nLab special linear group

Idea

Given a field $k$ and a natural number $n \in \mathbb{N}$, the special linear group $SL(n,k)$ (or $SL_n(k)$) is the subgroup of the general linear group $SL(n,k) \subset GL(n,k)$ consisting of those linear transformations that preserve the volume form on the vector space $k^n$. It can be canonically identified with the group of $n\times n$ matrices with entries in $k$ having determinant $1$.

This group can be considered as a subvariety of the affine space $M_{n\times n}(k)$ of square matrices of size $n$ carved out by the equations saying that the determinant of a matrix is 1. This variety is an algebraic group over $k$, and if $k$ is the field of real or complex numbers then it is a Lie group over $k$.

Properties

Proposition

The special linear group $SL_n(\mathbb{F})$ is a perfect group for any field $\mathbb{F}$ and any $n \geq 1$, except for the cases of the prime fields $\mathbb{F}_2$ and $\mathbb{F}_3$.

See for example here, or Lang 02, theorems XIII 8.3 and 9.2.

Examples

Remark

The first case admitted by Prop. is the binary icosahedral group (this Prop.):

$$SL(2,\mathbb{F}_5) \;\simeq\; 2I$$

• SL(2,Z)

• not quite an example: SL(2,O) (over the octonions)

Related concepts

• general linear group,

• projective special linear group

• special unitary group

• congruence subgroup

• special linear Lie algebra

References

• Serge Lang, Algebra, $3^{rd}$ edition, Springer 2002 (pdf)