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Representations of sl2, Part II
*July 18, 2009*

*Posted by Akhil Mathew in algebra, representation theory.*

Tags: highest weights, Lie algebras, linear algebra, representation theory, sl2

1 comment so far

Tags: highest weights, Lie algebras, linear algebra, representation theory, sl2

1 comment so far

This post is the second in the series on and the third in the series on Lie algebras. I’m going to start where we left off yesterday on , 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 -representations, the really interesting material here. By Weyl’s theorem, we can restrict ourselves to irreducible representations. Fix an irreducible .

So, we know that acts diagonalizably on , which means we can write

where for each , i.e. is the -eigenspace.

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Representations of sl2, Part I
*July 17, 2009*

*Posted by Akhil Mathew in algebra, representation theory.*

Tags: algebra, Jordan decomposition, Lie algebras, representation theory, semisimplicity, sl2

5 comments

Tags: algebra, Jordan decomposition, Lie algebras, representation theory, semisimplicity, sl2

5 comments

is a special Lie algebra, mentioned in my previous post briefly. It is the set of 2-by-2 matrices over of trace zero, with the Lie bracket defined by:

The representation theory of is important for several reasons.

- It’s elegant.
- It introduces important ideas that generalize to the setting of semisimple Lie algebras.
- Knowing the theory for is useful in the proofs of the general theory, as it is often used as a tool there.

In this way, is an ideal example. Thus, I am posting this partially to help myself learn about Lie algebras.