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I am looking for a package in Mathematica which can handle differential forms in a coordinate free manner. I am aware of several packages which do differential forms, but it seems that for all of them you are required to fix a coordinate chart $x_1,\ldots,x_n$ and then define your forms as things of the form $\omega = \sum \omega^1 dx_i$. The exterior derivative is then calculated formally.

What I need is a package where I can abstractly declare the exterior derivatives of forms without reference to coordinates. For example, in Maple there is the difforms package where on can assert such things as $\alpha,\beta,\gamma$ are 1-forms and that $d\alpha =\beta\wedge\gamma$. Then every time $d\alpha$ shows up it can replace it with $\beta\wedge\gamma$. Are there any packages in Mathematica which can do this?

As another example, given a Lie group $G$ one often defines the Maurer-Cartan form $\omega$, which is a left invariant $\mathfrak{g}$-valued 1-form on $G$ and which satisfies the Maurer-Cartan equation $d\omega = -\omega\wedge\omega$. Obviously in this case it would be inefficient to define coordinates on $G$ when we have a perfectly good abstract representation of the exterior derivative.

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xTerior handles tensor-valued differential forms in an abstract way:

http://xact.es/xTerior/

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There is this package from MathSource, which contains a number of similar packages.

Various packages dealing with Lie groups have also been proposed over the years.

Your mileage may vary.

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