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Lie algebras II
*July 20, 2009*

*Posted by Akhil Mathew in algebra.*

Tags: algebra, general theory, Lie algebras, linear algebra, quotients

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Tags: algebra, general theory, Lie algebras, linear algebra, quotients

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I’m going to get back eventually to the story about finite-dimensional modules, but for now, Lie algebras are more immediate to my project, so I’ll talk about them here.

From an expository standpoint, jumping straight to basically right after defining Lie algebras was unsound. I am going to try to motivate them here and discuss some theorems, to lead into more of the general representation theory.

** Derivations **

So let’s consider a not-necessarily-associative algebra over some field . In other words, is a -vector space, and there is a -bilinear map , which sends say , but it doesn’t have to either be commutative or associative (or unital). A Lie algebra with the Lie bracket would be one example.