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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

Author

Andruskiewitsch, N.; Schneider, H-J.

Published

1998

Language

EN