Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of Orderp3

We propose the following principle to study pointed Hopf algebras, or more generally, Hopf algebras whose coradical is a Hopf subalgebra. Given such a Hopf algebraA, consider its coradical filtration and the associated graded coalgebra gr A. Then gr Ais a graded Hopf algebra, since the coradicalA0ofAis a Hopf subalgebra. In addition, there is a projection π: gr A → A0; letRbe the algebra of coinvariants of π. Then, by a result of Radford and Majid,Ris a braided Hopf algebra and gr Ais the bosonization (or biproduct) ofRandA0: gr A R#A0. The principle we propose to studyAis first to studyR, then to transfer the information to gr Avia bosonization, and finally to lift toA. In this article, we apply this principle to the situation whenRis the simplest braided Hopf algebra: a quantum linear space. As consequences of our technique, we obtain the classification of pointed Hopf algebras of orderp3(pan odd prime) over an algebraically closed field of characteristic zero; with the same hypothesis, the characterization of the pointed Hopf algebras whose coradical is abelian and has indexporp2; and an infinite family of pointed, nonisomorphic, Hopf algebras of the same dimension. This last result gives a negative answer to a conjecture of I. Kaplansky.