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## Lie’s Theorem IJuly 26, 2009

Posted by Akhil Mathew in algebra, representation theory.
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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 ${L}$ over an algebraically closed field ${k}$ of characteristic zero.

Definition 1 The derived series of ${L}$ is the descending filtration ${D_n}$ defined by ${D_0 := L, D_n := [D_{n-1}, D_{n-1}]}$. The Lie algebra ${L}$ is solvable if ${D_M=0}$ for some ${M}$.

For instance, a nilpotent Lie algebra is solvable, since if ${\{C_n\}}$ is the lower central series, then ${D_n \subset C_n}$ for each ${n}$.