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 understand 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 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 $Ker \; c$ is an $L−$ submodule of $V$, where $c$ is an endomorphism of $V$ defined in 6.2.

The other parts of the proof are very hard, 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?



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