Download !FULL! Ebook Kimia Analitik Kuantitatifl 🚩

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Significance of this notation

I’ve been reading up on elliptic curves and I found this notation in a textbook:

This reminds me of the commutator, where $a^{b-c}=b^{ -1}ab^{ -1}$ and $\mathrm{commutator}(a,b)=ab-ba$

What does this notation mean? Thank you for your time.

A:

Short answer: It means the same thing as $[a,b]=ab-ba$, but with an extra “up to lower-order terms” qualification.
Longer answer:
First note that $[a,b]$ usually denotes the commutator, and we have some common examples:
$$
[a,b] =
\left\{
\begin{array}{c}
ab-ba &\text{if } b
eq 0\\
0 &\text{if } a=b\\
-b &\text{if } a=-b
\end{array}
\right.
$$
and
$$
[a,b,c] =
\left\{
\begin{array}{c}
abc-acb-bca+cab &\text{if } a,b,c
eq 0\\
0 &\text{if } a=b=c\\
-c &\text{if } a=b=-c\\
\end{array}
\right.
$$

So now, the upshot of all this is that $ab-ba$ is the “purely group-theoretic commutator” of $a$ and $b$, but that when we are talking about “terms at the highest order” we want to be able to handle a general commutator, not just $ab-ba$.
(Note that the term “purely group-theoretic commutator” is not standard at all, but I have been
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