New answers tagged principal-bundles
1
vote
Connection on associated bundle
Note that $\theta$ being the difference of two connection one-forms, it is also horizontal: for every $\xi\in\mathfrak{g}$, $\theta(X_\xi)=0$, where $X_\xi$ is the fundamental vector field of the $G$-...
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11
votes
Reduction of structure group and classifying spaces
To begin, I should mention that the proof of this equivalence is convincingly sketched in Stephen A. Mitchell's "Notes on principal bundles and classifying spaces", see Theorem 10.1 on page ...
- 371
2
votes
Non-semisimple Lie groups and Higgs bundles
There is a (related but not quite the same) construction which is valid for any Lie group $G$ and any closed (hence Lie) subgroup $H\subset G$ over any smooth base manifold $X$ which may be helpful.
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- 7,817
5
votes
Non-semisimple Lie groups and Higgs bundles
One can replace $\mathfrak m$ by $\mathfrak g/\mathfrak h$ where $\mathfrak g$ is the Lie algebra of $G$ and $\mathfrak h$ is the Lie algebra of $H$. We clearly have $[\mathfrak h,\mathfrak h ] \...
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