Representation of a Lie algebra

Remarks

Equivalently, a $\mathfrak{g}$-module is a vector space $V$ with a bilinear map $\mathfrak{g} \times V \to V$, $(x, v) \mapsto x \cdot v$, such that $[x, y] \cdot v = x \cdot (y \cdot v) - y \cdot (x \cdot v)$. The two notions match through $x \cdot v = \rho(x)(v)$; the dimension of $V$ is the dimension of the representation. Example: the adjoint representation $\operatorname{ad} \colon x \mapsto [x, \cdot\,]$ on $V = \mathfrak{g}$. The condition $\operatorname{ad}[x, y] = [\operatorname{ad} x, \operatorname{ad} y]$ is the Jacobi identity. Non-example: on $\mathfrak{gl}_n(F)$ with $n \ge 2$, the map $x \mapsto x^T$ is linear, but $([x, y])^T = y^T x^T - x^T y^T = -[x^T, y^T]$, which differs from $[x^T, y^T]$ when $[x, y] \ne 0$ and $F$ does not have characteristic $2$. The map $x \mapsto -x^T$ is a representation, the dual of the defining one. Representations of $\mathfrak{g}$ are the same as left modules over the universal enveloping algebra $U(\mathfrak{g})$.