The group ${\mathrm{SU}}_2\simeq U_1\left(ℍ\right)\simeq {\mathrm{Spin}}_3$ and the Lie algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 6 November 2011

The group ${\mathrm{SU}}_2\simeq U_1\left(ℍ\right)\simeq {\mathrm{Spin}}_3$ and the Lie algebra ,

The maximal compact subgroup of ${\mathrm{SL}}_2\left(ℂ\right)$ is

${\mathrm{SU}}_2=\{g=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)|g{\bar{g}}^t=1\text{and}\det \left(g\right)=1\}=\left\{\left(\begin{array}{cc}a& b\\ -\bar{b}& a\end{array}\right)\right|a,b\in ℂ,{\left|a\right|}^2+{\left|b\right|}^2=1\}$

since

$g^{-1}=\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)\text{and}{\bar{g}}^t=\left(\begin{array}{cc}\bar{a}& \bar{c}\\ \bar{b}& \bar{d}\end{array}\right).$

The Lie algebra is

where

${\sigma }^x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }^y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }^z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$

are the Pauli matrices and

$[{\sigma }^x,{\sigma }^y]=2i{\sigma }^z,[{\sigma }^y,{\sigma }^z]=2i{\sigma }^x,[{\sigma }^z,{\sigma }^x]=2i{\sigma }^y.$

Then is the complexification of ,

and the change of basis is given by

$\begin{array}{lll}{\sigma }^x=x+y,& {\sigma }^y=-ix+iy,& {\sigma }^z=h,\\ x=\frac{1}{2}({\sigma }^x+i{\sigma }^y),& y=\frac{1}{2}({\sigma }^x-i{\sigma }^y),& h={\sigma }^z.\end{array}$

(PtoC)

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

${ℍ}^{\times }={\mathrm{GL}}_1\left(ℍ\right)\text{and}{U}_1\left(ℍ\right)=\{x\in ℍ|\left|x\right|=x{\bar{x}}^t=1\}.$

The fundamental representation $\theta :U_1\left(ℍ\right)\stackrel{\sim }{⟶}{\mathrm{SU}}_2$

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

$\begin{array}{cccccc}\theta :& {ℍ}^{\times }& ⟶& {\mathrm{GL}}_2\left(ℂ\right)& \stackrel{\text{transpose}}{⟶}& {\mathrm{GL}}_2\left(ℂ\right)\\ & x_0+x_{1}i+x_{2}j+x_{3}k& ⟼& \left(\begin{array}{cc}{x}_0+x_{1}i& -x_2+x_{3}i\\ x_2+x_{3}i& x_0-x_{1}i\end{array}\right)& ⟼& \left(\begin{array}{cc}{x}_0+x_{1}i& -x_2+x_{3}i\\ x_2+x_{3}i& x_0-x_{1}i\end{array}\right)\end{array}$

This gives a group homomorphism

$\begin{array}{cccc}\theta :& {ℍ}^{\times }& ⟶& {\mathrm{GL}}_2\left(ℂ\right)\\ & a+cj& ⟼& \left(\begin{array}{cc}a& c\\ -\bar{c}& \bar{a}\end{array}\right)\end{array}\text{for}a=x_0+x_{1}i\text{and}c=x_2+x_{3}i\text{in}ℂ.$

The Pauli matrices are

$\theta \left(i\right)=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right),\theta \left(j\right)=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right),\theta \left(k\right)=\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right),$

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 $H=U_1\left(ℂ\right)$,

$\begin{array}{ccccccc}{U}_1\left(ℍ\right)& \supseteq & U_1\left(ℂ\right)=H& \stackrel{\sim }{⟶}& \theta \left(H\right)& \subseteq & {\mathrm{SU}}_2\\ & & a+bi& ⟼& \left(\begin{array}{cc}a+bi& 0\\ 0& a-bi\end{array}\right)\end{array}\text{with}|a+bi|=1.$

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

where

and

and

$\theta \left(i\right)=i{\sigma }^z,\theta \left(j\right)=i{\sigma }^y,\theta \left(k\right)=i{\sigma }^x.$

The adjoint representation is the 3-dimensional representation given by the action of $G$ on $𝔤$. If $G\subseteq {\mathrm{GL}}_n$ and then the action is given by

$g\cdot x=gx{g}^{-1},\text{for}g\in G\text{and}x\in 𝔤.$

In this case, $𝔤=ℂ{\otimes }_{ℝ}{\mathrm{𝔰𝔲}}_2=ℂ\text{-span}\{x,y,h\}$, 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

where ad is the adjoint representation of ${\mathrm{𝔰𝔩}}_2$ given by

$\mathrm{ad}\left(x\right)=\left(\begin{array}{ccc}0& -2& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right),\mathrm{ad}\left(y\right)=\left(\begin{array}{ccc}0& 0& 0\\ -1& 0& 0\\ 0& 2& 0\end{array}\right),\mathrm{ad}\left(h\right)=\left(\begin{array}{ccc}2& 0& 0\\ 0& 0& 0\\ 0& 0& -2\end{array}\right),$

(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

$\left\{1\right\}⟶\{\pm 1\}⟶U_1\left(ℍ\right)\stackrel{\rho }{⟶}{\mathrm{SO}}_3\left(ℝ\right)⟶\left\{1\right\}$

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.

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