A (particular case of a) representation of a Lie group $G$ is a homomorphism $G \rightarrow GL(n,\mathbb{C})$, so essentially the objective here would to represent group elements by some matrices that preserve some of the structure of the group itself. Now a Lie algebra representation is defined as a map:
$$\mathfrak{g} \rightarrow \mathfrak{gl}(\mathfrak{g})$$
that preserves the Lie bracket operation.
So this is a mapping that takes a vector in the Lie Algebra to an endomorphism of $\mathfrak{g}$. I'm struggling to see how an endomorphism "represents" a vector in the Lie Algebra. In other words, how does this mapping constitute a representation of the Lie Algebra in the same way that a matrix represent a group element of a Lie group? Can someone clarify this or give me some intuition behind this idea?
Best Answer
If you have a Lie algebra homomorphism $f\colon\mathfrak g\longrightarrow\mathfrak{gl}(\mathfrak g)$, then, if $X,Y\in\mathfrak g$,$$\bigl[f(X),f(Y)\bigr]=f\bigl([X,Y]\bigr).$$There are two brackets here: the one on the left, which is the Lie bracket of $\mathfrak g$, and the one on the right, which is simply$$[M,N]=M.N-N.M.\tag1$$So, each vector $X\in\mathfrak g$ is represented by a matrix and the Lie bracket is the standard operation defined in $(1)$.
You can also say that $X$ is represented by the linear map$$\begin{array}{rccc}\operatorname{ad}_X\colon&\mathfrak g&\longrightarrow&\mathfrak g\\&Y&\mapsto&[X,Y].\end{array}$$