Root graded groups revisited

Abstract

A group $G$ is called root graded if it has a family of subgroups $G_{\alpha }$ indexed by roots from a root system $\Phi$ satisfying natural conditions similar to Chevalley groups over commutative unital rings. For any such group there is a corresponding algebraic structure (commutative unital ring, associative unital ring, etc.) encoding the commutator relations between $G_{\alpha }$. We give a complete description of varieties of such structures for irreducible root systems of rank $\ge 3$ excluding ${\mathsf{H}}_3$ and ${\mathsf{H}}_4$. Moreover, we provide a construction of root graded groups for all algebraic structures from these varieties.