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Representations of sl2, Part II July 18, 2009

Posted by Akhil Mathew in algebra, representation theory.
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1 comment so far

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.

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Representations of sl2, Part I July 17, 2009

Posted by Akhil Mathew in algebra, representation theory.
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5 comments

{\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.

  1. It’s elegant.
  2. It introduces important ideas that generalize to the setting of semisimple Lie algebras.
  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.

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