  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 grad(A). Then grad(A) is a graded Hopf algebra, since the coradical
A_0 of A is a Hopf subalgebra. In addition, there is a projection \pi: grad(A)
\to 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 grad(A) is the bosonization
(or biproduct) of R and A_0: grad(A) is isomorphic to (R # A_0). The principle
we propose to study A is first to study R, then to transfer the information to
grad(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, non-isomorphic, Hopf
algebras of the same dimension. This last result gives a negative answer to a
conjecture of I. Kaplansky.
