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

[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!

 

[jvicens]

Hello! This is a great question. Gauge transformations in fiber bundles are closely related to automorphisms in Lie groups. In fact, gauge transformations can be thought of as a type of automorphism in the fiber bundle.

To answer your question, yes, there can be equivariant automorphisms that are not the result of left action by some element in the Lie group. These are known as outer automorphisms. They are essentially automorphisms that cannot be written as the composition of left and right actions.

In terms of relating this to fiber bundle gauge transformations, outer automorphisms can be thought of as gauge transformations that cannot be written in terms of a local gauge transformation. They are global transformations that affect the entire fiber bundle.

Hope this helps! Keep exploring and asking questions about gauge transformations – they are a fascinating topic in mathematics and have many applications in physics. Good luck!

 

[tacman]

Yes, there are right equivariant automorphisms that are not group left actions. These are known as outer automorphisms and they play an important role in understanding the structure of Lie groups. In fact, the outer automorphisms of a Lie group G can be thought of as gauge transformations of the associated fiber bundle with base space G and fiber G itself.

To see this, let's first define what a gauge transformation is. A gauge transformation of a fiber bundle is a bundle automorphism that maps each fiber to itself, but may permute the fibers in a nontrivial way. In other words, it is a transformation that preserves the local structure of the bundle, but allows for different global structures. This is similar to how outer automorphisms preserve the group structure locally, but can change the global structure of the group.

Now, let's consider the specific case of a principal G-bundle, where G is a Lie group. In this case, the fiber is the group G itself and the base space is also G. A gauge transformation of this bundle would be an automorphism of G that preserves the group structure locally, but may permute the elements of G in a nontrivial way. This is exactly what a right equivariant automorphism does. It preserves the group structure locally, but may change the global structure of the group by permuting the elements.

So, in summary, outer automorphisms of Lie groups can be thought of as gauge transformations of the associated principal G-bundle. This relationship between outer automorphisms and gauge transformations is a powerful tool in understanding the structure of Lie groups and their associated bundles.