Linkable Dynkin diagrams and Quasi-isomorphisms for finite dimensional pointed Hopf algebras

Linkable Dynkin Diagrams

In this article we develop some aspects of the construction of new Hopf algebras found recently by Andruskiewitsch and Schneider. There the authors classified (under some slight restrictions) all pointed finite dimensional Hopf algebras with coradical (Z/p)^s. We contribute to this work by giving a closer description of the possible ``exotic'' linkings.

Pointed Hopf algebras over non-abelian groups with non-simple standard braidings

We construct finite-dimensional Hopf algebras whose coradical is the group algebra of a central extension of an abelian group. They fall into families associated to a semisimple Lie algebra together with a Dynkin diagram automorphism. We show conversely that every finite-dimensional pointed Hopf algebra over a non-abelian group with non-simple infinitesimal braiding of rank at least 4 is of this form. We follow the steps of the Lifting Method by Andruskiewitsch--Schneider. Our starting point is the classification of finite-dimensional Nichols algebras over non-abelian groups by Heckenberger--V…

Multiparameter quantum groups, bosonizations and cocycle deformations

We describe quantum groups given by multiparametric deformations of enveloping algebras of Kac-Moody algebras as a family of pointed Hopf algebras introduced by Andruskiewitsch and Schneider associated to a generalized Cartan matrix. We show that under some assumptions, these Hopf algebras depend only on one parameter on each connected component of the Dynkin diagram, up to a cocycle deformation. In particular, we obtain in this way a known result of Hu, Pei and Rosso.

正特征下具有有限算术根系的对角型高阶Nichols代数

Higher rank Nichols algebras of diagonal type with finite arithmetic root system in positive characteristic

本文分类了正特征域上秩r≥5且具有有限不可约根系的对角型Nichols代数，通过Weyl群胚、算术根系和Cartan图证明其对应于表1中的广义Dynkin图。

Isomorphism classes and automorphisms of finite Hopf algebras of type A_n

In our previous paper math/0502157 we classified a large class of finite-dimensional pointed Hopf algebras up to isomorphism. However the following problem was left open for Hopf algebras of of type $A,D$ or $E_6$, that is whose Cartan matrix is connected and allows a non-trivial automorphism of the corresponding Dynkin diagram. In this case we described the isomorphisms between two such Hopf algebras with the same Cartan matrix only implicitly. The problem is whether it is possible to compute the isomorphisms in terms of the defining families of parameters. In the present paper we solve this …

Finite quantum groups over abelian groups of prime exponent

Since the discovery of quantum groups (Drinfeld, Jimbo) and finite dimensional variations thereof (Lusztig, Manin), these objects were studied from different points of view and had many applications. The present paper is part of a series where we intend to show that important classes of Hopf algebras are quantum groups and therefore belong to Lie theory. One of our main results is the explicit construction of a general family of pointed Hopf algebras from Dynkin diagrams. All the Frobenius-Lusztig kernels and their parabolic subalgebras belong to this family, but in addition we get many new ex…