Lifting of Quantum Linear Spaces

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 grA. Then grA is a graded Hopf algebra, since the coradical A0 of A is a Hopf subalgebra. In addition, there is a projection π : grA → 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 grA is the bosonization (or biproduct) of R and A0: grA ≃ R#A0. The principle we propose to study A is first to study R, then to transfer the information to grA 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 (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; 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. §0. Introduction. We assume for simplicity of the exposition that our groundfield k is algebraically closed of characteristic 0; many results below are valid under weaker hypotheses. Let A be a non-cosemisimple Hopf algebra whose coradical A0 is a Hopf subalgebra; for instance, A is pointed, that is all simple subcoalgebras are one dimensional. Let A0 ⊆ A1 ⊆ · · · ⊆ A be the coradical filtration of A, see [M, Chapter 5]. This is a coalgebra filtration and we consider the associated graded coalgebra grA = ⊕n≥0 grA(n), grA(n) = An/An−1, where A−1 = 0. Since A0 is a Hopf subalgebra, grA is a graded Hopf algebra and the zero term of its own coradical filtration is grA(0) = A0, which is a Hopf subalgebra of grA. Let us denote B = grA, H = grA(0). Let γ : H → B be the inclusion and let π : B → H be the projection with kernel ⊕n≥1 grA(n). Then π is a Hopf algebra retraction of γ. We can describe the situation in the following way: (0.1) R →֒ B π ⇄ γ H,