Nikolaus Conference 2024

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 commutative rings 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.