Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of order p³

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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 A₀ of A is a Hopf subalgebra. In addition, there is a projection π: grad(A) → A₀; let R be the algebra of coinvariants of π. 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 A₀: grad(A) is isomorphic to (R # A₀). 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 p³ (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 p²; 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.