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I would like to understand the left-invariant vector field of the additive group of real number. The left translation are defined by

\begin{equation}

L_a : x \mapsto x + a \; , \;\;\; x,a \in G \subseteq \mathbb{R}.

\end{equation}

The differential map is

\begin{equation}

L_{a*} = \frac{\partial (x + a)}{\partial x} = 1,

\end{equation}

and the left-invariant vector field is

\begin{equation}

X = \frac{\partial}{\partial x}.

\end{equation}

So

\begin{equation}

L_{a*} X|_x = \frac{\partial}{\partial x}|_x.

\end{equation}

But I don't understand why

\begin{equation}

L_{a*} X|_x = X|_{x+a} = \frac{\partial}{\partial x}|_{x+a}.

\end{equation}

This should be true if X is really a left-invariant vector field, right?

Thanks in advance for any help.

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# Left-invariant vector field of the additive group of real number

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