## Representations of sl2, Part IIJuly 18, 2009

Posted by Akhil Mathew in algebra, representation theory.
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This post is the second in the series on ${\mathfrak{sl}_2}$ and the third in the series on Lie algebras. I’m going to start where we left off yesterday on ${\mathfrak{sl}_2}$, and go straight from there to classification.  Basically, it’s linear algebra.

Classification

We’ve covered all the preliminaries now and we can classify the ${\mathfrak{sl}_2}$-representations, the really interesting material here. By Weyl’s theorem, we can restrict ourselves to irreducible representations. Fix an irreducible ${V}$.

So, we know that ${H}$ acts diagonalizably on ${V}$, which means we can write

$\displaystyle V = \bigoplus_\lambda V_\lambda$

where ${Hv_\lambda = \lambda v_{\lambda}}$ for each ${\lambda}$, i.e. ${V_\lambda}$ is the ${H}$-eigenspace.

## 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.