Local Moufang sets and PSL_2 over a local ring

TL;DR

This paper introduces local Moufang sets as a generalization of Moufang sets, providing a construction method and characterizing when they correspond to PSL_2 over a local ring.

Contribution

It presents a new construction method for local Moufang sets from a single root group and permutation, and characterizes those arising from PSL_2 over a local ring.

Findings

Describes how to construct local Moufang sets from minimal data

Provides necessary and sufficient conditions for a local Moufang set to be isomorphic to PSL_2 over a local ring

Establishes a link between local Moufang sets and algebraic groups over local rings

Abstract

We introduce local Moufang sets as a generalization of Moufang sets. We present a method to construct local Moufang sets from only one root group and one permutation. We use this to describe $\mathsf{PSL}_2$ over a local ring as a local Moufang set, and give necessary and sufficient conditions for a local Moufang set to be of this form.