On antipodes and integrals in Hopf algebras over rings and the quantum Yang-Baxter equation
type: Item
instance of: scholarly article
title: On antipodes and integrals in Hopf algebras over rings and the quantum Yang-Baxter equation
zbMATH Open document ID: 0880.16018
published in: Journal of Algebra
publication date: 1997-12-16T00:00:00Z
review text: In a previous paper [Trans. Am. Math. Soc. 349, No. 9, 3823-3836 (1997)] the authors showed that for every Frobenius algebra \(A\) over a commutative ring a pair of dual bases with respect to a Frobenius homomorphism of \(A\) defines a canonical solution of the quantum Yang-Baxter equation (QYBE). Let \(H\) be a Hopf algebra over a commutative ring \(K\) which is a Frobenius algebra over \(K\) such that the Frobenius homomorphism is an integral for the dual of \(H\). (By a result of \textit{B. Pareigis} [J. Algebra 18, No. 4, 588-596 (1971; Zbl 0225.16008)] the latter property is satisfied if \(H\) is a finitely generated projective \(K\)-module and the Picard group of \(K\) is trivial.) The first main result of the paper under review is a formula for a norm of \(H\) and its image under the comultiplication by using a pair of dual bases with respect to a Frobenius homomorphism of \(H\). From these formulas the authors obtain an explicit description of the canonical solution of the QYBE mentioned above in terms of the antipode and an arbitrary integral of \(H\). In particular, they prove a trace formula for the square of the antipode, which in the case that \(K\) is a field is due to \textit{R. G. Larson} and \textit{D. E. Radford} and was used in an essential way in their proof of a conjecture of Kaplansky [J. Algebra 117, No. 2, 267-289 (1988; Zbl 0649.16005), Am. J. Math. 110, No. 1, 187-195 (1988; Zbl 0637.16006)]. If in addition \(H\) is assumed to be unimodular (i.e., left and right integrals of \(H\) coincide), then they show that the antipode is the identity on the subspace of integrals of \(H\) and that the Nakayama automorphism (which measures the symmetry of the bilinear form associated to the Frobenius homomorphism) coincides with the square of the antipode. As a consequence, the fourth power of the antipode is the identity if \(H\) and its dual are unimodular, and the Drinfel'd double of \(H\) is symmetric if \(K\) is a field. The first result has been p…
reviewed by: Jörg Feldvoss
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 81R50
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16L60
zbMATH DE Number: 1053875
zbMATH Keywords: symmetric algebras
zbMATH Keywords: separable algebras
zbMATH Keywords: unimodular algebras
zbMATH Keywords: involutory Hopf algebras
zbMATH Keywords: dual bases
zbMATH Keywords: Frobenius homomorphisms
zbMATH Keywords: quantum Yang-Baxter equation
zbMATH Keywords: Frobenius algebras
zbMATH Keywords: integrals
zbMATH Keywords: finitely generated projective modules
zbMATH Keywords: Picard groups
zbMATH Keywords: norms
zbMATH Keywords: comultiplications
zbMATH Keywords: trace formula
zbMATH Keywords: antipodes
zbMATH Keywords: Nakayama automorphisms
zbMATH Keywords: Drinfel'd doubles
zbMATH Keywords: separability idempotents
zbMATH Keywords: Hopf algebras
author: K. I. Beidar
author: Y. Fong
author: A. Stolin
MaRDI profile type: Publication
full work available at URL: jabr.1996.7019
OpenAlex ID: W2006371894
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DOI: 10.1006/JABR.1996.7019
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description: scientific article; zbMATH DE number 1053875