I have started to learn about Lie Algebras from Humphreys book of Lie Algebras. Currently I am studying about nilpotent and solvable Lie Algebras. I think I understand the definition and I also understand the proofs of Engel's theorem and Lie's Theorem, but I don't understand the motivation for studying solvable and nilpotent Lie Algebras?

Is there any result that says that any 'good' Lie algebra decomposes into direct sum of solvable and nilpotent Lie Algebras? (or some similar result)

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    $\begingroup$ Lie theory is an inmmense theory, and any modern exposition is the result of years of collective experience in gauging the importance of its many parts. This usually means that you have to withhold your need for motivation till later in the book. It sort of sucks, yes, but you will need to learn to do this eventually. $\endgroup$ – Mariano Suárez-Álvarez Aug 28 '16 at 20:52

There is an important result, the Levi decomposition theorem that says that under good conditions a Lie algebra is a semidirect product of a semisimple algebra and a solvable one.

Nilpotent and solvable algebras appear quite naturally when one studies the structure theory of Lie algebras, leading for example to the theorem of Levi I mentioned. But quite independently of that, nilpotent and solvable algebras are interesting because lots of algebras are nilpotent or solvable, algebras that we encounter in real life. This may not be very obvious when one is starting to learn abut this, of course, and it can be appreciated only with experience and lots of examples.

  • $\begingroup$ Thank you sir! Is there any 'good' book of Lie Algebras which gives some motivation before defining the concepts? $\endgroup$ – Dontknowanything Aug 29 '16 at 5:27
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    $\begingroup$ Your motivation should be that you want to learn about Lie algebras and you should trust respected authors that they made a sensible choices of topics. If you browse a couple of books and they all touch on solvable algebras and nilpotent ones, then that should count as evidence that it is a useful subject. When you start learning advanced stuff, you need to exercise patience. $\endgroup$ – Mariano Suárez-Álvarez Aug 29 '16 at 6:12
  • $\begingroup$ In other words: presumably you do have already some motivation to learn about Lie algebras, for you are doing it. Given that, learn the subject, entrusting such great expositors as Humphreys to lead you into the subject. $\endgroup$ – Mariano Suárez-Álvarez Aug 29 '16 at 6:14
  • $\begingroup$ Thank you.I hope I will get much motivations with experience and by solving examples/exercises.Thank you for your great advice. $\endgroup$ – Dontknowanything Aug 29 '16 at 6:18
  • $\begingroup$ If you keep on reading the book you are reading, you'll see solvable and nilpotent algebras put to use. It just takes a bit of patience :-) $\endgroup$ – Mariano Suárez-Álvarez Aug 29 '16 at 6:24

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