PSL(1,R) - Knowlpedia

$\operatorname{PSL}(1,\mathbb R)$ is the projective special linear Lie group

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

with the quotient Lie group structure.

Dimensions and structure

This group has real dimension $0$ . It acts on projective space by applying a representative matrix to a line. In size one both numerator and scalar subgroup coincide (or are trivial), so the projective group is trivial.

The map induced by inclusion SL→GL is an isomorphism: normalize a representative by a 1th root of its determinant. See $\operatorname{PGL}(1,\mathbb R)$ .

References

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