More Examples of Lie Groups: 3-Sphere as a Lie group, The 3-Dimensional Heisenberg Group

3-Sphere S^3 as a Lie Group

Consider 4 elements 1,i,j,k satisfying the following relation
\begin{align*} 1^2&=1,\ i^2=j^2=k^2=-1,\\ 1i&=i1=i,\ 1j=j1=j,\ 1k=k1=k,\\ ij&=-ji=k,\ jk=-kj=i,\ ki=-ik=j. \end{align*}
Let \mathbb{H} be the algebra spanned by 1,i,j,k over \mathbb{R}
\mathbb{H}=\{a1+bi+cj+dk:a,b,c,d\in\mathbb{R}\}.
Then \mathbb{H}\cong\mathbb{R}^4 as a vector space over \mathbb{R}. Define ||q|| of q=a1+bi+cj+dk\in\mathbb{H} by
||q||^2:=q\bar q=a^2+b^2+c^2+d^2,
where \bar q=a1-bi-cj-dk.

Set S^3=\{q\in\mathbb{H}: ||q||=1\}. Then S^3 is a unit sphere in \mathbb{R}^4. S^3 is closed under the multiplication of \mathbb{H}. So, S^3 is a group. In fact, it is a Lie group.

The 3-Dimensional Heisenberg Groups

Set
G=\left\{(x,y,z):=\begin{pmatrix} 1 & y & z\\ 0 & 1 & x\\ 0 & 0 & 1 \end{pmatrix}: x,y,z\in\mathbb{R}\right\}.
Define a multiplication in G by
\begin{align*} (x,y,z)\cdot (a,b,c)&=\begin{pmatrix} 1 & y & z\\ 0 & 1 & x\\ 0 & 0 & 1 \end{pmatrix}\begin{pmatrix} 1 & b & c\\ 0 & 1 & a\\ 0 & 0 & 1 \end{pmatrix}\\ &=\begin{pmatrix} 1 & y+b & z+ya+c\\ 0 & 1 & x+a\\ 0 & 0 & 1 \end{pmatrix}\\ &=(x+a,y+b,z+ya+c)\in G. \end{align*}
The identity element is (0,0,0) and that (x,y,z)^{-1}=(-x,-y,xy-z). G is a Lie subgroup of \mathrm{GL}(3,\mathbb{R}).

Let \gamma:(-\epsilon,\epsilon)\longrightarrow G be a regular curve in G such that \gamma(0)=(0,0,0)=\begin{pmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 &1 \end{pmatrix}.  Then                                        \begin{align*}\dot\gamma(t)&=\frac{\partial\gamma}{\partial x}\frac{dx}{dt}+\frac{\partial\gamma}{\partial y}\frac{dy}{dt}+\frac{\partial\gamma}{\partial z}\frac{dz}{dt}\\ &=\begin{pmatrix}0 & 0 & 0\\0 & 0 & 1\\0 & 0&0\end{pmatrix}\frac{dx}{dt}+\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix}\frac{dy}{dt}+\begin{pmatrix}0&0&1\\0&0&0\\0&0&0\end{pmatrix}\frac{dz}{dt}.\end{align*}
Let
\mathfrak{e}_1:=\begin{pmatrix}0&0&0\\0&0&1\\0&0&0\end{pmatrix},\mathfrak{e}_2:=\begin{pmatrix}0 & 1 & 0\\ 0 & 0 & 0\\0 & 0 & 0\end{pmatrix},\mathfrak{e}_3:=\begin{pmatrix}0 & 0 & 1\\0 & 0 & 0\\0 & 0 & 0\end{pmatrix}.
Then \mathfrak{e}_1,\mathfrak{e}_2,\mathfrak{e}_3 generate the Lie algebra \mathfrak{g} of G. The Lie algebra \mathfrak{g} is called Heisenberg algebra. Note the commutation relation
[\mathfrak{e}_1,\mathfrak{e}_3]=[\mathfrak{e}_2,\mathfrak{e}_3]=0,\ [\mathfrak{e}_1,\mathfrak{e}_2]=-\mathfrak{e}_3
which resembles the commutation relation [\hat p,\hat x]=-i\hbar in quantum mechanics.

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