Hopf Algebras with Abelian Coradical

In a previous work [AS2] we showed how to attach to a pointed Hopf algebra A with coradical kΓ, a braided strictly graded Hopf algebra R in the category Γ Γ YD of Yetter-Drinfeld modules over Γ. 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 ⊗ V → V ⊗ V . We denote R = B(V ). We assume further that Γ is finite abelian. Then c is given by a matrix (bij) 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 Γ is a direct sum of copies of a group of prime order, the conditions hold and any matrix is of Cartan type. We apply this result to show that R = R, in the case when Γ is a group of prime exponent. In other words, we show that a finite dimensional pointed Hopf algebra whose coradical is the group algebra of an abelian group of exponent p is necessarily generated by grouplike and skew-primitive elements. As a sample, we classify all the finite dimensional coradically graded pointed Hopf algebras whose coradical has odd prime dimension p. We also characterize coradically graded pointed Hopf algebras of order p. §