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It says on Wikipedia:

"to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to covering."

We can turn $2 \times 2$ matricies over the reals into a Lie algebra by defining $[a, b] = ab - ba$. Which Lie group does this define? We have

$$[a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0$$

for any $a, b, c$.

Robert Lewis
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Stephen Wynn
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    You are not new here, so please learn how to use MathJax: https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference. – Batominovski Jun 10 '18 at 16:40
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    $GL_+(2,\mathbb{R})$ is a Lie group with this Lie algebra. By this I $2\times 2$ real matrices of positive determinant. To get uniqueness you need to say simply connected. – Charlie Frohman Jun 10 '18 at 16:44
  • I edited your post to $\LaTeX$ify it and also to replace your notation $a \times b = ab - ba$ with the more conventioal $[a, b] = ab - ba$. Hope my math edits are OK; if not, you can change them back. But as Batomiovski says, please stick to MathJax. It makes your questions so much more readable . . . Cheers! – Robert Lewis Jun 10 '18 at 16:46

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It is the Lie group $\mathrm{GL}(2,\mathbb R)$. The Lie algebra $\mathfrak{gl(2,\mathbb R)}$ is the space of all $2\times 2$ matrices.

Hugo
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  • Of course there are other choices. We could choose for example $G={A \in \mathrm{GL}(2,\mathbb R): \det(A)>0}$. – Hugo Jun 10 '18 at 16:43