Special linear group

For a field $\mathbb F$ equal to $\mathbb R$ or $\mathbb C$, the special linear group is

viewed as a matrix Lie group inside the general linear group. It is a closed Lie subgroup (see Lie subgroup and closed subgroup), hence a Lie group in its own right.

The dimension depends on the scalar field:

Consequently, the underlying real Lie group $SL(n,\mathbb C)_{\mathbb R}$ has real dimension $2(n^2-1)$. An unqualified statement that $SL(n,\mathbb C)$ has manifold dimension $n^2-1$ is correct only when “dimension” means complex dimension.

Its Lie algebra is the trace-zero matrices, the special linear Lie algebra $\mathfrak{sl}_n(\mathbb F)$, and the exponential map restricts to $\exp:\mathfrak{sl}_n(\mathbb F)\to SL(n,\mathbb F)$ (see exponential map). The determinant condition differentiates to the trace condition:

The groups $SL(n,\mathbb R)$ and $SL(n,\mathbb C)$ are basic examples of connected linear Lie groups, and they play a central role in semisimple theory (compare semisimple Lie algebras and the root-theoretic framework starting at root systems).