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Here is what I mean by "Cartan's semisimplicity criterion":

Let $\mathfrak g$ be a finite-dimensional Lie algebra over a field of characteristic $0$. Assume that the center of $\mathfrak g$ is trivial. Then, the following three assertions are equivalent:

(1) The Killing form on $\mathfrak g\times \mathfrak g$ is nondegenerate.

(2) Every short exact sequence of finite-dimensional representations of $\mathfrak g$ splits.

(3) Every subrepresentation of the adjoint representation of $\mathfrak g$ has a complementary subrepresentation.

EDIT: This is not true as stated, but what I need is (1) $\Longrightarrow$ (2) only, so we can WLOG assume that $\mathfrak g$ has trivial center, in which case probably the whole equiavlence is correct.

What I am looking for is a slick proof for this equivalence . (although the only thing I really need is a proof of (1) $\Longrightarrow$ (3)). I am aware of the proof in Fulton-Harris Appendix C, but this could fill an hour of talking and seems to involve many unmotivated ideas. Is there something more explanatory? Using cohomology perhaps? Is the whole thing obvious from an advanced viewpoint? (I don't mean using the classification of simple Lie algebras, of course...) Maybe newer ideas such as Lie algebroids, algebraic groups etc. can help?

3 added 19 characters in body

Here is what I mean by "Cartan's semisimplicity criterion":

Let $\mathfrak g$ be a finite-dimensional Lie algebra over a field of characteristic $0$. Then, the following three assertions are equivalent:

(1) The Killing form on $\mathfrak g\times \mathfrak g$ is nondegenerate.

(2) Every short exact sequence of finite-dimensional representations of $\mathfrak g$ splits.

(3) Every subrepresentation of the adjoint representation of $\mathfrak g$ has a complementary subrepresentation.

EDIT: This is not true as stated, but what I need is (1) $\Longrightarrow$ (2) only, so we can WLOG assume that $\mathfrak g$ has trivial center, in which case probably the whole equiavlence is correct.

What I am looking for is a slick proof for this equivalence. I am aware of the proof in Fulton-Harris Appendix C, but this could fill an hour of talking and seems to involve many unmotivated ideas. Is there something more explanatory? Using cohomology perhaps? Is the whole thing obvious from an advanced viewpoint? (I don't mean using the classification of simple Lie algebras, of course...) Maybe newer ideas such as Lie algebroids, algebraic groups etc. can help?

2 added 206 characters in body

Here is what I mean by "Cartan's semisimplicity criterion":

Let $\mathfrak g$ be a finite-dimensional Lie algebra over a field of characteristic $0$. Then, the following three assertions are equivalent:

(1) The Killing form on $\mathfrak g\times \mathfrak g$ is nondegenerate.

(2) Every short exact sequence of representations of $\mathfrak g$ splits.

(3) Every subrepresentation of the adjoint representation of $\mathfrak g$ has a complementary subrepresentation.

EDIT: This is not true as stated, but what I need is (1) $\Longrightarrow$ (2) only, so we can WLOG assume that $\mathfrak g$ has trivial center, in which case probably the whole equiavlence is correct.

What I am looking for is a slick proof for this equivalence. I am aware of the proof in Fulton-Harris Appendix C, but this could fill an hour of talking and seems to involve many unmotivated ideas. Is there something more explanatory? Using cohomology perhaps? Is the whole thing obvious from an advanced viewpoint? (I don't mean using the classification of simple Lie algebras, of course...) Maybe newer ideas such as Lie algebroids, algebraic groups etc. can help?

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