Special Moufang sets with Abelian Hua subgroup.
Moufang sets were introduced by J. Tits and are the Moufang buildings of rank 1. A Moufang set \((X,(U_x)_{x\in X})\) consists of a set \(X\) and a family of permutation groups \(U_x\) acting on \(X\) such that \(U_x\) fixes \(x\), acts regularly on \(X\setminus\{x\}\) and permutes the \(U_y\), \(y\in X\), by conjugation. The group generated by all the \(U_x\) is usually called the little projective group of the Moufang set. \textit{T. De Medts} and \textit{R. M. Weiss} [Math. Ann. 335, No. 2, 415-433 (2006; Zbl 1163.17031)] showed that every Moufang set can be obtained from a group \(U\) and a permutation \(\tau\) of \(U^\#=U\setminus\{0\}\) with certain properties where \(X\) becomes \(U\cup\{\infty\}\) with a new symbol \(\infty\) not in \(U\). Such a Moufang set \(M(U,\tau)\) is called special if \(\tau\) is such that \((-a)\tau=-(a\tau)\) for all \(a\in U^\#\). The map \(\tau\) is not uniquely determined by the Moufang set and the condition for a special Moufang set does not depend on \(\tau\). Examples of special Moufang sets are the projective lines \(\mathbb{M}(K)\) over commutative fields \(K\) with little projective group isomorphic to \(\text{PSL}_2(K)\). De Medts and Weiss [loc. cit.] characterized these special Moufang sets among those Moufang sets \(M(U,\tau)\) where \(U\) is Abelian and contains no involution by the property that its Hua subgroup, that is, the subgroup generated by all Hua maps \(h_a\) for \(a\in U^\#\), is Abelian. In an analogous way one obtains a special Moufang set \(\mathbb{M}(J)\) from a quadratic Jordan division algebra \(J\). In this case \(U\) is the additive group of \(J\) and \(\tau\) is defined by \(a\tau=-a^{-1}\) for \(a\in J^\#\). Conversely, \textit{T. De Medts} and \textit{Y. Segev} [Trans. Am. Math. Soc. 360, No. 11, 5831-5852 (2008; Zbl 1179.20030)] showed that under certain assumptions on \(U\) and Hua maps the special Moufang set is obtained from a quadratic Jordan division algebra. In the paper under review the author considers a special Moufang set \(M(U,\tau)\) with \(\tau=\mu_e\) for an element \(e\in U^\#\) such that the Hua subgroup \(H\) of \(M(U,\tau)\) is Abelian. Then \(U\) is also Abelian and, in fact, a vector space over a prime field \(F\). The author shows that under these assumptions the special Moufang set is embeddable in a projective line, and gives an explicit description of the Moufang sets depending on the characteristic of \(F\). If \(K\) is the subring of \(\text{End}_F (U)\) generated by \(H\), then one has the following. (1) If \(\text{char\,}F\neq 3\), then \(K\) is a field and the map \(\mathcal H\colon U\to K\) which takes \(a\) to its Hua map \(h_a\) is an anisotropic quadratic form. (2) If \(\text{char\,}F\neq 2, 3\), then \(M(U,\tau)\simeq\mathbb{M}(K)\). (3) If \(\text{char\,}F=2\), then there is a \(K\)-vector subspace \(J\supseteq K\) of the field \(\sqrt K\) of all elements in an algebraic closure of \(K\) whose square is in \(K\), \(J\) is a quadratic Jordan division algebra and \(M(U,\tau)\simeq\mathbb{M}(J)\). For the proof of (1) the author derives various identities for Hua maps and verifies that the map \((x,y)\mapsto h_{x+y}-h_x-h_y\) is \(K\)-bilinear. The condition that \(M(U,\tau)\) is special is essential. There are many known examples of non-special Moufang sets with Abelian Hua subgroups, for example the Suzuki Moufang sets or the Ree-Tits Moufang sets.
scientific article; zbMATH DE number 5687640