Root Graded Groups

Abstract: We define and study root graded groups, that is, groups graded by finite root systems. This notion generalises 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 satisfying a variation of the Chevalley commutator formula. 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 book are proven in a characteristic-free way. This is made possible by a new computational method that we call the blueprint technique.