The Unapologetic Mathematician

Mathematics for the interested outsider

The Submodule of Invariants

If V is a module of a Lie algebra L, there is one submodule that turns out to be rather interesting: the submodule V^0 of vectors v\in V such that x\cdot v=0 for all x\in L. We call these vectors “invariants” of L.

As an illustration of how interesting these are, consider the modules we looked at last time. What are the invariant linear maps \hom(V,W)^0 from one module V to another W? We consider the action of x\in L on a linear map f:

\displaystyle\left[x\cdot f\right](v)=x\cdot f(V)-f(x\cdot v)=0

Or, in other words:

\displaystyle x\cdot f(v)=f(x\cdot v)

That is, a linear map f\in\hom(V,W) is invariant if and only if it intertwines the actions on V and W. That is, \hom_\mathbb{F}(V,W)^0=hom_L(V,W).

Next, consider the bilinear forms on L. Here we calculate

\displaystyle\begin{aligned}\left[y\cdot B\right](x,z)&=-B([y,x],z)-B(x,[y,z])\\&=B([x,y],z)-B(x,[y,z])=0\end{aligned}

That is, a bilinear form is invariant if and only if it is associative, in the sense that the Killing form is: B([x,y],z)=B(x,[y,z])

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September 21, 2012 - Posted by | Algebra, Lie Algebras, Representation Theory

9 Comments »

  1. Hi John, don’t know whether it’s worth posting here but your blog has been mentioned in the list of mathematics blogs here-> http://www.talkora.com/science/List-of-mathematics-blogs_112 (look for entry #5 in the list)

    Comment by Will | January 3, 2013 | Reply

  2. Will you be posting anymore for the rest of time?

    Cheers,
    NS

    Comment by notedscholar | April 13, 2013 | Reply

  3. Reblogged this on Observer.

    Comment by Kamran | July 14, 2013 | Reply

  4. Do you feel a certain amount of arrogance is necessary in being a top-level mathematician? I feel it is necessary to have an inherent faith in your own learning capacity. This can sometimes come across as “arrogance”. Your thoughts?

    [--Apologies for posting this here. I'm having serious issues with logging in to my spring.me! account and hitting you up there.--]

    Comment by Matty Skeeler | July 18, 2014 | Reply

  5. It would be really nice if this blog could restart. I find it very valuable and miss the new posts.

    Comment by Rob Ryan | July 20, 2014 | Reply

    • It would be nice, sure. Unfortunately, with the end of my academic career I have a real full-time job to hold down. An occasional “gee, this is nice” doesn’t really justify all the work it took to maintain this weblog, and I find my new side-project (at drmathochist.wordpress.com) far more rewarding than this ever was.

      Comment by John Armstrong | July 20, 2014 | Reply

      • If you don’t mind me asking, how long did each post take you to do, overall? I know you wrote somewhere that it took you an hour(ish) to write. But that probably doesn’t include research & note-taking time (I’m guessing).

        Is it a LOT more work to maintain this math blog than the movie review one?

        I ask because I would possibly want to start one myself in the future.

        Comment by Matty Skeeler | July 20, 2014 | Reply

        • It’s hard to say. An hour a day just for the writing is a good start, but yes there was a lot of planning and research to make sure I said the right thing next. I wouldn’t say it’s a lot more to write this weblog than my reviews, but I’m not getting pay or prestige for either one, and the reviews are far more creatively rewarding than this ever was. So if I only have time for one or the other (again, on top of a 50-60 hour/week Real Job), I’m going with the one that I enjoy.

          Comment by John Armstrong | July 21, 2014 | Reply

  6. Very interesting, and thanks for taking the time to reply. I do certainly miss it. What you say is interesting though, because I got the sense that you found this one rewarding. And certainly, my “it would be really nice…” comment understates the value I find in it. Nevertheless, keeping it simply for me or me and some few others is certainly not worth a major investment of your time. I find that my own trivial blog, read by not a lot of people, rewards me in the thought that I put into it. There’s no pay or prestige in that for me either. Best of luck in your real job. If you don’t mind, two more questions. 1. Is your real full-time job in the field of mathematics? 2. Are you going to leave this blog up for a while? Otherwise, I’ll need to try to figure a simple way to keep the content.

    Comment by Rob Ryan | July 23, 2014 | Reply


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