## The group ${\mathrm{SU}}_{2}\simeq {U}_{1}\left(ℍ\right)\simeq {\mathrm{Spin}}_{3}$ and the Lie algebra ${\mathrm{𝔰𝔲}}_{2}$,

The maximal compact subgroup of ${\mathrm{SL}}_{2}\left(ℂ\right)$ is

 ${\mathrm{SU}}_{2}=\left\{g=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}g{\stackrel{‾}{g}}^{t}=1\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\mathrm{det}\left(g\right)=1\right\}=\left\{\left(\begin{array}{cc}a& b\\ -\stackrel{‾}{b}& a\end{array}\right)\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}a,b\in ℂ,\phantom{\rule{0.5em}{0ex}}{|a|}^{2}+{|b|}^{2}=1\right\}$
since
 ${g}^{-1}=\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{\stackrel{‾}{g}}^{t}=\left(\begin{array}{cc}\stackrel{‾}{a}& \stackrel{‾}{c}\\ \stackrel{‾}{b}& \stackrel{‾}{d}\end{array}\right).$
The Lie algebra ${\mathrm{𝔰𝔲}}_{2}$ is
 ${\mathrm{𝔰𝔲}}_{2}=\left\{x\in {\mathrm{𝔤𝔩}}_{2}\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}x+{\stackrel{‾}{x}}^{t}=0,\phantom{\rule{0.5em}{0ex}}\mathrm{tr}x=0\right\},=ℝ\mathrm{-span}\left\{i{\sigma }^{x},i{\sigma }^{y},i{\sigma }^{z}\right\},$
where
 ${\sigma }^{x}=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),\phantom{\rule{2em}{0ex}}{\sigma }^{y}=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),\phantom{\rule{2em}{0ex}}{\sigma }^{z}=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$
are the Pauli matrices and
 $\left[{\sigma }^{x},{\sigma }^{y}\right]=2i{\sigma }^{z},\phantom{\rule{0.5em}{0ex}}\left[{\sigma }^{y},{\sigma }^{z}\right]=2i{\sigma }^{x},\phantom{\rule{0.5em}{0ex}}\left[{\sigma }^{z},{\sigma }^{x}\right]=2i{\sigma }^{y}.$
Then ${\mathrm{𝔰𝔩}}_{2}\left(ℂ\right)$ is the complexification of ${\mathrm{𝔰𝔲}}_{2}$,
 ${\mathrm{𝔰𝔩}}_{2}\left(ℂ\right)=ℂ{\otimes }_{ℝ}{\mathrm{𝔰𝔲}}_{2}$
and the change of basis is given by
 $\begin{array}{lll}{\sigma }^{x}=x+y,\phantom{\rule{0.5em}{0ex}}& {\sigma }^{y}=-ix+iy,\phantom{\rule{0.5em}{0ex}}& {\sigma }^{z}=h,\\ x=\frac{1}{2}\left({\sigma }^{x}+i{\sigma }^{y}\right),\phantom{\rule{0.5em}{0ex}}& y=\frac{1}{2}\left({\sigma }^{x}-i{\sigma }^{y}\right),\phantom{\rule{0.5em}{0ex}}& h={\sigma }^{z}.\end{array}$ (PtoC)

Let $ℍ$ be the division algebra of Hamiltonians so that

 ${ℍ}^{×}={\mathrm{GL}}_{1}\left(ℍ\right)\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{U}_{1}\left(ℍ\right)=\left\{x\in ℍ\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}|x|=x{\stackrel{‾}{x}}^{t}=1\right\}.$

#### The fundamental representation $\theta :{U}_{1}\left(ℍ\right)\stackrel{\sim }{⟶}{\mathrm{SU}}_{2}$

The action of ${ℍ}^{×}$ on the 2-dimensional $ℂ$-vector space $ℍ$ by right multiplication provides

 $\begin{array}{cccccc}\theta :& {ℍ}^{×}& ⟶& {\mathrm{GL}}_{2}\left(ℂ\right)& \stackrel{\text{transpose}}{⟶}& {\mathrm{GL}}_{2}\left(ℂ\right)\\ & {x}_{0}+{x}_{1}i+{x}_{2}j+{x}_{3}k& ⟼& \left(\begin{array}{cc}{x}_{0}+{x}_{1}i& -{x}_{2}+{x}_{3}i\\ {x}_{2}+{x}_{3}i& {x}_{0}-{x}_{1}i\end{array}\right)& ⟼& \left(\begin{array}{cc}{x}_{0}+{x}_{1}i& -{x}_{2}+{x}_{3}i\\ {x}_{2}+{x}_{3}i& {x}_{0}-{x}_{1}i\end{array}\right)\end{array}$
This gives a group homomorphism
 $\begin{array}{cccc}\theta :& {ℍ}^{×}& ⟶& {\mathrm{GL}}_{2}\left(ℂ\right)\\ & a+cj& ⟼& \left(\begin{array}{cc}a& c\\ -\stackrel{‾}{c}& \stackrel{‾}{a}\end{array}\right)\end{array}\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{0.5em}{0ex}}a={x}_{0}+{x}_{1}i\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}c={x}_{2}+{x}_{3}i\phantom{\rule{0.5em}{0ex}}\text{in}\phantom{\rule{0.5em}{0ex}}ℂ.$
The Pauli matrices are
 $\theta \left(i\right)=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right),\phantom{\rule{2em}{0ex}}\theta \left(j\right)=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right),\phantom{\rule{2em}{0ex}}\theta \left(k\right)=\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right),\phantom{\rule{2em}{0ex}}$
and
 $\theta :{U}_{1}\left(ℍ\right)\stackrel{\sim }{⟶}{\mathrm{SU}}_{2}.$

The Cartan subalgebra of ${𝔲}_{1}\left(ℍ\right)$ is $𝔥=ℝ\text{-span}\left\{i\right\}$ and the Cartan subgroup of ${U}_{1}\left(ℍ\right)$ is the group $H={U}_{1}\left(ℂ\right)$,

 $\begin{array}{ccccccc}{U}_{1}\left(ℍ\right)& \supseteq & {U}_{1}\left(ℂ\right)=H& \stackrel{\sim }{⟶}& \theta \left(H\right)& \subseteq & {\mathrm{SU}}_{2}\\ & & a+bi& ⟼& \left(\begin{array}{cc}a+bi& 0\\ 0& a-bi\end{array}\right)\end{array}\phantom{\rule{2em}{0ex}}\text{with}\phantom{\rule{1em}{0ex}}|a+bi|=1.$

#### The isomorphism ${\mathrm{SU}}_{2}\simeq {\mathrm{Spin}}_{3}$

The differential of the isomorphism $\theta :{U}_{1}\left(ℍ\right)\stackrel{\sim }{\to }{\mathrm{SU}}_{2}\left(ℂ\right)$ is the Lie algebra isomorphism

 $\theta :{𝔲}_{1}\left(ℍ\right)\stackrel{\sim }{⟶}{\mathrm{𝔰𝔲}}_{2},$
where
 ${𝔲}_{1}\left(ℍ\right)=\left\{x\in {\mathrm{𝔤𝔩}}_{1}\left(ℍ\right)\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}x+{\stackrel{‾}{x}}^{t}=0\right\}=\left\{{x}_{1}i+{x}_{2}j+{x}_{3}k\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}{x}_{1},{x}_{2},{x}_{3}\in ℝ\right\}$
and
 ${\mathrm{𝔰𝔲}}_{2}=\left\{x\in {\mathrm{𝔤𝔩}}_{2}\left(ℂ\right)\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}x+{\stackrel{‾}{x}}^{t}=0\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\mathrm{tr}x=0\right\}=ℝ\text{-span}\left\{i{\sigma }^{x},i{\sigma }^{y},i{\sigma }^{z}\right\}$
and
 $\theta \left(i\right)=i{\sigma }^{z},\phantom{\rule{2em}{0ex}}\theta \left(j\right)=i{\sigma }^{y},\phantom{\rule{2em}{0ex}}\theta \left(k\right)=i{\sigma }^{x}.$
The adjoint representation is the 3-dimensional representation given by the action of $G$ on $𝔤$. If $G\subseteq {\mathrm{GL}}_{n}$ and $𝔤\subseteq {\mathrm{𝔤𝔩}}_{n}$ then the action is given by
 $g\cdot x=gx{g}^{-1},\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}g\in G\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}x\in 𝔤.$
In this case, $𝔤=ℂ{\otimes }_{ℝ}{\mathrm{𝔰𝔲}}_{2}=ℂ\text{-span}\left\{x,y,h\right\}$, and the representation $\rho :{U}_{1}\left(ℍ\right)\simeq {\mathrm{SU}}_{2}\left(ℂ\right)\to {\mathrm{GL}}_{3}\left(ℂ\right)$ coming from the adjoint action of ${\mathrm{SU}}_{2}\left(ℂ\right)$ has differential
 $d\rho :{\mathrm{𝔰𝔲}}_{2}\subseteq {\mathrm{𝔰𝔩}}_{2}\left(ℂ\right)\stackrel{\text{ad}}{⟶}{\mathrm{𝔤𝔩}}_{3}\left(ℂ\right)$
where ad is the adjoint representation of ${\mathrm{𝔰𝔩}}_{2}$ given by
 $\mathrm{ad}\left(x\right)=\left(\begin{array}{ccc}0& -2& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right),\phantom{\rule{2em}{0ex}}\mathrm{ad}\left(y\right)=\left(\begin{array}{ccc}0& 0& 0\\ -1& 0& 0\\ 0& 2& 0\end{array}\right),\phantom{\rule{2em}{0ex}}\mathrm{ad}\left(h\right)=\left(\begin{array}{ccc}2& 0& 0\\ 0& 0& 0\\ 0& 0& -2\end{array}\right),\phantom{\rule{2em}{0ex}}$ (adsl2)
The change of basis formulas in (PtoC) allow any favourite element of ${\mathrm{𝔰𝔲}}_{2}$ in any favourite basis of $𝔤$ to be worked out from (adsl2).

The adjoint representation $\rho :{U}_{1}\left(ℍ\right)\simeq {\mathrm{SU}}_{2}\left(ℂ\right)\to {\mathrm{GL}}_{3}\left(ℂ\right)$ gives rise to the exact sequence

 $\left\{1\right\}⟶\left\{±1\right\}⟶{U}_{1}\left(ℍ\right)\stackrel{\rho }{⟶}{\mathrm{SO}}_{3}\left(ℝ\right)⟶\left\{1\right\}$
which realizes ${U}_{1}\left(ℍ\right)$ as the 2-fold cover ${\mathrm{Spin}}_{3}\simeq {U}_{1}\left(ℍ\right)$ of ${\mathrm{SO}}_{3}\left(ℝ\right)$.

## Notes and References

These notes were influenced by the Wikipedia articles ????. They were prepared for lectures and working seminars in Representation Theory at University of Melbourne in 2008-2011.

## References

[St] R. Steinberg, Lectures on Chevalley groups, Notes prepared by John Faulkner and Robert Wilson, Yale University, New Haven, Conn., 1968. iii+277 pp. MR0466335.