Finite Dimensional Pointed Hopf Algebras with Abelian Coradical and Cartan matrices

In a previous work \cite{AS2} we showed how to attach to a pointed Hopf algebra A with coradical $\k\Gamma$, a braided strictly graded Hopf algebra R in the category $_{\Gamma}^{\Gamma}\Cal{YD}$ of Yetter-Drinfeld modules over $\Gamma$. In this paper, we consider a further invariant of A, namely the subalgebra R' of R generated by the space V of primitive elements. Algebras of this kind are known since the pioneering work of Nichols. It turns out that R' is completely determined by the braiding c:V\otimes V \to V \otimes V. We denote R' = B(V). We assume further that $\Gamma$ is finite abelian. Then c is given by a matrix (b_{ij}) whose entries are roots of unity; we also suppose that they have odd order. We introduce for these braidings the notion of "braiding of Cartan type" and we attach a generalized Cartan matrix to a braiding of Cartan type. We prove that B(V) is finite dimensional if its corresponding matrix is of finite Cartan type and give sufficient conditions for the converse statement. As a consequence, we obtain many new families of pointed Hopf algebras. When $\Gamma$ is a direct sum of copies of a group of prime order, the conditions hold and any matrix is of Cartan type. As a sample, we classify all the finite dimensional pointed Hopf algebras which are coradically graded, generated in degree one and whose coradical has odd prime dimension p. We also characterize coradically graded pointed Hopf algebras of order p^4, which are generated in degree one.