General linear group - Imported Knowlpedia Content

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

Definition (General linear group). The general linear group of $V$ is

\mathrm{GL}(V)=\{A:V\to V \text{ linear and invertible}\},

with group operation given by composition. After choosing a basis, $\mathrm{GL}(V)\cong \mathrm{GL}_n(\mathbb F)$ where $\mathbb F=\mathbb R$ or $\mathbb C$ and

\mathrm{GL}_n(\mathbb F)=\{A\in M_n(\mathbb F):\det(A)\ne 0\}.

Remarks

Lie group structure. Viewed as a subset of the affine space $M_n(\mathbb F)$ , $\mathrm{GL}_n(\mathbb F)$ is open (since $\det$ is continuous and $\mathbb F^\times$ is open), hence it is a smooth manifold and a Lie group. Its Lie algebra is the general linear Lie algebra $\mathfrak{gl}_n(\mathbb F)$ , identified with $T_I\mathrm{GL}_n(\mathbb F)$ (compare Lie algebra of a Lie group).

Basic structure. Over $\mathbb C$ , $\mathrm{GL}_n(\mathbb C)$ is connected. Over $\mathbb R$ , $\mathrm{GL}_n(\mathbb R)$ has two connected components distinguished by the sign of $\det$ . The exponential map is the matrix exponential $\exp:\mathfrak{gl}_n(\mathbb F)\to \mathrm{GL}_n(\mathbb F)$ .

Context. Many linear representations of Lie groups are concretely homomorphisms into some $\mathrm{GL}(V)$ ; special subgroups such as SL_n, O(n), and U(n) are defined by additional algebraic constraints.