When I read the paper [Universal approximations of invariant maps by neural networks](https://arxiv.org/abs/1804.10306) of *Dmitry Yarotsky*, it happens on *page 36* that he used some concepts about the representation of Lie algebra of the Lie group $\operatorname{SE}(2)$. <br/>
Describe rigid motions of $\mathbb{R}^2$ by identifying it with $\mathbb{C}$. An element in $\operatorname{SE}(2)$ can be written as $\left(\gamma,\theta\right)=\left(x+iy,e^{i\phi}\right)$ with some $x,y\in\mathbb{R}$ and $\phi\in\left[0,2\pi\right)$. The action of $\operatorname{SE}(2)$ on $\mathbb{R}^2\cong\mathbb{C}$:
    \begin{equation*}
        \mathcal{A}_{\left(x+iy,e^{i\theta}\right)}z=x+iy+e^{i\theta}z,\quad z\in \mathbb{C}.
    \end{equation*}
    Consider the generators of the representation:
    \begin{equation*}
        J_x=i\lim_{\delta x\to 0}\dfrac{R_{\left(\delta x,1\right)}-1}{\delta x}, \quad J_y=i\lim_{i\delta y\to 0}\dfrac{R_{\left(\delta y,1\right)}-1}{\delta x}, \quad J_\phi=i\lim_{\delta \phi\to 0}\dfrac{R_{\left(0,e^{i\delta \phi}\right)}-1}{\delta \phi}
    \end{equation*}
where $R_{\left(\gamma,\theta\right)}$ is the action of $\operatorname{SE}(2)$ given by
$$R_{\left(\gamma,\theta\right)}\Phi=\Phi\circ\mathcal{A}^{-1}.$$
    The generators can be explicitly written as
    \begin{equation*}
        J_x=-i\partial_x, \quad J_y=-i\partial_y,\quad J_\phi=-i\partial_\phi=-i\left(x\partial_y-y\partial_x\right)
    \end{equation*}
    and obey the commutation relations
    \begin{equation}\label{eq41}
        \left[J_x,J_y\right]=0,\quad \left[J_x,J_\phi\right]=-iJ_y, \quad \left[J_y,J_\phi\right]=iJ_x.
    \end{equation}


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I am a newbie in Lie representation. Can you explain to me the definition of $J_x, \partial_x$, the Lie brackets, and how these equations hold here? Or can you give me some books/papers defining these concepts? I looked upon the internet about the representation of Lie algebra but met nothing like these.<br/>

  [1]: https://arxiv.org/pdf/1804.10306.pdf