Radford's formula for generalized weak bi-Frobenius algebras.
type: Item
instance of: scholarly article
title: Radford's formula for generalized weak bi-Frobenius algebras.
zbMATH Open document ID: 1314.16020
DOI: 10.1216/RMJ-2014-44-2-419
published in: Rocky Mountain Journal of Mathematics
publication date: 2014-09-04T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1407154907
reviewed by: Alan Koch
Mathematics Subject Classification ID: 16T05
zbMATH DE Number: 6339709
zbMATH Keywords: GWBF-algebras
zbMATH Keywords: Radford formula
zbMATH Keywords: weak Hopf algebras
zbMATH Keywords: generalized weak bi-Frobenius algebras
author: Quan-guo Chen
author: Shuanhong Wang
MaRDI profile type: Publication
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review text: Let \(H\) be a finite dimensional Hopf algebra over a field \(\mathbf K\), and let \(H^*\) be its linear dual. Then \(H\) has the structure of a left and right \(H^*\)-module by \(f\rightharpoonup h=\text{mult}(f\otimes 1)\Delta(h)\) and \(h\leftharpoonup f=\text{mult}(f\otimes 1)\Delta(h)\), respectively, where \(h\in H\), \(f\in H^*\), and \(\Delta\) is the comultiplication on \(H\). Radford's formula states that \(S^4(h)=g(\alpha\rightharpoonup h\leftharpoonup\alpha^{-1})g^{-1}\) for some distinguished grouplike elements \(g\in H\), \(\alpha\in H^*\).NEWLINENEWLINE Recently, adaptations of Radford's formula have appeared in other categories of generalized Hopf algebras, such as weak Hopf algebras, biFrobenius Hopf algebras, co-Frobenius Hopf algebras, algebraic quantum groups, and bornological quantum groups.NEWLINENEWLINE The work under review marks the introduction of generalized weak biFrobenius algebra (hereafter, GWBF-algebra). Let \(H\) be finite dimensional algebra with unit \(1\) which also has a coalgebra structure with counit \(\varepsilon\). Pick \(\Psi\in H^*\), \(\Psi\neq 0\). Then there is a bijective map \(S\colon H\to H\) such that \(\text{mult}(\Psi\otimes S)(h\otimes 1)\Delta(g)=\text{mult}(\Psi\otimes 1)\Delta(h)(g\otimes 1)\) for all \(g,h\in H\). If \(S\) satisfies a sequence of axioms, including \(S\) is an anti-algebra map and anti-coalgebra map; and \((H,\Psi)\) is a Frobenius algebra, then \((H,\Psi,S)\) is said to be a GWBF-algebra. As an example, a weak Hopf algebra \(H\) gives a GWBF-algebra \((H,\Psi,S)\) where \(S\) is the antipode and \(\Psi\) is a right integral. Examples of GWBF-algebras which are ``new'' (in the sense that they are not weak Hopf algebras nor biFrobenius Hopf algebras) are also given.NEWLINENEWLINE The main result of the paper is to establish Radford's formula, which is done quite explicitly.
description: scientific article; zbMATH DE number 6339709