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I'm currently reading Fulton and Harris' Representation Theory text. In particular, I am looking at Chapter 8 where they introduce the definition of the Lie bracket. A question they raise is: Why could we not simply define the Lie bracket to be $$[X,Y] = XY - YX?$$ Why did we have to consider the maps Ad and ad?

The reason for this is that for any embedding of a Lie group $G$ into a general linear group $GL(V)$, we have a corresponding embedding of its Lie algebra $\mathfrak{g}$ into $\text{End}(V)$. If we simply defined $[X,Y] = XY - YX$, we need to define what is meant by the composition $XY$ and $YX$. These however depend on the embedding and may not even be an element of the Lie algebra.

Can someone provide a concrete example of when the composition $X \cdot Y$ may be not even be an element of the Lie algebra?

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  • $\begingroup$ Actually the Lie bracket on a Lie algebra comes from the Lie bracket of vectorfields on smooth manifolds, because in general the Lie algebra is the space of left-invariant vectorfields on $G$. From this point of view the composition $XY - YX$ is already determined. $\endgroup$ – T'x Apr 17 '17 at 9:47
  • $\begingroup$ Just pick any easy-to-work with matrix lie algebra and check if it's closed under composition. $\endgroup$ – arctic tern Apr 18 '17 at 4:12
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For example, let $\mathfrak{g}$ consists of matrices in $\text{End}(V)$ of trace zero. But the product of two matrices of trace zero might not be of trace zero.

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There are multiple questions.

  1. Why could we not simply define the Lie bracket to be $[X,Y] = XY - YX?$

Answer: In an abstract Lie algebra $L$, there is no product $XY$ defined for $X,Y\in L$, only a Lie bracket $[X,Y]$. The elements of $L$ are not necessarily matrices.

  1. Can someone provide a concrete example of when the composition $X \cdot Y$ may be not even be an element of the Lie algebra?

Yes, take $L=\mathfrak{sl}_n(K)$, the Lie algebra of trace zero matrices of size $n$.

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