Is there any way of directly proving that the lie algebra $\mathfrak{sl}(n,\Bbb C)$ is simple? I am not asking for a complete proof, but could somebody please give me a hint on how I can proceed?
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I think you can play around with nthe $e_{i,j}$ matrices- i.e. matrices with $0$ everywhere, and $1$ in $(i,j)$th entry, and look at the commutation relations. – voldemort Dec 23 '14 at 18:11
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Look at the way the elements of the form $e_{i,i}-e_{i+1,i+1}$ act (through the adjoint action) on the algebra: they are diagonalizable; find their eigenvalues. Using that, show that an ideal has to be generated by elementary matrices. Then compute commutations as suggested by voldermort in a comment above.
Mariano Suárez-Álvarez
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