$SO(n)$ groups are 'special' subgroups of $O(n)$, the orthogonal group in dimension of $n$. $SO(n)$ groups are called 'special' because they are consisted of orthogonal matrices of determinant $1$. We are familiar with $SO(n)$ groups, especially when $n$ is 2 or 3. $SO(2)$ and $SO(3)$ groups represent the rotations in dimension of 2 and 3 each, so they are also called as 'rotation groups'. In this post, only $SO(2)$ will be discussed and $SO(3)$ will be discussed later. Discussing $SO(2)$ is not so difficult, when we focus on the definition and properties of $SO(2)$ as a 'rotation' in dimension of 2.
As the title of this post mentions, we need to find the 'generators' of this group.
But wait.... what IS a generator?
Considering small finite groups would help us understand generators. Think about the finite cyclic group
\[ Z_{4}=\left(G,\times \right) \]
when $G=\left \{ i,-1,1,-i \right \}$. We can easily catch the element $i$ of $G$, which 'generates' the set $G$ by the binomial operator $ \times $ of $ \left(G,\times \right) $. It's same in the compact group. There'll be a generator ( or generators ), on $SO(2)$ and it will 'generate' the set of special orthogonal $2 \times 2$ matrices.
Let's take a closer look on $SO(2)$. From the definition of $SO(2)$, we can define the set of $SO(2)$ as
\[ \left \{ R\in GL(2,R) | R^{T}R=RR^{T}=I, det(R)=1 \right \} \]
and we know that every rotation matrices
\[R(\theta)=\begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} \]
are in it as they satisfy the properties of orthogonality and unimodularity of $+1$. There's no problem with $R(\theta)$ as a representative element of $SO(2)$, so it would be convinient to represent $SO(2)$ by $R(\theta)$,
\[SO(2)=\left \{ \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} : \theta\in R (mod_{2\pi}) \right \} \]
When compact groups are represented by a matrix, the components of the matrix are denoted as a function of a certain variable, and we call that a 'parameter'. Hence, $SO(2)$ has a single parameter $\theta$. We learned that every group needs a identity element, and so does $SO(2)$, which is
\[ I=\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}=R(0) \].
Now, everything is ready. Compact groups have infinite elements, that is why they are 'compact' and we assume that there is a certain element of that infinite elements, that has very little difference with the identity element $I=R(0)$. Let's call it as $\mathcal{J}$, and $\mathcal{J}$ is defined as
\[\mathcal{J}\equiv \frac{1}{i}\frac{\partial }{\partial\theta}R(\theta)|_{\theta=0}=\frac{1}{i}\frac{\partial }{\partial \theta}\begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}_{\theta=0} \]
Further explanations about the reason of differentiation and related subjects are required but I'll explain them on later posts, so just relate the differentiation of $R(\theta)$ with finding an element having infinitesimal difference with $R(0)$ for a while. Differentiating a matrix is not different. Just do it component-wisely.
\[\mathcal{J} = \frac{1}{i} \frac{ \partial } {\partial \theta}\begin{pmatrix}\cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}_{\theta=0} = \frac{1}{i}\begin{pmatrix}0 & 1 \\-1 & 0\end{pmatrix}\]
\[\mathcal{J}=\begin{pmatrix}0 &-i \\i &0\end{pmatrix} \]
So this is the generator of $SO(2)$, and as $SO(2)$ has a single parameter on it's representation ( actually this is not an accurate explanation ) the generator is single too. Then how does this $\mathcal{J}$ generates $SO(2)$?? The answer lies on the exponential function. Let's define a new matrix called $A$
\[A=i\mathcal{J}\theta=\begin{pmatrix}0 & \theta \\-\theta & 0\end{pmatrix}\]
And take $A$ as an index,
\[e^{A}=\sum_{n=0}^{\infty }\frac{1}{n!}A^{n}\]
Expanding the above equation,
\[\sum_{n=0}^{\infty }\frac{1}{n!}A^{n}=\sum_{n=0}^{\infty }\frac{1}{n!}(iJ\theta)=1+iJ\theta+\frac{1}{2!}(iJ\theta)^{2}+\frac{1}{3!}(iJ\theta)^{3}+...\]
Multiplying and summing, we can derive the below matrix expression.
\[e^{A}=\begin{pmatrix}\sum_{n=0}^{\infty} \frac{(-1)^n}{2n!} \theta^{2n} &\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} \theta^{2n+1} \\-\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!} \theta^{2n+1} & \sum_{n=0}^{\infty}\frac{(-1)^n}{2n!} \theta^{2n}\end{pmatrix}\]
And this looks familiar because each components seem to be some Maclaurin expansions of trigonometric functions. And actually they are.
\[e^{A}=\begin{pmatrix}\cos\theta & \sin\theta \\-\sin\theta & \cos\theta\end{pmatrix}\ = R(\theta) \]
Surprisingly, we discovered that taking $A=i\mathcal{J}\theta$ as an index of an exponential function generates the representation of $SO(2)$.
$SO(2)$ is also special among its $SO(n)$ siblings, because it has a single generator $\mathcal{J}$. If there are lots of generators, we can research about the relationships between each generators. But there's something different with this 'relationship' as we usually know. Between generators we research the 'commutation relation', which is denoted as
\[\left [ \mathcal{J}_{i},\mathcal{J}_{j} \right ]=\mathcal{J}_{i}\mathcal{J}_{j}-\mathcal{J}_{j}\mathcal{J}_{i}\]
These commutation relations between generators are called the Lie Algebra of a group. Surprisingly, results of these commutations are something related with generators itself.
\[\left [ \mathcal{J}_{i},\mathcal{J}_{j} \right ]=f_{ijk}\mathcal{J}_{k}\]
We call the $f_{ijk}$ as the 'structure constant' of the Lie algebra of $SO(2)$.
Unfortunately, we only have 1 generator $\mathcal{J}$ in $SO(2)$, which means that the only Lie algebra we can derive is,
\[\left [ \mathcal{J},\mathcal{J}\right ]=0\]
which is super-trivial.
Think about rotations in the $R^2$ and $R^3$. Obviously rotations in $R^2$ has commutativity, which means that the results are identical when first rotating $\theta_1$ then $\theta_2$ and first rotating $\theta_2$ then $\theta_1$. But in $R^3$, there has no commutativity.
Quite interesting....? Seems there has to be something with generators and Lie algebra.
Later posts will discuss about the generator's' and Lie Algebra of $SO(3)$.
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