I am doing some complicated and tedious calculation on iterated commutators. A typical term in my calculation looks like $$[x_a,[[[x_b,x_c]-x_d,x_e],[x_f,x_g]]]\text{.}$$

(I am considering commutators on Lie algebras so the minus sign $-$ is well defined here.) In the notation above there is no ambiguity; but when I have dozens of terms like that it looks horrible and a mistake in calculation is almost inevitable. I have simplified the notation a little bit by forgetting the $x$'s and write the commutator as $$[a,[[[b,c]-d,e],[f,g]]]\text{,}$$ but it still looks confusing. Not sure if I should continue to simplify it and simply write $$[a[[([bc]-d)e][fg]]]$$ instead. I tried to use physical papers as my draft papers and also opened a rich text document to note down the calculation; but both ways look awkward and it is difficult for me to "see" when should I cancel two terms (under some conditions) or use the Jacobi's identity.

So my question is:

Is there any "smart" notation that I can use for calculation like this and make the calculation less fallible?

I am not referring to writing for publishing, but just for calculation on one's own draft paper.

  • $\begingroup$ I wonder if it would be difficult to try to get the computer to simplify expressions of this type. $\endgroup$
    – MJD
    Dec 12 '14 at 15:55
  • $\begingroup$ @MJD, are you referring to using some softwares such as GAP? $\endgroup$
    – Zuriel
    Dec 12 '14 at 15:56
  • $\begingroup$ How about $[x_a,x_b]=ab$? You'll still have to keep track of parentheses, but I think it uses about as little notation as possible. $\endgroup$
    – user134824
    Dec 12 '14 at 15:57
  • $\begingroup$ I suggest to use larger brackets. $\big[a,[b,c]\big]$ $\endgroup$ Dec 12 '14 at 15:58
  • 1
    $\begingroup$ @user134824, sounds good! I am also considering using different colours to keep track of parentheses; if I do so, I thinking using pens/pencils should be easier than using LaTex/Microsoft Word. $\endgroup$
    – Zuriel
    Dec 12 '14 at 16:01

I think that you're going to have some cluttered-looking formulas no matter what.

However, instead of $[X, Y]$, one could write, say, $\mathcal{L}_X(Y)$.

You could obviously even shorten this to $X(Y)$, like function notation. If you then switch to exponent notation for functions (as, $f(x) = x^f$), and especially if you're willing to mix these notations, then (assuming I parsed it right!) your sample formula could become, for instance: $$ ([f, g]^{[[b, c] - d, e]})^a $$ which to my eye is slightly more readable than what you had.

Obviously this could cause a bunch of other problems for you! For instance: (1) you may want to reserve exponential notation for, well, exponents. (2) Visually, it switched the order. So even though $a$ started out way on the left, now it's way on the right.

We could perhaps try to lessen the confusion in both (1) and (2) at once: Write ${}^xy = [x, y]$, so now your formula is $$ {}^a({}^{[[b, c] - d, e]}[f, g]) $$ You could also use subscripts instead of superscripts, etc. It's a matter of taste whether something like this is preferable. Obviously, do whatever works best for you!

  • $\begingroup$ Thank you for the suggestion! So you mean we write ${}^xy=[x,y]$ sometimes but not always, right? Since otherwise the notation will look like $${}^a({}^{^{{}^bc-d}e}({}^fg)).$$ And when to write ${}^xy$ and when to write $[x,y]$, there is no rule; right? $\endgroup$
    – Zuriel
    Dec 12 '14 at 16:26
  • $\begingroup$ Yeah, my suggestion would be to mix notations, assuming you're comfortable with that. So you could still use the $[x,y]$ notation sometimes. There wouldn't be have to be any rule for when to use each notation; you'd just use whatever you thought looked more clear. (Like how sometimes people write $2*3$ and sometimes people write $(2)(3)$; either is acceptable, and you just use whatever is most convenient at the time). $\endgroup$
    – mathmandan
    Dec 12 '14 at 16:36

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