Equivariant Hopf structures on a sphere (Q1912818)

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.