Representations of sl2, Part II July 18, 2009Posted by Akhil Mathew in algebra, representation theory.
Tags: highest weights, Lie algebras, linear algebra, representation theory, sl2
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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.
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.
Representations of sl2, Part I July 17, 2009Posted by Akhil Mathew in algebra, representation theory.
Tags: algebra, Jordan decomposition, Lie algebras, representation theory, semisimplicity, sl2
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.