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I was required to compute the Killing form of $\mathfrak{sl}(n,\mathbb{C})$ by using it is simple to avoid tedious computations.

However, here is my insignificant attempt:

Consider the ideal $$\{~[y,x]~|~y\in \mathfrak{sl}(n,\mathbb{C})\}.$$ For $\mathfrak{sl}(n,\mathbb{C})$ is simple, so the above ideal is $\mathfrak{sl}(n,\mathbb{C})$. In other words, every elements in $\mathfrak{sl}(n,\mathbb{C})$ can be written in the form $y=[y_x,x]$ for some $y_x\in \mathfrak{sl}(n,\mathbb{C})$. Then consider the Killing form $$\kappa(x,y)=\kappa(x,[x,y_x])$$

Then I got stuck... Could you give me any hint? Thank you.

Aolong Li
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1 Answers1

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On a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ the space of invariant bilinear forms is at most $1$-dimensional by Schur's lemma, so if you find any nonzero such form then any other such form is a scalar multiple of it. On $\mathfrak{sl}_n$, the form

$$\mathfrak{sl}_n \times \mathfrak{sl}_n \ni (X, Y) \mapsto \text{tr}(XY)$$

(where by $\text{tr}$ I just mean the ordinary trace on $n \times n$ matrices) is a nonzero invariant bilinear form, so the Killing form must be a scalar multiple of it. Now it suffices to compute any nontrivial example to see what scalar you need.

Qiaochu Yuan
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