Classical, Algebraic and Mixed Buildings

Abstract

All Moufang spherical buildings can be found “in nature” as geometries associated with classical groups, with algebraic groups and with the mixed groups introduced in (10.3.2) of [101]. In this last chapter, we describe (without proofs) the Moufang spherical buildings of rank at least two as given in (16.1)–(16.9), (40.22) and (40.63)–(40.64) in terms of these three kinds of groups. Our main purpose is merely to provide a kind of lexicon between spherical buildings as organized from the “local” point of view taken in this book and spherical buildings as they are found “in nature.” In the appendix, we take a much more detailed look at spherical buildings as they arise in the theory of algebraic groups.