# Definition
The [[Special Orthogonal Group]] in 2 [[dimension|dimensions]], SO(2), is the [[group]] of all [[Rotation|rotations]] in 2 spatial dimensions. It is a [[Lie group]] of [[dimension]] 1.
# Elements
We write an [[group element|element]] as $R(\theta)$ for a *counterclockwise* rotation by an angle $\theta$. $SO(2)$ can be written as:
$
SO(2) \equiv \left\{\begin{pmatrix}\cos\theta & -\sin\theta\\\sin\theta & \cos\theta\end{pmatrix}\Bigg| \, \theta \in [0, 2\pi)\right\}
$
![[Pasted image 20210702094952.png|500]]
From this geometrical picture it is clear that
1. $R(\theta)R(\phi) = R(\theta + \phi)$
2. $R(\theta)^{-1} = R(-\theta)$
From (1) we conclude that rotations in $2D$ are *composable* and from (2) that they are *invertible*. Both properties can be verified algebraically from the definition of $R(\theta)$. Thus, $SO(2)$ forms a [[group]].
Specifically, it is a [[Lie Group|Lie group]] since its elements $R(\theta)$ are completely, uniquely and smoothly determined by the continuous parameter $\theta$.
# Infinitesimal Generators
Infinitesimal generators are elements of the [[Lie Algbera|Lie algebra]]. We discuss them from a very elementary perspective as they apply to $SO(2)$ below.
Infinitesimal generators are just derivatives of elements of the Lie group. This concept only applies to [[Lie group|Lie groups]] since they have a smooth parameter.
Differentiating $R(\theta)$ and evaluating at $\theta = 0$, we get:
$
\frac{dR}{d\theta}\bigg|_{\theta = 0} =
\begin{pmatrix}
0 & -1 \\ 1& 0
\end{pmatrix} \equiv X
$
we cal the matrix $X$ a generator because it generates rotations in the following sense. Consider the [[vector]] $\pmb{r}_0 = (1,0)$, then we get the curve
$
\pmb{r}(\theta) \equiv R(\theta) \cdot \pmb{r}_0
$
which is traced out when $\pmb{r}_0$ is rotated by an angle $\theta$. The tangent vector to this curve at $\theta = 0$ is given by:
$
\frac{d\pmb{r}}{d\theta}\bigg|_{\theta = 0} = \frac{dR}{d\theta}\bigg|_{\theta=0}\cdot \pmb{r}_0 = X \pmb{r}_0 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}
$
Thus, $X$ takes in a vector and gives out the direction in which it will change as a rotation is applied.
# Lie Algebra
The Lie Algebra of $SO(2)$, denoted $\mathfrak{so}(2)$, consists of all $2\times2$ antisymmetric matrices, as noted in [[Special Orthogonal Group#Lie Algebra|here]]. These have the form
$
\begin{pmatrix}
0 & -a \\ a & 0
\end{pmatrix}
$
and we can pick a basis:
$
\begin{pmatrix}
0 & -1 \\ 1 & 0
\end{pmatrix}
$
It is straightforward to show that
$
e^{\theta X} =
\begin{pmatrix}
\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta
\end{pmatrix}, \quad \theta \in \mathbb{R}
$
so that $X$ is the generator of rotations in the XY plane as discussed [[#Infinitesimal Generators|above]].
Moreover, since $X\pmb{r}$ is the direction in which the [[vector]] $\pmb{r}$ would be rotated, then a vector field $X^\sharp$ is generated by $X$, given by:
$
X^\sharp(\pmb{r}) \equiv X \pmb{r}
$
It is shown below.
![[Pasted image 20210707070254.png|450]]
# Isomorphism to $U(1)$
$U(1)$ is the [[Unitary Group]] in 1-dimension, i.e. the set of complex $1\times1$ matrices that obey the condition:
$
U^\dagger = U^{-1}
$
writing an arbitrary complex number as $re^{i\phi}$ (i.e. a $1\times 1$ complex [[matrix]]) and enforcing this condition, we have:
$
r e^{-i\phi} = r^{-1} e^{-i \phi}
$
and we get $r = 1$.
Thus
$
U(1) = \left\{z \in \mathbb{C} \big| \,\, |z| = 1\right\}
$
which can be parameterized as
$
z = e^{i\phi}, \quad \phi \in [0, 2\pi)
$
One can show that the [[types of maps|bijective]] mapping:
$
f(e^{i\phi}) = \begin{pmatrix}\cos\phi & -\sin\phi\\\sin\phi & \cos\phi\end{pmatrix}
$
is a homomorphism using the usual trignometric identities. Thus, since $f$ is also a bijection, we have
$
U(1) \cong SO(2)
$