Projective special linear Lie group

Definition

Let $\mathbb F=\mathbb R$ or $\mathbb C$ , and let $n\ge2$ . The projective special linear Lie group is

\operatorname{PSL}_n(\mathbb F) =\operatorname{SL}_n(\mathbb F)/ \{\lambda I_n:\lambda\in\mathbb F^\times,\ \lambda^n=1\}.

Its denominator is the finite center of the special linear group, so this is a quotient Lie group and the projection is a finite covering homomorphism.

Lie algebra and dimension

Quotienting by a discrete subgroup does not change the Lie algebra. Hence

\operatorname{Lie}(\operatorname{PSL}_n(\mathbb F)) \cong\mathfrak{sl}_n(\mathbb F).

The real group has real dimension $n^2-1$ ; the complex group has complex dimension $n^2-1$ , or real dimension $2(n^2-1)$ after forgetting its complex structure.

Connectedness and comparison with

\operatorname{PGL}

Both $\operatorname{SL}_n(\mathbb R)$ and $\operatorname{SL}_n(\mathbb C)$ are connected for $n\ge2$ , and a continuous quotient of a connected space is connected. Therefore both corresponding $\operatorname{PSL}$ groups are connected.

Over $\mathbb C$ , every nonzero complex number has an $n$ -th root, so

\operatorname{PSL}_n(\mathbb C)\cong \operatorname{PGL}_n(\mathbb C)

as complex Lie groups. Over $\mathbb R$ , $\operatorname{PSL}_n(\mathbb R)$ is the identity component of $\operatorname{PGL}_n(\mathbb R)$ . The two groups agree for odd $n$ , while $\operatorname{PSL}_n(\mathbb R)$ has index $2$ for even $n$ . In particular,

\operatorname{PSL}_2(\mathbb R) =\operatorname{PGL}_2(\mathbb R)^\circ \ne\operatorname{PGL}_2(\mathbb R).

Center versus simple terminology

The quotient removes the scalar center of $\operatorname{SL}_n$ , but the word “simple” depends on whether one means a Lie algebra, a connected Lie group, an algebraic group, or an abstract group of points. Low-dimensional and small-finite-field exceptions should not be suppressed by a blanket simplicity claim.

References

Brian C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed., Springer, 2015. Publisher record. Relevant: classical matrix groups and covering homomorphisms.
Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002. Publisher record. Relevant: Chapter I, linear Lie groups and their centers.