Definition 1

Let $g$ be a Lie algebra over $K$ and $M$ a $K$-module. A homomorphism of $g$ into the Lie Algebra $gl(M)$ is called a representation of $g$ on the module $M$. An injective representation is called faithful. If $K$ is a field, the dimension (finite or infinite) of $M$ over $K$ is called the dimension of the representation. The representation $x \mapsto \operatorname{ad} x$ of $g$ on the $K$-module $g$ is called the adjoint representation of $g$.

A representation of $g$ on $M$ is thus a $K$-linear mapping $\rho$ of $g$ into the endomorphism module of $M$ such that

for all $x \in g, y \in g, m \in M$.