There're some nomenclatures from Michael Artin's Algebra to explain. 3-Sphere, or $\mathbb S^3$, is the locus of $x_0^2+x_1^2+x_2^2+x_3^2=1$, where $(x_0,x_1,x_2,x_3)\in\mathbb R^4$. $SU_2$ is a special unitary group, i.e, the group of complex $2\times 2$ matrices of the form \begin{bmatrix} a&b\\ -\bar b&\bar a \end{bmatrix} with $\bar aa+\bar bb=1$. Rewrite $a=x_0+x_1i,b=x_2+x_3i$, there's an obvious 1-1 correspondence of $SU_2$ with $\mathbb S^3$, therefore when we say matrix $A$ is on $\mathbb S^3$, we actually mean that $A\in SU_2$. Some similar terminologies apply for the subsets of $\mathbb S^3$, as well.

Latitudes of $\mathbb S^3$ are the loci of $x_0=c$, where $-1<c<1$. When $c=0$, it's so-called the equator $\mathbb E$. They are 2-spheres. Longitudes of $\mathbb S^3$ are the intersections of 2D subspaces of $\mathbb R^2$ (which contains $0$) and $\mathbb S^3$. They are circles.


In Algebra, there's a group action of $SU_2$ on $\mathbb S^3$: conjugation, namely, $\gamma_P\colon\mathbb S^3\to\mathbb S^3;U\mapsto PUP^*$, where $P^*=\bar P^t$, the transpose of the complex conjugation. It's not hard to show that the same operation acts on the equator $\mathbb E$, too. The effect of this action is rotation. Suppose that $P=\cos\theta I+\sin\theta A$ where $A\in\mathbb E$, viz. $\operatorname{tr}A=0$, $\gamma_P$ is a $2\theta$-spin around the axis $A$. It's not obvious.

Now I want to consider another group action, which might be used to interpret the preceding group action: act by left-multiplication (when the left-multiplication is interpreted, the right-multiplication might follow from some slight modifications). Suppose $P\in SU_2$, $A\mapsto PA$ is another natural group action. It's not hard to show that the longitudes of $\mathbb S^3$ are conjugate subgroups of $SU_2$, therefore if $A,P$ lie on the same longitude, so does $PA$, and the group action is apparent, just like the multiplication on the circle $\mathbb S^1$. However, I don't know the geometrical interpretation for the whole action on the whole set. Precisely, my question is:

How can we interpret the group action of left-multiplication of $SU_2$ on $\mathbb S^3$ geometrically, which is highly related to the geometrical facts on $\mathbb R^4$ and $\mathbb E$ as a subset of $\mathbb R^3$, so that we can more clearly see the reason that group actions by conjugation are just rotations in $\mathbb E\subset\mathbb R^3$?


  1. It's natural to correspond the inner product $x_0y_0+x_1y_1+x_2y_2+x_3y_3$ in the matrix form. Suppose that $P,Q$ are the corresponding matrices of $(x_0,x_1,x_2,x_3)$ and $(y_0,y_1,y_2,y_3)$, it's not hard to check that $\operatorname{tr}(P^*Q)/2$ is the required expression, therefore it's easy to see that left-multiplication is orthogonal operator on $\mathbb S^3\subset\mathbb R^4$.
  2. $\mathbb S^3$ is also related to quaternions of length $1$. In fact, these groups are isomorphic.
  • 1
    $\begingroup$ You're aware that the 1-1 correspondence in your first paragraph is an isomorphism of groups...? That is, you may as well view $S^3$ as $SU_2$ for the purposes of your question. $\endgroup$ – Andrew D. Hwang Aug 7 '13 at 10:45
  • $\begingroup$ Could you be more precise about what exactly you are asking? $\endgroup$ – Marc van Leeuwen Aug 7 '13 at 14:56
  • $\begingroup$ @MarcvanLeeuwen I hope it's clearer now. $\endgroup$ – Yai0Phah Aug 7 '13 at 15:04

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