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I have an exercise of Lie group as follows: "Let $G,H$ be closed connected subgroup of $GL_n(\mathbb{R})$, and $H$ be subgoup of $G$. Suppose that $Lie(H)$ is an ideal of $Lie(G)$. Prove that $H$ is a normal subgroup of $G$." I get stuck to solve this problem. Also I have no idea to use the connectedness of $G$ and $H$. Some one can help me? Thanks a lot!

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3 Answers 3

This is essentially an application of the Lie subalgebra-subgroup correspondence:

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Suppose that $\mathfrak{h}$ is a Lie subalgebra of $\mathfrak{g}$. Then there is a unique connected immersed Lie subgroup $H\subseteq G$ whose Lie algebra corresponds to $\mathfrak{h}$.

You are given $H$ a closed connected Lie subgroup of the Lie group $G$. Choose $g\in G$, and let $H' = gHg^{-1}$. Then $H'$ is a Lie group with corresponding Lie algebra $Lie(H')\subseteq Lie(G)$. However, the assumption that $Lie(H)$ is an ideal of $Lie(G)$ says exactly that $Lie(H') = Lie(H)$. You then have that $H$ and $H'$ are two connected Lie subgroups of $G$ with the same Lie algebra. By the uniqueness in the above theorem, it follows that $H = H'$, and hence that $H$ is normal.

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Thank you so much. But I don't know about the theorem you said. Can you tell me the name of the theorem? By chance, if the Lie group $G$ is linear, i.e a closed subgroup of $GL_n{\mathbb{R}}$, do we have simple proof for this theorem? –  mapping Nov 3 '12 at 15:11
    
@mapping: I don't know if the theorem has a name. It is essentially the Frobenius Theorem‌​. The idea is as follows. Take a basis $X_1,\ldots, X_r$ for $\mathfrak{h}$. These are invariant vector fields. Since $\mathfrak{h}$ is a subalgebra, these vector fields involutive. Frobenius' Theorem says there is a unique connected integral submanifold $H\subseteq G$ through the identity element. This is the subgroup $H$. Of course, you have to prove it is a Lie subgroup. –  froggie Nov 3 '12 at 16:44
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Thanks a lot! I will try to prove by this way. However I have trouble to prove $Lie(H')=Lie(H)$: If I take $X\in Lie{H'}$ then we have $\exp(tX)\in H'\forall t$. Then I can deduce that $g^{-1}Xg=Z\in Lie(H)$. So I need to prove $gZg^{-1}\in Lie(H)$. Using the condition $Lie(H)$ is an ideal of $Lie(G)$, we obtain $gZ-Zg\in Lie(H)$, hence $[gZ-Zg,g^{-1}]\in Lie(H)$. Finally I get $gZg^{-1}+g^{-1}Zg\in Lie(H)$. Until here, I don't know how to continue. Can you help me? –  mapping Nov 3 '12 at 17:53

There seems to be a full proof contained in lemma 0.1 here http://math.berkeley.edu/~ianagol/261A.F09/Simplegroups.pdf

Here is the argument:

Claim: For $X \in \frak{g}$, $Y \in \frak{h}$, we have $e^Xe^Ye^{-X}\in H$.

Proof of claim: Denote by $exp$ the exponential map $End(\frak{g}) \rightarrow $ $Aut(\frak{g}) $ (where Aut and End are the vector space automorphisms and endomorphisms) . Since $\frak{h}$ is an ideal, $ad_X^n$ preserves $\frak{h}$ for all $n \in \mathbb{N}$, and therefore so does $exp(ad_X)$. We therefore have $$e^Xe^Ye^{-X}=e^{Ad_{e^X}(Y)}=e^{exp(ad_X)Y} \in e^{\frak{h}} $$

This establishes the claim.

Now the fact that $G$ normalizes $H$ follows from the fact that $$H= \bigcup_{n\in \mathbb{N}} (e^\frak{h})^n$$ $$G= \bigcup_{n \in \mathbb{N}}(e^\frak{g})^n$$ Since $G$ and $H$ are connected.

In particular, this doesn't seem to use the fact that $G$ is a closed subgroup of $GL(n, \mathbb{R})$, although this is a hypothesis of lemma 0.1. Nor does it require even that $H$ be closed in $G$.

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I will two theorems that can be found in Lie groups and Algebraic group book by Vinberg.

$Theorem:1$ A a homomorphism from a connected lie group H to a lie group G is uniquely determined by tangent algebra homomorphism.

$Theorem:2$ Let $f:H \rightarrow G$ where is H is connected.If $G_1\subset G$ such that $df(Lie(H))\subset Lie(G_1)$. Then $f(H)\subset G_1$.

Let take arbitary $g\in G$ then $a(g):x\rightarrow gxg^{-1}$. The differential of this map say Ad(g) coincide with the adjoint representation of lie algebra hence this map Lie(H) to Lie(H). So $gHg^{-1}\subset H$ for all g hence H is normal.// You can find all the proofs in the book I mentioned in chapter 1 section 2 I hope.

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I'm not sure I agree. This is essentially the reasoning given in the other answer. The missing piece is the connection between Ad(g) and ad. ad is the derivative of Ad(exp(tv)), but a priori it doesn't tell you how Ad(exp(tv)) relates to ad, much less how Ad(g) relates to ad for some g outside the image of exp. –  Tim kinsella Oct 2 at 17:10
    
Please read Lie groups and Algebraic group book by Vinber. They explain that in full detail. Chapter 1 section 2. –  GGT Oct 6 at 11:25

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