Some questions about differential forms

• If $A$ is a differential one form then $A\wedge A .. (more\text{ }than\text{ }2\text{ }times) = 0$ Then how does the $A\wedge A \wedge A$ make sense in the Chern-Simon's form, $Tr(A\wedge dA + \frac{2}{3} A \wedge A \wedge A)$ ?

  I guess this anti-commutative nature of wedge product does not work for Lie algebra valued one-forms since tensoring two vectors is not commutative.

• If the vector space in which the vector valued differential form is taking values is $V$ with a chosen basis ${v_i}$ then books use the notation of $A = \Sigma _{i} A_i v_i$ where the sum is over the dimension of the vector space and $A_i$ are ordinary forms of the same rank.

I would like to know whether this $A_i v_i$ is just a notation for $A_i \otimes v_i$ ?

• Similarly say $B$ is a vector valued differential form taking values in $W$ with a chosen basis ${w_i}$ and in the same notation, $B = \Sigma _j B_j w_j$. Then the notation used is that, $A \wedge B = \Sigma _{i,j} A_i \wedge B_j v_i \otimes w_j$

  I wonder if in the above $A_i \wedge B_j v_i \otimes w_j$ is just a notation for $A_i \wedge B_j \otimes v_i \otimes w_j$ ?

• $A$ and $B$ are vector bundle valued differential forms (like say the connection-1-form $\omega$ or the curvature-2-form $\omega$) then how is $Tr(A)$ defined and why is $d(Tr A) = Tr( d A)$ and $Tr(A \wedge B) = Tr(B \wedge A)$ ?

• Is $A \wedge A \wedge A \wedge A = 0$ ? for $A$ being a vector bundle valued $1$-form or is only $Tr(A \wedge A \wedge A \wedge A) = 0$ ?

• If A and B are two vector bundle valued $k$ and $l$ form respectively then one defines $[A,B]$ as , $[A,B] (X_1,..,X_{k+l}) = \frac{1}{(k+l)!} \Sigma _{\sigma \in S_n} (sgn \sigma) [A (X_{\sigma(1)},X_{\sigma(2)},..,X_{\sigma(k)}) , B (X_{\sigma(k+1)},X_{\sigma(k+2)},..,X_{\sigma(k+l)})]$

  This means that if say $k=1$ then $[A,A] (X,Y) = [A(X),A(Y)]$ and $[A,A] = 2A \wedge A$. The Cartan structure equation states that, $d\Omega = \Omega \wedge \omega - \omega \wedge \Omega$.

  But some people write this as, $d\Omega = [\Omega,\omega]$.

  This is not clear to me. Because if the above were to be taken as a definition of the $[,]$ then clearly $[A,A]=0$ contradicting what was earlier derived.

I've seen some of this stuff before, but I'm definitely not an expert. Here's my take (although I'm likely wrong on at least some of it, hopefully not too much):

• You can think of this as a matrix of 1-forms. Then, $A \wedge A$ means "matrix multiplication, where multiplication is wedge product".
• Yes, I think $A_i v_i$ really means $A_i \otimes v_i$.
• I think also $A_i \wedge B_j v_i \otimes w_j$ means $A_i \wedge B_j \otimes v_i \otimes w_j$, although there might be some issue of skew-symmetrization.
• A vector-bundle-valued differential k-form locally looks like $\omega \otimes s$, where $\omega$ is a usual k-form and $s$ is a section. Then $d$ should act by the Leibniz rule (once you've fixed a connection). I'd imagine that the trace is just trace in the usual sense, and then it's obvious that this commutes with $d$ and that $Tr$ admits cyclic permutations of the factors in its argument.
• I think that whether $A\wedge A\wedge A\wedge A$ must be zero depends on the dimension of the vector bundle where it takes its values. If for example it's the constant 1-dimensional bundle, then this is no different from usual 1-forms.
• Your last question is a little unclear, sorry. Could you clarify exactly what you're referring to?

EDIT

I'll respond to your comments in order.

1. If $A$ takes a vector field and spits out a matrix, we can consider each of the entries of the matrix (in a fixed basis, of course) as an ordinary 1-form. If you want to wedge two matrix-valued forms, you just write out the matrices, act like you're multiplying matrices like you learned a long time ago (i.e. take the dot product of rows with columns), but then when you want to simplify the expression, you consider the product to be the wedge. So, e.g. if $$ A = \begin{pmatrix}a&b\\c&d\end{pmatrix} $$ is a 2x2 matrix-valued 1-form, this means that $a$, $b$, $c$, and $d$ are just ordinary 1-forms. So evaluating on a vector $v$ gives the 2x2 matrix (of numbers) $$ A(v) = \begin{pmatrix} a(v) & b(v) \\c(v) & d(v) \end{pmatrix}. $$ If you want to calculate $A\wedge A$, this is the 2x2 matrix-valued 2-form $$ A\wedge A = \begin{pmatrix}a&b\\c&d\end{pmatrix} \wedge \begin{pmatrix}a&b\\c&d\end{pmatrix} $$ $$ = \begin{pmatrix} a\wedge a + b\wedge c & a\wedge b + b\wedge d \\ c\wedge a + d\wedge c & c\wedge b+d\wedge d\end{pmatrix} $$ $$ = \begin{pmatrix} b\wedge c & b\wedge (d-a) \\c\wedge (a-d) & c\wedge b \end{pmatrix}. $$
2. This should be more or less answered by (1). The decomposition of the form $\omega \otimes s$ is just saying that a vector-bundle-valued form can be thought of as an ordinary form (that's $\omega$), except that you need to decide which direction in the fiber it takes you (that's $s$). This specializes to the case where the bundle is the trivial $\mathbb{R}$-bundle. Then $s$ is actually just a function, and we're recovering the fact that $n$-forms are a module over $C^\infty(M,\mathbb{R})$.
3. Like you said in your first comment, a connection takes in tangent vectors to the total space. From here, this is exactly what I said in (2).
4. Well, any form is nilpotent -- if you wedge it with itself enough times, you eventually get 0 for dimensional reasons. (This assumes that the (co)tangent space is finite-dimensional, of course. I have no idea what happens in the case that it isn't.) The only obvious reason (to me) that $A\wedge A\wedge A\wedge A$ would be traceless is if it's 0. But then again, I don't know anything about Chern-Simons theory. Or maybe it's a trick along the lines of the proof that the commutator of two square matrices is always traceless (? I'm pretty sure that's true, and follows from the fact that $Tr(AB)=Tr(BA)$, which you should note does not imply that $Tr(ABC)=Tr(ACB)$, only that you can cyclically permute the matrices so e.g. $Tr(ABC)=Tr(CAB)=Tr(BCA)$).
5. All I know is that back in linear algebra, you learn that the trace of a matrix is invariant under basis change. In other words, we can have a well-defined notion of the trace of a linear map. Of course, this requires that the dimensions of the domain and codomain agree, which doesn't seem like it'll be true in the example you gave (of a vector-valued differential form). However, if they do agree, then (in a basis) you can consider a form to be matrix-valued, and then the trace is in all likelihood just the sum of the diagonal entries, which are themselves ordinary differential forms.

Let me note that http://en.wikipedia.org/wiki/Vector-valued_form seems to be highly relevant. They essentially define the bracket. Note that in the case of a 1-form A valued in $\mathfrak{g}$ you get $[A,A]=[A\wedge A (v \otimes w)]= A(v)\otimes A(w)- A(w)\otimes A(v)$, which has value in $\mathfrak{g} \otimes \mathfrak{g}$. To combine them you use the bracket in $\mathfrak{g}$, so you get $2 [A(x), A(y)]$.

Your definition for $[A, B]$ has a factor of $\frac{1}{(k+l)!}$, which in the case of your vector bundle being over a point does not reproduce the bracket on $\mathfrak{g}$. Maybe it should be $\frac{1}{(k!l!)}$, as in Wikipedia?

Edit: Mmm, this was nonsense. Over a point you'd be doing k=l=0, so it would not matter. Anyhow, Wikipedia definition seems to give constants more consistent with the rest. But I can't do arithmetic, so no guarantees.

On the other hand, as suggested by Aaron, you can view A as a matrix of 1-forms, and wedge-multiply the matrices. Then $A\wedge A (x, y) = A(x)A(y)-A(y)A(x)= [A(x), A(y)]$ and indeed, $2 A\wedge A= [A,A]$.

I am a bit confused by this wedge notation you give in the question. $A_i$ and $B_j$ differential forms - components of A and B. We then tensor things up, using wedge instead of product, getting a vector in forms valued in the tensor product. We would then need a map $V \otimes V=\mathfrak{g} \otimes \mathfrak{g}$ to $V=\mathfrak{g}$ to get a form with values in $V$ again. Presumably this is the Lie bracket on V, and the result is the 'wedge multiplication of matrices' as above. The 'issue with skew symmetrization' is that this wedge operation is not the bracket operation on forms, but it's less skewsymmetrized version.

The Cartan equation should work out fine. We have 2-form $\Omega$ and 1-form $\omega$. Their bracket on the triple $a,b,c$ is a sum of 6 terms, with a factor of 1/2 in front. These 6 terms pair up into 3 pairs of 2 equal terms each(as $\Omega$ is antisymmetric). So you get 3 terms with coefficient 1, which we then bracket. On the other hand, wedge multiplying the matrices for $\Omega$ and $\omega$ in different order results in same 3 bracket terms (wedging 2 form and 1 form produces 3 distinct terms). One just needs to write down the formulas. (I'm sure there is a general formula for bracket in terms of antisymmetrization of the wedge).

Maybe another point is that $[A,A]=0$ for 'ordinary' $\mathbb{R}$ valued forms is a consequence of the Lie bracket on $\mathbb{R}$ being trivial.

Now, trace is evidently defined as before - viewing A as a matrix of forms, trace is the summ of the diagonal forms. It is then clear why it commutes with the differential. However, $Tr(A\wedge B) = (-1)^{deg A \times deg B} Tr(B\wedge A)$. This follows from the usual formulas for trace and commuting the forms $A_{i,j}$ past $B_{j,k}$.

Finally, the trace of $A\wedge A\wedge A\wedge A$. It is $\Sigma_i A^4_{ii}= \Sigma_i \Sigma_j A^2_{ji} \wedge A^2_{ij}$. $A^2_{kl}=\Sigma_m A_{km} \wedge A_{ml}$ is in general non-zero (just take A a 3 by 3 matrix with 0 in $A_{12}$ entry and $A_{2,3}\wedge A_{3,1} \wedge A_{1,3} \wedge A_{3,2}$ non 0). BUT, you are on a 3-manifold (or you'd be doing a differnt C-S form), so there is no 4-form at all!