Is $(\mathbb{C},+)$ isomorphic to a matrix Lie group?

Question

I'm wondering if $(\mathbb{C},+)$ is isomorphic to a matrix Lie group; I mean, if it is isomorphic to a closed subgroup of $GL(n,\mathbb{C})$. If it is, what is the isomorphism? And if it is not, why?

Answer 1

Yes, it is isomorphic to the subgroup of $GL(2,\Bbb C)$ which consists of the matrices of the form$$\begin{bmatrix}1&z\\0&1\end{bmatrix},$$because$$(\forall z_1,z_2\in\Bbb C):\begin{bmatrix}1&z_1\\0&1\end{bmatrix}.\begin{bmatrix}1&z_2\\0&1\end{bmatrix}=\begin{bmatrix}1&z_1+z_2\\0&1\end{bmatrix}.$$

Answer 2

It's isomorphic to the subgroup of $\operatorname{GL}(2,\mathbb C)$ consisting of matrices of the form

$$\begin{pmatrix}1&0\\z&1\end{pmatrix}$$

via the isomorphism

$$\varphi:z\mapsto\begin{pmatrix}1&0\\z&1\end{pmatrix},$$

since $\varphi(0)=E_2$ and

$$\varphi(a)\varphi(b)=\begin{pmatrix}1&0\\a&1\end{pmatrix}\begin{pmatrix}1&0\\b&1\end{pmatrix}=\begin{pmatrix}1&0\\a+b&1\end{pmatrix}=\varphi(a+b).$$

Injectivity and surjectivity should be clear.