## Lie’s Theorem IIJuly 27, 2009

Posted by Akhil Mathew in algebra, representation theory.
Tags: , , , , ,
Yesterday I was talking about Lie’s theorem for solvable Lie algebras. I went through most of the proof, but didn’t finish the last step. We had a solvable Lie algebra ${L}$ and an ideal ${I \subset L}$ such that ${I}$ was of codimension one.
There was a finite-dimensional representation ${V}$ of ${L}$. For ${\lambda \in I^*}$, we set
$\displaystyle V_\lambda := \{ v \in V: Yv = \lambda(Y) v, \ \mathrm{all} \ Y \in I \}.$
We assumed ${V_\lambda \neq 0}$ for some ${\lambda}$ by the induction hypothesis. Then the following then completes the proof of Lie’s theorem, by the “fundamental calculation:”
Lemma 1 If ${V_\lambda \neq 0}$, then ${\lambda([L,I])=0}$.