Definition 1. A Lie algebra $L$ is abelian if $[L,L]=0$, which means for all $x,y\in L,\ [x,y]=0$.

Representations

Definition 1. Let $M_n(k)$ be the algebra of $n\times n$ matrices over $K$ and let
$$
\mathfrak{gl}_n(K):=[M_n(K)]
$$

be the corresponding Lie algebra, which is also called the general linear Lie algebra.

Lie Algebra

Definition 1. A Lie algebra is a vector space $L$ over a field $K$ with a Lie bracket
$$
\begin{aligned}
\left[ \cdot,\cdot \right] &: L \times L \to L\\\
(x,y) &\mapsto [x,y],
\end{aligned}
$$
such that