Semi-Simple Lie Algebra Representations

[lion8172]

I'm trying to prove that any representation of a semisimple Lie algebra can be uniquely decomposed into irreducible representations.

I have seen some sketches of proofs that show that any representation \phi of a semisimple Lie algebra which acts on a finite-dimensional complex vector space V is completely reducible (i.e. V = V_1 \oplus V_2 \oplus \cdots \oplus V_k, such that the restriction of \phi to each V_i is irreducible). But how do we know that this decomposition is unique?

 

[matt grime]

You show that in two decompositions

V\cong \oplus V_i \cong W_j

that one of the summands V_r is isomorphic to one W_s, wlog V_1 and W_1, then use induction on V/V_1.

 

[lion8172]

question

So two representations \phi and \phi ' for \mathfrak{g} are said to be equivalent if there is an isomorphism E between the underlying vector spaces such that E \phi (X) = \phi' (X) E, \forall X \in \mathfrak{g}. How do we know that the restriction of a representation to a given V_i above is equivalent to a restriction of that representation to a given W_i?
Note that, by irreducible, I mean a vector space with no invariant subspaces under the given representation (except zero and itself).

 

[matt grime]

The identity map on V restricts to the identity map on V_1. This must then factor through one of the W_i, wlog W_1, hence V_1 is isomorphic to W_1.

 

[lion8172]

Could you be a little more explicit?
Note that I'm trying to prove this for an arbitrary representation.

 

[matt grime]

I know what you're trying to do; I was explicit; where do you see me do something not for an arbitrary representation? You want to show that V_1 is isomorphic to W_1 (after reordering the indices of the W_i). That is what I did.

1. You want to find an isomrphism from V_1 to W_1, that is maps f,g such that gf= Id on V_1 and fg=Id on W_1. That is the definition of isomorphism.

2. Consider Id on V. This maps \oplus V_i to \oplus W_j, and back again.

3. Look at V_1 mapping under Id to \oplus W_j.

4. The image must lie in at least one of the W_j. WLOG W_1.

5. So we have a map, Id restricted to V_1, call it f, that maps to W_1.

6. Call Id restricted to W_1 g.

7. What is gf? It is Id on V_1.

I really can't be any more explicit. In fact that is precisely what I wrote before.