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

  2. can we use quaternions to represent both groups?

  3. what about their connectedness?

  • $\begingroup$ What do you mean by "parameter space"? $\endgroup$ – ACuriousMind Mar 24 '19 at 17:58
  • $\begingroup$ Parameter space usually refers to the space of parameters of a physical model, so the question doesn't make sense. $\endgroup$ – InertialObserver Mar 24 '19 at 18:07
  • $\begingroup$ parameter space is a manifold every point in that manifold corresponds to a group element $\endgroup$ – user212422 Mar 24 '19 at 18:17
  • $\begingroup$ 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. $\endgroup$ – ACuriousMind Mar 24 '19 at 18:29
  • $\begingroup$ Related: physics.stackexchange.com/q/399546/2451 , physics.stackexchange.com/q/394152/2451 , physics.stackexchange.com/q/271419/2451 and links therein. $\endgroup$ – Qmechanic Mar 24 '19 at 18:49

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 $-{\bf q}$.

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