Talk:Maurer–Cartan form

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


 * It's Ludwig Maurer. Charles Matthews 06:06, 30 September 2005 (UTC)

If G is embedded n GL(n)
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)

A Simpler Characterization of the Cartan-Maurer Form
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)


 * 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)

Properties
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)


 * 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)
 * Yes, thank you (really, it was trivial). 89.135.19.155 (talk) 22:02, 11 August 2008 (UTC)

1/2 is missing?
--刻意(Kèyì) 00:02, 28 November 2012 (UTC)

Do you mean that there is 1/2 in the formula $$d\omega + \frac{1}{2}[\omega,\omega]=0$$ but not in $$d\omega(X,Y)+[\omega(X),\omega(Y)]=0?$$ There is no contradiction: by the definition of bracket of Lie algebra-valued forms, $$\frac{1}{2}[\omega,\omega](X,Y) = [\omega(X),\omega(Y)].$$ (I elaborated on that definition a little and added a link to it in the Properties section). Jaan Vajakas (talk) 02:46, 10 December 2012 (UTC)


 * These equations are OK. I suppose I mean the equation
 * $$d\theta^i(E_j,E_k) = -\theta^i([E_j,E_k]) = -\sum_r c_{jk}^r\theta^i(E_r),$$

by duality yields
 * $$d\theta^i=-\sum_{jk} c_{jk}^i\theta^j\wedge\theta^k$$ (2),

the equation (2) is missing a constant 1/2 on the right. But this probably depends on the definition of wedge product of Differential forms.--刻意(Kèyì) 12:08, 10 December 2012 (UTC)


 * You are right. If we do not require j < k then there should be a coefficient 1/2 by the usual definition of wedge product in differential geometry. Now I fixed it. Jaan Vajakas (talk) 13:01, 10 December 2012 (UTC)


 * Hah hah. See also Talk:Connection form This is an endemic problem that one must be careful about. 67.198.37.16 (talk) 00:07, 24 October 2016 (UTC)

Examples needed
This stackexchange post discusses a good example of the Mauer-Cartan form: math exchange — Preceding unsigned comment added by 161.98.8.1 (talk) 20:59, 12 August 2017 (UTC)


 * Hmm. I dunno. It's actually a terrible example. It's a trick question from someone's homework, attached to a mis-understanding. The mis-understanding of the student is that they failed to treat the one-form as a one-form; they just stuck random 'd's in front of things. Worse, the teacher posing the homework problem made it tricky, by intentionally breaking it at x=0, thus forcing the student to think "hmm how do I write a one-form not at the origin?". So not only is it a trick question, but even without the trick, the student still completely bungled the answer. So .. its actually a terrible example.


 * However, point taken: this article could provide one or two worked examples. The canonical example is always SU(2), but some tricky example would be good too. The meta-problem is that wikipedia is an encyclopedia, not a textbook; examples that are worked out in detail tend to become very long, much longer than the base article, and quite tedious to read. Which is why examples are rare in Wikipedia. Hmmm. 67.198.37.16 (talk) 03:53, 3 May 2019 (UTC)

Problematic contradiction
The section Motivation and interpretation contains this passage:

"''A question of importance to Cartan and his contemporaries was how to identify a principal homogeneous space of G. That is, a manifold P identical to the group G, but without a fixed choice of unit element. ... ... ...

"A principal homogeneous space of G is a manifold P abstractly characterized by having a free and transitive action of G on P.''"

But there are plenty of Lie groups G that have manifolds M on which they act freely and transitively, without M being "a manifold ... identical to the group G". For instance if p: Spin(3) → SO(3) is the double covering from S3 → P3, then letting G = Spin(3) and M = SO(3) there is such an action given by the formula g∙m = p(g)m ∊ SO(3).

So these two characterizations of "principal homogeneous space" are contradictory.

I hope someone knowledgeable about this subject can fix this.