The Lifting Method, which was introduced by  Andruskiewitsch and Schneider in the late 1990s, has been very fruitful for the problem of classification of Hopf algebras (of finite dimension). In fact, it has led to the classification of all pointed Hopf algebras whose coradical is the group-algebra of an abelian group and several cases where the groups are not abelian.  The Method starts by associating to a Hopf algebra a graded Hopf algebra, whose existence assume that the coradical of the original Hopf algebra is a subalgebra, which is not always the case.

When the coradical is not a subalgebra, Andruskiewitsch and Cuadra introduced a generalized version of the Method: it is based on replacing the coradical by the Hopf coradical, which is always a subalgebra, though it is not semi-simple. In the present talk, we shall show how to proceed in the case that the Hopf coradical is a basic algebra. The answer rely on results of classification of pointed Hopf algebras.