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Lie’s Theorem II
*July 27, 2009*

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

Tags: eigenvalues, Lie algebras, Lie's theorem, linear algebra, representation theory, solvability

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Tags: eigenvalues, Lie algebras, Lie's theorem, linear algebra, representation theory, solvability

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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 and an ideal such that was of codimension one.

There was a finite-dimensional representation of . For , we set

We assumed for some by the induction hypothesis. Then the following then completes the proof of Lie’s theorem, by the “fundamental calculation:”

Lemma 1If , then .

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Lie’s Theorem I
*July 26, 2009*

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

Tags: derived series, Lie algebras, Lie's theorem, solvability

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Tags: derived series, Lie algebras, Lie's theorem, solvability

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I talked a bit earlier about nilpotent Lie algebras and Engel’s theorem. There is an analog for *solvable* Lie algebras, and the corresponding Lie’s theorem.

So, first the definitions. Solvability is similar to nilpotence in that one takes repeated commutators, except one uses the *derived series* instead of the lower central series.

In the future, fix a Lie algebra over an algebraically closed field of characteristic zero.

Definition 1Thederived seriesof is the descending filtration defined by . The Lie algebra issolvableif for some .

For instance, a nilpotent Lie algebra is solvable, since if is the lower central series, then for each .