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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
    Commented Aug 28, 2016 at 20:52
  • $\begingroup$ @Mariano Suárez-Álvarez this sort of attitude mostly comes from the lack of familiarity with the origins of the theory. Solvability was called "integrability" by Lie himself and had to do with literally being able to solve a system of first-order PDE's in quadratures if it admits a symmetry described by a solvable Lie algebra, by analogy with Galois theory. This is extremely easy to see and would be very useful for all students of Lie theory to see early on. It's also no accident that Lie formalized these concepts just 2 years into his studies, long before abstract Lie groups even existed. $\endgroup$
    – Alex Bogatskiy
    Commented Jul 11 at 15:10

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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.

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  • $\begingroup$ Thank you sir! Is there any 'good' book of Lie Algebras which gives some motivation before defining the concepts? $\endgroup$
    – Dontknowanything
    Commented Aug 29, 2016 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
    Commented Aug 29, 2016 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
    Commented Aug 29, 2016 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
    Commented Aug 29, 2016 at 6:18
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    $\begingroup$ Infact your last comment is also a motivation for keep on reading the fantastic book of Humphreys.Thanks again : ) $\endgroup$
    – Dontknowanything
    Commented Aug 29, 2016 at 6:28
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I know it's been over 7 years, but I'll post this excerpt from the book "Emergence of the theory of Lie groups" by Thomas Hawkins because I find the accepted answer quite dismissive of the original motivation behind these concepts, which is in fact quite natural and simple. This "teleological" attitude of treating fundamental concepts as a mere means to an end (such as proving some highbrow theorems) is unfortunately plaguing exact and natural sciences, whereas I actually find the ability to clearly answer questions like yours very important for mathematics education. As explained in the excerpt, "solvability" referred literally to the complete integrability of a system of linear PDE's. Lie himself exclusively thought of continuous groups not as abstract objects, but as collections of invertible transformations acting on the independent variables in a system of PDE's. It turns out that if the symmetry is solvable, then you can construct a complete set of solutions by solving a sequence of ODE's.

page 90 page 91

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  • $\begingroup$ Can you give any examples of such a system of PDEs? I've always been curious about Lie's original motivations for studying Lie theory but I don't know a single example of the types of systems to which it was supposed to apply. $\endgroup$
    – Qiaochu Yuan
    Commented Jul 11 at 15:41
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    $\begingroup$ @QiaochuYuan not anything I can type in a comment, but page 156 of this book describes a system of two 4-th order ODE's with a solvable 3-dimensional symmetry. $\endgroup$
    – Alex Bogatskiy
    Commented Jul 11 at 17:35
  • $\begingroup$ @QiaochuYuan also every 2-dimensional Lie algebra is solvable, so, for example, a system of 3-rd order ODE's with a 2-dimensional symmetry group is always solvable. $\endgroup$
    – Alex Bogatskiy
    Commented Jul 11 at 17:38
  • $\begingroup$ @QiaochuYuan the exercises at the end of the chapter I linked also contain a lot more elementary examples that are very instructive. $\endgroup$
    – Alex Bogatskiy
    Commented Jul 11 at 17:42
  • $\begingroup$ Nice, thanks for the reference! $\endgroup$
    – Qiaochu Yuan
    Commented Jul 11 at 17:47

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