TY - THES
T1 - Integrability of Moufang foundations : a contribution to the classification of twin buildings
A1 - Weiß,Sebastian
Y1 - 2014/09/24
N2 - 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.
It is a (not completely trivial) fact that a 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. In particular, the foundation is independent of the chosen chamber.
Therefore, the classification of 2-spherical twin buildings reduces to the classification of all foundations which can be realized as the local structures of a twin building. We call such a foundation integrable. In order to determine the integrable foundations, one proceeds in two steps.
Step 1: Exclude Non-Integrable Foundations
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. Moreover, one can reduce the list of possibly integrable foundations by considering certain automorphisms of the twin building, the so-called Hua automorphisms, which are closely related to the double mu-maps of the appearing Moufang sets.
Step 2: Existence / Integrability Proof
Finally, one has to prove that each of the remaining candidates is in fact integrable, i.e., realized by a twin building, which is then unique up to isomorphism.
Goals and Main Results
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 (which is an alternative division ring in the simply laced case). In order to make the different glueings visible, a crucial question is how to parametrize sequences of Moufang polygons, more precisely their root group sequences with the usual commutator relations.
As mentioned above, excluding non-integrable foundations is closely related to the investigation of Moufang sets and their isomorphisms. Therefore, a large part deals with the introduction of underlying parameter systems and, in the sequel, with the solution of the isomorphism problem for Moufang sets. One of the main results in this context is the solution of the isomorphism problem for Moufang sets of pseudo-quadratic spaces. Moreover, many further questions have already been answered, but we need to refine and extend the existing results for our purposes and translate their proofs into our setup.
KW - Zwillingsgebäude
KW - Fundamente
KW - Moufang-Mengen
CY - Gießen
PB - Universitätsbibliothek
AD - Otto-Behaghel-Str. 8, 35394 Gießen
UR - http://geb.uni-giessen.de/geb/volltexte/2014/11089
ER -