Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Poor Phrasing of a Lie Group Theorem

  1. Mar 21, 2014 #1
    I found what might be the worst written book on Lie Groups. Ever. Until I find one I like better, I'm going to see if I can persevere through the sludge. I'll write out the theorem word for word and then explain what I can. Hopefully someone can decipher it.

    Typically, I use the term "chart" to mean the inverse of what the author uses here. Also, in case it wasn't clear from the title, ##G## is a Lie group.

    What does this mean, explicitly using the chart (preferably using ##\varphi_\alpha^{-1}## for his ##\varphi_\alpha##)?
     
    Last edited: Mar 21, 2014
  2. jcsd
  3. Mar 23, 2014 #2
    For those of you who might be looking for this in the future, I think I've figured out what the author is trying to say.

    Pretend ##G## is an abelian Lie group. Then, the multiplication map ##\mu:G\times G\to G, (a,b)\mapsto ab## is a homomorphism of Lie groups. Consider ##\mu_1=\left.\mu\right|_{\{\mathrm{e}\}\times G}##. Identifying ##\{\mathrm{e}\}\times G\cong G##, we note that ##\mu_1## is the identity. Thus, evaluating the pushforward of ##0\oplus y## by ##\mu_1##, we get ##\mathrm{d}(\mu_1)_{(\mathrm{e},\mathrm{e})}(0\oplus y)=y##. Similarly, for ##\mu_2=\left.\mu\right|_{G\times\{\mathrm{e}\}}##, we get ##\mathrm{d}(\mu_2)_{(\mathrm{e},\mathrm{e})}(x\oplus 0)=x##. By linearity, we get our result: $$\mathrm{d}\mu_{(\mathrm{e},\mathrm{e})}(x\oplus y)=\mathrm{d}\mu_{(\mathrm{e},\mathrm{e})}(x\oplus 0 + 0\oplus y)=x+y.$$

    This is used, notably, to prove that ##\exp## is a homomorphism of groups (with an appropriate forgetful functor slapped onto the domain) if and only if the underlying Lie group is abelian.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Poor Phrasing of a Lie Group Theorem
  1. Lie Groups (Replies: 3)

  2. Lie Groups (Replies: 7)

Loading...