# Lie Subalgebras and Lie Subgroups

This post will be shorter than usual. I thought it might be fun to write up some neat small results that have come up in my classes. I'll start today by talking about Lie subalgebras and Lie subgroups.

Recall that we can use the group structure of a Lie group $G$ to define a product on $T_eG$ (the tangent space to the identity). We call $T_eG$ with this product structure the *Lie algebra* of $G$, and denote it $\g$. The Lie algebra encodes a lot of significant information about the group - the Baker-Campbell-Haussdorf formula lets us relate the group product to the Lie bracket (at least in the image of the exponential map).

A Lie subgroup $H \subseteq G$ is a Lie group $H$ along with an injective Lie group homomorphism $\iota:H \inj G$. The differential of this homomorphism gives us a map between their Lie algebras $d\iota: \h \to \g$. The image of $d\iota$ is a Lie subalgebra of $\g$ (i.e. a linear subspace which is closed under the Lie bracket). This gives us a nice way of associating Lie subalgebras of $\g$ to Lie subgroups of $G$.

A natural follow-up question to ask is whether this correspondence works the other way as well: given a Lie subalgebra $\h \subseteq \g$, does it necessarily come from a Lie subgroup $i:H \inj G$? It turns out that the answer is yes! The proof is pretty neat, and not too long, although that's largely because I'll use a powerful theorem without proof.

The general idea of the proof is fairly intuitive. We can view the subalgebra $\h \subseteq \g$ as a linear subspace of $T_eG$. Using left-multiplication, we can translate this subspace to get a subspace of $T_xG$ for all $x \in G$. Then, we can essentially "integrate up" these planes to get a submanifold which is tangent to these planes. To make this argument more formal, we will look at distributions, which are just assignments of planes to each point in a manifold.

A $k$-plane distribution on a manifold $M^n$ is a rank-$k$ subbundle of the tangent bundle $TM$. Explicitly, this means that for each point $x \in M$, we assign a $k$-dimensional subspace $\Delta_x \subseteq T_xM$, and we make these choices in a smooth way. We denote the distribution by $\Delta$. Now, suppose we have a submanifold $N \subseteq M$ such that for every $x \in N$, $\Delta_x$ is the tangent space to $N$ at $x$. In this case, we call $N$ an *integral manifold* of $\Delta$.

We call a distribution $\Delta$ *involutive* if for any vector fields $X,Y$ whose vectors all lie in $\Delta$, then the Lie bracket $[X,Y]$ also lies in $\Delta$. Frobenius' Theorem tells us that if a distribution is involutive, then we can find a unique maximal integral manifold passing through any point $x \in M$ (Frobenius' theorem is actually stronger than this, but this is enough for us). This is great for us!

We can use Frobenius' theorem to find our subgroup $H$. Suppose we have a Lie subalgebra $\h$. We can construct a distribution $\Delta$ be defining $\Delta_x := dL_x\h$. Since $\h$ is a Lie subalgebra, it is closed under the Lie bracket. So $\Delta$ is involutive. Thus, we can find a maximal integral manifold of $\Delta$ passing through the identity $e$. Suggestively, we'll call this submanifold $H$.

Now, we just need to show that $H$ is a subgroup. This sounds like it might be difficult, but there's actually a clever trick that makes it really easy!. Let $h \in H$. Consider the translated submanifold $h^{-1}H$. $h^{-1}H$ is an integral manifold of $h^{-1}\Delta$. Since we constructed $\Delta$ by left-translating a subspace of $T_eG$, $\Delta$ must be left-invariant. So $h^{-1}\Delta$ is just $\Delta$. Thus, $h^{-1}H$ is a maximal integral submanifold of $\Delta$. And since $h \in H$, $h^{-1}h = e \in h^{-1}H$. By uniqueness of maximal integral submanifolds, we conclude that $h^{-1}H = H$. Thus, $H$ is a subgroup.

Welcome to the future! Financing made easy with Prof. Mrs. DOROTHY JEAN INVESTMENTS

ReplyDeleteHello, Have you been looking for financing options for your new business plans, Are you seeking for a loan to expand your existing business, Do you find yourself in a bit of trouble with unpaid bills and you don’t know which way to go or where to turn to? Have you been turned down by your banks? MRS. DOROTHY JEAN INVESTMENTS says YES when your banks say NO. Contact us as we offer financial services at a low and affordable interest rate of 2% for long and short term loans. Interested applicants should contact us for further loan acquisition procedures via profdorothyinvestments@gmail.com

We invest in all profitable projects with cryptocurrencies. I'm here to share an amazing life changing opportunity with you. its called Bitcoin / Forex trading options, Are you interested in earning a consistent income through binary/forex trade? or crypto currency trading. An investment of $100 or $200 can get you a return of $2,840 in 7 days of trading and you get to do this from the comfort of your home/work. It goes on and on The higher the investment, the higher the profits. Your investment is safe and secured and payouts assured 100%. if you wish to know more about investing in Cryptocurrency and earn daily, weekly OR Monthly in trading on bitcoin or any cryptocurrency and want a successful trade without losing Contact MRS.DOROTHY JEAN INVESTMENTS profdorothyinvestments@gmail.com

categories of investment

Cryptocurrency

Loan Offer

Mining Plan

Business Finance Plan

Binary option Trade Plan

Forex trade Plan

Stocks market Trade Plan

Return on investment (ROI) Plan

Gold and Silver Trade Plan

Oil and Gas Trade Plan

Diamond Trade Plan

Agriculture Trade Plan

Real Estate Trade Plan

YOURS IN SERVICE

Mrs. Dorothy Pilkenton Jean

Financial Advisor on Bank Instruments,

Private Banking and Client Services

Email Address: profdorothyinvestments@gmail.com

Operation: We provide Financial Service Such As Bank Instrument

From AA Rate Banks, Cash Loan,BG,SBLC,BOND,PPP,MTN,TRADING,FUNDING MONETIZING etc.

I never thought that I could be this wealthy after all I've been through trying to meet ends and take care of my family, I used to play lottery but has never be lucky to win until I saw some comments online how he had helped a lot of people win, after first I didn't believe him but still I gave it a try, I wrote him a message on email after few hours he replied back so I told him what I want he assured me success. he also told me what I should do and I did all he requested from me, after some hours he gave me the lucky numbers and showed me where to play so i did as instructed to my biggest surprise I won 27 million pounds (Euromillions). Now I'm rich and happy all thanks to Dr Benjamin for the help, I'm nothing without you. if you need his help to win you can reach him via his email drbenjaminlottospell711@gmail.com and his whatsapp +13344539570 or visit his website Drbenjamintemple.com

ReplyDeleteI have been unlucky playing the Lottery over the years. I couldn't take the pain anymore and I searched for help online. I saw random positive reviews of DR AMBER saying how he has been helpful with his spells. I visited his webpage: ( https://amberlottotemple.com ) and he told me what was required to get what I seek for done and I accepted. He did a reading for me that made it clear to him that I was going to become a MEGA JACKPOT winner and he gave me the right numbers to play the Lottery. I had faith in him and I went back to the store a few days later to confirm if I had won. While the clerk was checking my ticket, I heard her say ‘Oh my God! I looked up and saw the Big Winner screen of $36,449,852.60. I was completely shocked, but calm. It feels amazing – I can’t imagine this shift in my life and I want to appreciate this man DR AMBER for his help. It’s not everybody that is naturally lucky to win the Lottery but the solution to win is DR AMBER. For more info Call/Text/Telegram +1 808 481 5132 or E-mail: amberlottotemple@yahoo.com

ReplyDeleteOral herpes is a bothering virus, but overcoming is possible,I was diagnosed with genital herpes type 1 (hsv1),i contacted dr.voodoospelltemple66@gmail.com for treatment where i heard and read about Dr. voodoo on a forum. After Three weeks of herbal medication i went for test and i was tested negative from Herpes thank you Doctor,Here is Dr voodoo WhatsApp's number +2348140120719

ReplyDeleteNothing kills fast than cheating partner giving a man your all will only kill you because men can never be trusted, I almost lost my life yesterday after seeing a lot on my husband cell phone with the help of schwartzsoftwarehackingprogram AT Gmail DOT com I decided to run a quick hack on my husband so I had to contact : schwartzsoftwarehackingprogram AT Gmail DOT com for help and anonymously we broke into my husband phone and I saw a lot of rubbish starting from his WhatsApp text, text messages, Messages, and many more I was so disappointed in this man after giving my all to him he ended up cheating on me with different women and still come to sleep with me am so blessed to work with you schwartzsoftwarehackingprogram AT Gmail DOT com message him on +1 704-313-9661

ReplyDelete