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Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Let and $P$ a $G$-bundle with a Riemannian base space $(M,g)$. Denote $\mathfrak{h}$ the Lie subalgebra formed by the Killing vector fields within the vector fields of $M$. This subalgebra acts on the space of vector fields via the adjoint representation. I would like to know if there is natural representation of $\mathfrak{h}$ over the space $\Omega^1(M, \mathfrak{g})$ of Lie algebra valued forms.

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1 Answer 1

Sure: the entire Lie algebra of vector fields acts on forms and so vector-valued forms and, in particular, $\mathfrak{g}$-valued forms. Now restrict this action to $\mathfrak{h}$.

However, yr principal bundle $P$ plays absolutely no role at all in this so I suspect I am not answering the question you really meant to ask.

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Perhaps by $\Omega^1(M,\mathfrak{g})$ the OP means $\Omega^1(M,\operatorname{ad}P)$? – José Figueroa-O'Farrill Jul 3 '13 at 0:34
Perhaps, José. But then the answer to the question is surely no unless $P$ is somehow tied to the Riemannian structure of the base... – Fran Burstall Jul 5 '13 at 22:30

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