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Engel’s Theorem and Nilpotent Lie Algebras
*July 23, 2009*

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

Tags: algebra, Engel's theorem, Lie algebras, linear algebra, nilpotent

1 comment so far

Tags: algebra, Engel's theorem, Lie algebras, linear algebra, nilpotent

1 comment so far

Now that I’ve discussed some of the basic definitions in the theory of Lie algebras, it’s time to look at specific subclasses: nilpotent, solvable, and eventually semisimple Lie algebras. Today, I want to focus on nilpotence and its applications.

** Engel’s Theorem **

To start with, choose a Lie algebra for some finite-dimensional -vector space ; recall that is the Lie algebra of linear transformations with the bracket . The previous definition was in terms of matrices, but here it is more natural to think in terms of linear transformations without initially fixing a basis.

Engel’s theorem is somewhat similar in its statement to the fact that commuting diagonalizable operators can be simultaneously diagonalized.