Question about the notation of Lie algebra valued differential forms

Question

In mathematical differential geometry, one generally writes the structure equation as:

$$\Omega = d\omega + \omega \land \omega.$$

However in the following Physics Forums post, user romsofia writes the structure equation as

$$\Omega ^i_j = d\omega^i_j +\omega ^i_k\land \omega ^k_j.$$

I have seen this notation elsewhere, including in the following notes for F.P. Schuller's course on differential geometry (particularly lecture 22). Why do we use this notation for differential forms in physics? How do calculations with it work, e.g. how do we even compute wedge products? Essentially, I am asking for an explanation of the index notation for Lie algebra valued differential forms.

Answer 1

To add to the confusion there we have also the invariant notation for Lie-algebra valued forms:

$\alpha [\wedge] \beta := [\alpha \wedge \beta]$

This is usually defined on simple forms and then extended by linearity. More precisely, say $\alpha = \alpha' \otimes X$ and $\beta = \beta' \otimes Y$ where $\alpha, \beta$ are ordinary forms and $X,Y$ are elements of the Lie algebra. Then

$\alpha [\wedge] \beta = \alpha' \otimes X [\wedge] \beta' \otimes Y := \alpha' \wedge \beta' \otimes [X,Y]$

Now we see the rationale behind the symbol $[\wedge]$; it's because it is a combination of the wedge $\wedge$ and the commutator $[-,-]$.

The above is the global description of the product. We can also write it locally in components. For your particular situation we need to be more precise about the Lie algebra in question.

Recall that the frame bundle of the tangent bundle is a principal bundle and that any principal connection here can be written as a Lie-algebra 1-form.

Now, what is the Lie algebra in question here?

Well, it turns out to be $gl(\mathbb{R}^m)$ where $m$ is the dimension of the base manifold. And this is the Lie algebra of the Lie group $GL(\mathbb{R}^m)$. Now there is a more precise description of this Lie algebra, it turns out to be $End (\mathbb{R}^m )\simeq \mathbb{R}[m]$ and the last is just notation for the $m \times m$ matrices with entries in $\mathbb{R}$ - that is ordinary matrices.

Ordinary matrices in components, correctly written, have one upper and one lower index. And this is why in your post $\omega$, written locally is $\omega^i_j$.

This is one more index than one would expect for a Lie algebra valued form because any Lie algebra is vector space and a vector in components has only one (upper) index. The difference comes from the specific form of the Lie algebra here which turns out to be the full algebra of matrices. Basically, our vector turns out to be a matrix, and these have two indices (one upper and one lower). And the reason why we have this particular Lie algebra turn up here is because we don't have just any old principal bundle - we have the frame bundle of the tangent bundle. And its structure group (which some physicists called the gauge group, but properly isn't - it might be worth calling it the gauge structure group as a helpful bridge between mathematical and physical terminology) is $GL(\mathbb{R}^m)$ whose Lie group is $gl(\mathbb{R}^m)$ and which is none other the full algebra of matrices.