special linear group

Definition

The subgroup of the 202406151143 with determinant 1, $\text{SL}(n,\mathbb{F})=\{A\in {\text{Mat}}_{n\times n}(\mathbb{F})|\det (A)=1\}$ is called the special linear group

Lie group structure

The determinant function $\det (A)$ is continuous since it is a polynomial of entries of the matrix. Therefore, if we look at the inverse image of the closed set $\{1\}$,

$\text{SL}(n,\mathbb{C})={\det }^{-1}(\{1\})$

which is closed. Hence, by the closed subgroup theorem it is a Lie group. (both $\text{SL}(n,\mathbb{C})$ and $\text{SL}(n,\mathbb{R})$ )

Lie algebra

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

$\mathfrak{sl}(n,\mathbb{R})=\{A\in {\text{Mat}}_{n\times n}(\mathbb{R})|\text{tr}(A)=0\}$

and

$\mathfrak{sl}(n,\mathbb{C})=\{A\in {\text{Mat}}_{n\times n}(\mathbb{C})|\text{tr}(A)=0\}$