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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https://math.stackexchange.com/questions/24601/classsifying-1-and-2-dimensional-algebras-up-to-isomorphism?noredirect=1&lq=1 – Moishe Kohan Jun 10 '19 at 22:20
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The link is about classification of 2 dim Lie algebras not Lie groups!!! – ali_ns Jun 10 '19 at 22:27
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3Yes, and now you should think about the correspondence between Lie algebras send simply connected Lie groups. – Moishe Kohan Jun 10 '19 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 only four connected $2$-dimensional Lie groups:
- $\mathbb R^2$;
- $\mathbb R\times S^1$;
- $S^1\times S^1$;
- $\displaystyle\left\{\begin{bmatrix}a&b\\0&\frac1a\end{bmatrix}\,\middle|\,a>0\wedge b\in\mathbb R\right\}$.
José Carlos Santos
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