special linear group

Definition

The subgroup of the general linear group with determinant 1, is called the special linear group

Lie group structure

The determinant function is continuous since it is a polynomial of entries of the matrix. Therefore, if we look at the inverse image of the closed set ,

which is closed. Hence, by the closed subgroup theorem it is a Lie group. (both and )

Lie algebra

Let () denote the Lie algebra of ( respectively).

and