Weyl's theorem and Representations

Question

Let $L$ be a semisimple Lie algebra and let $(V,\varphi)$ be a finite-dimensional $L$-module representation. Our main goal is to prove that $\varphi$ is completely reducible. Consider an $L$-submodule of $V$ of codimension one, let $0 \longrightarrow W \longrightarrow V \longrightarrow F \longrightarrow 0$ be an exact sequence (where $F$ is an $L$-module). From the book of James Humphreys called "Introduction to Lie Algebras and Representation Theory", I have understood the following steps:

1. We take another proper submodule of $W$ denoted by $W'$ such that the exact sequence $0 \longrightarrow W/W' \longrightarrow V/W' \longrightarrow F \longrightarrow 0$ splits, so there exists a one dimensional $L$-submodule of $V/W'$ (say $\tilde{W}/W'$) complementary to $W/W'$.
2. We proceed by induction on the dimension of $W$, so we get an exact sequence $0 \longrightarrow W' \longrightarrow \tilde{W} \longrightarrow F \longrightarrow 0$ which splits. It follows easily that $V=W \oplus X$, where $X$ is a submodule complementary to $W'$ in $\tilde{W}$.
3. We suppose that $W$ is irreducible, so we may use Schur's lemma on $c \vert_{W}$ to say that $\operatorname{Ker} c$ is an $L$-submodule of $V$, where $c$ is the endomorphism of $V$ defined in 6.2.

The other parts of the proof are very hard, and I didn't understand them. Can someone help me to figure out those parts? If there is another comprehensible method, can someone share it with us?

Answer 1

Maybe it would clarify matters if I gave a little more background, in community wiki format.

The basic idea of this algebraic proof goes back to a short paper by Richard Brauer (1936) in German in Mathematische Zeitschrift, an interesting time in his life when he had been expelled from his professorship in Berlin and took up a position in Toronto. (This particular paper has no marginal additions, as do some other papers; I bought a used copy of his collected papers in three volumes which used to belong to him.)

Brauer's proof is of course less natural than the original proof by Weyl, but it applies to all semisimple Lie algebras over an algebraically closed field of characteristic 0 and may be the simplest algebraic proof. An attempt was made to simplify the proof, in a more recent textbook by K. Erdmann and her recent student M. Wildon Introduction to Lie algebras (MSN), published by Springer in 2006 as an undergraduate text. This result is not essential for their further work but occurs in Theorem 9.16 with an unconvincing proof. This was acknowledged by them in a list of errata, and affects Exercises 9.15 (cf. 9.16) as well perhaps as Lemma 10.7. (A good exercise is to track down the specific fault in the proof.) The book itself is perhaps an attractive alternative to mine, having a more leisurely pace and more examples but covering much less material in a similar number of pages.

The Brauer method is used in Bourbaki's Chapter 1 (of their Groupes de Lie et algèbres de Lie) as well as Jacobson's 1962 book Lie algebras (now in a Dover reprint). But as stated above, Weyl's 1925 proof is more natural in the Lie group context, imitating the finite group method.

Answer 2

Since you also ask for another method, perhaps you may try Hans Samelson's approach in his textbook "Notes on Lie Algebras" (I have an older edition but it has been republished by Springer). It is in Chapter III Section 4. Roughly, the idea is to prove first a lemma of Whitehead that for the Lie algebra $L$ acting on a finite dimensional vector space $V$ and a linear function $f:L\rightarrow V$ satisfying that $f[x,y]=xf(y)-yf(x)$ (a cocycle condition) there exists a vector $v\in V$ such that $f(x)=xv$ for all $x\in L$ (a coboundary condition). Then, the proof of Weyl's complete reducibility proceeds to split a canonical epimorphism in a more direct fashion. You may enjoy reading this approach.