Talk:Maurer–Cartan form

message: Who is Maurer in the Maurer-Cartan form? --romanm (talk) 21:22, 19 August 2005 (UTC)
thread_title: Who is Maurer?

message: We know that \omega:T_gG\rightarrow T_eG.  Quoting from the article, "If G is embedded in GL(n), then \omega = g^{-1}dg." 
This definition confused me for a while.  As one would correctly assume, the first g is really L_g, left multiplication by g.  However, the g in dg is not L_g, but rather a (local) function R^k\rightarrow R^{n^2}, where k is the dimension of G.  Thus dg is essentially the identity map T_gG\rightarrow T_gG, since g in this case takes any point in G (viewed in R^k) to itself (now viewed in R^{n^2}).
If we were to interpret (incorrectly, as I had) the second g also as L_g, then dg would denote a map dg:T_hG\rightarrow T_{gh}G, in which case the composite g^{-1}dg, evaluated at the point g, would be a map from T_gG\rightarrow T_gG \neq T_eG (unless g=e).
:You can regard it as a formal identity in Rn x n, so that g = (xij) and dg = (dxij).  This is useful for concrete calculations.  More formally, g-1 is (L_{g^{-1}})_*, and dg is the identity map of the tangent space. Silly rabbit 23:37, 16 June 2006 (UTC)
thread_title:  If G is embedded n GL(n) 

message: A much simpler way of describing (and understanding) the Cartan-Maurer form should be incorporated into the article. The group quotient (Q_g(h) = g^{-1}h) extends to a quotient operation on the tangent spaces through its differential map g^{-1}v = {Q_g}_*(v). This is the algebraic generalization of the Cartan-Maurer Form; which is the special case of this operation restricted to tangent vectors v \in T_g(G).
This should also address the issue raised by the previous comment. If the product operation L_g(h) = gh is similarly extended to a tangent space operation by gv = {L_g}_*(v), then an invariant field X is characterized by X(g) = g X(e), and the application of the Cartan-Maurer form to it by
:g^{-1} X(g) = g^{-1} (g X(e)) = X(e).
These characterizations apply independently of any question of an embedding into GL(n), though they reduce to the corresponding matrix operations in GL(n), when an embedding exists. —Preceding unsigned comment added by 4.159.174.19 (talk • contribs) 
:I agree that the statement in the article is awkward, and I suppose I assume some responsibility for it.  I will see what I can do to make it more palatable. Silly rabbit 14:52, 19 June 2007 (UTC)
replies: text: I think I know why I had introduced the Maurer-Cartan form in this strange fashion.  At the time, I was working on a circle of articles dealing with integrability conditions and Cartan connections.  From this point of view, it was desirable to have a version of the MC form which imitated the definition of a Cartan connection by using the right action rather than the left action.  I suppose I never came around to finishing off my revisions here, and the article now needs some rather significant organizational changes. Silly rabbit 15:20, 19 June 2007 (UTC)
thread_title:  A Simpler Characterization of the Cartan-Maurer Form 

message: By the definition of the differential, if X and Y are arbitrary vector fields then
:d\omega(X,Y)=X(\omega(Y))-Y(\omega(X))-\omega([X,Y]).  
What is the sense of X(\omega(Y)) here? If \omega were a real-valued 1-form then it would be OK, because  \omega(Y) would be a real-valued function and X acts on real-valued functions. But \omega(Y) is now a \mathfrak g-valued function. How acts X on it?
89.135.19.155 (talk) 21:09, 11 August 2008 (UTC)
replies: text: I have added an explanation.  The bottom line is that you can still calculate the Lie derivative of a function with values in a fixed vector space.  Hopefully this addresses your question satisfactorily. siℓℓy rabbit (talk) 21:29, 11 August 2008 (UTC)
replies: text: Yes, thank you (really, it was trivial). 89.135.19.155 (talk) 22:02, 11 August 2008 (UTC)
thread_title:  Properties