# 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) $