# Classification of two dimensional Lie Groups

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?!

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

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:

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