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A How Do Engel and Lie Arise Naturally?

  1. Oct 16, 2016 #1
    I am wondering how Lie's theorem and Engel's theorem fit into the theory of Lie algebras naturally, perhaps they are motivated by the Levi decomposition and the Jordan normal form decomposition for operators?

    I find it jarring to prove Engel for nilpotent Lie algebras for no real reason, or even the concept of nilpotency (Serre mentions ##T^n(v) = 0## being equivalent to ##T^{n-1}(v) = (I + \varepsilon T)T^{n-1}(v)## as another way of looking at it, that's about it!) but nilpotent operators arise naturally in the JNF, so perhaps this motivates Engel's theorem? It would be great to run into the necessity of needing these theorems without realizing it, as can be done for the Levi decomposition!

    Overall, my best attempt so far is:

    In a general real/complex Lie algebra ##L##, if you write a JNF decomposition $$x = n + s,$$ with ##s## diagonalizable, ##n## nilpotent, what happens to ##x## when you do a Levi decomposition $$L = N \rtimes S,$$ ##N## solvable, ##S## semi-simple? On the one hand it kind of looks like a Levi decomposition is motivated by the JNF decomposition, but Erdmann's Lie Algebras book seems to only apply the JNF to the semi-simple part ##S## of ##J##, so it seems like they are different things?

    If the JNF applies only to the semi-simple part, then am I right in saying, given ##L##, you first do a Levi decomposition $$L = N \rtimes S,$$ apply Lie's theorem on solvable Lie algebras to ##N## to decompose some of ##x## into upper triangular form ##u## and the rest, ##x'##, then just needs to be dealt with, $$x = u + x',$$ so for ##x'## we can decompose it using the JNF into $$x' = n + s,$$ so that $$x = u + n + s,$$ and then on the nilpotent part ##n## we apply *Engel's theorem* to bring ##n## into strictly-upper-triangular form?
     
  2. jcsd
  3. Oct 16, 2016 #2

    fresh_42

    Staff: Mentor

    You should reset your priorities. Lie, Engel (and Cartan) where basically the founding fathers of Lie Theory. Not the other way around.
    The two theorems arise naturally, if one tries to classify Lie Algebras. The natural operation is the adjoint operation, i.e. the left multiplication in the Lie algebra. So when one thinks about criteria for Lie algebras, the radical and nilradical, along with its center automatically show up.
    If one considers, e.g. the resulting Levi-Malcev decomposition, one sees why.

    Engel's thesis in 1883 (the year Levi was born) has been: About the theory of tangent transformations.
     
  4. Oct 16, 2016 #3
    What???

    Seems like your answer is tantamount to 'because it works'.
     
  5. Oct 16, 2016 #4

    fresh_42

    Staff: Mentor

    Yep, that has been unlucky, sorry. I recognized it a little late. I hope the rest of my answer is clearer.
     
  6. Oct 16, 2016 #5
    No problem, but if you check p. 208 of Hawkins Emergence of the Theory of Lie Groups you'll see Cartan was basically using the Levi decompositon and Engel's theorem before either of them, so the history of when X appeared is irrelevant, the internal logic motivating the necessity of such concepts before even defining them is the issue.
     
  7. Oct 16, 2016 #6

    fresh_42

    Staff: Mentor

    The decomposition into diagonalizable and nilpotent linear transformations is basic linear algebra. The idea to do it simultaneously is the essential point here, together with the representation ##X \mapsto ad X##. Linear transformations, which were first called linear substitutions originate in publications by Lagrange and Gauß, i.e. a century earlier than Engels and Levi. E.g. the first time a solution of a system of linear equations was presented was in 1748 by MacLaurin.
    However, this doesn't change the fact that Lie, Engels and Cartan founded the theory of Lie algebras, as the title of Engel's thesis shows.

    Edit: In my textbooks, this decomposition of linear transformation is called Jordan-Chevalley decomposition. And Lie, Engel, Cartan, Jordan, Levi and Chevalley all lived in the second 19th and early 20th century. Unfortunately my sources don't tell who has proven what in which year and one may assume they all knew the publications of the others, if available. The foundations in terms of linear transformation, however, have already been laid.

    Edit: H. Weyl has to be added to the list as well.
     
    Last edited: Oct 16, 2016
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