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Let $\mathfrak{g}$ be a finite dimensional Lie algebra. The Killing form $K:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathbb{C}$ is given by $$K(x,y) = tr(ad_xad_y)$$ I have two questions about the Killing form:

  1. How can it be computed? And more generally, how can $ad_x$ be expressed for a general $x\in\mathfrak{g}$? In particular, if $\mathfrak{g}\subset\mathfrak{gl}(V)$ for some vector space $V$ (and it is always the case, by Ado's theorem) then $ad_x$ can be expressed as a matrix. How can this matrix be determined?
  2. I don't really see why $K([x,y],z)\neq 0$. Indeed we have $$K([x,y],z) = tr(ad_{[x,y]}ad_z) = tr(ad_xad_yad_z) - tr(ad_yad_xad_z)$$ and since we can commute matrices in the trace, this should give zero. Could someone help me see what I'm overlooking?
Shaun
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Daniel Robert-Nicoud
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    Surprisingly, you cannot actually commute matrices in traces. If you start from the formula $tr(AB)=tr(BA)$ and you try to rearrange $tr(X_1 X_2 \cdots X_n)$ by parenthesizing the $X_i$ into two blocks, you will get $tr(X_k X_{k+1}\cdots X_n X_1 \cdots X_{k-1})$, and so the order can only be permuted cyclically. It is an instructive exercise to find $A,B,C$ such that $tr{ABC}\neq 0$ but $tr(BAC)=0$. – Aaron Aug 03 '13 at 00:14
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    @Aaron is correct; see, e.g., http://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics/23481#23481 – Amitesh Datta Aug 03 '13 at 00:18
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    @Aaron Ah, I was not aware of this fact. Thank you very much. – Daniel Robert-Nicoud Aug 03 '13 at 09:34

1 Answers1

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I'm not sure what you mean by your first question. If you have a lie algebra, with basis $ \{ e_1, \cdots, e_n \} $ with structure constants $ C^{i}_{jk} $, then it is easy to write the matrix $ ad_{e_j} $: $ C^{i}_{jk} $ with $ j $ fixed. You should also keep in mind that, in the case the lie algebra is embedded in $ \mathfrak{g}\subset\mathfrak{gl}(V)$, the dimension of the matrix $ ad_x $ will be different than the dimension of $ V $. Maybe you should try with the case $ g = \mathfrak{gl}(2,\mathbb{R}) $ and use the basis of four matrices each with 1 in one spot and 0 everywhere else.

In the second question, the $ \text{tr} $ is not invariant under arbitrary permutations, only cyclic permutations.

user88684
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  • Thank you. I have seen that if we take for example $\mathfrak{gl}(2,\mathbb{C})$ with the standard basis (the four matrices with one one and three zeros), then we can write $ad_x=x\otimes 1-1\otimes x$. Do you know if this can be generalized to all matrix Lie algebras by taking the standard basis in $\mathfrak{gl}(n)$ and considering only a subspace? – Daniel Robert-Nicoud Aug 05 '13 at 22:00