General linear Lie algebra

Let $V$ be a finite-dimensional real or complex vector space.

Definition (General linear Lie algebra).
The general linear Lie algebra is the vector space

equipped with the commutator Lie bracket

After choosing a basis, $\mathfrak{gl}(V)\cong \mathfrak{gl}_n(\mathbb F)=M_n(\mathbb F)$ with the same bracket.

Relation to the group $\mathrm{GL}(V)$.
If $G=\mathrm{GL}(V)$ is the general linear group , then $\mathfrak{gl}(V)$ is naturally the Lie algebra of G , identified with $T_I G$; under this identification, the Lie bracket on $\mathfrak g$ agrees with the commutator bracket.

Useful subalgebras.
The trace map $\mathrm{tr}:\mathfrak{gl}_n(\mathbb F)\to\mathbb F$ is a Lie algebra homomorphism to the abelian Lie algebra $(\mathbb F,0)$, and its kernel is sl_n . The center is the scalar matrices, matching the center description $Z(\mathfrak{gl}_n)=\mathbb F\cdot I$.

Context.
Representations of Lie groups and Lie algebras are, by definition, maps into some $\mathfrak{gl}(V)$ (see representation of a Lie algebra and representation of a Lie group ).