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I am learning about vector-valued differential forms, including forms taking values in a Lie algebra. On Wikipedia there is some explanation about these Lie algebra-valued forms, including the definition of the operation $[-\wedge -]$ and the claim that "with this operation the set of all Lie algebra-valued forms on a manifold M becomes a graded Lie superalgebra". The explanation on Wikipedia is a little short, so I'm looking for more information about Lie algebra-valued forms. Unfortunately, the Wikipedia page does not cite any sources, and a Google search does not give very helpful results.

Where can I learn about Lie algebra valued differential forms?

In particular, I'm looking for a proof that $[-\wedge -]$ turns the set of Lie algebra-valued forms into a graded Lie superalgebra. I would also appreciate some information about how the exterior derivative $d$ and the operation $[-\wedge -]$ interact.

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1 Answer 1

up vote 7 down vote accepted

A $\frak g$-valued differential form is , as far as I know, just a section $\alpha$ of the tensor product of the exterior power of the cotangent bundle $\Lambda^{\bullet}T^*M$ of some manifold $M$ with the trivial vector bundle $M\times\frak{g}$. As such, locally over some chart domain $U$, $\alpha$ can be cast in the follwing form $$\alpha\equiv\alpha_1\otimes x_1+\cdots+\alpha_n\otimes x_n$$ where $\alpha_1,\dots,\alpha_n$ are local differential forms on $M$ defined over the chart domain $U$, and $x_1,\dots,x_n$ is a basis of $\frak g$. The differential is then calculated by ignoring the Lie algebra terms: $$d\alpha\equiv (d\alpha_1)\otimes x_1+\cdots+(d\alpha_n)\otimes x_n$$ Similarly, the product is defined by treating the differential forms and the Lie algebra elements as separate entities: $$[\alpha\wedge\beta]=\sum_{1\leq i,j\leq n}\alpha_i\wedge\beta_j\otimes[x_i,x_j]$$ For instance, for a pure form $\alpha$ of degree $p$, what you know about the exterior differential immediately implies that $$d[\alpha\wedge\beta]=[(d\alpha)\wedge\beta]+(-1)^p[\alpha\wedge(d\beta)]$$ Also, if $\alpha$ has degree $p$, and $\beta$ has degree $q$, then $$[\beta\wedge\alpha]=(-1)^{pq+1}[\alpha\wedge\beta]$$


I think the algebraic questions that arise are easy enough that I'm sure you can find all the relations you want on your own. However, you can always take a look at Peter W. Michor's Topics in Differential Geometry, in particular his chapter IV, §19, or Morita's Geometry of Characteristic Classes.

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