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\begin{document}
\section*{Rough notes for Maths 445\footnote{{\tt www.maths.tcd.ie/\char126 houghton/445.html}}}
Please send corrections and comments to Conor Houghton: {\tt houghton@maths.tcd.
ie}.

\subsection*{1.2 One-dimensional subgroups and generators}
Consider rotations about the $z$-axis
\begin{equation}
R_3(\theta)=\left(\begin{array}{ccc}\cos{\theta}&-\sin{\theta}&0\\\sin{\theta}&\cos{\theta}&0\\0&0&1\end{array}\right)
\end{equation}
This is a subgroup of SO$(3)$: multiplying any two rotations about the
$z$-axis gives another rotation about the $z$-axis. The subgroup is
one-dimensional: the elements are parameterized by $\theta$. This is
an example of a {\sl one-parameter subgroup}. This subgroup can be
rewriten in terms of an exponential: 
\begin{equation}
R_3(\theta)=e^{-i\theta J_3}
\end{equation}
where 
\begin{equation}
J_3=\left(\begin{array}{ccc}0&-i&0\\i&0&0\\0&0&0\end{array}\right)
\end{equation}
The exponetial of a matrix is defined by the power series
\begin{equation}
e^{-i\theta J_3}=1_3+(-i\theta)J_3+\frac{1}{2}(i\theta)^2J_3^2+\ldots
\end{equation}
and the series is split into even and odd parts because
$J_3^2=$diag$(1,1,0)$. 

Now, there are two similar one parameter families corresponding to
rotations about the $x$ and $y$ axis given by 
\begin{equation}
R_i(\theta)=e^{-i\theta J_i}
\end{equation}
where 
\begin{equation}
J_1=\left(\begin{array}{ccc}0&0&0\\0&0&-i\\0&i&0\end{array}\right)
\end{equation}
and
\begin{equation}
J_2=\left(\begin{array}{ccc}0&0&i\\0&0&0\\-i&0&0\end{array}\right).
\end{equation}

This exhausts the list in independant one-parameter subgroups in the following sense. Say 
\begin{equation}
R(\theta)=e^{-\theta iJ}
\end{equation}
is a one-parameter subgroup where $iJ$ is some real matrix.\footnote{I am following Tung in considering pure imaginary generators, this is slightly confusing at this point. It is simpler to think about the real matrix $iJ$.} Since
\begin{equation}
e^{-\theta iJ}e^{\theta iJ}=1_3
\end{equation}
uniqueness of the group inverse implies
\begin{equation}
R(\theta)^{-1}=e^{\theta iJ}
\end{equation}
and, since 
\begin{equation}
R(\theta)^{-1}=R(\theta)^T=e^{-\theta iJ^T}
\end{equation}
by orthogonality, we conclude that the generators of SO$(3)$ are
skewsymmetric $iJ^T=-iJ$. Now, all real skewsymmetic $3\times 3$
matrices lie in the vector space generated by $iJ_1$, $iJ_2$ and
$iJ_3$.

In fact, the vector space of generators has an additional
structure. By calculation it is easy to see that
\begin{equation}
[J_i,J_j]=i\epsilon_{ijk}J_k
\end{equation}
where $[A,B]=AB-BA$ is the usual matrix commutator. By the linearity
of the commuator, this extends to imply that the commutator of any two
matrices in the vector space generated by $J_1$, $J_2$ and $J_3$ is itself a vector in this space. In other words, commutation is closed on the vector space. This makes the vector space an example of a Lie algebra.

A {\sl Lie algebra} is a vector space $V$ equipped with a skewsymmetric
bilinear map
\begin{eqnarray}
[\cdot,\cdot]:V&\times& V\rightarrow V\nonumber\\
              (v_1,v_2)&\mapsto&[v_1,v_2]=-[v_2,v_1]
\end{eqnarray}
which satisfies the Jacobi idenity
\begin{equation}
[v_1,[v_2,v_3]]+[v_2,[v_3,v_1]]+[v_3,[v_1,v_2]]=0
\end{equation}
It is easy to see that the normal matrix commutator is a skewsymmetric
bilinear map satisfying the Jacobi identity.

The idea is that a Lie group always has an assosiated Lie algebra and
the Lie algbra bracket encodes the group structure while ignoring the
global properties of the group manifold. The fact every Lie group has an assosiated algebra follows from the expansion of 
\begin{equation}
e^Ae^Be^{-A}e^{-B}=e^{C}
\end{equation}
for small $A$, $B$ and $C$:
\begin{eqnarray}
(1+A)(1+B)(1-A)(1-B)&=&(1+A+B+AB)(1-A-B+AB)\nonumber\\
                    &=&1+[A,B]+O(3)=1+C+O(2)
\end{eqnarray}
which shows that the matrix commutator closes on the space of generators. The Lie algebra for the group $G$ is often called $\mathfrak{g}$.





\end{document}

