Structure group - Knowlpedia

The structure group of a G-bundle with model fiber $F$ is the specified group $G$ , acting on $F$ , in which its transition maps $g_{ij}:U_i\cap U_j\to G$ take values. The change of fiber coordinates is

(b,f)\longmapsto(b,g_{ij}(b)\cdot f).

Thus the structure group records which transformations of the model fiber are permitted by the chosen bundle structure.

Examples and reductions

A real rank- $n$ vector bundle has structure group $\mathrm{GL}(n,\mathbb R)$ . A bundle metric permits orthonormal local frames, whose transitions lie in $\mathrm O(n)$ . This gives a reduction of structure group of the associated frame bundle.

For a principal G-bundle, the model fiber is $G$ itself; the transition functions act by left multiplication and commute with the principal right action.

What the choice specifies

The underlying fiber bundle need not determine a unique structure group or a unique reduction. Nor is the structure group generally the gauge group, whose elements are automorphisms of the whole principal bundle over its base.