  We derive necessary and sufficient conditions for an Ore extension of a Hopf
algebra to have a Hopf algebra structure of a certain type. This construction
generalizes the notion of Hopf-Ore extension, called a generalized Hopf-Ore
extension. 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.
