What is the exact relationship between principal bundles, representations, and vector bundles?



Principal bundles, representations, and vector bundles
Let G be an algebraic group (or, since the question was tagged as differential geometry, a Lie group). Then if we're given a principal G-bundle $E\_G$ and a representation V of G, we get a vector bundle out of this through the associated bundle construction: $(E\_G \times V)/G$ is a vector bundle with generic fiber V. Here, G acts on $E\_G \times V$ as $g(x,v) = (xg^{-1},gv).$


This shows that fixing a G-bundle determines an exact tensor functor from the category of representations of G to the category of vector bundles. There's a converse to this which says that giving an exact tensor functor from representations of G to vector bundles is equivalent to a G-bundle.


