PSL(n,C) - Knowlpedia

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

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

with the quotient Lie group structure.

Dimensions and structure

This group has real dimension $2(n^2-1)$ , complex dimension $n^2-1$ . It acts on projective space by applying a representative matrix to a line. Every nonzero complex determinant has an $n$ th root. Rescaling a representative therefore identifies $\mathrm{PSL}(n,\mathbb C)$ with $\mathrm{PGL}(n,\mathbb C)$ .

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

References

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