Lie’s Theorem II July 27, 2009Posted by Akhil Mathew in algebra, representation theory.
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 1 If , then .
Lie’s Theorem I July 26, 2009Posted by Akhil Mathew in algebra, representation theory.
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 1 The derived series of is the descending filtration defined by . The Lie algebra is solvable if for some .
For instance, a nilpotent Lie algebra is solvable, since if is the lower central series, then for each .