# Meaning of a Lie Algebra representations

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?

• A Lie algebra representation is defined as any Lie algebra homomorphism $\mathfrak g \rightarrow \mathfrak{gl}(V)$, where $V$ is any vector space over an appropriate base field. The case where $V=\mathfrak g$ itself is kind of special. Commented Apr 10, 2021 at 15:44
• And since $\mathfrak{gl}(V)$ as a vector space is just $End(V)$, which for $V$ of dimension $n$ (over $\mathbb C$, say) can be identified with $M_n(\mathbb C)$, the basic idea is the same: represent elements of $\mathfrak g$ as matrices (or more abstractly, as linear maps). Commented Apr 10, 2021 at 16:14

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$$\begin{array}{rccc}\operatorname{ad}_X\colon&\mathfrak g&\longrightarrow&\mathfrak g\\&Y&\mapsto&[X,Y].\end{array}$$
• Ok so here my question would be: why does the map $ad_X$ represent the vector $X$? I think this is the point I'm struggling to wrap my head around. Commented Apr 10, 2021 at 10:21
• Because it represents the element $X$ of the Lie algebra $\mathfrak g$ as a linear map from a vector space into itself. Commented Apr 10, 2021 at 10:25
• I see that point. Perhaps a better way of asking my question is "what properties does $ad_X$ have that represent $X$? Is it the unique linear map that represents X? For example, in the case of a Lie group representation of $g,h$ then $R(g) \circ R(h) = R(g \circ h)$ so composition of representation obey the same rules as the composition of the corresponding group elements. Commented Apr 10, 2021 at 10:40
• And $[\operatorname{ad}_X,\operatorname{ad}_Y]=\operatorname{ad}_{[X,Y]}$. Commented Apr 10, 2021 at 10:42