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])

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


  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-> (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?


    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! 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 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

    • It was rewarding, but not nearly so much after being washed out of academic mathematics. I’m now in the private sector as a software engineer, focusing on a lot of functional programming, which is intimately related to the category theory I used to study, so it’s not been quite a total waste.

      The movies, however, allow me to explore many more interests, and the critical process inspires a lot more creative growth than recapping and explaining subjects that, ultimately, have already been written down by many other authors before I ever came along.

      As for your other question, yes, I have no intention of taking down the content, but I don’t see myself having much time to add more.

      Comment by John Armstrong | July 23, 2014 | Reply

  7. Thanks for the effort you put in and the depth and breadth you covered and for taking the time to answer. All the best in your career and in your film reviews.

    Comment by pa32r | July 24, 2014 | Reply

  8. I’ve always wanted to know. Are modern mathematicians able to follow [specifically] the postulates and proofs set out by Euclid in the Elements? Have you ever read it?

    You obviously have more than enough knowledge to actually do the math. But when you read the Elements, does it make sense to you? The actual way it is written, and are you able to follow the proofs easily?

    Do mathematicians bother going through that stuff anymore, or is it a waste of time? Here, I don’t mean the fundamentals of Euclidean geometry; I mean specifically the way it is set out in the Elements. Or is the format just simply too out of date to be useful?

    Comment by Rakkal | August 15, 2015 | Reply

    • Modern mathematicians, sure, and pretty much every one of them would recognize the value of Euclid’s work. The axiomatic method he follows is basic to our whole understanding of how good mathematics is done, and it’s the jumping-off point for all the revisions the 20th century brought in the philosophy of mathematics, starting with Hilbert’s challenges at the 1900 International Congress of Mathematics.

      The only major difference between the way Euclid seems to do geometry and they way most mathematicians now would think about it is that he conceived of his geometrical entities as describing “real” things, but now we’re a lot more careful about what “real” means.

      In the fine details, Euclid tends to be more wordy than modern proofs might be, but that’s more of a stylistic difference, and nothing that a working mathematician wouldn’t be able to figure out given a little thought.

      As to the snarky comment below, claiming to come from the writer of the nonexistent “professorcurmudgeon” blog, this question was asking about mathematicians rather than students. He also doesn’t mention what level of students at what sort of school he’s talking about. That said, it wouldn’t surprise me that students in general know less about the history and philosophy of mathematics than professional mathematicians do. On the other hand, the fault is far more likely to lie with their teachers who have evidently spent 35+ years bitching about the situation rather than working to fix it.

      Comment by John Armstrong | August 17, 2015 | Reply

      • Professor Curmudgeon is the pseudonym of Emeritus Associate Professor of Computer Science at Clemson. I have a PhD in mathematics and taught most course in CS plus research in verification and validation of simulation. I also mentor middle and high school students. Who’s being snarky? I simple observe that modern students over the past 35 years arrive at mathematics programs in universities without ever seeing Euclid as written in the Elements and have little or no introduction to the methods of Euclid. Modern views of geometry started with Hilbert in “Grundlagen der Geometrie” in1899. Hilbert set the standard for the axiomatic development and presentations of such. Euclid is important historically and the history is fascinating.

        BTW, university professors have no say in how SC runs its K-12 programs. Believe me, we’ve tried. And my blog is ComputationalThought.

        Comment by ComputationalThought | August 17, 2015 | Reply

      • @ John Armstrong

        Thank you for this response. I found it very illuminating.

        To clear up: yes, I was asking specifically about [working] mathematicians, as you correctly interpreted.

        Comment by Rakkal | August 20, 2015 | Reply

  9. In my 35+ years of teaching I believe that no students has ever seen the Elements nor do they have any clue on the history/philosophy of mathematics.

    Comment by ComputationalThought | August 15, 2015 | Reply

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