PSL(2,C) - Knowlpedia

Example

The complex projective special linear group is

PSL(2,\mathbb C):=SL(2,\mathbb C)/\{\pm I\}.

It is a connected complex Lie group of complex dimension $3$ . Its underlying real Lie group has real dimension $6$ , and its Lie algebra is $\mathfrak{sl}_2(\mathbb C)_{\mathbb R}$ .

Equivalent matrix description

Because every nonzero complex number has a square root, every class in $PGL_2(\mathbb C)$ has a determinant-one representative, unique up to sign. Hence

PSL(2,\mathbb C)\cong PGL_2(\mathbb C).

This equality is special to fields for which the required determinant roots exist and must not be transferred to arbitrary fields.

Its principal geometric actions are recorded by focused theorem knowls: the Möbius action on the celestial sphere, the proper Lorentz isomorphism, and the action on hyperbolic three-space. Its separate role as the complex points of the adjoint algebraic group of type $A_1$ is treated in the Langlands-dual-group context.

References

Alan F. Beardon, The Geometry of Discrete Groups, Springer, 1983, Chapters 3 and 7. Publisher record.
John G. Ratcliffe, Foundations of Hyperbolic Manifolds, 3rd ed., Springer, 2019, Chapters 3–4. Publisher record.