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I am trying to classify all connected two dimensional Lie groups up to isomorphisms. In the compact case, I proved it is isomorphic to torus, but I do not know what to do for the non-compact case. I read in some website that we can classify all simply connected two dim Lie groups with their Lie algebra. Do you know how we can do that?!

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  • $\begingroup$ math.stackexchange.com/questions/24601/… $\endgroup$ Jun 10, 2019 at 22:20
  • $\begingroup$ The link is about classification of 2 dim Lie algebras not Lie groups!!! $\endgroup$
    – ali_ns
    Jun 10, 2019 at 22:27
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    $\begingroup$ Yes, and now you should think about the correspondence between Lie algebras send simply connected Lie groups. $\endgroup$ Jun 10, 2019 at 23:35

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Up to isomorphism, there are only two simply connected $2$-dimensional Lie groups:

  • $(\mathbb R^2,+)$
  • $\displaystyle\left(\left\{\begin{bmatrix}a&b\\0&\frac1a\end{bmatrix}\,\middle|\,a>0\wedge b\in\mathbb R\right\},\times\right)$

The connected $2$-dimensional Lie groups are the quotients of these groups by normal discrete subgroups. In the case of $(\mathbb R^2,+)$, we get $(\mathbb R^2,+)$, $\mathbb R\times S^1$ and $S^1\times S^1$ (the torus). The other group has no non-trivial normal discrete subgroups. So, up to isomorphism, there are ony $4$ connected $2$-dimensional Lie groups:

  • $\mathbb R^2$
  • $\mathbb R\times S^1$
  • $S^1\times S^1$
  • $\displaystyle\left\{\begin{bmatrix}a&b\\0&\frac1a\end{bmatrix}\,\middle|\,a>0\wedge b\in\mathbb R\right\}$
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