Séminaire Algèbre ICJ

Root Graded Groups

by Torben Wiedemann (Giessen)

112 (ICJ (Braconnier))


ICJ (Braconnier)

Université Lyon 1

Let \Phi be a finite root system. A \Phi-graded group is a group G together with a family of subgroups (U_\alpha)_{\alpha \in \Phi} satisfying some purely combinatorial axioms. The main examples of such groups are the Chevalley groups of type \Phi, which are defined over a commutative ring and which satisfy the well-known Chevalley commutator formula. We show that if \Phi is of rank at least 3, then every \Phi-graded group is defined over some algebraic structure (e.g. a ring, possibly non-commutative or, in low ranks, even non-associative) such that a generalised version of the Chevalley commutator formula is satisfied. A new computational method called the blueprint technique is crucial in overcoming certain problems in characteristic 2. This method is inspired by a paper of Ronan-Tits.