Generalized Hopf-Ore extensions

We investigate some Hopf algebra structures over an Ore extension of a Hopf algebra. Necessary and sufficient conditions for an Ore extension of a Hopf algebra to have a Hopf algebra structure of a certain type are given. This construction, called a generalized Hopf-Ore extension, generalizes Hopf-Ore extensions. We describe the generalized Hopf-Ore extensions of the enveloping algebras of Lie algebras. For some Lie algebras g, the generalized Hopf-Ore extensions of U(g) are classified.