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I am trying to work through the section in Humphreys Lie algebras about Jordan–Chevalley decomposition but am finding it difficult to parse. The main meat of this section is the following proposition. enter image description here

I am having trouble understanding what $(b)$ is stating and how it is useful. I know that $x_s$ and $x_n$ are endomorphisims, but I am confused what kind of object $p(T)$ and $q(T)$ are. If we evaluate them on an element in $F$ we get an endomorphism of $V$? I have read on and it does not seem to me that $(b)$ gets used in the rest of this section/subsequent sections. Is this only here for completion or is this a useful remark in the context of Lie algebras.

I am also confused why all of this doesn't simply follow form the Jordan-Normal form. Given the Jordan normal form we can let $x_s$ be the diagonal and $x_n$ be the remaining. I think the Jordan-Normal form requires stricter conditions to exist, but it seems with Lie algebras and what we are interested in those conditions will be met.

Lastly if this does not follow form the Jordan-Normal form, I am unsure how one should think about the semisimple part of an endomorphism. What do semisimple endomorphisms do or act like?

(1) What kind of objects are $p(T)$ and $q(T)$? I don't understand the explanation given

(2) Can you give a use for proposition $(b)$ outside of this proof

(3) Why do we care about Jordan-Chevalley decompositions? How should one think about semisimple endomorphisms?

Mark Murray
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  • Doesn't Humphreys use (b) to prove (c)? – Angina Seng Jul 23 '20 at 10:54
  • I think you are right but I must admit I am still struggling with the proof. It still seems strange to me to state it in the proposition if it's only use is to be an intermediate step in the proof. – Mark Murray Jul 23 '20 at 10:58
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    I've never actually seen anyone compute and write $x_s = 3x^4-4x^2+x$ or something like that; in fact, I've never seen the middle part used for anything except proving that $x_s$ and $x_n$ commute with each other and $x$ and anything that commutes with it (this, however, is an important and useful fact). Of course it might be that there is some nice theory involving the actual polynomials, I've just never encountered them again in structure or representation theory of Lie algebras. – Torsten Schoeneberg Jul 23 '20 at 16:53
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    Re third question, one crucial tool in structure theory (at least for semisimple LAs) are Cartan subalgebras, which kind of are the abstract analogue of diagonal matrices; to understand them, one has to understand semisimple elements. Cf. https://math.stackexchange.com/q/3314046/96384. Showing that two common characterisations of CSAs are equivalent is one thing where decomposition in semisimple and nilpotent parts is used, cf. https://math.stackexchange.com/q/3480970/96384, https://math.stackexchange.com/q/3537312/96384. – Torsten Schoeneberg Jul 23 '20 at 17:06

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