Integrability of Moufang Foundations - A Contribution to the Classification of Twin Buildings

TL;DR

This paper advances the classification of twin buildings by analyzing integrable Moufang foundations, focusing on simply laced and 443 triangle diagrams, and solves the isomorphism problem for Moufang sets of pseudo-quadratic spaces.

Contribution

It provides complete lists of integrable foundations for specific diagram types, refining existing classification techniques for twin buildings.

Findings

Classified integrable foundations for simply laced diagrams.

Classified integrable foundations for 443 triangle diagrams.

Solved the isomorphism problem for Moufang sets of pseudo-quadratic spaces.

Abstract

Buildings have been introduced by J. Tits in order to study semi-simple algebraic groups from a geometrical point of view. One of the most important results in the theory of buildings is the classification of irreducible spherical buildings of rank at least 3. About 25 years ago, M. Ronan and J. Tits defined the class of twin buildings, which generalize spherical buildings in a natural way. The motivation of their definition is provided by the theory of Kac-Moody groups. A 2-spherical twin building is uniquely determined by its local structure in almost all cases: The so-called foundation is the union of the rank 2 residues which contain an (arbitrary) chamber. Therefore, the classification of 2-spherical twin buildings reduces to the classification of all foundations which can be realized as the local structure of such a twin building. We call such a foundation "integrable". By a result of Tits, an integrable foundation is Moufang, which means that the rank 2 buildings in the foundation are Moufang polygons, and that the glueings are compatible with the Moufang structures induced on the rank 1 residues. As a consequence, the classification of Moufang polygons and the solution of the isomorphism problem for Moufang sets are essential to work out which Moufang polygons fit together in order to form a foundation. The present thesis contributes to establish complete lists of integrable foundations for certain types of diagrams, namely for simply laced diagrams and for 443 triangle diagrams. In this process, we closely follow the approach for the classification of spherical buildings. However, we have to refine the techniques used there, since in general, foundations don't only depend on the diagram and the defining field. Moreover, one of the main results in the context of Moufang sets is the solution of the isomorphism problem for Moufang sets of pseudo-quadratic spaces.