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 {\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.