Hopf Images of Hopf algebra Coactions

We introduce the Hopf image of a right coaction of a Hopf algebra on an algebra as the smallest Hopf subalgebra through which the coaction factors. We establish its basic properties and show that the induced coaction on the Hopf image is inner-faithful. For coactions on Nichols algebras arising from Yetter–Drinfeld modules, we prove that the Hopf image is the smallest Hopf subalgebra detected by the underlying coaction on the Yetter–Drinfeld module. We give explicit examples of proper non-trivial Hopf images arising from bosonizations.