Some Properties and Examples of Triangular Pointed Hopf Algebras

TL;DR

This paper investigates properties and classifications of finite-dimensional triangular pointed Hopf algebras over algebraically closed fields of characteristic zero, establishing structural results and construction methods for minimal triangular structures.

Contribution

It proves the order of the antipode in such algebras is four, classifies minimal triangular structures, and introduces a construction method for these algebras.

Findings

The antipode's fourth power is the identity in triangular pointed Hopf algebras.

Group algebra of grouplike elements admits a minimal triangular structure.

Any algebra generated by grouplike and skew primitive elements is isomorphic to a constructed minimal triangular Hopf algebra.

Abstract

A fundamental problem in the theory of Hopf algebras is the classification and construction of finite-dimensional (minimal) triangular Hopf algebras (A,R) introduced by Drinfeld. Only recently Etingof and the author completely solved this problem for semisimple A over algebraically closed fields of characteristics 0 and p>>dim(A) (any p if one assumes that A is also cosemisimple). In this paper we take the first step towards solving this problem for finite-dimensional pointed Hopf algebras over an algebraically closed field k of characteristic 0. We first prove that the fourth power of the antipode of any triangular pointed Hopf algebra A is the identity. We do that by focusing on minimal triangular pointed Hopf algebras (A,R) (every triangular Hopf algebra contains a minimal triangular sub Hopf algebra) and proving that the group algebra of the group of grouplike elements of A (which must be abelian) admits a minimal triangular structure and consequently that A has the structure of a biproduct. We also generalize our result on the order of the antipode to any finite-dimensional quasitriangular Hopf algebra A whose Drinfeld element u acts as a scalar in any irreducible representation of A (e.g. when A^* is pointed). Second, we describe a method of construction of finite-dimensional pointed Hopf algebras which admit a minimal triangular structure, and classify all their minimal triangular structures. We conclude the paper by proving that any minimal triangular Hopf algebra which is generated as an algebra by grouplike elements and skew primitive elements is isomorphic to a minimal triangular Hopf algebra constructed using our method.