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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.