# Parameter space of $SO(3)$ and $SU(2)$

1. Is it parameter space of $$SO(3)$$ and $$SU(2)$$ are same?

2. can we use quaternions to represent both groups?

• Lie groups are manifolds. I'm not sure I understand the difference between asking whether the "parameter space" of these groups is the same and asking whether the groups are the same. The properties of the group manifolds of $\mathrm{SO}(3)$ and $\mathrm{SU}(2)$ should be easy to find, e.g. on their Wikipedia pages. Mar 24 '19 at 18:29
The group manifold of $${\rm SU}(2)$$ is the three sphere $$S^3$$. The group manifold of $${\rm SO}(3)$$ is the three-sphere with antipodal points identified. The two spaces have different connectectness (measured by homotopy) because $$\pi_1({\rm SU}(2))= \{0\}$$ and $$\pi_1({\rm SO}(3)={\mathbb Z}_2$$. $${\rm SU}(2)$$ is a double cover of $${\rm SO}(3)$$. You can identify $${\rm SU}(2)$$ with the group of unit length quaternions by the homomorphism $$U= x_0{\mathbb I}-ix_1 \sigma_1-ix_2 \sigma_2-ix_3 \sigma_3 \leftrightarrow {\bf q}=x_0+x_1{\bf i}+x_2{\bf j}+x_3 {\bf k}.$$ Here $$U\in {\rm SU}(2)$$ and $${\bf q}\in {\mathbb H}$$ and $$x_0^2+x_1^2+x_2^2+x_3^2=1$$. To get $${\rm SO}(3)$$ you identify $${\bf q}$$ with $$-{\bf q}$$.