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This is a bit of an odd question perhaps but I found myself asking the "why" question when studying "Lie Algebras in particle physics" by Howard Georgi concerning the previously mentioned objects.

Why do we introduce and study simple roots, Dynkin diagrams and Cartan matrices? What I gather from the book is that they offer a way of "compressing" the "information" of a Lie Algebra immensely in the sense that all the properties of a Lie Algebra can be encoded within Dynkin diagrams/simple roots and then the entire root system and the commutation relations of the algebra can then be reconstructed using the Cartan matrix and other tools. But of what use is all that?

It's not like one couldn't just draw the whole root systems or give the commutation relations of the Lie Algebra when one wants to communicate the structure of an algebra, right?

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    $\begingroup$ Earlier in my career, when I used to think of a compact semisimple Lie group, I used to think mostly about their definitions which are geometric/algebraic (and I used to think mostly of the $4$ infinite families). Historically, root systems allowed mathematicians to classifiy complex semisimple Lie algebras (I think this was mostly done by Killing and E. Cartan, though it was later simplified by Dynkin and others). However, even when dealing with representation theory, it also helps to think in terms of Dynkin diagrams and thus root systems. $\endgroup$
    – Malkoun
    Aug 12, 2020 at 21:03
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    $\begingroup$ I do not quite understand your last line / question, which really might be a language issue (double negation and all). Could you clarify, please? $\endgroup$ Aug 12, 2020 at 22:10

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The crucial point is that we consider a finite-dimensional complex simple or semisimple Lie algebra. Then a classification is based on "compressing the information" via root systems, Cartan matrices and Dynkin diagrams. This is the famous classification by Cartan (and Killing).

Related post: Motivation for Killing Form and Root Diagrams

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