Lie Algebra of real numbers

Question

Let $G$ be the set of real numbers with the addition as group multiplication. What is the associated Lie Algebra of $G$ if an exponential map is considered?

Answer 1

The exponential map is not the usual one defined on $\mathbb{R}$. One could check by definition that under a natural parametrization the exponential map $\operatorname{exp}: Lie(\mathbb{R})=\mathbb{R}\rightarrow \mathbb{R}$ is the identity map.

A nicer view from matrix Lie group is that $(\mathbb{R},+)$ is a matrix Lie group by identifying $\mathbb{R}=\{\begin{pmatrix} 1 &a\\ 0 & 1 \end{pmatrix}|a\in\mathbb{R}\}$. The Lie algebra contains all the strictly upper triangular matrices. The exponential map is the matrix exponential map: $$\operatorname{exp}(\begin{pmatrix} 0 &a\\ 0 & 0 \end{pmatrix})=\sum_{n\geq 0} \frac{1}{n!} \begin{pmatrix} 0 &a\\ 0 & 0 \end{pmatrix}^n=\begin{pmatrix} 1 &a\\ 0 & 1 \end{pmatrix}$$.