Torsors and ternary Moufang loops arising in projective geometry

TL;DR

This paper interprets torsors and Moufang loops within classical projective geometry, revealing new insights especially in the Moufang case and clarifying the link between lattice and algebraic structures.

Contribution

It provides a new geometric interpretation of torsors and Moufang loops, including novel results for the Moufang case and insights into lattice-algebra relations in projective spaces.

Findings

Reformulation of known results in the Desarguesian case

New results on Moufang loops in projective geometry

Enhanced understanding of lattice and algebraic structure relations

Abstract

We give an interpretation of the construction of torsors from preceding work (Bertram, Kinyon: Associative Geometries. I, J. Lie Theory 20) in terms of classical projective geometry. For the Desarguesian case, this leads to a reformulation of certain results from lot.cit., whereas for the Moufang case the result is new. But even in the Desarguesian case it sheds new light on the relation between the lattice structure and the algebraic structures of a projective space.