Hua structures and proper Moufang sets with Abelian root groups. (Q635875)

Moufang sets are 2-transitive permutation groups that have a split BN-pair of rank 1. \textit{T. De Medts} and \textit{R. Weiss} [Math. Ann. 335, No. 2, 415-433 (2006; Zbl 1163.17031)] showed that every Moufang set can be obtained in the form \(M(U,\tau)\) from a group \(U\) and a permutation \(\tau\) of \(U^*=U\setminus\{0\}\) with certain properties. Every root group of \(M(U,\tau)\) is isomorphic to \(U\). Such a Moufang set is called proper if its little projective group is not sharply 2-transitive. The author investigates proper Moufang sets with Abelian root groups and their connection to quadratic Jordan division algebras. To do so he introduces what he calls Hua structures. A Hua structure \((U,\mathcal H)\) consists of an Abelian group \(U\) with at least four elements together with a map \(\mathcal H\colon U^*\to\Aut(U)\) that satisfies five axioms, which reflect essential properties found by \textit{T. De Medts} and the author [Trans. Am. Math. Soc. 360, No. 11, 5831-5852 (2008; Zbl 1179.20030)] for Hua maps of Moufang sets \(M(U,\tau)\). The main result of the paper is that if \((U,\mathcal H)\) is a Hua structure and \(\tau\colon U^*\to U^*\) is defined by \(x\tau=-x^{-1}\), then \(M(U,\tau)\) is a proper Moufang set. Conversely, if \(M(U,\tau)\) is a proper Moufang set with \(U\) being Abelian and such that \(\tau=\mu_e\) for some \(e\in U^*\), then \((U,\mathcal H)\) is a Hua structure with an identity and \(x^{-1}=-x\tau\) where \(\mathcal H\colon a\mapsto h_a\) and \(h_a=\tau\mu_a\in\Aut(U)\) is the Hua map corresponding to \(a\in U^*\). Moreover, if \((J,\mathcal W,e)\) is a quadratic Jordan division algebra, then it is also a Hua structure with \(U\) the additive group of \(J\) and \(\mathcal H=\mathcal W\) given by the structure maps. On the other hand, T. De Medts and the author [loc. cit.] essentially showed that under certain assumptions on \(U\) and Hua maps the Hua structure \((U,\mathcal H)\) gives rise to a quadratic Jordan division algebra.