Local Moufang sets and \(\mathsf{PSL}_2\) over a local ring (Q311048)

Moufang sets are doubly transitive permutation groups with a conjugacy class of subgroups, each fixing one point and acting regularly on the other points. The standard examples are the groups \(\mathrm{PGL}_2 (k)\) acting on the projective line determined by the field \(k\).NEWLINENEWLINEThe authors generalize this concept to local Moufang sets; here, some equivalence relation on the set of points is preserved by the group. Examples are obtained by taking for \(k\) a local (commutative) ring. The authors provide sufficient conditions which ensure that a special local Moufang set originates from a local ring. These conditions require that the root groups and the Hua group are abelian, and that the characteristic is not \(2\).NEWLINENEWLINEIt seems that there is some relation to the work of \textit{C. Bartolone} and \textit{A. G. Spera} [Ann. Mat. Pura Appl. (4) 149, 297--309 (1987; Zbl 0638.20027)], where non-commutative local rings are admitted and conditions on the involutions are used.