# Any right equivariant automorphisms that aren't group left actions?

1. Apr 1, 2007

### pensano

Hello, I'm trying to get a feel for gauge transformations of a fiber bundle, and hitting a tricky (for me) question. I'll posit the equivalent question for Lie groups, since it's more straight forward that way:

For a Lie group, $G$, are their any equivariant automorphisms, $\phi : G \to G$ satisfying
$$\phi(g \, h) = \phi(g) \, h$$
for all $h \in G$ acting on the right, that are NOT the result of left action by some $f \in G$? i.e.
$$\phi(g) \neq f \, g$$

This is a similar question to whether there are group automorphisms that are not inner automoriphisms, to which the answer is "yes, there are outer automorphisms." I'm wondering if there is some equivalent construction, or some way to relate the two, for these right equivariant automorphisms that arise as fiber bundle gauge transformations?

Thanks to any math god who sheds light on this!