On pointed ribbon Hopf algebras
type: Item
instance of: scholarly article
title: On pointed ribbon Hopf algebras
zbMATH Open document ID: 0857.16036
author: Shlomo Gelaki
published in: Journal of Algebra
publication date: 1996-07-09T00:00:00Z
review text: This paper deals mainly with the families of pointed Hopf algebras constructed by Radford: \(H_{n,q,N,\nu}\), which generalize Sweedler's 4-dimensional noncommutative non-cocommutative Hopf algebra, and \(U_{(N,\nu,\omega)}\), which is a family of finite dimensional pointed unimodular ribbon Hopf algebras, and which generalizes the well known quantum group \(U_q(\text{sl}_2)'\) (\(q\) is a root of unity.) The paper focusses on three aspects of these families of Hopf algebras. First, the author proves the uniqueness of these families as quasitriangular Hopf algebras, with respect to certain properties of the grouplikes and the skew primitives. Next, he finds all the possible quasitriangular structures (up to equivalence) of \(U_{(N,\nu,\omega)}\) and shows that they are all ribbon; in the process he describes its group of automorphisms and all its Hopf subalgebras. In the last section, he applies these results in two contexts. First he applies them to the quantum group \(U_q(\text{sl}_2)'\) (\(q\) a root of unity), giving, for example, all possible \(R\)-matrices that render it quasitriangular. Then he uses the 8-dimensional \(U_{(2,1,-1)}\) to compute invariants of lens spaces; it turns out that this Hopf algebra distinguishes between any two lens spaces, and it also shows that Hennings' 3-manifold invariant is not trivial.
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 57T05
zbMATH DE Number: 883785
zbMATH Keywords: pointed Hopf algebras
zbMATH Keywords: unimodular ribbon Hopf algebras
zbMATH Keywords: quasitriangular Hopf algebras
zbMATH Keywords: groups of automorphisms
zbMATH Keywords: Hopf subalgebras
zbMATH Keywords: quantum groups
zbMATH Keywords: \(R\)-matrices
zbMATH Keywords: invariants of lens spaces
reviewed by: Davida Fischman
MaRDI profile type: Publication
full work available at URL: jabr.1996.0145
OpenAlex ID: W2044890428
DOI: 10.1006/JABR.1996.0145
description: scientific article; zbMATH DE number 883785