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If there are two matrix Lie groups $G_1$ and $G_2$ and $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be the corresponding Lie algebras. If it is known that $\mathfrak{g}_1=\mathfrak{g}_2$ and additionally that $\mathfrak{g_1}$ and $\mathfrak{g_2}$ are simple lie algebras then what can we comment about $G_1$ and $G_2$ ? I don't have much intuition as to what happens to the lie groups when their corresponding lie algebras are the same.

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    $\begingroup$ Do you actually mean for the Lie algebras to be equal, or do you just want them to be isomorphic? $\endgroup$ – Tobias Kildetoft Sep 20 '16 at 6:15
  • $\begingroup$ I know that the lie algebras are actually equal. That is, they are the same $\endgroup$ – Srini Sep 20 '16 at 6:18
  • $\begingroup$ The same in what sense? So you mean you have identified them as subalgebras of a common set of matrices? $\endgroup$ – Tobias Kildetoft Sep 20 '16 at 6:18
  • $\begingroup$ They are the same set of matrices.$\mathfrak{g}_1 = \mathfrak{g}_2 = S \subseteq \mathbb{F}^{n\times n}$ $\endgroup$ – Srini Sep 20 '16 at 6:20
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Recall the following important fact:

Theorem. Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. For each subalgebra $\mathfrak{h}$ in $\mathfrak{g}$ there is a unique connected Lie subgroup $H$ of $G$ with Lie algebra $\mathfrak{h}$.

(However, $H$ is in general not necessarily a closed subgroup of $G$.) Any way, what is important here is uniqueness. If you have two connected matrix Lie groups in $\mathrm{GL}(n,\mathbb{R})$, say $H_1,H_2\subseteq\mathrm{GL}(n,\mathbb{R})$, such that $\mathfrak{h}_1=\mathfrak{h}_2\subseteq\mathrm{Mat}(n,\mathbb{R})$, then $H_1=H_2$. If they are not necessarily connected, then you can still say that their identity components are equal: $(H_1)_0=(H_2)_{0}$.

However, if $H_1\subseteq\mathrm{GL}(n_1,\mathbb{R})$ and $H_2\subseteq\mathrm{GL}(n_2,\mathbb{R})$ for $n_1\neq n_2$, then you cannot necessarily conclude that $H_1=H_2$.

In general, if $\mathfrak{h}_1\cong\mathfrak{h}_2$ and $H_1,H_2$ are connected, all you can say is that their universal cover are equal: $\tilde{H}_1=\tilde{H}_2$.

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  • $\begingroup$ If I know that 𝔥1=𝔥2 is a simple lie algebra is there anything I can comment about H1,H2 $\endgroup$ – Srini Apr 21 '17 at 6:48
  • $\begingroup$ @Srini They are locally isomorphic but not necessarily globally isomorphic. More precisely, all you can say is that there is a simple Lie group $G$ and two discrete subgroups $\Gamma_1,\Gamma_2\subseteq G$ such that the connected components $(H_1)_0$ and $(H_2)_0$ of $H_1$ and $H_2$ containing the identity elements satisfy $(H_1)_0\cong G/\Gamma_1$ and $(H_2)_0\cong G/\Gamma_2$. $\endgroup$ – Spenser Apr 21 '17 at 6:49
  • $\begingroup$ Is there any condition on $\mathfrak{h_1}$ and $\mathfrak{h_2}$ that forces $H_1=H_2$ assuming $\mathfrak{h_1}$ and $\mathfrak{h_2}$ are simple lie algebras ? I don't think there is any but I don't have a characterization either. $\endgroup$ – Srini Apr 21 '17 at 6:55
  • $\begingroup$ @Srini There is none. But there are conditions on $H_1$ and $H_2$, namely if they are simply connected and $\mathfrak{h}_1=\mathfrak{h}_2$, then $H_1=H_2$. $\endgroup$ – Spenser Apr 21 '17 at 7:04
  • $\begingroup$ $H1$ in the case that I am working on contains the alternating group. Hence $H1$ is not connected. Isn't $H1$ disconnected when it contains the alternating group ? $\endgroup$ – Srini Apr 21 '17 at 8:13
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Well, the third Lie theorem states that for each Lie algebra $\mathfrak{g}$, there exist a unique connected and simply connected Lie group $G$ such that the Lie algebra of $G$ is isomorphic to $\mathfrak{g}$.

Hence, if your groups are connected and simply connected, they shall be equal. Otherwise, anything could happen. For example, the Lie groups $SO(n)$ and $O(n)$ have the same Lie algebra, consisting of skew symmetric matrices. Note that in this case, $SO(n)$ is the connected component of $O(n)$ at the identity.

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