Exceptional groups and Moufang sets of polar type.

01 November 2019 → Ongoing
Research Foundation - Flanders (FWO)
Research disciplines
  • Natural sciences
    • Group theory and generalisations
    • Geometry not elsewhere classified
    • Combinatorics
Project description

The notion of a Moufang set originates from the theory of semi-simple algebraic groups of relative rank 1 (algebraic groups are matrix groups and among the most fundamental algebraic structures). A Moufang set is a 2-transitive permutation group subject to some natural conditions. The theory has many applications in and outside group theory. The most important problem in this theory is the long standing conjecture that any Moufang set is either sharply 2-transitive or related to a semi-simple algebraic group of relative rank 1 or a slight variation of it. We aim to contribute to a solution by zooming in on the large and important subclass containing Moufang sets of “polar type”, which are among the least well-understood. By naturally extending the notion of a Moufang set to that of a Tits set, we make the classification accessible to geometric methods, in addition to the group theoretic ones. The geometry permits us to use the rich theory of parapolar spaces (geometric structures introduced to interpret semi-simple algebraic groups of higher relative rank). On the group theoretic side we want to use Timmesfeld’s theory of abstract root subgroups, a theory whose true potential has not yet been fully explored in this context. This way we hope to classify the class of Tits sets of polar type. This would not only be an important result on its own, but also provide new tools, possibilities and insights for the classification problem of Moufang sets, in particular of polar type.