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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

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

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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.

** Irreducibility **

The notion of irreducibility parallels that in the theory of group representations:

Definition 1Let be a Lie algebra. A -representation isirreducibleif it contains no nontrivial subrepresentations, i.e. there is no , , such that “multiplication” by maps into itself.

Since we are working in an “artinian” category, we have, as a general fact:

Example 1Every representation contains an irreducible representation. We prove this by induction on ; if this is trivial, since itself is irreducible. Suppose we know it for representations of smaller dimension than . If contains no subrepresentations, then we’re done. Otherwise, it contains such that , and contains an irreducible subrepresentation.

There is in fact an analog of Maschke’s theorem:

Theorem 2 (Weyl’s Theorem)Any (finite-dimensional) representation of decomposes uniquely (up to isomorphism) as a direct sum of irreducible representations.

Here this result is more difficult than Maschke’s theorem, but it can be proved more generally for semisimple Lie algebras, using either algebraic methods or the theory of compact groups. I shall therefore omit the proof, at least for now.

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** A Presentation of **

Now we want to classify the irreducible -modules. First, we need information on itself:

Choose the matrices (I’m following Fulton and Harris’s conventions)

There are three key relations:

The first two should be thought of in the following sense: There is a linear map sending . Then is an eigenvector with eigenvalue 2, and is an eigenvector with eigenvalue -2. is an eigenvector with value . This is the way these techniques generalize to semisimple Lie algebras, except that is replaced with a “Cartan subalgebra.”

There is another theorem I will quote without proof:

Theorem 3In any (finite-dimensional) representation of , the map , is diagonalizable.

The result follows from general facts about the Jordan-Chevalley decomposition in semisimple Lie algebras. I’d rather prove it in the abstract setting though.

We’re just about ready to get to the classification theorem itself. But I’ll leave that for another post.

[...] 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 [...]

One of the main reasons for the importance of the representation theory of sl(2) is that it helps you understand all other (say, simple, complex) Lie algebras. You pick 3 elements spanning an sl(2) inside the Lie algebra at hand and let them act on it. Then the Lie algebra breaks into irreducible reps and this decompositions characterises the Lie algebra. This is a strengthened version of your reason #3.

Yes, that’s what I was referring to.

And in that line of thought, it’s better to think that you have a map

sl(2) ->End(sl(2)), v\mapsto [v,.] . That’s how sl(2) becomes a reprsentation of itself. Then H,X,Y span the eigenspaces for the endomorphism [H,.] (aka adH) for eigenvalues 0,1,-1. This is the basis in which [H,.] becomes diagonal (as Thm.3 predicts, for V=sl(2)).

It might be also fun to write the (3×3) matrices of the linear operators [X,.], [Y,.], [H,.] in the basis {H,X,Y}. Not that it’s deep or anything, but just gives you a feeling of what’s going on.