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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?

Newstudent
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  • The tangent space of any vector space at any point is canonically isomorphic to the vector space itself (you have the identity chart which gives the isomorphism). – peek-a-boo May 16 '23 at 00:14
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    What have you tried on this question? In what context did it arise? What ways do you know to get the Lie Algebra of a Lie Group, which of them have you tried on this example, where did you get stuck? – Torsten Schoeneberg May 16 '23 at 03:06

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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: $$\operatorname{exp}(\begin{pmatrix} 0 &a\\ 0 & 0 \end{pmatrix})=\sum_{n\geq 0} \frac{1}{n!} \begin{pmatrix} 0 &a\\ 0 & 0 \end{pmatrix}^n=\begin{pmatrix} 1 &a\\ 0 & 1 \end{pmatrix}$$.

Yunsong WEI
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