SU(2) as a normal subgroup of SL(2, C)

[TrickyDicky]

SU(2) matrices act isometrically on the Riemann sphere with the chordal metric. At the same time the group of automorphisms of the Riemann sphere is isomorphic to the group SL(2, C) of isometries of H 3(hyperbolic space) i.e. every orientation-preserving isometry of H 3 gives rise to a Möbius transformation on the Riemann sphere and viceversa.
Does this make SU(2) a normal(or even characteristic) subgroup of SL(2, C) when acting on H 3 with the chordal metric in its conformal boundary(Riemann sphere) ?

 

[TrickyDicky]

Maybe giving some more context to my question would help getting replies.
I'm referring always to the special case of the groups acting on H³ or CP¹, I'm aware than when acting on Rⁿ the only normal subgroup of SL(2, C) is {+/-I}.

 

[lpetrich]

It may be easier to do it with Lie algebras. Let's see how a normal subgroup translates into Lie algebras. Here's the definition of a normal subgroup. H is a normal subgroup of G if for every g in G and h in H, g.h.g^-1 is also in H. Furthermore, the g.h.g^-1 values are all of H.

If G and H are Lie groups, we can go to Lie algebras. g = 1 + ε*L + ..., h = 1 + ε*M + ... where L and M are members of algebras A and B. Turning normal-subgroupishness into algebra language, B is a subalgebra of A and for all L in A and M in B, [L,M] is in B and spans B. Or for short, [A,B] = B. Thus, B is an ideal of A.

So we have to find some ideal of the algebra SL(2,C) that is isomorphic to the algebra SU(2). They are all real combinations of these basis sets, where the σ's are Pauli matrices:
SU(2): {i*σ_x, i*σ_y, i*σ_z}
SL(2,C): {σ_x, σ_y, σ_z, i*σ_x, i*σ_y, i*σ_z}

I suspect that SL(2,C) has no nontrivial ideals, and thus that SU(2) is not a normal subgroup of SL(2,C).

 

[TrickyDicky]

It may be easier to do it with Lie algebras...

Yes but Lie groups global structure is in general not uniquely determined by their Lie algebras, except for some cases like the simply connected real groups Lie algebras, SL(2, C) is complex.

 

[blue_raver22]

I would agree with the statement that SU(2) is a normal subgroup of SL(2, C) when acting on H 3 with the chordal metric in its conformal boundary (Riemann sphere). This is because normal subgroups are those that are preserved under conjugation by all elements of the larger group. In this case, since every orientation-preserving isometry of H 3 gives rise to a Möbius transformation on the Riemann sphere and vice versa, the elements of SU(2) are preserved under conjugation by all elements of SL(2, C). Therefore, SU(2) can be considered a normal subgroup of SL(2, C) in this context.