# Definition of the Lie bracket

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?

• 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. – T'x Apr 17 '17 at 9:47
• Just pick any easy-to-work with matrix lie algebra and check if it's closed under composition. – arctic tern Apr 18 '17 at 4:12

## 2 Answers

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.

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 $n$.