## Representations of sl2, Part IJuly 17, 2009

Posted by Akhil Mathew in algebra, representation theory.
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${\mathfrak{sl}_2}$ is a special Lie algebra, mentioned in my previous post briefly. It is the set of 2-by-2 matrices over ${\mathbb{C}}$ of trace zero, with the Lie bracket defined by:
$\displaystyle [A,B] = AB - BA.$
The representation theory of ${\mathfrak{sl}_2}$ is important for several reasons.
3. Knowing the theory for ${\mathfrak{sl}_2}$ is useful in the proofs of the general theory, as it is often used as a tool there.
In this way, ${\mathfrak{sl}_2}$ is an ideal example. Thus, I am posting this partially to help myself learn about Lie algebras.