Lifting of quantum linear spaces and pointed Hopf algebras of order p3

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 algebra A, consider its coradical filtration and the associated graded coalgebra gr A. Then gr A is a graded Hopf algebra,since the coradical A(0) of A is a Hopf subalgebra. In addition, there is a projection pi: gr A --> A(0); let R be the algebra of coinvariants of pi. Then, by a result of Radford and Majid, R is a braided Hopf algebra and gr A is the bosonization (or biproduct) of R and A(0): gr A similar or equal to R#A(0). The principle we propose to study A is first to study R, then to transfer the information to gr A via bosonization, and finally to lift to A. In this article, we apply this principle to the situation when R is the simplest braided Hopf algebra: a quantum linear space. As consequences of our technique, we obtain the classification of pointed Hopf algebras of order p(3) (p an 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 index p or p(2); 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. (C) 1998 Academic Press.