Representation of Lie algebra $\operatorname{SE}(2)$

Question

When I read the paper Universal approximations of invariant maps by neural networks of Dmitry Yarotsky, it happens on page 36 that he used some concepts about the representation of Lie algebra of the Lie group $\operatorname{SE}(2)$.
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.

Answer 1

The representation $R$ can be seen as a map: $$R:\operatorname{SE} (2)\to\operatorname{Diff}(\mathbb R^2) $$

and the derivations above correctly allows to compute the corresponding infinitesimal action

$$\mathfrak{se} (2)\to \mathfrak X(\mathbb R^2) $$

which is an injective map having as image the three-dimensional vector space of vector fields on the plane spanned by $J_x$, $J_y$ and $J_\phi$. Here the Lie bracket is just the commutator of vector fields $$ [X, Y] =X\circ Y-Y\circ X$$ and $\partial_x$ (resp. $y$) is just the vector field of constant horizontal (resp. vertical) translations that acts, as derivations on functions, exactly as the partial derivative along $x$ (resp. $y$).