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The Unapologetic Mathematician
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All Derivations of Semisimple Lie Algebras are Inner
It turns out that all the derivations on a semisimple Lie algebra 
If we set 
![[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)
&=\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 
![[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
![\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)
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September 11, 2012 - Posted by John Armstrong | Algebra, Lie Algebras
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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.