Root Graded Groups

Abstract

We define and study root graded groups, that is, groups graded by finite root systems. These provide a uniform framework to investigate several existing concepts in the literature, including in particular Jacques Tits’ notion of RGD-systems. The most prominent examples of root graded groups are Chevalley groups over commutative associative rings. Our main result is that every root graded group of rank at least 3 is coordinatised by some algebraic structure such that a variation of the Chevalley commutator formula is satisfied. This result can be regarded as a generalisation of Tits’ classification of thick irreducible spherical buildings of rank at least 3 to the case of non-division algebraic structures. All coordinatisation results in this thesis are proven in a characteristic-free way. This is made possible by a new computational method that we call the blueprint technique.