PSL(1,C) - Knowlpedia

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

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

with the quotient Lie group structure.

Dimensions and structure

This group has real dimension $0$ , complex 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.

Its complex Lie group structure uses holomorphic multiplication and inversion. Forgetting the complex structure gives the underlying real Lie group; real and holomorphic homomorphisms belong to different categories.

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

References

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