On the antipode of a quasitriangular Hopf algebra
type: Item
instance of: scholarly article
title: On the antipode of a quasitriangular Hopf algebra
zbMATH Open document ID: 0767.16016
DOI: 10.1016/0021-8693(92)90128-9
author: David E. Radford
published in: Journal of Algebra
publication date: 1993-03-29T00:00:00Z
review text: Let \((A,R)\) be a quasi-triangular Hopf algebra with antipode \(s\) over a field, where \(R=\sum R^{(1)}\otimes R^{(2)}\) in \(A\otimes A\). Let \(u=\sum s(R^{(2)})R^{(1)}\). The author shows that \(u\) is invertible and \(s^ 2\) is conjugation by \(u\), i.e., \(s^ 2(a)=uau^{-1}\) for \(a\) in \(A\). This implies that \(s\) is bijective. The conjugacy result had been established by \textit{V. G. Drinfel'd} [Algebra Anal. 1, No. 2, 30-46 (1989; Zbl 0718.16035)] assuming that \(s\) was bijective. If \((A,R)\) is also finite-dimensional, the author shows that \(s^ 4\) is conjugation by a group-like element \(h\) of \(A\). In fact, \(h=uv\), where \(v=s(u)^{-1}\). This result had also been proved by Drinfel'd, using different methods. The author's methods are based on his previous work on integrals and trace functions.
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16W20
zbMATH DE Number: 130645
zbMATH Keywords: quasi-triangular Hopf algebras
zbMATH Keywords: antipodes
zbMATH Keywords: conjugation
zbMATH Keywords: group-like elements
zbMATH Keywords: integrals
zbMATH Keywords: trace functions
reviewed by: Earl J. Taft
MaRDI profile type: Publication
full work available at URL: 0021-8693(92)90128-9
OpenAlex ID: W1968394200
cites work: Q3819193
cites work: Q5749435
cites work: QUASITRIANGULAR HOPF ALGEBRAS AND YANG-BAXTER EQUATIONS
cites work: The Order of the Antipode of a Finite Dimensional Hopf Algebra is Finite
cites work: On a coradical of a finite-dimensional Hopf algebra
cites work: Q5590089
description: scientific article; zbMATH DE number 130645