Representations of a Lie algebra

DEFINITION 2.1   Let $k$ be a field. A finite dimensional representation of a Lie algebra $\mathfrak{g}$ over $k$ is a Lie algebra homomorphism

$$\rho : \mathfrak{g}\to M_n(k).$$

Note: The full matrix algebra $M_n(k)$ , when regarded as a Lie algebra equipped with the commutator product, is commonly denoted as $\mathfrak{gl}_n(k)$ .

EXAMPLE 2.2   Let $k$ be a field. Let $\mathfrak{g}$ be a finite dimensional Lie algebra over $k$ . We then have an adjoint representation

$$\mathfrak{g}\ni X \mapsto \operatorname{ad}(X)=(Y\mapsto [X,Y]) \in \operatorname{End}_{k-\operatorname{linear}}(\mathfrak{g}).$$