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Given the definition of a Matrix Lie Group as a subgroup of $GL(n,\mathbb{C})$ such that for every sequence $A_n$ its limit $A$ is either in the subgroup or it is not invertible, what are the known examples for which the limit $A$ is invertible but is not contained in the subgroup, hence subgroups of $GL(n,\mathbb{C})$ that are not matrix Lie groups?

Daniele Bernardini
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    Here is another example: Let $x_1,\dots,x_n\in\mathbb{R}$ be linearly independent over $\mathbb{Q}$ and consider the 1-parameter subgroup $\exp(it\cdot\operatorname{diag}(x_1,\dots,x_n))\subset GL(n,\mathbb{C})$. – user10354138 Feb 14 '21 at 17:26
  • Yes, thanks. I did not find that question when I was searching, and I came to the same conclusion a few minutes after I wrote the question here. Will accept as duplicate ;-) – Daniele Bernardini Feb 14 '21 at 20:46

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Well it dawned on me. the group of invertible matrices with rational entries. The limit can have values that are not rational but still be invertible.

Daniele Bernardini
  • 149