Finite Dimensional Pointed Hopf Algebras with Abelian Coradical and Cartan matrices

TL;DR

This paper classifies finite-dimensional pointed Hopf algebras with abelian coradical by linking their structure to Cartan matrices and braidings of Cartan type, providing new families and classifications.

Contribution

It introduces the notion of braidings of Cartan type for pointed Hopf algebras and characterizes when the associated Nichols algebra is finite dimensional, leading to new classifications.

Findings

Finite dimensionality of Nichols algebra B(V) linked to Cartan type matrices.

Classification of pointed Hopf algebras with prime order coradical.

Conditions for Hopf algebras of order p^4 to be generated in degree one.

Abstract

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.