General linear Lie algebra

Example

Let $k$ be a field and $n \ge 1$. The general linear Lie algebra $\mathfrak{gl}_n(k)$ is the vector space of all $n \times n$ matrices over $k$ with the bracket $$[X, Y] = XY - YX.$$ It is the Lie algebra of the associative algebra $M_n(k)$, so it is a Lie algebra. More generally, for a vector space $V$, $\mathfrak{gl}(V)$ is the space of linear endomorphisms of $V$ with the same bracket.

Check in a small case: in $\mathfrak{gl}_2(k)$, with the matrix units $E_{ij}$, one has $[E_{12}, E_{21}] = E_{11} - E_{22}$ and $[E_{11}, E_{12}] = E_{12}$. In general $[E_{ij}, E_{kl}] = \delta_{jk} E_{il} - \delta_{li} E_{kj}$. The $n^2$ matrix units form a basis, so $\dim \mathfrak{gl}_n(k) = n^2$.