isomorphism of the group PSL_2(C) with the group of Möbius transformations
We identify the group G of Möbius transformations with the projective special linear group
PSL2(ℂ). The isomorphism
Ψ (of topological groups) is given by Ψ:[(abcd)]↦az+bcz+d. (Here, the notation [M] means the equivalence class
[M]={Mt∣t∈ℂ})
This mapping is:
- Well-defined:
-
If [(abcd)]=[(a′b′c′d′)] then (a′,b′,c′,d′)=t(a,b,c,d) for some t, so z↦az+bcz+d is the same transformation as z↦a′z+b′c′z+d′.
- A homomorphism
:
-
Calculating the composition
az+bcz+d|z=ew+fgw+h=aew+fgw+h+bcew+fgw+h+d=(ae+bg)w+(af+bh)(ce+dg)w+(cf+dh) we see that Ψ([(abcd)])⋅Ψ([(efgh)])=Ψ([(abcd)]⋅[(efgh)]).
- A monomorphism
:
-
If Ψ([(abcd)])=Ψ([(a′b′c′d′)]), then it follows that (a′,b′,c′,d′)=t(a,b,c,d), so that [(abcd)]=[(a′b′c′d′)].
- An epimorphism
:
-
Any Möbius transformation z↦az+bcz+d is the image Ψ([(abcd)]).
Title | isomorphism of the group PSL_2(C) with the group of Möbius transformations |
---|---|
Canonical name | IsomorphismOfTheGroupPSL2CWithTheGroupOfMobiusTransformations |
Date of creation | 2013-03-22 12:43:30 |
Last modified on | 2013-03-22 12:43:30 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 9 |
Author | rspuzio (6075) |
Entry type | Result |
Classification | msc 57S25 |