Principle Bundles: Right or Left Action?

[pensano]

Hey, I'm a little confused on the definition of a principle bundle. The basic question:

"Do elements of the structure group, G, have to act on elements of the fiber, G, from the right?"

I've read a bunch of papers that seem to imply that the fiber bundle structure group elements could act from the left and it's still called a principle bundle, but the wikipedia definition and a few others insist on the structure group elements acting on the right. What gives?

I guess it's kind of nit picky, but I'd like to know what's going on here.

 

[Doodle Bob]

It does not matter as long as you stay consistent within your own work and context.

 

[Hurkyl]

To elaborate, there are simply left and right G-bundles. The theories are dual to each other, so it doesn't hurt to study one kind of bundle, and then dualize if you need to know something about the other.

In many contexts, it simply boils down to notation: one choice might be notationally more convenient than another.

 

[pensano]

Hmm, so what would one call a G-bundle with G as fiber, with left action on G?

The left action dual of a principle bundle?

Thanks.

 

[matt grime]

For groups, or any Hopf algebra structure, left and right actions are identical concepts (a right action is the same as a left G^op action, and there is the involution turning G into G^op).