Special Moufang sets with Abelian Hua subgroup.
type: Item
instance of: scholarly article
title: Special Moufang sets with Abelian Hua subgroup.
zbMATH Open document ID: 1197.20026
author: Matthias Grüninger
published in: Journal of Algebra
publication date: 2010-03-29T00:00:00Z
review text: 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 a…
reviewed by: Günter F. Steinke
Mathematics Subject Classification ID: 20E42
Mathematics Subject Classification ID: 17C60
Mathematics Subject Classification ID: 51E24
zbMATH DE Number: 5687640
zbMATH Keywords: special Moufang sets
zbMATH Keywords: Hua subgroups
zbMATH Keywords: quadratic Jordan division algebras
zbMATH Keywords: Moufang buildings
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.12.016
OpenAlex ID: W2011666822
cites work: Identities in Moufang sets
cites work: Special Moufang sets, their root groups and their μ-maps
cites work: Moufang sets and Jordan division algebras
cites work: Q2731785
cites work: Q5289098
DOI: 10.1016/J.JALGEBRA.2009.12.016
description: scientific article; zbMATH DE number 5687640