general linear group

The general linear group is one of the most important groups in group theory.

Not only is $\text{GL}(n,\mathbb{R})$ one of the first groups people encounter, it comes built in with a lot of matrix theory. In addition, it is a Lie group.

Definition

For ${\mathbb{F}}^n$ for some field $\mathbb{F}$, the group of invertible $n\times n$ matrices is called the general linear group denoted

$\text{GL}(n,\mathbb{F})\subset {\text{Mat}}_{n\times n}(\mathbb{F})$

For an arbitrary vector space $V$, $\text{GL}(V)=\{\phi \in \text{End}(V)|\phi \text{is invertible}\}$

Group structure

We can take group multiplication to be composition, which for matrices in $\text{GL}(n,\mathbb{F})$ this is simply matrix multiplication.

Lie group structure

We know that $\text{GL}(n,\mathbb{R})$ and $\text{GL}(n,\mathbb{C})$ are both open subsets of ${\text{Mat}}_{n\times n}(\mathbb{R})$ and ${\text{Mat}}_{n\times n}(\mathbb{C})$ respectively, since (informally) we can “wiggle” the entries of an invertible matrix by a small epsilon and still have an invertible matrix.

More formally, the determinant function is continuous, so $\text{GL}(n,\mathbb{F})={\det }^{-1}((-\infty ,0)\cup (0,\infty ))$ is open, so thus it is an open submanifold of ${\text{Mat}}_{n\times n}(\mathbb{F})$.

For matrix multiplication as the group operation, then we see that it is continuous, since matrix multiplication is a polynomial of entries of the matrix. So $\text{GL}(n,\mathbb{R})$ is a Lie group

Lie algebra $\mathfrak{gl}(n,\mathbb{R})$, $\mathfrak{gl}(n,\mathbb{C})$

Let $\mathfrak{gl}(n,\mathbb{R})$ ($\mathfrak{gl}(n,\mathbb{C})$) denote the Lie algebra of $\text{GL}(n,\mathbb{R})$ ($\text{GL}(n,\mathbb{C})$ respectively).

$\mathfrak{gl}(n,\mathbb{R})={\text{Mat}}_{n\times n}(\mathbb{R})\text{and }\mathfrak{gl}(n,\mathbb{C})={\text{Mat}}_{n\times n}(\mathbb{C})$