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Lie’s Theorem II July 27, 2009

Posted by Akhil Mathew in algebra, representation theory.
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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}.