- #1
bolbteppa
- 300
- 41
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?
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?