representations, which turns out to be of use on quite a lot of theoretical physics, for one thing.
To me this seems like breaking large rocks for small change, but I guess you have to enjoy it.
Still — to contradict myself — I actually do find this tempting. I wish someone would give me a few hints about those uses in physics. QCD or something like that?
I’ve mentioned before — though quite a while ago, now — that Lie algebras arise as the “infinitesimal” versions of Lie groups. That is, if you look at a continuously-varying collection of symmetries, if you want to do calculus on it you’re going to end up using Lie algebras. Since quite a lot of modern physics is about symmetries, this comes up a lot.
As for large rocks and small change, I understand the frustration given how hard I’ve twisted your arm to force you to read this stuff.
As for me, I find the algebraic approach much easier to wrap my brain around than the other presentations of Lie theory I’ve struggled with. (Still not _easy_, just considerably easier 🙂 I’ve seen all the topics that have been covered so far in this series many times before, but until now have never had any clue what the heck they meant. I *really* appreciate the way John has presented this material, it’s finally starting to make a bit of sense to me.
I apologize. I certainly did not mean to denigrate your work. On the contrary, I admire it. Otherwise, of course, I would not be reading and asking questions.
I do get the proof, but do you have some reference for this?
I don’t have my library immediately at hand, but it’s a pretty standard result in the representation theory of Lie Algebras, and I’d think most textbooks would contain it. Humphreys probably does, but I’d have to look it up to be certain.