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Can anyone kindly give some reference on taking trace of vector valued differential forms?

Like if $A$ and$B$ are two vector valued forms then I want to understand how/why this equation is true?

$dTr(A\wedge B) = Tr(dA\wedge B) - Tr(A\wedge dB)$

One particular case in which I am interested in will be when $A$ is a Lie Algebra valued one-form on some 3-manifold. Then I would like to know what is the precise meaning/definition of $Tr(A)$ or $Tr(A\wedge dA)$ or $Tr(A\wedge A \wedge A)$?

In how general a situation is a trace of a vector valued differential form defined?

It would be great if someone can give a local coordinate expression for such traces.

Any references to learn this would be of great help.

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    I assume by a vector valued form you mean a section of some $\Lambda^iT^*_M\otimes \mathcal{E}$ where $\mathcal{E}$ is a trivial vector bundle on $M$ with fiber a vector space $E$, say. So for your question to make sense, it seems to me $E$ needs to be equipped with a trace map $\mathrm{Tr}:E\to \mathbf{R}$. This will be the case, for example, when $E$ is the Lie algebra of a Lie subgroup $G\subset \mathrm{GL}_n(\mathbf{R})$. In such a case (and in any reasonable case I can imagine) the trace map will be linear, and so the argument in Mariano's answer works. It's easy to write in coordinates2010-08-25
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    (cont.) because the trace is all happening on $\mathcal{E}$. For example, if $e_{ij}$ is the standard basis for $\mathfrak{gl}_n$ and if $x_k$ are coordinates on your manifold, a $\mathfrak{gl}_n$-valued $n$-form locally looks like $\sum f_{ijk_1\ldots k_n} e_{ij}dx_{k_1}\wedge\cdots\wedge dx_{k_n}$, and the trace is then locally the (usual) $n$-form $\sum f_{iik_1\ldots k_n} dx_{k_1}\wedge\cdots dx_{k_n}$.2010-08-25
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    @Sam Thanks for this answer. Can you tell me what is the domain and range space of the Tr map of a vector valued k-form? If $\omega$ is such a form then at the point $p \in M$ $\omega$ is mapping $T_pM \times T_pM \times..k-times..T_pM \rightarrow E$. Now what is the $Tr(\omega)$ map? Your coordinate expression seems to indicate that $Tr(\omega)$ is an ordinary k-form?2010-08-25
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    @Sam I guess the rank of the form $k$, the rank of the bundle and the dimension of the manifold are independent quantities.2010-08-25
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    In those terms, $Tr(\omega)$ should (I think) just be the composition of $\omega_p:T_pM\times\cdots\times T_pM\to E$ with the map $Tr:E\to \mathbf{R}$. So yeah, it's an ordinary form.2010-08-25
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    @Sam I guess it would completely convincing if say given two vector valued forms $A$ and $B$ of say different ranks one can write down $Tr(A\wedge B)$ in terms of what we know $A$ and $B$ does to vector fields. Say $A$ is a vector valued 1-form and B is a vector valued 2-form then $Tr(A\wedge B)$ would be an ordinary 3-form.Then given 3 vector fields $X_1,X_2,X_3$ I would like to know an expression for the number $Tr(A\wedge B)[X_1,X_2,X_3]$ in terms of what we know $A[X_i]$ and $B[X_i,X_j]$ are. It would be great if you can give that.2010-08-31

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