Representation of a Lie algebra

Let $\mathfrak{g}$ be a Lie algebra over $\mathbb{R}$ (or $\mathbb{C}$), and let $V$ be a vector space over the same field. A representation of $\mathfrak{g}$ on $V$ is a linear map

such that for all $X,Y\in\mathfrak{g}$,

i.e. $\rho$ is a Lie algebra homomorphism from $\mathfrak{g}$ to the Lie algebra $\mathfrak{gl}(V)$ with commutator bracket.

When $\mathfrak{g}$ is the Lie algebra of a Lie group $G$ and $\pi:G\to \mathrm{GL}(V)$ is a smooth group representation, the induced map $\mathrm{d}\pi_e:\mathfrak{g}\to\mathfrak{gl}(V)$ (the differential at the identity ) is a Lie algebra representation.

Examples

1. Standard matrix action. For $\mathfrak{g}=\mathfrak{gl}(n,\mathbb{R})$ and $V=\mathbb{R}^n$, the map $\rho(A)(v)=Av$ is a representation.
2. Adjoint representation. The map $\mathrm{ad}:\mathfrak{g}\to \mathrm{End}(\mathfrak{g})$ given by $\mathrm{ad}_X(Y)=[X,Y]$ is a representation of $\mathfrak{g}$ on itself.
3. Trivial representation. The map $\rho:\mathfrak{g}\to \mathrm{End}(V)$ with $\rho(X)=0$ for all $X$ is always a representation (since both sides of the bracket-preservation identity vanish).