Lie algebras IIJuly 20, 2009

Posted by Akhil Mathew in algebra.
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From an expository standpoint, jumping straight to ${\mathfrak{sl}_2}$ 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.
So let’s consider a not-necessarily-associative algebra ${A}$ over some field ${F}$. In other words, ${A}$ is a ${F}$-vector space, and there is a ${F}$-bilinear map ${A \times A \rightarrow A}$, which sends say ${(x,y) \rightarrow xy}$, but it doesn’t have to either be commutative or associative (or unital). A Lie algebra with the Lie bracket would be one example.