Equivariant Hopf structures on a sphere
type: Item
instance of: scholarly article
title: Equivariant Hopf structures on a sphere
zbMATH Open document ID: 0866.55014
author: Kouyemon Iriye
published in: Journal of Mathematics of Kyoto University
publication date: 1996-05-21T00:00:00Z
review text: Let \(G\) be a compact Lie group and \(X\) a pointed \(G\)-space. A map \(\mu: X\times X\to X\) which is a pointed \(G\)-map is called an equivariant Hopf structure on \(X\), if the restriction of \(\mu\) to \(X\vee X\) is (equivariantly) homotopic to \(1\vee 1\). Such structures have been considered by \textit{K. Iriye} [J. Math. Kyoto Univ. 22, 719-727 (1983; Zbl 0518.55010)] and by various other authors including \textit{A. L. Cook} and \textit{M. C. Crabb} [J. Lond. Math. Soc., II. Ser. 48, 365-384 (1993; Zbl 0795.55009)]. The standard Hopf structures on \(S^1\), \(S^3\) and \(S^7\) are equivariant with respect to the actions of 0(t), SO(3) and \(G_2\). The present work has several goals: first the author ties these questions to questions of the existence of certain representations, secondly the author obtains results on possible actions, of elementary Abelian groups on spheres. Details are too complex to state here.
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 57S25
zbMATH DE Number: 880137
zbMATH Keywords: Lie group
zbMATH Keywords: \(G\)-space
zbMATH Keywords: equivariant Hopf structure
reviewed by: Donald W. Kahn
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518704
OpenAlex ID: W1597459992
DOI: 10.1215/KJM/1250518704
description: scientific article; zbMATH DE number 880137