PSL(3,C) - Knowlpedia

$\operatorname{PSL}(3,\mathbb C)$ is the projective special linear Lie group

\operatorname{PSL}(3,\mathbb C)=\operatorname{SL}(3,\mathbb C)/\{\lambda I:\lambda\in \mathbb C^\times,\ \lambda^{3}=1\},

with the quotient Lie group structure.

Dimensions and structure

This group has real dimension $16$ , complex dimension $8$ . It acts on projective space by applying a representative matrix to a line. Every nonzero complex determinant has an $3$ th root. Rescaling a representative therefore identifies $\mathrm{PSL}(3,\mathbb C)$ with $\mathrm{PGL}(3,\mathbb C)$ .

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 3th root of its determinant. See $\operatorname{PGL}(3,\mathbb C)$ .

References

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