PSL(n,R) - Knowlpedia

For every integer $n\ge1$ , $\operatorname{PSL}(n,\mathbb R)$ is the projective special linear Lie group

\operatorname{PSL}(n,\mathbb R)=\operatorname{SL}(n,\mathbb R)/\{\lambda I:\lambda\in \mathbb R^\times,\ \lambda^{n}=1\},

with the quotient Lie group structure.

Dimensions and structure

This group has real dimension $n^2-1$ . It acts on projective space by applying a representative matrix to a line. For odd matrix size, a real determinant has a real root of that degree and PSL equals PGL. For even matrix size, PSL is the identity component of PGL and PGL has two components. The real scalar quotient is taken on real matrices; replacing it by a statement about complex points changes the object.

References

Pavel Etingof, Lie Groups and Lie Algebras (MIT lecture notes), §6.1, pp. 38–39; Proposition 6.3 and Exercise 6.5.