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

Differentiation Problem on Lie Groups

  1. Sep 20, 2013 #1


    User Avatar
    Science Advisor
    Gold Member

    Suppose θ is a differential 1 form defined on a manifold and with values in the Lie algebra of a Lie group,G.

    On MxG define the 1 form, ad(g)θ ,where θ is extended by letting it be zero on the tangent space to G

    How do you compute the exterior derivative, dad(g)θ ?

    BTW: For matrix Lie groups this is straightforward. What is the abstract calculation?
  2. jcsd
  3. Oct 7, 2013 #2


    User Avatar
    Science Advisor
    Gold Member

    Here is some computation for matrix groups

    dad(g)θ(x + h, y + k) = (x + h).ad(g)θ(y + k) - (y + k).ad(g)θ(x + h) -ad(g)θ[x + h, y + k]

    where Y and X are tangent to the manifold and h and K are left invariant vector fields.


    dad(g)θ(x + h, y + k) = (x + h).ad(g)θ(y) - (y + k).ad(g)θ(x) -ad(g)θ[x, y]

    = ad(g)dθ(x,y) + h.ad(g)θ(y) - k.ad(g)θ(x)

    h.ad(g)θ(y) = h. gθ(y)g[itex]^{-1}[/itex] = dg(h)θ(y)g[itex]^{-1}[/itex] - gθ(y)g[itex]^{-1}[/itex]dg(h)g[itex]^{-1}[/itex]

    = dg(h)g[itex]^{-1}[/itex]gθ(y)g[itex]^{-1}[/itex] - gθ(y)g[itex]^{-1}[/itex]dg(h)g[itex]^{-1}[/itex]

    = ω(h)ad(g)θ(y) - ad(g)θ(y)ω(h) where ω is the right invariant Maurer-Cartan form.

    So h.ad(g)θ(y) - k.ad(g)θ(x) = ω(h)ad(g)θ(y) - ad(g)θ(y)ω(h) - ω(k)ad(g)θ(x) + ad(g)θ(x)ω(k)

    Last edited: Oct 7, 2013
  4. Oct 7, 2013 #3

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    It looks like you are writing about the Baker–Campbell–Hausdorff formula, which does have a somewhat simplified representation in the case of SO(n).
  5. Oct 7, 2013 #4


    User Avatar
    Science Advisor
    Gold Member

    This doesn't simplify anything but succinctly rewrites the formula as

    ad(g)dθ +[ω,ad(g)θ] where the second term is the Lie bracket of the two differential forms ω and ad(g)θ
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Differentiation Problem on Lie Groups
  1. Lie Groups (Replies: 3)

  2. Lie Groups (Replies: 7)