On the unexpected geometrical origin of the algebra of symmetries

Abstract

The fundamental symmetries in gravity and gauge theories, formulated using differential forms, are gauge transformations and diffeomorphisms. Although these symmetries act differently on regular fields and connections, their commutator yields the same closed algebra in both cases, hinting at an underlying mathematical structure. This work uncovers a natural correspondence between this algebra and the Lie bracket of some vector fields on the principal fiber bundle associated with the physical theory, providing a geometric interpretation of the symmetry algebra. Furthermore, we investigate how the previously identified structure changes when the connection on the principal fiber bundle is modified, and we analyze an example, using our methods, illustrating the relationship between Lorentz-Lie transformations and spacetime Killing vector fields.