General linear Lie algebra - Imported Knowlpedia Content

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

\mathfrak{gl}(V)=\mathrm{End}(V)

equipped with the commutator Lie bracket

[X,Y]=XY-YX.

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

Remarks

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).